Mad About Modern Physics - F. Potter_ C. Jargodzki by xero.loka

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									  Mad about
Modern Physics
  Mad about
Modern Physics
Braintwisters, Paradoxes, and Curiosities




             Franklin Potter
                   and
         Christopher Jargodzki




          John Wiley & Sons, Inc.
This book is printed on acid-free paper.

Copyright © 2005 by Franklin Potter and Christopher Jargodzki. All rights reserved

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Library of Congress Cataloging-in-Publication Data:

Potter, Frank, date
 Mad about modern physics : braintwisters, paradoxes and curiosities / Franklin Potter and
Christopher Jargodzki.
  p. cm.
 Includes index.
 ISBN 0-471-44855-9
1. Physics--Popular works. I. Jargodzki, Christopher II. Title
QC24.5.P68 2004
530—dc22
                                                                              2004014941

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1
To my late parents, who nourished my formative years and
     have now crossed that portal to another world.

                           F. P.


       To my late grandmother—Zofia Lesinska,
    who instilled in me the idea that the visible world
           owes its being to the invisible one.

                           C. J.
Contents

               Preface . . . . . . . . . . . . . . . . . . . . . . . . . .    ix

               Acknowledgments. . . . . . . . . . . . . . . . . .             xii

               To the Reader . . . . . . . . . . . . . . . . . . . . .       xiii

 Chapter 1     The Heat Is On . . . . . . . . . . . . . . . . . . . .           1

 Chapter 2     Does Anybody Really Know What
               Time It Is? . . . . . . . . . . . . . . . . . . . . . . . .     11

 Chapter 3     Crazy Circles . . . . . . . . . . . . . . . . . . . . .        19

 Chapter 4     Fly Me to the Moon. . . . . . . . . . . . . . . . .           29

 Chapter 5     Go Ask Alice . . . . . . . . . . . . . . . . . . . . . .      39

 Chapter 6     Start Me Up . . . . . . . . . . . . . . . . . . . . . .       49

 Chapter 7     A Whole New World. . . . . . . . . . . . . . . . .            63

 Chapter 8     Chances Are . . . . . . . . . . . . . . . . . . . . . .       75

 Chapter 9     Can This Be Real? . . . . . . . . . . . . . . . . .           91

 Chapter 10    Over My Head. . . . . . . . . . . . . . . . . . . . .         105

 Chapter 1 1   Crystal Blue Persuasion . . . . . . . . . . . . .             117


                                                                                    vii
                  Answers
                     The Heat Is On . . . . . . . . . . . . . . . . . . . . . .     125
                     Does Anybody Really Know What
                        Time It Is? . . . . . . . . . . . . . . . . . . . . . . .   139
                     Crazy Circles . . . . . . . . . . . . . . . . . . . . . . .    151
                     Fly Me to the Moon . . . . . . . . . . . . . . . . . .         164
                     Go Ask Alice . . . . . . . . . . . . . . . . . . . . . . .     181
                     Start Me Up . . . . . . . . . . . . . . . . . . . . . . . .    192
                     A Whole New World . . . . . . . . . . . . . . . . . .          206
                     Chances Are. . . . . . . . . . . . . . . . . . . . . . . .     224
                     Can This Be Real? . . . . . . . . . . . . . . . . . . .        24 1
                     Over My Head . . . . . . . . . . . . . . . . . . . . . .       257
                     Crystal Blue Persuasion . . . . . . . . . . . . . .            27 7

                  Index . . . . . . . . . . . . . . . . . . . . . . . . . . 287




viii   Contents
Preface
T
     his book of almost 250 puzzles begins where our first book, Mad
     About Physics: Braintwisters, Paradoxes, and Curiosities (2001)
     ended—with the physics of the late nineteenth and early twenti-
eth centuries. The Michelson-Morley experiment of 1887, the
challenges posed by atomic spectra and blackbody radiation, the
unexpected discoveries of X-rays in 1895, radioactivity in 1896, and
the electron in 1897 all loosened the protective belt of ad hoc hypothe-
ses around the mechanistic physics the nineteenth century had so
laboriously built. Anomalies and paradoxes abounded, ultimately
necessitating a radical rethinking of the very foundations of physics
and culminating in the theory of relativity and quantum mechanics.
Numerous applications of these new and strange concepts followed
very quickly as atomic and nuclear physics led to semiconductor
devices on the small scale and nuclear energy on the large scale. There-
fore we have developed a whole new set of challenges to tickle the
minds of our scientifically literate readers, from science students to
engineers to professionals in the sciences.
    The challenges begin with the classical problem of getting a cooked
egg into a bottle through a narrow bottleneck and back out again and
progress gradually to the famous aging-twin paradox of the theory of
special relativity and eventually reach problems dealing with the large-
scale universe. In between, we explore the nature of time and of space
as well as how the world of films and television tends to sacrifice
physics for the sake of entertainment. We also consider some of the
more startling questions in relativity. For example, we ask whether a
person can go on a space journey out to a star 7,000 light-years distant
and return while aging only 40 years! And we certainly want to
emphasize the practical applications of microphysics through an exam-
ination of some properties of exotic fluids, unusual motors running on
air or on random motion, as well as thermal, electrical, and photonic
properties of materials in a challenging journey into the atomic world.


                                                                           ix
              Particularly important microworld challenges include: What happened
              to Schrödinger’s cat? Can a cup of coffee be the ultimate quantum
              computer? Why is a Bose-Einstein condensate a new state of matter?
              Why is quantum mechanical coherent scattering so important in devel-
              oping new detectors for neutrinos and gravitational waves? When we
              reach the nucleus, there are challenges about the accuracy of carbon-14
              dating, the reason for neutron decay, and the amount of human
              radioactivity. Then our journey reverses as we reach for the stars to con-
              sider Olbers’ paradox about why the night sky is dark instead of burst-
              ing with light, how gravitational lensing by galaxies works, and what
              the total energy in the universe might be. This book finishes with a pot-
              pourri of challenges from all categories that ranges from using bicycle
              tracks in the mud to determine the direction of travel, to analyzing
              water-spouting alligators, and ending with a space-crawling mechanical
              invention that seems to defy the laws of physics.
                  The puzzles range in difficulty from simple questions (e.g., “Will
              an old mechanical watch run faster or slower when taken to the
              mountains?”) to subtle problems requiring more analysis (e.g., “Is the
              Bragg scattering of X-rays from an ideal crystal a coherent scattering
              process?”) Solutions and more than 300 references are provided, and
              they constitute about two-thirds of the book.
                  As these examples demonstrate, most of the puzzles contain an ele-
              ment of surprise. Indeed, one finds that commonsense conjecture and
              proper physical reasoning often clash throughout this volume. Ein-
              stein characterized common sense as the collection of prejudices
              acquired by age eighteen, and we agree: at least in science, common
              sense is to be refined and often transcended rather than venerated.
              Many of the challenges were devised to undermine physical precon-
              ceptions by employing paradoxes (from the Greek para and doxos,
              meaning “beyond belief”) to create cognitive dissonance. Far from
              being simply amusing, paradoxes are uniquely effective in addressing
              specific deficiencies in understanding. Usually the contradiction
              between gut instinct and physical reasoning for some people will be so
              painful that they will go to great lengths to escape it even if it means
              having to learn some physics in the process.
                  Philosopher Ludwig Wittgenstein considered paradoxes to be an
              embodiment of disquietude, and as we have learned, these disqui-
              etudes often foreshadow revolutionary developments in our thinking

x   Preface
about the natural world. The counterintuitive upheavals resulting
from relativity theory and quantum mechanics in the twentieth cen-
tury only enhanced the reputation of the paradox as an agent for
change in our understanding of physical reality.
    Such disquietudes, rather than unexplained experimental facts,
writes Gerald Holton in Thematic Origins of Scientific Thought, were
what led Einstein to rethink the foundations of physics in his three
papers of 1905. Each begins with the statement of formal asymmetries
of a predominantly aesthetic nature, then proposes a general postu-
late, not derivable directly from experience, that removes the asym-
metries. For example, in the paper on the quantum theory of light,
formal asymmetry existed between the discontinuous nature of parti-
cles and the continuous functions used to describe electromagnetic
radiation. As Holton notes, “The discussion of the photoelectric
effect, for which this paper is mostly remembered, occurs toward the
end, in a little over two pages out of the total sixteen.” Consistent
with this approach is Einstein’s statement in Physics and Reality
(1936), “We now realize . . . how much in error are those theorists
who believe that theory comes inductively from experience,” and later
in The Evolution of Physics (1938), coauthored with the Polish physi-
cist Leopold Infeld, “Physical concepts are free creations of the human
mind, and are not, however it may seem, uniquely determined by the
external world.”
    As another sore point, the term “quantum mechanics” is really a
misnomer: quantum systems cannot be regarded as made up of sepa-
rate building blocks. In the helium atom, for instance, we do not have
electron A and electron B but simply a two-electron pattern in which
all separate identity is lost. This indivisible unity of the quantum world
is paralleled by another kind of unity—between subject and object. Is
light a wave or a particle? The answer seems to depend on the experi-
mental setup. In the double-slit experiment, the observations of light
yield characteristics of the box and its slits as much as of light itself.
Is reality then observer-dependent? And would this justify Einstein’s
insistence on the power of pure thought in the construction of physi-
cal reality? Modern physics seems particularly adept at generating such
disquietudes. If that’s the case, then perhaps the word Mad in the title
of our book should not be construed as a mere metaphor!



                                                                             Preface   xi
      Acknowledgments

      W
               e all “stand on the shoulders of giants” as we develop our
               minds to become individuals living today on our planet Earth.
               And we owe so much to so many people that we cannot
      acknowledge all of them.
          Franklin Potter would like to express appreciation to his wife,
      Patricia, and their two sons, David and Steven, for their love and
      inspiration through many wonderful years of family adventures. He
      also treasures the numerous inspiring physics discussions over the
      decades with many friends and colleagues: Howard G. Preston,
      Gregory Endo, Fletcher Goldin, David M. Scott, John Priest, Lowell
      Wood, Julius S. Miller, George E. Miller, Leigh H. Palmer, Charles W.
      Peck, Myron Bander, Joseph Weber, Richard Feynman, Willard Libby,
      Edward Teller, and Kamal Das Gupta.
          Christopher Jargodzki would like to express appreciation to
      Myron Bander of the University of California at Irvine; Stephen Reu-
      croft of Northeastern University in Boston; and James H. Taylor of
      Central Missouri State University in Warrensburg. His interactions
      with close to twenty thousand students (and counting!) in his classes
      at UC Irvine, Northeastern University, and CMSU have been, over the
      years, never-ending sources of stimulation, as well as occasional exas-
      peration. In fact, the present volume got its start in 1975 when one of
      us (C. J.), still a graduate student at UC Irvine, put together a proposal
      for a book of paradoxes in modern physics, partly to allay his own
      exasperation with the koanlike conundrums that abound in modern
      physics. Alas, the project had to wait several decades for the author to
      mature and join forces with Franklin Potter in our joint inquiry into
      the nature of physical reality. The authors hope that physical reality is
      duly impressed with their efforts.
          Both authors sincerely thank Kate C. Bradford, senior editor at
      John Wiley & Sons, Inc., who continues to support our paradoxical
      adventures into the world of physics.


xii
To the Reader

T
      hese puzzles are meant to be fun. How many puzzles you solve is
      not as important as how many you enjoy thinking about. Some
      of them are even challenging to research physicists, and some
were generated by research articles that have appeared only recently in
physics journals, so these topics may not have been part of physics just
10 years ago! It would be a rare reader who could provide detailed
solutions to all the puzzles. Indeed, sometimes you may need to think
a bit to even understand the answer. If we included all the steps, this
book would double its present size. We offer no apologies, but we do
try to provide all the key steps to make each answer complete on its
own. If you find the puzzles perplexing and intriguing, we have suc-
ceeded in our mission.
    Mad about Modern Physics can be read with profit by anyone who
has had some exposure to a year of introductory physics and is eager
to learn more about its applications and its more recent discoveries.
Most puzzles are nonmathematical in character and require only a
qualitative application of fundamental physics principles. Many
physics concepts are defined directly or indirectly in the questions or
in the answers, so they can be found with the aid of the index. How-
ever, even someone who knows the subject will quickly realize that the
application of physics to the real world can be quite challenging, and
in this sense this is not an elementary book.
    More than three hundred follow-up references provide further
resources for interested readers. These references—to journal research
papers, books, and magazine articles—are included with only some of
the puzzles, typically those that are either controversial or that involve
relatively new concepts. There was no space to include a more com-
plete list of references. Consequently we had to make choices, and we
apologize to the authors whose work may have been left out or inad-
vertently overlooked.


                                                                             xiii
                 Any errors are solely those of the authors, and we would appre-
              ciate your communications via e-mail to Franklin Potter (see
              www.sciencegems.com) with regard to the puzzles and their
              answers.




xiv   To the Reader
1 The Heat
  Is On
  S        CIENCE IN THE HOME CONTRIBUTES IMMENSELY

           to our everyday repertoire of activities, although
  most of us are unaware of exactly how science does so.
  Physics, in particular, is all around us and plays a crucial role
  in determining what we can and cannot do. One enjoyable
  activity for many people is cooking, which is an application
  of physics and chemistry to satisfy our gastronomical tastes.
  Or are physics and chemistry just other modes of cooking?
  We’ll let you decide. Most of the challenges in this chapter
  involve physics from a high-school-level course. But be care-
  ful. Quick responses may be correct occasionally, but you
  should not rely on your intuition very much, for Nature,
  particularly in the kitchen, is nonintuitive for the most part.
  Anyone who has tried to make a soufflé can attest to how
  limited a recipe can be!




                                                                      1
We can detect five               1. Egg into a Bottle
basic tastes—four are                                                              Egg
very familiar: sweet,            Perhaps the most intriguing
sour, bitter, and salty.         physics-in-the-kitchen demon-
The fifth, while familiar        stration for all ages is get-
in East Asia, is less well       ting a hard-boiled egg with
known in Western                 the shell removed into a
cuisine—it is called             bottle that has an opening
 Umami and is the taste          diameter smaller than the
of monosodium gluta-             minimum diameter of the
mate, MSG. MSG is                egg. One solution is to very
used widely in Eastern
                                 carefully drop some bits of
cooking and that is
                                 burning paper into
probably why it is rec-
                                 the upright bottle and
ognized as a separate
taste sensation more             then place the egg at
                                 the opening. Soon, if      Burning
readily by those familiar                                   paper
with that cuisine. How-          the sequence is done
ever, many common                with the correct tim-
western foods contain            ing, the egg will have the urge to go inside. What is the
large amounts of MSG,            correct timing, and why does the egg have this urge?
notably tomatoes and
parmesan cheese.
            —P ETER B ARHAM ,
     T HE S CIENCE OF C OOKING   2. Egg out of a Bottle
                                 Perhaps the most challenging physics-in-the-kitchen
                                 demonstration for all ages is getting a hard-boiled egg
I was raised in Alabama          with the shell removed out of a bottle that has an open-
and Florida a Southern           ing diameter smaller than the minimum diameter of
Baptist, a lad given             the egg. Of course, one could cut up the egg with a
simple answers to pro-           knife inserted into the bottle and then pour out the
found questions. At the          pieces. However, we want the egg out whole and
same time I came to              undamaged.
love science, which                  Long ago, physics professor Julius Sumner Miller,
seeks profound answers           (who was Professor Wonderful on the early Mickey
to simple questions.             Mouse Club shows) was on the Tonight Show with
      —E DWARD O. W ILSON        host Johnny Carson and showed first how to get the
                                 egg into the bottle and then, taking no more than three


2     Mad about Modern Physics
seconds, had the same egg back in his hand. What is the    Why is it that tea made
procedure? (Hint: the same physics principles that put     with microwave-heated
the egg into the bottle can get the egg out.)              water doesn’t taste as
                                                           good as tea made with
                                                           teakettle water? The
3. Sugar                                                   main reason is that
                                                           microwaves heat only
Add two cups of sugar to one cup of water in a             the outer inch or so of
saucepan and stir while heating slightly. All the sugar    the water all around the
will dissolve. About how much total sugar will dissolve    cup, because that’s as
in one cup of water? What is the physics?                  far as they can pene-
                                                           trate. The water in the
                                                           middle of the cup gets
4. Kneading Bread                                          hot more slowly, through
                                                           contact with the outer
Bread made with yeast is usually kneaded—that is,          portions. When the
drawn out and pressed together to create a distribution    outer portions of the
of the ingredients. Then the bread dough is set aside to   water have reached boil-
“rise.” Why is some bread then kneaded a second time       ing temperature and
and sometimes even a third time before baking?             start to bubble, you can
                                                           be tricked into thinking
                                                           that all the water in the
5. Measuring Out Butter                                    cup is that hot. But the
                                                           average temperature
Suppose you have a solid chunk of butter and a meas-       may be much lower, and
uring cup in the kitchen. You desire to accurately         your tea will be short-
measure one-half cup of butter chunks without melting      changed of good flavor.
them. What is a quick, easy way to do so? Often one          —ROBERT L. WOLKE , W HAT
encounters the statement in cookbooks that Archi-                E INSTEIN TOLD H IS C OOK :
                                                             K ITCHEN S CIENCE E XPLAINED
medes’ principle is being used. What is this principle,
and why is the statement erroneous?

                                                           If there were one drop
6. Milk and Cream                                          of water less in the
                                                              universe,
You are given two identical bottles, one with milk and
                                                           the whole world would
the other with cream, both filled to the top. Quick now,
                                                              thirst.
which is heavier? And is light cream lighter than heavy                    —U GO B ETTI ,
cream?                                                              I TALIAN P LAYWRIGHT



                                                             The Heat Is On             3
CALORIC REQUIREMENT             7. Straw and Potato
BASED ON BODY WEIGHT
The basal calorie               A paper or plastic drinking straw can be pushed
requirement of the              through an uncooked potato. Explain the physics. If
average adult is ten            you plan to try this demonstration, be sure that you
times the ideal weight          take appropriate safety precautions—keep your hands
in pounds (e.g., 1,270          and body out of harm’s way.
for 127 lbs.), plus 30
percent for light activity
(i.e., 1,650 kcal), 50                                      Push down
percent for moderate                            Straw
activity (1,905 kcal),
and 100 percent for
                                                                Potato
heavy activity (2,540
kcal). Expressed slightly
differently, the basal
energy requirement is
about 1 kilocalorie per
hour for every kilogram
(2.2 lbs.) of ideal body
weight. Of course, any
                                8. Blueberry Muffins
estimate of calorie             Marion loves to bake warm, fresh blueberry muffins,
requirements based on           with the blueberries almost uniformly distributed
such formulas is just           throughout the muffin. She knows that if one simply
that—an estimate .              prepares the batter and mixes in the blueberries, they
Individual requirements         may be uniformly distributed before entering the oven,
vary widely with age,
                                but upon baking they will gravitate to lodge in the
health, body size,
                                lower part of the muffin. How does she prevent this
and environmental
                                natural downward drift?
temperature.

                                9. Can of Soup
                                Some people buy canned soup and store the cans in the
When men reach their
                                cupboard. Some people even turn these soup cans
sixties and retire, they
go to pieces. Women go          upside down for storage. If we open a can of soup that
right on cooking.               was stored in the upright position by removing the top,
    —G AIL S HEEHY, A MERICAN   quite often all the concentrated ingredients are on the
                  J OURNALIST   bottom and must be scooped out with a spoon. Even

4      Mad about Modern Physics
then, not all the concentrate is removed. Suppose,           The boiling temperature
instead, we turn the same can upside down and open           of water decreases
the bottom. Upon turning the can over, the soup simply       about 1.9°F for every
rushes out into the pot. Why so?                             1,000 feet above sea
                                                             level. So in Denver, the
                                                             mile-high city, water will
10. Salt and Sugar                                           boil at 202°F—that is,
                                                             at 94.4°C. Tempera-
Salts have been used for thousands of years to preserve
                                                             tures above 165°F are
meats, and sugar has been used to preserve fruits and
                                                             generally thought to be
berries. How do they work?
                                                             high enough to kill most
                                                             germs, so there is no
11. Defrosting Tray                                          danger on this account
                                                             until you get to about
In catalogs and cookware stores one can buy a “mira-         25,000 feet.
cle” defrosting tray advertised as made of an
“advanced, space-age super-conductive alloy” that
“takes heat right out of the air.” How does this defrost-
ing tray work?
                                                             On the average we get
                                                             about 9 (food) calories
12. Ice Cream Delight                                        (kcal) of energy from
Most of us have made ice cream or seen ice cream being       each gram of fat and
made. Milk, eggs, sugar, and flavorings are slowly           4 calories from each
                                                             gram of protein or
chilled. Terri likes to make ice cream in a simpler and
                                                             carbohydrate. To lose a
more efficient way. Practicing proper safety precautions,
                                                             pound (454 g) of fat,
she pours liquid nitrogen directly into the ingredients in
                                                             we have to cut the food
a metal bowl. About equal volumes of liquid nitrogen         intake by 3,500 calo-
and the mixture are used for ice cream or sorbet, and        ries. The discrepancy in
she stirs while adding the coolant until the ice cream is    numbers is due to the
nicely stiff. Why does this method produce absolutely        fact that body fat is
marvelous ice cream, and what is the physics here?           only about 85 percent
                                                             actual fat, the rest
                                                             coming from connective
13. Cooking a Roast                                          tissue, blood vessels,
For many types of meat—beef, pork, lamb, etc.—one            and other things.
can buy a roast from the butcher with or without the
bone inside. Suppose we have two beef roasts of the same

                                                               The Heat Is On        5
Light bounces off mir-                weight of 4.4 pounds (2 kg) and cook them in identical
rors; microwaves bounce               ovens at the same temperature. One roast has the bone in
off metal. If what you                and the other does not. Which roast cooks faster? Why?
put in the microwave
oven reflects too many
microwaves back                       14. Cooking Chinese Style
instead of absorbing                  Estimates of Chinese meals include more than 3,000
them, the magnetron
                                      varieties, possibly more meal types than the total num-
tube that generates the
                                      ber of meals by all other cultures combined. Many of
microwaves can be
                                      the Chinese dishes use meats cut into small cubes or
damaged. There must
always be something in                other small volumes. Certainly, these small volumes are
the oven to absorb                    much easier to eat with chopsticks. Are there any sig-
microwaves. That’s why                nificant scientific reasons for cutting up the meats into
you should never run it               small volumes?
empty.
    Metals in microwave
ovens can behave
                                      15. Baked Beans
unpredictably. Micro-                 If you buy dry beans in bulk, they must be soaked in
waves set up electrical               water overnight in a covered container before they are
currents in metals, and               ready to be baked. To bake them without soaking
if the metal object is too            would require an enormous amount of cooking time.
thin it may not be able               An alternative preparation is to “parboil” them in a
to support the current
                                      cooking pot—that is, simmer them. Simmer means “to
and will turn red hot and
                                      be on the verge of boiling.”
melt. And if it has sharp
                                           How does one know that the beans have simmered
points, it may even act
like a lightning rod and              enough? The test involves good physics. Take up a few
concentrate so much                   beans in a spoon and, after making sure that no liquid
microwave energy at the               is in the spoon, blow a stream of air gently with pursed
points that it will send              lips against the beans. If the bean skin cracks, the beans
off lightning-like                    are ready for baking. Why must the lips be pursed, and
sparks.                               why do the bean skins then crack open?
    —ROBERT L. WOLKE , W HAT
        E INSTEIN TOLD H IS C OOK :
    K ITCHEN S CIENCE E XPLAINED      16. Ice Water
                                      Normally, to cool a pitcher of water quickly, one adds
                                      ice. The ice floats at the top. Suppose one could add the
                                      same amount of ice so it could be held in the water at

6       Mad about Modern Physics
the bottom of the pitcher. Which technique would lead       A standard 12-ounce
to faster cooling of the water?                             aluminum can, whose
                                                            wall surfaces are
                                                            thinner than two pages
17. Peeling Vegetables                                      from this book (about
A friend of ours peels ripe tomatoes by impaling the        0.00762 cm), with-
tomato on a fork, then holding it over a gas flame and       stands more than 90
                                                            pounds of pressure per
rotating gently. If you try this procedure, use appropri-
                                                            square inch—three
ate safety procedures to protect your eyes and body.
                                                            times the pressure in an
    Peeling fresh beets is also a messy chore. Their col-
                                                            automobile tire.
ored liquid stains everything, including your fingers.          —W ILLIAM H OSFORD AND
Another friend of ours peels fresh beets by first boiling            J OHN D UNCAN , “T HE
them, then immediately holding them under cold water          A LUMINUM B EVERAGE C AN ,”
                                                                      S CIENTIFIC A MERIC AN ,
with a fork. What is the physics in both of these meth-                    S EPTEMBER 1994
ods used for preparing vegetables for peeling?


18. Igniting a Sugar Cube
Sugar burns in air. But igniting
a sugar cube is much more dif-
ficult than expected. Put a
sugar cube on the end of a                                  Decaffeinated coffee
toothpick and bring a lighted                               still contains caffeine!
match flame under a remote                                  A regular cup of
                                                            coffee has 80 to 135
corner. The sugar melts
                                                            milligrams of caffeine.
instead of burning, and the
                                                            For a coffee to be con-
brown, gooey stuff is caramel.
                                                            sidered decaffeinated,
    However, we wish to burn                                at least 97 percent of
the sugar, not melt it! We want to see it on fire with a     the coffee’s caffeine
flame of its own. Why is this process so difficult to         must be removed. Test-
achieve? How can we succeed in lighting the sugar cube      ing shows that decafs
with the burning match?                                     typically have 2 to 6
                                                            milligrams of caffeine
                                                            per cup.
19. Water Boiling
An open pot of water is boiling on the kitchen stove.
Sprinkle some room-temperature table salt (which

                                                               The Heat Is On               7
An object at room tem-     contains mostly NaCl and some KCl) into the clear
perature (20°C) emits      boiling water, and the boiling ceases. Isn’t it amazing
radiation with a peak at   how the water ceases its boiling as the salt warms up!
the wavelength 9.89        Can you explain the physics? What is the surprise here?
micrometers, roughly
.01 mm, in the infrared
region of the electro-
magnetic spectrum.         20. Put the Kettle On
                           Bring water to a boil in a teakettle with a spout. Let it
                           cook! Now watch the mouth of the spout carefully.
                           What do you see? Can you see the water vapor come
                           out?



                           21. The Watched Pot
                           You have probably heard the expression “A watched
                           pot never boils.” Is this statement correct physics? That
For an isolated water
                           is, when would this statement be good physics? (Hint:
molecule the H-O-H
angle is 104.5°. In ice    One should interpret the phrase “never boils” here to
each water molecule        mean that the cooking takes a longer time.)
forms hydrogen bonds
to four nearest neigh-
bors in a tetrahedral
arrangement. The tetra-    22. Ice in a Microwave
hedral bond geometry       The microwave oven emits microwaves that are
explains the openness      absorbed by water molecules in food. Microwaves
and relatively low den-    make the polar water molecules rotate or oscillate, and
sity of ice (i.e., why
                           their “friction” within the material converts some of
water expands upon
                           this kinetic energy into thermal energy to raise the tem-
freezing). In ice the
                           perature of the food.
H-O-H angles are
nearly the same as the         Suppose you made an ice block that had liquid
perfect tetrahedral        water trapped in a large cavity inside and then you
angle of 109.5°.           placed the block into a microwave oven. Could the
                           trapped water be brought to a boil while the ice
                           remained ice?


8     Mad about Modern Physics
23. The Glycemic Index                                     Night cooling by evapo-
                                                           ration of water and heat
The glycemic index is an important number for anyone       radiation had been per-
concerned with the conversion of food to blood sugar       fected by the peoples
(sucrose), for the gylcemic index gives the measured       of Egypt and India, and
rate of this conversion process. The higher the glycemic   several ancient cultures
index value, the faster the conversion rate to sucrose.    had partially investi-
There are types of sugar molecules other than sucrose.     gated the ability of
Glucose, for example, is normally the standard reference   salts to lower the freez-
for the conversion rate to sucrose, with a value of 100.   ing temperature of
    Some sample values of the glycemic index for foods     water. Both the ancient
                                                           Greeks and Romans
are: brown rice, 59; white rice, 88; table sugar,
                                                           had figured out that
65; grapefruit, 25; spaghetti, 25 to 45; potato, boiled,
                                                           previously boiled water
55; potato, baked, 85; and dates, 103. Brown rice has
                                                           will cool more rapidly
more outer layer intact than white rice, so its lower      than unboiled water, but
value is evident. But why would a baked potato have a      they did not know why;
much higher glycemic index than a boiled potato? And       boiling rids the water
how could the value for dates, or any food, be higher      of carbon dioxide and
than 100?                                                  other gases that other-
                                                           wise retard the lowering
                                                           of water temperature.
24. Electric Pickle                                        —TOM S HACHTMAN , A BSOLUTE
                                                               Z ERO AND THE C ONQUEST
Some specialty and novelty stores sell an electrical
                                                                               OF C OLD
“appliance” that cooks hot dogs between two metal
electrodes. A protective cover with a safety interlock

                                                           Interestingly, microwave
                                                           ovens are not very good
                                                           at melting ice. The
                                                           water molecules in ice
                                                           are bound pretty tightly
                                                           together into a crystal
                                                           lattice, so they can’t flip
                                                           back and forth under
                                                           the influence of micro-
                                                           waves’ oscillation.




                                                              The Heat Is On        9
Although it flies in the           closes over the device before electrical energy in the
face of common sense,              form of a standard AC current can be applied. Suppose
people with more insula-           that instead of a hot dog one places a pickle between the
tion—fat—whose body                electrodes. When the room lights are dimmed, the
core is better protected           pickle glows impressively, predominantly at one end.
from the cold, may feel            What is the physics, and what might the glow look like?
cold more quickly than
thinner people with less
protection. The reason             25. Space-Age Cooking
is that insulation keeps
                                   Microwave ovens were probably the first new method
heat in the core, away
from the skin, which               for making heat for cooking in more than a million
gets cold. When the                years. In addition, two newer methods have become
skin gets cold, you feel           available for the kitchen. Magnetic induction cooktops
cold. Paradoxically,               have been available for about fifteen years in Europe
women may feel colder              and Japan and are now becoming known in the United
than men because                   States. And for the modern chef, cooking with light in
women are better                   a “light oven” has been done since the mid-1990s and
insulated.                         may become a fad in the immediate future. How do
  —J AMES G ORMAN , “B EYOND       both of these cooking sources work?
   B RR : T HE E LUSIVE S CIENCE
   OF C OLD,” T HE N EW YORK
   T IMES , F EBRUARY 10, 2004



The main compartment
of a refrigerator should
always be below 40°F
(4.4°C). Above that
temperature, bacteria
can multiply fast
enough to be dangerous.




10      Mad about Modern Physics
2 Does Anybody
  Really Know
  What Time It Is?
               is?
   W         HAT IS TIME ?”      S T. A UGUSTINE     FAMOUSLY

             wrote. “If no one asks me, I know. But if I
   wanted to explain it to one who asks me, I plainly do not
   know.” Time itself is a strange quantity to some people. To
   many of us, time never seems to be going at the right rate—
   sometimes too fast, sometimes too slow. In some parts of the
   world, promptness and being on time are important aspects
   of the local culture. In other regions, time is almost irrele-
   vant. In this chapter, we have created a mixture of familiar
   challenges and many new ones in preparation for later chap-
   ters in which time shares its role with space as a major ingre-
   dient of motion, chapters that look at concepts such as the
   space-time of the special theory of relativity and the world of
   astrophysics.


                                                                     11
How did the day get to be         26. January Summer
divided into 24 hours?
The night appears to have         Contrary to the popular belief that Earth is closest to the
been divided first, by the        Sun on about June 23 or possibly December 22 each
ancient Egyptians.                year, the date of perihelion actually falls between January
According to Prof. Owen           2 and January 5! In the Northern Hemisphere, we expe-
Gingerich, a historian of         rience winter on this January date because the North
science at Harvard Uni-           Polar axis is tilted away from the Sun. The Southern
versity, they divided the         Hemisphere enjoys a warm summer at this time. Will the
heavens into intervals of         Northern Hemisphere ever enjoy summer in January?
10 degrees of arc, making
it possible to squeeze 12
hours, each of 10                 27. Proximity of Winter
degrees, into the shortest        Solstice and Perihelion
night. When the day also
                                  Earth reaches perihelion—the point in its orbit when
became divided, the
hours of night and day            it’s closest to the Sun—between January 2 and 5,
were of unequal length,           depending on the year. That’s about two weeks after
and the system of so-             the December solstice, December 21 or 22. Thus win-
called “unequal hours,”           ter begins in the Northern Hemisphere at about the
12 each for night and             time that the Earth is nearest the Sun. Is there a reason
day, lasted well into the         why the times of solstice and perihelion are so close, or
Middle Ages, coexisting           is this a coincidence?
with another reckoning
of the “equal hours.”
                                  28. Earth’s Speed
     —Q & A, S CIENCE T IMES ,
       T HE N EW YORK T IMES ,    The time interval required for Earth to travel from the
        D ECEMBER 13, 1983
                                  autumnal equinox to the vernal equinox (approximately
                                  179 days) is less than the time interval from the vernal to
You must remember this,
                                  the autumnal equinox (roughly 186 days). Why?
A kiss is still a kiss,
A sigh is just a sigh,
The fundamental things            29. The Equinox Displaced
  apply,
                                  At the time of the spring equinox (usually March 20) or
As time goes by.
                                  the fall equinox (September 22 or 23), night and day
         —H ERMAN H UPFELD,
         “A S T IME G OES B Y ”   are supposed to be of equal duration. But according to
                                  the almanacs of sunrise and sunset times, on the dates
                                  of the equinoxes, daytime is longer by 8 to 10 minutes.
                                  How come?

12     Mad about Modern Physics
30. The Dark Days of December                                The minute first
                                                             appeared as a division of
At latitude 40 degrees north, earliest sunset occurs on      the hour about A.D. 1320
about December 8 and latest sunrise on about January         in Paris editions of the
5. The shortest day of the year, the winter solstice, is     so-called Alfonsine
December 21 or 22. Why are all these dates not the           Mean Motion Tables,
same?                                                        sponsored by King
                                                             Alfonso the Wise of
                                                             Spain. But the idea of
31. Days of the Year                                         the minute was implicit
                                                             all the time in a method
The length of the year (i.e., the interval of time between   of reckoning used by
two successive passages of Earth through the same            early astronomers. They
point in its orbit) is about 365.2422 days. How many         employed a system of
entire rotations on its own axis does Earth execute dur-     sexagesimal fractions,
ing that time?                                               first devised by the
                                                             Babylonians, based on
                                                             successive powers of
32. Leap Years                                               60. Any unit could be
                                                             divided into 60 parts;
Every four years, in years divisible by four, is a leap
                                                             these were called in
year, when an extra day is added to the month of Feb-        Latin “partes minutae
ruary, except years divisible by 100. For example,           primae,” or “first very
1700, 1800, and 1900 were not leap years, yet 2000           small parts,” yielding the
was a leap year. Why?                                        word “minute”. A minute
                                                             in turn was eventually
                                                             divided into 60 “partes
33. Full Moons                                               minutae secundae,”
                                                             hence the word “second.”
Is the interval of time between one full Moon and the
                                                                 —Q & A, S CIENCE T IMES ,
next equal to 28 days?                                             T HE N EW YORK T IMES ,
                                                                    D ECEMBER 13, 1983


34. Moon Time
Cheryl is sitting at a desk in an office and the clock
                                                             When it comes to
shows 12:20 and the Moon is seen through the window          procrastinating, I do it
as a thin crescent with the open side pointing down-         right away!
ward to the right. What do you make of this scene?                        —A NONYMOUS
Where could the Sun be?

                                    Does Anybody Really Know What Time It Is?         13
In Wicca, February 2
(Groundhog Day) is one
of the four “greater
sabbats” that divide the
year at the midpoints
between the solstices
and equinoxes.
                                      12:20
Sundials tell Sun time
while clocks tell mean
time. The true Sun
leads or lags the mean
Sun, crossing the
meridian from 16 min-         35. Lunar Calendar
utes, 25 seconds ear-
lier than the mean Sun        Although there have been numerous calendars over the
(in early November) to        millennia of civilizations, they fall into two basic types,
14 minutes, 20 seconds        solar and lunar calendars. Today, while practically every-
later (in February). Only     one uses the solar calendar with 365.2422 days per trop-
on or about April 16,         ical year, rice farmers in many parts of the world
June 14, September 2,         continue to use the lunar calendar based on a 29.53-day
and December 25 are           lunar month. Can you figure out a scientific reason why?
the true and mean Suns
together as they cross
the meridian.                 36. The Sandglass
                              For a sandglass timer one could simply have a straight
                              glass or plastic tube with equally spaced markings and
The angle between the
                              then the whole tube would be inverted to start the time
Equator and the ecliptic
                              measurement. Why do ruled sandglasses have a tapered
(i.e., the plane of Earth’s
orbit), also known as the     “hourglass” shape instead?
tilt of the globe, was
23° 26' 32" in 2002.          37. Old Watch
Through the ages, this
value varies between          Lenni has an old mechanical watch in pristine condi-
21° and 28°. At present       tion that has an internal balance wheel that operates
it goes down by 0.47"         perfectly. She takes a drive into the mountains. Will the
per year.                     watch run fast or slow?


14     Mad about Modern Physics
38. Reading a Digital Timer                                   More people are born on
                                                              October 5 in the United
Many digital timers show the elapsed time to one-             States than on any
hundredth of a second. What is the minimum uncer-             other day. Not so sur-
tainty in the value? What value should be reported?           prising, as conception
                                                              would have fallen on
                                                              New Year’s Eve.
39. Eternal Clocks?
There are laser and atomic clocks in special laboratory
environments that are accurate to one second in 300
                                                              If 23 students are in a
million years! Yet their lifetimes are typically less than    classroom and you pick
30 years. Some wristwatches run longer! There are             two at random, the
mechanical clocks in development that could last about        probability that their
10,000 years! But they would need periodic winding.           birthdays (month and
Why do the laser and atomic clocks have such short            day) match is about
lifetimes? How might one build a mechanical clock that        1/365. The probability
would survive so long?                                        that at least two of the
                                                              23 have the same birth
                                                              date, however, is a trifle
40. Room Light                                                better than 1⁄2. The
Suppose there is a photodetector with a flash lamp at          reason is that now there
the exact center of a 3 m × 3 m × 3 m dark, barren            are 1 + 2 + 3 + . . . + 22
                                                              = 253 possible match-
room with reflective walls. The flash lamp flashes for
                                                              ing pairs.
one nanosecond. For simplicity, assume that the light is
                                                              —M ARTIN G ARDNER , “M ATHE -
emitted isotropically in all directions when the lamp          MATICAL G AMES ,” S CIENTIFIC
flashes. If the photodetector simply sums the light from        A MERIC AN (O CTOBER 1972)
all directions, what is its recorded intensity versus time?
If the photodetector is an array capable of discerning
different angular directions, what is the intensity versus    There are 365 days in
time for several different directions? Suppose the lamp       the year. Note the
flashes for one microsecond. What now?                         following:
                                                                365 = 102 + 112 + 122
41. Right to Left Driving Switch                                    = 132 + 142
                                                              Coincidence? Preestab-
Suppose you live in a country in which the driving is
                                                              lished harmony? You be
on the right and there is to be a change to driving           the judge!
on the left. If highways with on-ramps and off-ramps,


                                    Does Anybody Really Know What Time It Is?           15
Studying mid-                   and so on are built for driving on the right, will they
twentieth-century               work equally well for driving on the left? Of course,
scientists, psychologist        we must assume the same patterns of driving speeds
Bernice T. Eiduson              as before.
found a disproportionate
number who had been
confined to their beds          42. Light Clock
for large amounts of
time by childhood ill-          Some museums and labora-
                                tories have a light clock with            Mirror
nesses. During these
travails, they “searched        two parallel mirrors and a
for resources within            pulse of light bouncing back
themselves and became           and forth repeatedly, retrac-
comfortable being by            ing the same path over and
themselves”; most               over, keeping very accurate
turned to reading, and          time as each complete tran-
through reading they            sit of the light pulse is
developed a bent for                                                      Mirror
                                detected and counted. The
intellectual work. Not          mirror separation is usually
very good at sports,
                                about a meter or less, so a
unfit by illness to com-
                                very large number of reflections occur during each sec-
pete in childhood games,
                                ond of time. Suppose this light clock is moved sideways
they remained emotion-
ally fragile throughout         parallel to the mirrors at a constant velocity, and
life, deriving satisfaction     assume that the light will continue to reflect off both
mostly from intense             mirrors during this sideward movement. Will the clock
involvement in science.         continue to keep accurate time?
 —TOM S HACHTMAN , A BSOLUTE
     Z ERO AND THE C ONQUEST
                     OF C OLD
                                43. Time Reversal
                                    Frame 1                              Frame 4




16      Mad about Modern Physics
A movie is made showing successive frames for an           In a Proustian moment
object accelerating downward. If the sequence is run       an unexpected smell or
backward, the object accelerates (a) upward or (b)         taste or perhaps a song
downward. Explain.                                         from your past can
                                                           unleash in you a raging
                                                           torrent of realistic and
44. Molecular Clock                                        graphic memory. The
                                                           phrase recalls a scene
Different species of organisms have enormous regions       in Marcel Proust’s
of DNA that are the same or very similar. Humans and       Remembrance of
chimpanzees, for example, share about 98 percent of        Things Past when a
their DNA. We share much less of our DNA with              madeleine cake (a
rodents and amphibians and insects.                        small, rich cookie-like
     In a general way, the percentage of shared DNA        pastry) enables the
might be a means to establish a molecular clock—that       narrator to experience
is, the more DNA that is shared, the more recent was       the past completely as
the separation of the family tree. And, if by accident,    a simultaneous part of
the changes in the DNA happened to proceed at a com-       his present existence:
                                                           “And suddenly the mem-
mon rate, then one could set up a timeline also.
                                                           ory revealed itself: The
     However, the genetic changes do not occur with
                                                           taste was that of the
any regularity. Why not?                                   little piece of madeleine
                                                           which on Sunday morn-
                                                           ings at Combray
45. SAD                                                    (because on those
Most animals experience dramatic seasonal cycles: they     mornings I did not go
migrate, hibernate, mate, and molt at specific times of     out before mass), when
the year. These cycles appear to be hardwired; they        I went to say good
occur even when the temperature is held constant and       morning to her in her
                                                           bedroom, my aunt
the light and dark periods are varied. But humans are
                                                           Leonie used to give me,
among the least seasonally sensitive creatures, having
                                                           dipping it first in her
only a vestige of seasonal effects known as seasonal       own cup of tea or
affective disorder (SAD), an extremely mild version of     tisane.”
the cyclical responses animals experience. Only about 5
percent of adults overtly sense the seasonal changes and
suffer from SAD during the winter days of longer dark-
ness. Amazingly, light therapy—looking into a light that
mimics sunlight—or merely sleeping until dawn helps


                                   Does Anybody Really Know What Time It Is?     17
Now he has departed                 the people with SAD in northern latitudes. Would these
from this strange world             therapies be effective on people living at the Equator?
a little ahead of me.
That signifies nothing.
For us believing physi-             46. Two Metronomes
cists, the distinction
                                    Suppose the timekeeping abilities of two identical
between past, present,
and future is only a                metronomes are compared over several hours. They
stubbornly persistent               will drift faster or slower at different rates. When both
illusion.                           metronomes are placed on a skateboard that moves
  —A LBERT E INSTEIN ON LIFE -      freely horizontally, their drifts change gradually as they
 LONG FRIEND  M ICHELE B ESSO,      tend to synchronize. Each metronome has been sub-
 IN A LETTER OF CONDOLENCE
 TO THE B ESSO FAMILY, M ARCH
                                    jected to the driving force of the other, the result being
21, 1955, LESS THAN A MONTH         the phenomenon called “phase-locking” or “mode-
BEFORE HIS OWN DEATH . A LICE
                                    locking.” Suppose now that each metronome on the
      C ALAPRICE , T HE E XPANDED
             QUOTABLE E INSTEIN     skateboard begins with different initial conditions, but
                                    one of the two metronomes is driven by perturbations
                                    that fluctuate randomly in time. Can the metronomes
                                    become synchronized?
HOW TO FIND NORTH
USING A WATCH
In the Northern Hemi-               47. Time Symmetry
sphere, hold the watch              The fundamental equations of physics—at least those
horizontal and point the            that derive from symmetries in nature—all exhibit time
hour hand at the Sun.               symmetry because they are second-order differential
Bisect the angle
                                    equations. Newton’s second law and Maxwell’s equa-
between the hour hand
                                    tions are immediate examples. However, one can con-
and the 12 o’clock mark
                                    sider time running forward or backward. Even
to get the north-south
line. If your watch is set          general-relativity equations formulated in tensor math-
on daylight saving time,            ematics exhibit time symmetry. Assuming that all these
use the midway point                equations are correct, must nature at its most funda-
between the hour hand               mental level obey time symmetry? (Note: Entropy rela-
and one o’clock. The                tions are not derived from a fundamental symmetry
farther you are from                and therefore are excluded.)
the Equator, the more
accurate this method
will be.


18       Mad about Modern Physics
3 Crazy
  Circles
   S        PACE IS RELATED TO POSITION , DISTANCE , AND

            size and has its own paradoxes and influences. We
   live in a space of three dimensions, but our ability to visual-
   ize three-dimensional relationships among objects is not as
   easy as judging distance. Our brain activity relies on neural
   connections in a 3-D biomass that would probably become
   moronic if limited to two dimensions. However, robots usu-
   ally operate in our 3-D space by following computer pro-
   grams that maneuver in multidimensional configuration
   spaces that often far exceed three dimensions. Recent theo-
   retical research in quantum physics hints that the natural
   world may be as large as 11-dimensional, with seven dimen-
   sions curled up too small for our senses, leaving the four
   dimensions of space-time. In this chapter we have created a
   mixture of familiar challenges and many new ones regarding
   space in preparation for a later chapter on the space-time of
   the special theory of relativity.

                                                                     19
                                      48. Spider and Fly
A Pythagorean triplet is              On a plane the shortest distance between two points
a set of three numbers                is a straight line. Suppose a spider sits on a cube and
that describes the sides              wants to catch a fly sitting on the opposite face. How
of a right triangle.                  would you determine the path of shortest distance for
Pythagoras invented his               the spider to crawl on the surface to catch the fly?
theorem around 550
B.C., but the Babyloni-
ans had catalogued per-               49. Moon Distance
haps hundreds of                      In measuring the length of a 1-meter table with a
triplets by 2000 B.C.,                meterstick to within 0.1 millimeter, the uncertainty in
long before Pythagoras.
                                      the measurement is one part in ten thousand. Meter-
One of the triplets the
                                      sticks, however, are usually inconvenient for measuring
Babylonians found is
                                      the distance to the Moon. Instead, a laser light pulse
the enormous
3,367:3,456:4,825.                    can be reflected from a stationary corner reflector on
            —D ICK T ERESI , L OST    the Moon similar to the reflectors on bicycles, and the
D ISCOVERIES : T HE A NCIENT R OOTS   total duration of the pulse from Earth to Moon and
OF M ODERN S CIENCE — FROM THE
                                      back to Earth again is timed. What do you estimate for
       B ABYLONIANS TO THE M AYA
                                      the uncertainty in the measurement for the Moon’s dis-
                                      tance? Which determination would you expect to have
                                      the greater distance uncertainty, the table length or the
In the hallowed groves                distance to the Moon?
of the academe they
whisper the tale of a
physicist who spent
                                      50. Ideal Billiards Table
the first half of his life
trying to become
famous, at which he
failed; then spent the
second half of his life
trying to convince him-
self it wasn’t important
to be famous, at which
he also failed.




20         Mad about Modern Physics
Suppose you have an ideal rectangular billiards table on   In the Conics of Apollo-
which a ball collides with any wall (called the cushion)   nius, the words “ellipse”
so that the angles of incidence and reflection are equal.   (defect), “parabola”
Let there be pockets at the corners only. Describe how     (equality), and “hyper-
to shoot a given ball into a specific corner pocket with    bola” (excess) were
either zero, one, two, or three banks of the ball.         applied to the three
                                                           curves now known by
                                                           these names because of
51. Wallpaper Geometry                                     the relationships y 2 <
                                                           px , y 2 = px , and y 2 >
Some of the old video games used an interesting but
                                                           px , respectively, where
simple visual technique to extend the playing field. A
                                                           p is the parameter
character running off the right side of the screen then    (latus rectum) of the
entered the left side while the background scenery         curve which is so placed
remained fixed. That is, the right side edge is matched     upon a coordinate sys-
to the left side edge, and the top and bottom are          tem that a vertex is at
matched also. One could even have a rectangular array      the origin, and the axis
of video screens, each right edge matched to a left        of the curve lies along
edge, etc., each screen showing the same image. Faster     the axis of abscissas.
systems later came along, and the scenery moved            Hence one can see that
instead, and these 2-D views were eventually replaced      Apollonius applied the
by 3-D views.                                              name “ellipse” to indicate
    Consider now a 3-D regular array of cubes touch-       not a defective circle but
                                                           a defective parabola.
ing face to face and top to bottom, the 3-D space ana-
                                                            —C ARL B. B OYER , “L ETTERS ,”
log to the old style 2-D video game. Let opposite cube              S CIENTIFIC A MERIC AN
faces be matched and imagine that these face surfaces                  (F EBRUARY 1960)
are invisible. You are standing in one cube inside this
space and look to your right. Behold! You see yourself!
                                                           I put tape on the mirrors
                                                           in my house so I don’t
                                                           accidentally walk
                                                           through into another
                                                           dimension.
                                                           —S TEVEN W RIGHT, C OMEDIAN
      You are in
      this cube.



                                                              Crazy Circles            21
The Chinese mathe-                  What exactly do you see? What do you see when look-
matician Liu Hui calcu-             ing upward?
lated a value for π
(3.1416) in A.D. 200
that remained the most              52. Space-Filling Geometry
accurate estimation for
a thousand years.
          —D ICK T ERESI , L OST
     D ISCOVERIES : T HE A NCIENT
 R OOTS OF M ODERN S CIENCE —
        FROM THE B ABYLONIANS
                    TO THE M AYA




FOUR-DIMENSIONAL
GEOMETRY IN THE BIBLE?
St. Paul’s Letter to the
                                    Cubes can be placed next to each other in three direc-
Ephesians contains the
                                    tions to fill all of 3-D space. Regular octahedrons can
following passage: “that
you, being rooted and               fill 3-D space also. Spheres of the same radius cannot.
grounded in love, may               Can regular tetrahedrons fill all of 3-D space and leave
be able to comprehend               no gaps? Can regular dodecahedrons and regular icosa-
with all the saints what            hedrons?
is the width and length
and depth and height”
(Ephesians 3: 17–18).               53. Archimedes’ Gravestone
—M ARTIN G ARDNER , “M ATHE -       Archimedes’ gravestone is said to have a sphere inside
  MATICAL G AMES ,” S CIENTIFIC
 A MERIC AN (S EPTEMBER 1975)
                                    a cylinder etched into the stone as well as the symbol π.
                                    How are the two 3-D objects related if they have the
                                    same radius? And why are they on his gravestone?
A SILLY SYLLOGISM
Nothing is better than
  eternal life;
A salami sandwich is
  better than nothing;
Therefore, a salami
  sandwich is better
  than eternal life!



22       Mad about Modern Physics
54. Brain Connections                                       THE LATE APPEARANCE IN
                                                            ENGLISH OF THE WORD
The human brain has more than 100 billion neurons,          “SCIENTIST”
with each neuron receiving input signals from 10 to         In 1840 William Whewell
1,000 other neurons. Schematic representations of           noted that there was no
these connections in the brain always show an incredi-      simple and natural way
ble web of lines representing the neurons, either as a      to refer to “a cultivator
2-D or a 3-D image. Suppose you created a scaled-           of science in general.”
down computer model of this human brain using only          He was, he concluded,
1 million neurons in a 3-D space. On average, how           inclined to call him “a
many input connections would each neuron have?              scientist.” Before
What is the surprise here?                                  Whewell scientists
                                                            tended to refer to each
                                                            other as philosophers,
55. Configuration Space                                     or more fully, as natural
                                                            philosophers. For this
Suppose we have a robotic arm that mimics the move-
                                                            reason Newton’s treatise
ments of a person’s arm. The arm exists in the familiar
                                                            on mathematical physics
3-D physical space. Consider a simplification of the         was given the title The
robotic arm that assumes just three connected parts:        Mathematical Principles
upper arm, forearm, and hand, all in the shape of           of Natural Philosophy
straight rods that are connected. The body of the robot,    (1687).
including the shoulder, remains fixed in position. We
wish to have the robotic arm touch a particular point-
like object in the room. How many numbers are
required in a computer program to describe the arm
position?                                                   The Indian mathematician
                                                            Srinivasa Ramanujan
                                                            (1887–1920) discovered
56. Farmer Chasing a Goose                                  an approximation to π
                                                            that is remarkable for its
Farmers know that to catch a stray goose one does not
                                                            precision and concise-
run after the goose in an open field. A better strategy is
                                                            ness: (2143/22)1/4 =
to corner the goose. However, suppose the farmer and
                                                            3.14159265258 . . .
the goose are in an open field and they both run with        (to be compared with π =
the same speed, V, to provide us with some semblance        3.14159265358 . . .).
of fair play. Furthermore, restrict the farmer to chasing
the goose along the instantaneous line of sight to the
goose. When will the farmer catch the goose?


                                                              Crazy Circles      23
Stigler’s law of                 57. A Spooky Refrigerator
eponymy, formulated by
statistician Stephen             Christina notices that food is disappearing from her
Stigler, states that no          refrigerator and yet her surveillance camera shows that
scientific discovery is          no one is opening the door. Suppose our 3-D spatial
named after its original         world were really a 4-D spatial world, but we did not
discoverer. Journalist           know anything about the existence of the fourth spatial
Jim Holt points out that         dimension. There is still the single time dimension.
Stigler’s law itself is          Could a 4-D being remove food from her 3-D refriger-
self-confirming, given           ator without opening the refrigerator door?
that Stigler admits that
it was discovered by
someone else: Robert             58. Fractional Dimensions?
Merton, a sociologist of
                                 A point has zero dimensions. A line has one dimension.
science. The most noto-
                                 A plane has two dimensions. Space has three dimensions.
rious example of
Stigler’s law is probably        Can something have 1.585 . . . spatial dimensions?
the Pythagorean theo-
rem, widely known by             59. Platonic Solids
the ancient Egyptians,
Babylonians, and Indi-
ans long before
Pythagoras.
   —A DAPTED FROM J IM H OLT,
“M ISTAKEN I DENTITY T HEORY,”
L INGUA F RANC A (M ARCH 2000)



DIVINE MADNESS
The word “theory” comes
from the Greek word
theoria, meaning
“ecstatic contemplation
of the truth,” as exem-
plified in Plato’s belief
that “the greatest truths        There are five 3-D regular polyhedrons called the
are those that come to           Platonic solids: the regular tetrahedron (4 faces), the
us through divine                regular hexahedron (cube), the regular octahedron (8
madness.”                        faces), the regular dodecahedron (12 faces), and the


24       Mad about Modern Physics
regular icosahedron (20 faces). All these solids have a      The size of the Moon
twofold rotational symmetry axis through the center of       compared to the Earth
each edge—that is, a rotation about this axis by 180         is 3:11 (with accuracy of
degrees leaves the object looking the same as the initial    99.9 percent). This
view. But the regular tetrahedron does not have inver-       Earth–Moon proportion
sion symmetry.                                               is also precisely invoked
                                                             by our two planetary
    If we intersect two identical regular tetrahedrons so
                                                             neighbors, Venus and
their centers coincide, can the composite object have a
                                                             Mars. The closest : far-
twofold rotational symmetry axis? Can it have inver-
                                                             thest distance ratio that
sion symmetry?                                               each experiences of the
                                                             other is, incredibly, 3:11
60. Intersecting Spheres                                     (with accuracy of 99.9
                                                             percent). Quite by
                                                             chance, 3:11 is 27.3
                                                             percent, and the Moon
                                                             orbits the Earth every
                                                             27.3 days, also the
                                                             average rotation period
                                                             of a sunspot.
                                                                       —J OHN M ARTINEAU ,
                                                             A L ITTLE B OOK OF C OINCIDENCE




If in 2-D we intersect two circles (called one-spheres by
mathematicians), the intersection is either a point, two     The Roman numeral
                                                             representing “five,”
points, or a circle. In 3-D the intersection of two
                                                             symbolized by the letter
spheres (each called a two-sphere) will be either a
                                                             V, derives from the
point, a circle, or a sphere. What can the intersection of
                                                             shape of the space
two three-spheres be? And three three-spheres?               between the open
                                                             thumb and index finger.
61. Arm Contortions                                          The Roman numeral for
                                                             “ten,” the letter X, is
Normally, the rotation of an object about a fixed axis        actually two V’s.
by 360 degrees brings the object back to its initial ori-
entation. However, Barbara has the agility to do the
following double rotation. She places a small object or
book in her right hand, holding the object horizontal


                                                               Crazy Circles          25
The biblical approxima-        and noting its orientation in the room. While imagining
tion of π is given in          a vertical axis from floor to ceiling, the book is moved
1 Kings 7:23 and is            inward first and then under the upper arm, keeping the
repeated in 2 Chroni-          book horizontal and rotating the object completely
cles 4:2. Both verses          around this vertical axis back to its initial position. Her
speak of a circular “sea       arm is now twisted. Can she untwist by rotating her
of cast bronze” with a
                               arm a second time in the same direction?
diameter of 10 cubits
and a circumference of
30. The Greeks used a
more accurate value of
22/7 (error 0.04 per-
cent), and the Egyptians
used a ratio of two
squares 256/81 (error
0.6 percent).


The other day, I was
walking my dog around
my building . . . on the
ledge. Some people are
afraid of heights. Not
me, I am afraid of
widths.
—S TEVEN W RIGHT, C OMEDIAN



R. G. Duggleby, a bio-
chemist at the Univer-
sity of Ottawa, found
that the sum of π to the
fourth power (97.40909
. . .) and π to the fifth
power (306.01968 . . .)
is e (i.e., 2.7182818 . . .)
to the sixth power
(403.42879 . . .),
correct to four decimal
places!


26       Mad about Modern Physics
62. The Rotating Cup                                        The ancient composers
                                                            of Vedic literature in
Place a cup with a handle on a shelf at eye height. Now     India had to develop a
walk in a straight line at a nearly constant speed past     method of evaluating
the cup, all the while rotating your head to observe the    square roots. The tech-
orientation of the cup. Notice what you see. The cup        nique apparently
appears to rotate in the direction opposite your walk-      evolved from a need to
ing direction, at first very slowly, then quickly, then      double the size of a
slowly again. Now consider yourself to be stationary        square altar. One needs
and imagine the cup itself moving past in a straight line   a square whose sides
with constant speed. You could try to demonstrate this      are the square root of
                                                            2. In the Sulbasutras, a
with the cup in your hand. What do you see now?
                                                            collection that dictates
                                                            the shapes and areas of
63. Space and Time Together                                 altars and the location
                                                            of the sacred fires, the
To explain Einstein’s 1905 special theory of relativity     square root of 2 is
and Minkowski’s 1908 unification that combines three         stated as 1.414215 . . . ,
space dimensions and one time dimension into a four-        an amazingly accurate
dimensional space-time continuum, most introductory         value! The Sulbasutras
physics textbooks use a four-dimensional coordinate         were written between
system, with three real coordinates for space and one       800 and 500 B.C.,
imaginary coordinate for the time coordinate. Why not       making them at least as
four real coordinates? Why not have three imaginary         old as the earliest
space coordinates and one real-time coodinate?              Greek mathematics.
                                                            The Greeks, however,
                                                            had no positional nota-
64. Space > 3-D?                                            tion system. Hence
                                                            their approximations of
Can you provide arguments for why space has three
                                                            the square roots were
dimensions? Hint: Are planetary orbits stable in a space
                                                            rather crude.
of n dimensions, where n > 3? Is the hydrogen atom
                                                                —G EORGE G HEVERGHESE
stable when n > 3?                                              J OSEPH , T HE C REST OF THE
                                                                  P EACOCK : N ON -E UROPEAN
                                                                    R OOTS OF M ATHEMATICS




                                                               Crazy Circles           27
4 Fly Me to
  the Moon
   W         E LIVE IN A WORLD OBEYING THE RULES OF

             nature. But this natural world described by physics
   and the other sciences can be superseded and replaced by the
   imagination of the human mind. The artificial worlds cre-
   ated in the many forms of literature and in audio and visual
   renderings today cast a powerful influence on the minds of
   everyone in the modern world. In fact, more people prefer to
   live in these artificial fantasy worlds than in the real world
   than are willing to admit. In these challenges we focus on
   some of the “fuzzy science” prevalent in movies and televi-
   sion shows. In certain ways, the awareness of the correct sci-
   ence can enhance your enjoyment of the entertainment
   product, just like knowing how a bee communicates to the
   other bees in its hive enhances the beauty of the bee itself.




                                                                    29
The anagram of              65. Gunfight
MOON STARERS is
ASTRONOMERS.                Some TV programs and films have high drama scenes
                            based on a victim being shot by the pursuer and being
                            “blown backward” a meter or two by the projectile
                            impact. Is this dramatic response Hollywood hype, or
Jonathan Swift’s            is there good physics here?
Gulliver’s Travels,
published in 1726,          66. Body Cushion
describes Gulliver’s
many adventures,            A fall from a height of several stories onto pavement or
including his “Voyage to    even onto a lawn will produce serious injuries or even
Laputa.” Gulliver learns    death. Yet we have seen the movie hero going over the
that the scientists there   edge of a roof holding another human body in position
discovered two moons        just beneath to cushion the fall on impact. Certainly,
of Mars, which revolve      collision with this second body is better than direct col-
around the planet at        lision with the ground. What do you think about the
distances from its cen-     advantages here?
ter equal to 3 and 5
Martian diameters.
When the moons of           67. Cartoon Free Fall
Mars were discovered        So many of us in our youth learned the laws of nature
by Asaph Hall in 1877, it   from cartoons. Some of us are still learning from
turned out that Swift       cartoons! The cartoon character steps forward off
not only had the number     a cliff and remains there in
of the moons right, but
                            suspension until realizing the
he also placed them
                            situation, then the acceleration
close to the actual
                            downward begins. As you
distances: 1.4 and 3.5
diameters of Mars.          recall the scene, what violations
The two moons, named        of physics can you discern?
Phobos and Deimos, are
tiny. Phobos, which         68. Silhouette
measures 27 by 19 kilo-
                            of Passage
meters, is shaped rather
like a potato. Deimos,      When a cartoon character
too, is oddly shaped,       smashes through a solid wall or
and measures 15 by 11       other object, we see the perfo-
kilometers.                 ration as the crisp outline of the


30      Mad about Modern Physics
character. What would a condensed matter physicist            Jules Verne’s From the
say about this cookie cutter type of material response?       Earth to the Moon was
                                                              published in 1865.
                                                              Breaking with literary
69. Artificial Gravity                                        tradition, which called
                                                              for recounting such a
                                                              voyage only as an imagi-
                                                              nary undertaking, Verne
                                                              based his account on an
                                                              extrapolation of contem-
                                                              porary scientific princi-
                                                              ples. The resulting
                                                              prophetic qualities of
                                                              this novel are uncanny.
                                                              For instance, Verne
We all know that a body will tend to “float” around in         chose a launch site not
a space station orbiting Earth or in a spaceship cruising     far from Cape Canaveral
                                                              in Florida; he also gave
at a constant velocity with respect to the stars. Some
                                                              his readers the initial
films depict a dumbbell-shaped space station rotating
                                                              velocity required for
about an axis through its middle perpendicular to the         escaping the earth’s
long axis in order to provide artificial gravity. What         gravitation. In the sequel,
interesting behavior patterns might be experienced by         Around the Moon, Verne
an astronaut who walks across the axis from one end           correctly described the
to the other?                                                 effects of weightless-
                                                              ness, and he even
                                                              pictured the space-
70. Small                                                     craft’s fiery reentry and
Wings                                                         splashdown in the
                                                              Pacific Ocean—
Space heroes who
                                                              amazingly, at a site just
visit other planets                                           three miles from where
have encountered alien                                        Apollo 11 landed on its
beings who suspend them-                                      return from the Moon in
selves in the air with two                                    1969.
small beating wings, each                                       —A RTHUR E VANS AND RON
about 40 centimeters long,                                            M ILLER , “J ULES V ERNE ,
                                                               M ISUNDERSTOOD V ISIONARY,”
attached to their backs. These                                          S CIENTIFIC A MERIC AN
characters are less than a                                                      (A PRIL 1997)


                                                            Fly Me to the Moon             31
The first public suggestion        meter tall but probably have a mass of at least 20 kilo-
that Abraham Lincoln be            grams. Could these wings suffice?
the Republican candidate
for president is believed to
be a letter written Novem-         71. Shrunken People
ber 6, 1858, and published         Suppose someone is shrunken by some gimmick in the
in the Cincinnati Gazette.
                                   movies. Let’s say that you suffer this consequence and
The writer, Israel Green
                                   are now 100 times smaller in all dimensions. Actually,
(a druggist in Findlay,
Ohio), proposed a ticket           there is a lot of space between the atoms and molecules
of Lincoln for president,          of our bodies, but let’s ignore any increase in repulsive
and John Kennedy for               forces, etc., and assume that this shrinkage can be
vice president. The pro-           done. What does physics tell you will be a major prob-
posed Kennedy was John             lem as you walk?
Pendleton Kennedy of
Maryland, a prominent
author and politician who          72. Spaceship Designs
had been Millard Fillmore’s        The simple but effective spaceships of Buck Rogers and
Secretary of the Navy.
                                   Flash Gordon have been superseded by flashy new
         —M ARTIN G ARDNER ,
       T HE M AGIC N UMBERS OF
                                   designs with interesting shapes, sizes, and abilities. The
                   D R . M ATRIX   advent of the space age in the 1950s brought about a
                                   heightened awareness of the practical physics charac-
According to NASA, there           terizing a successful rocket or spaceship. Yet today,
is only one proven case of         more than 50 years later, the ingenuity of the movie
a human who was hit by an          industry continues to defy the laws of physics. We see
object from outer space,           the latest nuclear-powered spaceships operating in
Lannie Williams of Tulsa,          space coming in for a landing on Earth (or other com-
Oklahoma. In 1997, when            parable planet) at a spaceport and then taking off for
Ms. Williams was power             space a little while later in the same ship from the same
walking, she felt a tap on
                                   spaceport. Why can’t we do this feat with present-day
the back of her shoulder.
                                   space vehicles?
As something rolled off her
shoulder, she heard it hit
the sidewalk with a metallic       73. Warp Speed
thud. When she looked
back, she saw a mangled            Spaceships are known for their ability to turn on their
lump of metal, which later         warp drives to accelerate to speeds beyond the speed of
turned out to be a piece           light. Can present-day physics conceptually explain this
of a U.S. rocket.                  capability?

32      Mad about Modern Physics
74. North Pole Ice Melt                                           H. G. Wells’s 1914 novel
                                                                  The World Set Free, a
Environmental disasters have always been popular                  speculative history of
with filmmakers. In recent years, the trend has been               the future, contains the
toward disasters on a global scale because the public             following sentence:
has become more aware of global environmental                     “Nothing could have
problems. If there is a global warming trend, there               been more obvious to
could be much ice melting at the poles of Earth. Some             the people of the early
films have portrayed seacoasts being inundated by the              twentieth century than
rising water level. What would you predict for the sea            the rapidity with which
level change if the ice only at the North Pole melted             war was becoming
                                                                  impossible. And as cer-
completely?
                                                                  tainly they did not see
                                                                  it. They did not see it
                                                                  until the atomic bombs
75. Lightning and Thunder                                         burst in their fumbling
                                                                  hands.”
We see the flash of distant lightning and hear its thun-
der roll simultaneously in the movie. But we all know
that the lightning flash arrives before the thunder in the
real world, there being about five seconds of sound
delay for each mile of distance to the lightning. Suppose         How would you suspend
you were in charge of a battle scene in a war movie.              500,000 pounds of
When editing the scenes of the explosions on the bat-             water in the air with no
tlefield, how would you ensure the correct experience              visible means of sup-
for the theater patron?                                           port? (Answer: build a
                                                                  cloud!)
                                                                         —B OB M ILLER , A RTIST


76. Explosions in Outer Space
Explosions in outer space on the big screen are magnif-
icent to behold. Brilliant colors of stuff shooting               On November 10, 1907,
outward in all directions, decreasing their density as            the magazine section of
the inverse square of the distance. The sound of the              the New York Times
explosion rocks the spaceship with a thundering roar              included the headline
just as the light flash is first seen. Finally, bits of debris      “Martians Probably
scream past. What do you think about this space                   Superior to Us.”
physics?

                                                               Fly Me to the Moon         33
The American socialist       77. Space Wars
writer Edward Bellamy
(1850–1898), in his          One space battlecruiser after another shoots powerful
best-selling 1888 novel      laser beams that destroy the enemy’s space battlecruiser.
Looking Backward:            We see the powerful red laser beams strike the oppo-
2000–1887, antici-           nent, and we hear the explosion as the object blows
pated, by seven years,       apart. What wonderful physics can be learned here?
the motif of time travel
H. G. Wells was to use
in his Time Machine          78. Security Lasers
(1895). Bellamy’s vision
of the future included,      Quite often the drama in a crime movie or an adventure
among other things,          movie is enhanced by having crisscrossed visible laser
shopping malls and           beams around the item to be protected from theft. The
“credit cards” (his own      thief must avoid intersecting these beams during the
expression).                 escapade to steal the item; otherwise a security alarm
                             will notify the appropriate authorities and the thief will
                             be caught. If you were the movie’s director, how would
                             you make this scene to ensure good physics?

PERVERSITY OF
INANIMATE OBJECTS?           79. Bullet Fireworks
As many a scientist          Bullets bounce everywhere. The bad guys shoot a
concluded with con-          lengthy burst of submachinegun fire as the hero runs
sternation, there is a       through an industrial plant. The bullets impacting on
reason why there is          steel railings, for example, give off bright flashes of
a word demon in
                             light. This scene is a dramatic event for almost anyone
demonstrations.
                             watching the hero in a time of great peril. What can
                             you say about the physics here?


                             80. Internet Gaming
Were the sun and moon
  to doubt                   For years people have been playing “live time” games
They’d immediately go        over the Internet. If the game is checkers or poker, for
  out.                       example, each player must take his or her turn in
          —W ILLIAM B LAKE   proper order, so short delays are not a problem. Even


34      Mad about Modern Physics
when the game is a world-domination board game with             PERIODIC CLEANUP
multiple players, each player can submit moves at any           OPERATION
time before the deadline. But many video games require          Every time the solar
simultaneous play by several players, so delays can             cycle peaks, it causes
mean life or death for a player’s combatant in a shoot-         Earth’s atmosphere to
’em-up type of action game. One can hear comments               expand and pull in low-
by some action game players that they tried to make             orbiting debris, which
their move but the Internet was too slow. What is the           burn up on reentry.
possible truth here?
                                                                Jules Verne died in
                                                                1905. A memorial
81. Cartoon Stretching                                          sculpture placed over
Objects in cartoons are stretched and squeezed into             his grave depicts Verne
amazing distortions and then released. Some of the              rising from his tomb,
characters suffer the same fate. When the body material         one arm reaching
of a cartoon character is being pulled, we often see the        toward the stars. Some
                                                                two decades later an
part closer to the applied force stretch first and then the
                                                                American periodical
rest follow with a small time delay. For example, a car-
                                                                called Amazing Stories
toon dog may pull on a character’s leg, which we see
                                                                —the first magazine
being stretched while the torso remains normal, until           exclusively to feature
finally the torso stretches, the arms stretch, and the           tales of science and
character releases his or her handgrip from the door-           adventure—used a rep-
way. Using some physics concepts, what can you say              resentation of Verne’s
about the speed of sound in a cartoon character’s body?         tomb as a logo. To
                                                                describe these narra-
                                                                tives, the publisher,
82. Infrared Images                                             Hugo Gernsback, coined
                                                                the term “scientifiction,”
In crime dramas and in adventure films the result of an          which was later changed
infrared vision device is often reconstructed and shown         to science fiction.
in a sharp greenish or black-and-white format. We see           —A RTHUR B. E VANS AND RON
the infrared faces of people as if they were originally                 M ILLER , “J ULES V ERNE ,
                                                                 M ISUNDERSTOOD V ISIONARY,”
color images seen normally with one’s eyes, but now
                                                                          S CIENTIFIC A MERIC AN
these color images have been converted to black and                               (A PRIL 1997)
white. Is there any physics violation here in depicting
the infrared images via this process?



                                                             Fly Me to the Moon             35
One of the big uncer-            83. Light Sabers
tainties in calculating
the path of an asteroid          Enemies dueling with light sabers have graced the silver
involves how much sun-           screen for several decades now. Isn’t this type of
light it absorbs and then        weapon the most ridiculous thing you’ve ever seen?
reradiates as thermal
energy. Such radiation
can, over the centuries,         84. Force Fields
gently push the asteroid         In battle scenes of many science fiction movies we see
into a different orbit,          the baddies roll up with their giant laser guns to shoot
much as a tiny rocket
                                 the good guys, who are protected by a visibly transpar-
would. So, if scientists
                                 ent force field. Why do the laser beams suffer deflection
in future years should
                                 at the force field?
conclude that a collision
looks ever more likely,
then they can probably
                                 85. Cold Silence of Space
find ways to alter the
asteroid’s radiation pat-        In the “cold silence of space” begins many a description
tern by dusting its sur-         of space between planets. Can this statement survive a
face with soot or                physics analysis?
powdered chalk or drap-
ing it with reflective
Mylar. Such tinkering            86. Nuclear Submarine
could be enough to
                                 Several movies have involved an out-of-control nuclear
nudge the asteroid
                                 reactor aboard a nuclear submarine. We are told that
safely away.
                                 the containment vessel is about to fail and that the best
          —E DITORIAL D ESK ,
      “E NCOUNTER W ITH AN       action is to move the sub several hundred meters
  A STEROID,” N EW YORK T IMES   underwater. When the explosion occurs down there,
              (A PRIL 8, 2002)
                                 what might happen?


If you are a scientist,          87. Plutonium vs. Uranium
you’re more likely to be
killed in a film than a          Suppose you find a nuclear bomb and decide to trans-
member of any other              port the device to a safe hiding place. Would there
profession, including a          be any difference with regard to your safety as to
Mafia hit man.                   whether the device is made of uranium-235 or
              —C ARL S AGAN      plutonium-239?


36      Mad about Modern Physics
88. Nuclear Detonation                                        In a sense, 1:Sun = Moon,
                                                              and 1:Moon = Sun!
The threat of the detonation of a hydrogen nuclear            1:365.242 = 0.0027379,
warhead by striking one with another object such as a         which in days is 3 min-
missile or the shrapnel from a nearby explosion lurks in      utes and 56 seconds, the
a scriptwriter’s creative mind for many war and adven-        difference between side-
ture films. Suppose the nuclear warhead is aboard an           real and solar days, while
ICBM and is struck by an interceptor missile or the           1:27.322 = 0.0366,
warhead itself is penetrated by fast-moving BBs. What         which in days is 52 min-
will happen?                                                  utes, the difference
                                                              between lunar and solar
                                                              days.
89. Fabric of Space-time                                             —ROBIN H EATH , S UN ,
                                                                         M OON , & E ARTH
Conjectures about the “fabric of space-time” and
“tears in the space-time continuum” abound in science
fiction movies. An entertaining 2001 film involved a            On the average there is
protagonist who derived an equation for the time and          one catalogued satellite
place of a temporary tear in the fabric of space-time.        that falls back to Earth
Several characters jumped off the Brooklyn Bridge             uncontrolled every single
through the temporary space-time tear, acting as a            day and has been since
portal to another dimension to the year 1876, and then        the early 1960s. Most of
returned through the next temporary tear by jumping           them vaporize high in the
off the bridge again days later. In addition, film charac-     atmosphere.
ters have used the phrase “speed of gravity” in an
ambiguous way. What can you say about the physics
                                                              The first sentence in
here?
                                                              H. G. Wells’s 1908 novel
                                                              The War in the Air reads:
                                                              “Lower Manhattan was
                                                              soon a furnace of crimson
                                                              flames, from which there
                                                              was no escape.”




                                                            Fly Me to the Moon       37
5 Go Ask
  Alice
  P       ERHAPS NO OTHER ASPECT OF TWENTIETH -

          century physics has captured the imagination of the
  general public more than the concepts of the special theory
  of relativity (STR). Absolute time and absolute space are
  forever cast aside in favor of a union of space and time into
  one important entity called space-time. This four-dimen-
  sional world of space-time has spawned an enormous num-
  ber of conjectures about the behavior of nature. Among
  these conjectures are time travel, two people aging at differ-
  ent rates when one remains on Earth and the other travels on
  a space journey, the ability to see the back side of an
  approaching cube, and the conversion of mass into energy.
  As you know, the STR is based on the idea that two
  observers in different inertial reference frames must each
  experience physics described by the same basic laws. Even
  though these two inertial reference frames are moving with



                                                                   39
                        a constant velocity with respect to each other, the speed
                        of light in a vacuum is the same for both observers. The
                        important quantities in STR are the invariants. For
                        many people, the most useful invariant is the space-
                        time interval τ, defined by τ 2 = c2 ∆t 2 – ∆x2 – ∆y2 – ∆z 2.
                        For others, the four-momentum invariant E2 – p2c2 =
                        m2c4 is the most useful because E0 = mc2 can be derived
                        directly, where the mass m is a constant, the same at all
                        speeds, all places, and all times. Many challenges in this
                        chapter test your ability to use these invariants.




40   Mad about Modern Physics
90. Spotlight                                                Joseph Larmor in 1900,
                                                             stimulated directly by
Can a spot of light move faster than c, the speed of         the Michelson-Morley
light? For example, if a lighthouse light beacon spins       experiment, gave for
around at very high speed, will the spot of light seen far   the first time the full
from the beacon cut across the sky with a speed greater      Lorentz transformations
than 3 × 108 m/s?                                            for coordinates and
                                                             time, as well as electric
                                                             and magnetic field com-
91. Quasar Velocity                                          ponents, and showed
Quasars have been detected that have recessional veloc-      that the Maxwell equa-
ities greater than the speed of light c based on the cos-    tions remain exactly
                                                             invariant under these
mological relationship for the redshift z, namely, l + z =
                                                             transformations. It
exp[v/c]. That is, there are quasars with z > 3, for
                                                             has long appeared a
example. Also, to explain the present state of the uni-
                                                             historical anomaly that
verse, the inflationary big bang model requires a faster-     Larmor’s work, which
than-light expansion of space in the young universe.         preceded Lorentz’s by
Are these examples violations of the special theory of       four years, is so little
relativity?                                                  known among physi-
                                                             cists. Earlier still, in
                                                             1897, Larmor had
92. Spaceship Approach                                       discovered time dilation.
A spaceship is approaching Stephanie at the relativistic     Woldemar Voigt’s paper,
speed of v/c = 0.98974. What does she see as the space-      published in 1887, con-
ship nears and then passes? Hint: for simplicity, con-       tains an early version of
sider a cube approaching in place of the spaceship.          the Lorentz transforma-
                                                             tions that appear to be
                                                             almost the same except
93. Mass and Energy                                          for a scale factor.
                                                               —C. K ITTEL , “L ARMOR AND
A symbol of the twentieth century is the famous Ein-                THE P REHISTORY OF THE
stein relation between mass and energy. Here are four        L ORENTZ T RANSFORMATIONS ,”
possible equations: (1) E0 = mc2 (2) E = mc2 (3) E0 =          A MERIC AN J OURNAL OF P HYSICS
                                                                           (S EPTEMBER 1974)
m0c2 (4) E = m0c2. In the equations c is the velocity of
light, E is the total energy of a free body, E0 its rest
energy, m0 its rest mass, and m its mass.
    Which of these equations expresses one of the main
consequences of the STR? Which equation was first


                                                                 Go Ask Alice            41
Albert Michelson was              written by Einstein and was considered by him a con-
born in Strzelno,                 sequence of STR?
Poland, in 1852. The
town, about 150 miles
northwest of Warsaw,              94. Strain Gauge
was then under Pruss-             A long rectangular bar of metal sits at rest in my refer-
ian rule. His parents
                                  ence frame. The strain gauge attached to its middle
were Samuel Michelson
                                  reads zero. Now I run in the direction parallel to the
and Rozalia Przylubska.
                                  length of the bar at an enormous constant speed V at
Four years later the
family immigrated to              nearly light speed. I measure the bar length to determine
California. Michelson,            that Lorentz-Fitzgerald contraction has occurred—that
famous for the 1887               is, that the bar measures shorter than before. What
Michelson-Morley ether            should the strain gauge show?
drift experiment, in
1907 became the first
naturalized American
                                  95. Mass/Energy
citizen to win a Nobel            Under certain conditions, mass can be converted into
Prize in physics. The             energy à la E0 = mc2. Under certain restricted condi-
Nobel committee                   tions, energy can materialize as mass. What is wrong in
awarded the prize for             these statements?
an investigation to
determine whether
wavelengths of light              96. System of Particles
could provide a standard
                                  A system of particles is composed of n freely moving
unit of length. The ether
drift experiment was not          particles. Is the mass of this system equal to the sum of
even mentioned.                   the masses of the individual particles?
  —A DAPTED FROM D OROTHY
    M ICHELSON L IVINGSTON ,
       T HE M ASTER OF L IGHT :   97. Light Propagation
               A B IOGRAPHY OF
         A LBERT A. M ICHELSON    Suppose Patricia is driving her car at nearly the speed
                                  of light and turns on her headlights. For simplicity in
                                  calculations, in the rest frame of an observer on the
Are not gross bodies              ground the light takes one second to reach the stop sign
and light convertible             3 × 108 meters away. This ground observer then sees
into one another?                 the car reach the stop sign very soon after the initial
            —I SAAC N EWTON       light reaches the stop sign.


42       Mad about Modern Physics
     Patricia sees the light moving forward at 3 × 108        The notion of the
m/sec also, but she sees the stop sign approaching her        dependence of mass on
at nearly light speed. Therefore she sees the arrival of      velocity according to
the light flash at the stop sign and her arrival there in      m/(1 – v 2/c2)1/2 was
quick succession.                                             introduced by Lorentz in
                                                              1899 and then devel-
     Call the initial arrival of the light at the stop sign
                                                              oped by him and others
event A and the car’s arrival event B. Will the elapsed
                                                              in the years preceding
time between events A and B be the same for the driver
                                                              Einstein’s formulation
as for the observer on the ground? No, because the            of special relativity in
ground observer sees both events occur at the same            1905.
location, at the stationary stop sign, so ∆x = 0. As seen            —L EV B. O KUN , “T HE
by Patricia, these two events occur at two different            C ONCEPT OF M ASS ,” P HYSICS
locations separated by ∆x ≠ 0.                                          TODAY (J UNE 1989)

     Who measures the longer time interval between
events A and B? Can you provide a conceptual argu-
ment for this nonintuitive result? If the speed of the car
is closer to the speed of light, how does the difference in   I’ll be so happy and
elapsed times measured by driver and ground observer          proud when we are
change?                                                       together and can bring
                                                              our work on relative
                                                              motion to a successful
98. Sagnac Effect                                             conclusion!
                                                              —A LBERT E INSTEIN IN A LETTER
Suppose two identical clocks are in motion on Earth’s                                ´
                                                               TO M ILEVA M ARIC , HIS FUTURE
Equator with constant speed v relative to Earth, one          WIFE , M ARCH 27, 1901. J ÜRGEN

moving east and one moving west around the Equator.           RENN AND ROBERT SCHULMANN,
                                                                 EDS ., A LBERT E INSTEIN , M ILEVA
Do they tick at the same rate? What do their elapsed                  M AR I ´ : T HE L OVE L ETTERS
                                                                             C
times reveal when they meet again?


99. Light Flashes
                                                              A student riding in a
Suppose that a spaceship travels at constant velocity         train looks up and sees
between two planets, A and B. The spaceship sends out         Einstein sitting next to
a light flash in all directions every 10 minutes by its        him. Excited, he asks,
own clock reading. Traveling toward B, its light flashes       “Excuse me, Professor.
are seen at 5-minute intervals on planet B. What is the       Does Boston stop at
flash interval time as seen on planet A? One of these          this train?”



                                                                 Go Ask Alice                43
In 1905 Einstein wrote
twenty-one reviews for
the Beiblätter zu den
Annalen der Physik, a                                  Sends flash
“journal about journals.”          Sees flash          every 10 min            Sees flash
In addition to publica-            every ? min                                 every 5 min
tions written in German,
he also reviewed French                  A                                         B
and Italian papers, being
familiar with both lan-
guages. Without his
work for the Beiblätter,
he might have easily
missed the Festschrift
on the occasion of
Ludwig Boltzmann’s                possibilities is correct: 5-minute intervals; 10-minute
sixtieth birthday, which          intervals; 15-minute intervals; 20-minute intervals.
included 117 contribu-
tions by prominent
authors and thus                  100. Forces and Accelerations
offered an exceptionally
                                  In Newtonian physics, an applied contact force acting
broad panorama of
                                  on a rigid object will accelerate the object in the same
physics at the beginning
of the twentieth cen-             direction as the applied force. Does this behavior hold
tury. Einstein discussed          for applied contact forces in relativity physics (STR)?
three papers from this            For example, if an applied contact force pushes on the
volume, and he probably           same rigid object in the direction perpendicular to the
read the rest.                    direction of motion, will the resulting acceleration be in
 —A LBRECHT F ÖLSING , A LBERT    the direction of the applied contact force?
       E INSTEIN : A B IOGRAPHY


                                  101. Uniform Acceleration
Academic disputes are
vicious because so little         Suppose an object starts at rest with respect to the lab
is at stake.                      frame and undergoes a uniform acceleration a′ as
               —A NONYMOUS        measured by an observer on a spaceship moving at a
                                  uniform velocity v with respect to the lab. In Newton-
                                  ian mechanics, for speeds where v << c, the velocity
                                  after t ′ seconds in the moving frame has elapsed is


44       Mad about Modern Physics
V′ = a′ t ′ as measured by the observer on the moving          Henri Poincaré, building
object. This velocity is V = v + a′ t after the elapsed time   on Lorentz’s work but
t in the lab frame, because in Newtonian physics the           removing, at least for-
clocks in the different frames run at the same rates.          mally, certain inconsis-
What is the velocity value in the lab frame when the           tencies, arrived in 1905,
speed is allowed to become relativistic? Can the prod-         and more fully in 1906,
                                                               at the expressions
uct at be greater than c in either reference frame?
                                                               E = m c2/(1 – v 2/c2)1/2
                                                               and p = mv /(1 – v 2/c2)1/2.
102. Long Space Journey                                        Einstein obtained the
                                                               same relations, at the
Can a person go with a 1-g acceleration to a distant           same time, on purely
location 7,000 light-years away and return without             kinematic grounds. These
aging more than 40 years? That is, the bathroom scale          are the well-tested and
in the spaceship must show a person’s correct weight           familiar expressions of
for the whole journey. Is this feat within the realm of        today.
science or science fiction?                                               —J. DAVID J ACKSON ,
                                                                     “T HE I MPACT OF S PECIAL
                                                                 R ELATIVITY ON T HEORETICAL
                                                                      P HYSICS ,” P HYSICS TODAY
103. Head to Toe                                                                    (M AY 1987)

Can relativitic effects make your feet age more slowly
than your head?

                                                               Einstein simply
104. Neutrino Mass                                             postulates what we
                                                               have deduced, with
Since their proposed existence in the 1930s, neutrinos
                                                               some difficulty and not
and antineutrinos of all three lepton families have been       altogether satisfactorily,
thought to have zero mass and travel at light speed to         from the fundamental
conserve energy and angular momentum in nuclear                equations of the electro-
decays. In 1969 came the first hints that at least one          magnetic field.
type of neutrino can become another type of neutrino,                —H. A. L ORENTZ , 1906,
and a neutrino oscillation scheme was proposed. We                     Q UOTED IN A LBRECHT
                                                                    F ÖLSING , A LBERT E INSTEIN :
now know that muon neutrinos created in Earth’s                                     A B IOGRAPHY
atmosphere can oscillate into electron neutrinos and
tau netrinos before reaching an underground detector.
Why cannot all three neutrino types still have zero
mass?


                                                                 Go Ask Alice              45
Einstein’s special rela-             105. Spaceship Collision
tivity paper (“On the
Electrodynamics of                   Two spaceships, A and B, move toward one another on
Moving Bodies”), pub-                courses for a head-on collision. According to an
lished in 1905,                      observer at rest in an inertial reference frame, both
attracted very little                have speed V along the x-axis. At the time of observa-
attention, perhaps                   tion, spaceship A is coincident with the observer—that
partly because it was                is, has the same x value. Spaceship B is at a distance L
one of a number of con-              away. One would like to know how much later the col-
tributions by many dif-              lision occurs according to the observer and according
ferent authors in the                to an observer aboard spaceship A.
general field of the
                                          Let us propose a solution method. According to the
electrodynamics of
                                     observer, the collision occurs when spaceship A or B
moving bodies. In the
                                     travels L/2, half the distance between them, which
Annalen der Physik
alone there are eight                requires the elapsed time T = L/2V. Put into a better
papers from 1902 up to               format, three events occur:
1905 concerned with
this general problem.                   Event 1:             X1 = 0               T1 = 0
Einstein himself always
                                        Event 2:             X2 = L               T2 = 0
insisted on this aspect
of continuity, “With
                                        Event 3:             X3 = L/2             T3 = L/2V
respect to the theory of
relativity it is not at all a
                                         These same events can be specified in the inertial
question of a revolution-
                                     (primed) frame of spaceship A as:
ary act, but of a natural
development of a line
                                        Event 1′:            X1′ = 0              T1′ = 0
which can be pursued
through centuries.”                     Event 2′:            X2′ = ?              T2′ = ?
—G ERALD H OLTON , T HEMATIC
O RIGINS OF S CIENTIFIC T HOUGHT :
             K EPLER TO E INSTEIN       Event 3′:            X3′ = ?              T3′ = ?


                                     106. Twin Paradox
                                     On their twenty-first birthday, Peter leaves his twin
                                     brother, Paul, behind on Earth and goes off in a straight
                                     line for 7 years on his own wristwatch time (2.2 × 108
                                     seconds) at 0.96 c with respect to an inertial reference


46        Mad about Modern Physics
frame at rest with respect to Earth, then reverses direc-    From a historical
tion, and in another 7 years of his time returns at the      perspective, Einstein’s
same constant speed. Paul sees Peter’s wristwatch run-       recognition of E = m c2
ning slower, so Peter ages v2/ 2) = 0.28 as much, or
                            √(1 – c                        (where c is for “celeri-
1.96 years for each direction. But Peter looks back to       tas,” from the Latin for
see Paul’s clock running slower than his own wrist-          “swiftness”) did not
                                                             quite come “out of the
watch, so Paul should be aging slower by 0.28 as
                                                             blue.” Already in 1881,
much—that is, 1.96 years for each direction. On his
                                                             J. J. Thomson had cal-
return, Peter is surprised: “I know that I aged 14 years,
                                                             culated that a charged
but Paul should have aged only 3.92 years. Why is Paul       sphere behaves as if it
an old man with gray hair?”                                  had an additional
                                                             mass of amount 4/3c–2
                                                             times the energy of its
                                                             Coulomb field. That set
                                                             off a quest for the
                                                             “electromagnetic mass”
                                                             of the electron—an
                                                             effort to explain its
                                                             inertia purely in terms
                                                             of the field energy. In
                                                             1900, Poincaré made
                                                             the simpler observation
                                                             that since the electro-
                                                             magnetic momentum of
                                                             radiation is 1/c2 times
                                                             the Poynting flux of
                                                             energy, radiation seems
                                                             to possess a mass
                                                             density 1/c2 times its
                                                             energy density.
                                                                   —WOLFGANG R INDLER ,
                                                               R ELATIVITY : S PECIAL , G ENERAL ,
                                                                          AND C OSMOLOGIC AL




                                                                Go Ask Alice                47
6 Start
  Me Up
  E      NGINEERING PHYSICS IS REALLY APPLIED PHYSICS,

         but more general, with social, political, financial, and
  aesthetic issues to be considered that are often beyond the
  immediate concerns of the applied scientist. We have
  included the ability to understand the microscopic behavior
  of atoms and their components in solid and liquids, a knowl-
  edge that has begun to reap huge benefits in improving the
  materials and devices around us. In fact, we have entered the
  era of ingenious devices and designer materials. A very small
  sampling of the vast array of these advances is included in
  the challenges and puzzles considered in this chapter.




                                                                   49
The earliest objection on           107. Air-Driven Automobile Engine
record to Aristotle’s
theory of falling bodies,           Can a normal four-cylinder gasoline engine actually
based on observing the              operate on compressed air instead of gasoline as its
actual fall of two bodies,          energy source?
is that of John Philo-
ponus (ca. 490–570),
also known as John of               108. Coin Tosses
Alexandria, a Christian             The behavior of many systems and materials can be bet-
philosopher, scientist,             ter understood by considering the random walk of par-
and theologian. He
                                    ticles in the system. To get some “feeling” for a random
writes, “If you let fall
                                    walk, consider the following exercise. Divide a group of
from the same height
two weights of which one            people into two groups. Have each individual in one
is many times as heavy              group toss a fair coin 256 times and write down in
as the other, you will              sequence the outcome of each toss. Have each individual
see that the ratio of the           in the other group write down what they would imagine
times required for the              a typical sequence of 256 random tosses to be but not
motion does not depend              actually do the tossing. Collect all the papers and mix
on the ratio of the                 them up thoroughly. Can you determine with reasonable
weights, but that the
                                    accuracy which sets of data were obtained experimen-
difference in time is a
                                    tally? How accurate should your selection be?
very small one.” How-
ever, he was in no sense
a precursor of Stevin               109. More Coin Tosses
and Galileo, believing as
he did that in a vacuum             Suppose we are really ambitious about tossing a fair
the speeds of falling               coin. Indeed, suppose we toss a fair coin 1000 times,
bodies would indeed be              and for each head we step one unit distance radially
in the ratio of their               away from a lamppost, and for each tail we step back
weights, and ascribing              radially the same unit distance. About how many times
the near equality of                would you expect to be at the lamppost?
their speeds in air
entirely to the resist-
ance provided by the                110. Brownian Motor
medium.                             In his famous lectures, physicist Richard Feynman
—J. L. R EDDING , “A RISTOTLE ’ S
T HEORY OF FALLING B ODIES ,”
                                    discussed the impossibility of violating the second law of
 A MERIC AN J OURNAL OF P HYSICS    thermodynamics by a ratchet mechanism. The simplest
                  (J UNE 1978)      model for a ratchet is an overdamped Brownian particle
                                    in an asymmetric but spatially periodic potential (with

50        Mad about Modern Physics
                   Movable wall                             It’s important to learn
                                                            classical mechanics
                                                            before learning modern
                                                            physics so that you will
                                                            know how to wave your
                                                            hands correctly when
                                                            discussing things you
                                                            don’t quite understand.
                                                                —H OWARD G EORGI , U.S.
                                                                            P HYSICIST




          Asymmetric periodic potential
                                                            DISCOVERY OF
                                                            KINETIC ENERGY
asymmetry and period L). Due to the fluctuating force        Huygens’ (and, inde-
caused by the pushing molecules of the surrounding          pendently, Christopher
fluid or gas, the Brownian particle may overcome the         Wren’s) studies of rigid
potential barrier moving to the left or to the right. The   colliding balls around
probabilities for both directions are equal, and on aver-   1655 led them to con-
age the particle does not move. Hence building a motor      clude that there was
that turns thermal energy into mechanical work from a       something special about
single heat bath is impossible.                             the product of mass and
    But the ratchet can be turned into a so-called          velocity-squared.
Brownian motor that seems to violate the second law         Remarkably, adding
of thermodynamics. The idea is to turn the ratchet          values of mv 2 for each
                                                            ball prior to a collision
potential on and off periodically. Under certain cir-
                                                            yielded a total that was
cumstances, this action may yield directed motion even
                                                            essentially the same
against an applied force f. Indeed, this device does
                                                            after the collision, even
work. (No pun intended!)                                    though the velocities
    Recall that a perpetuum mobile of the first kind         had changed.
violates the law of conservation of energy, while a per-           —E UGENE H ECHT, “A N
petuum mobile of the second kind uses the “free”                     H ISTORICO -C RITICAL
                                                                 ACCOUNT OF P OTENTIAL
energy around us in the form of heat—that is, random
                                                             E NERGY: I S PE REALLY REAL ?,”
thermal motion of molecules and atoms—to run an                       T HE P HYSICS T EACHER
engine without fuel. Why isn’t a Brownian motor a                        (N OVEMBER 2003)

perpetuum mobile of the second kind?

                                                                 Start Me Up            51
                                       111. Magnetocaloric Engine

                                                    Magnet              Ferrofluid
What is the origin of
the 260-day cycle in
the Mayan calendar?
According to Anthony                                                          Heat
Aveni, the 260-day                              Heat source                   sink
cycle has meaning only
in tropical latitudes,
being connected with                   A ferrofluid is a fluid containing small magnetic
the interval the noonday
                                       particles that respond to an applied magnetic field, so
sun spends north as
                                       a ferrofluid becomes magnetized in the presence of the
opposed o south of the
                                       magnet. The diagram shows a closed tube loop con-
overhead position. These
intervals vary depending               taining a ferrofluid, a heat source, a strong magnet,
on the latitude, but in                and a heat sink all working together to act as an engine
latitude 141⁄2°N , close to            transporting the ferrofluid around the closed loop. Its
the locations of the                   thermal efficiency approaches the efficiency of a Carnot
great Maya city of                     cycle, so demands for this device should increase.
Copan and the pre-                     Exactly how does this engine maintain the fluid move-
Classic city of Izapa,                 ment around the loop? Can a solar heating system
the annual cycle divides               operate in this way?
up neatly into 105-
and 260-day periods.
 —A DAPTED FROM A NTHONY               112. Magnetorheological Fluid
AVENI , E MPIRES OF T IME : C ALEN -
   DARS , C LOCKS , AND C ULTURES      In a beaker is 250 milliliters of corn oil to which has
                                       been added about 0.5 kilogram of iron filings about 1
                                       millimeter long. The mixture is stirred thoroughly and
                                       a strong horseshoe magnet is brought up to straddle the
                                       beaker. The iron filings align with the magnetic field as
If you go against the                  expected to magnetize the fluid mixture. What other
grain of the universe,                 physical property of the fluid changes?
you’re liable to get
splinters.
                 —A NONYMOUS           113. Binary Fluids
                                       The two possible phase diagrams show the miscible and
                                       immiscible phases of a binary fluid, a mixture of two

52         Mad about Modern Physics
                                                              A technician named
           Miscible                     Miscible              Richard Woodbridge III
 T                             T                              coined the phrase
                                                              “acoustic archaeology”
                                                              in the August 1969
                                       Immiscible             issue of Proceedings
                                                              of the I.E.E.E. Wood-
                                                              bridge theorized that
          Immiscible
                                                              there were many occa-
                                                              sions when sound might
     0       50        100         0       50       100       innocently get scooped
                                                              out of the air and pre-
          % Fluid B                     % Fluid B
                                                              served. For example,
                                                              when an ancient potter
kinds of fluid, in a plot with axes of temperature versus      typically held a flat
concentration. For example, coffee and cream are mis-         stick against a rotating
cible at room temperature but oil and water are not.          pot, he was accidentally
    Consider the 50 percent mixture in each phase             (and crudely) recording
diagram and start at a high temperature in the miscible       into the clay the sounds
phase. The diagram to the left reveals that the binary        around him. Woodbridge
fluids become immiscible upon being cooled, while             wrote about experi-
the diagram to the right tells us that the fluids become       ments he performed
immiscible as the cooling proceeds but that even              pulling basic noises off
further cooling brings back the miscible phase. Can           a pot. Another experi-
                                                              ment involved setting up
both phase diagrams represent a real binary fluid, or is
                                                              a canvas and then
one false?
                                                              talking while making
                                                              different brush strokes,
114. Baseball Bats                                            hoping to record a
                                                              spoken word in an oil
Hitting a baseball well is not easy. Even professional
                                                              portrait. In this fashion,
baseball players have difficulty consistently making          for instance, the word
solid contact with a pitched baseball. Once hit, the dis-     “blue” was pulled off a
tance of flight of the ball is determined by its initial       blue paint stroke.
velocity—that is, the initial speed and direction—which       —J ACK H ITT, “E AVESDROPPING
depends on how hard the ball has been hit by the bat.         O N H ISTORY,” N EW YORK T IMES
                                                              M AGAZINE (D ECEMBER 3, 2000)
All other factors being held constant, the initial velocity
can be said to depend on the speed of the bat just before
collision. A quicker swing would mean a faster bat


                                                                  Start Me Up          53
Many experts believe           speed during the collision to add distance to each hit
that Egyptian pyramids         and also allow the batter more time to judge the pitch.
are aligned with true              There have been proposals to put shallow, pea-sized
north because the              depressions—dimples—in the surface of a baseball bat
more stationary stars          to allow a greater swing speed. Another sports object,
near the North Celestial       the golf ball, already is made with dimples on its sur-
Pole represented
                               face. How would these dimples affect the bat’s swing
permanency and
                               speed?
eternal life.

                               115. Old Glass
                               In old castles and houses in Europe can be found win-
                               dows with old glass in which many of the panes are
                               slightly thicker on the bottom than at the top. What are
                               some possible reasons for this result, and what is the
The adult brain as a
                               most likely reason?
whole consumes some
twenty-five watts of
power when in full             116. Ferromagnetism
action.
                               Why are so few substances ferromagnetic, yet practi-
                               cally all materials exhibit paramagnetic behavior?


                               117. Coupled Flywheels
                               Conservation of angular momentum does not always
                               help in understanding the behavior of rotating devices.
The more I have studied
                               The diagram shows two flywheels, 1 and 2, of
him, the more Newton
                               moments of inertia I1 and I2, mounted on parallel hor-
has receded from me.
      —R ICHARD W ESTFALL ,
                               izontal shafts along with pulleys of diameters D1 and
      H ISTORIAN OF S CIENCE   D2. The belt is slack at first, and the two flywheels are




                                             1                   2




54      Mad about Modern Physics
running at angular velocities ω10 and ω20. Suddenly the    MV 2 GAINS A
belt is tightened. One can write out the torque equa-      FACTOR OF 1/2
tions and the angular momentum equation to get the         It was not until 1807
relation I1 ω1 + I2 ω2 = k – (N – 1) I1 ω1. Here, k is a   that Thomas Young, an
constant of integration and N = D2 / D1, the ratio of      English physicist and
pulley diameters. When N = 1, angular momentum is          physician, spoke of
conserved. If N ≠ 1 and ω1 changes, the angular            mv 2 for the first time
momentum is not conserved! Why not?                        as energy. Then in a
                                                           textbook published in
                                                           1829 Gustave Coriolis,
118. Superconductor Suspension                             a French physicist, was
                                                           the first to give the
A popular physics demonstration since the late 1980s
                                                           exact modern definition
involves floating a small piece of high-temperature
                                                           to kinetic energy and
superconductor, such as yttrium barium cuprate
                                                           work. He carried out a
(YBa3Cu3O7), over a strong permanent magnet. The           calculation of the work
levitation is easy to see, and the suspended supercon-     done in accelerating a
ductor rectangular solid spins rapidly about its long      body and arrived at the
axis. The demonstration is done by first cooling the        change in the quantity
superconductor in liquid nitrogen and then using tongs     1/2mv 2. By the end of
to place the piece in the air above the permanent mag-     the 19th century, most
net. The repulsive force between the magnet and the        scientists were avoiding
superconductor is a demonstration of the Meissner          Leibniz’s old phrase
effect. Or is it?                                          vis viva (living force),
                                                           and using instead
                                                           “kinetic energy,” a term
119. Nanophase Copper                                      introduced in 1849 by
The hardness and strength of a metal are measured by       Lord Kelvin to better
                                                           distinguish between
studying its deformation in response to an applied
                                                           force and energy.
force. A metal is deformed when its crystalline atomic
                                                                 —E UGENE H ECHT, “A N
planes slide over each other. An analogy may be the                 H ISTORICO -C RITICAL
bump in a rug that can be pushed across the floor. In            ACCOUNT OF P OTENTIAL
                                                           E NERGY: I S PE R EALLY R EAL ?,”
other words, a dislocation in a plane of atoms is moved
                                                                      T HE P HYSICS T EACHER
until a barrier is reached, such as a grain boundary,                   (N OVEMBER 2003)
where the micron-sized grains are differently oriented.
    One interesting advance in metal technology is the
ability to assemble nanometer-size clusters of atoms in
grain sizes of less than 100 nanometers in diameter

                                                              Start Me Up             55
                                   instead of having the micron-size grains found in a
                                   typical metal. A graph of hardness versus grain size is
                                   shown.

Jean Buridan (ca.                      Hardness (GigaPascals)
1295–ca. 1358), rector                 3
of the University of
Paris in 1327, in his
impetus theory intro-                  2
duced the prescient
notion that the true
measure of the motion                   1
of an object was not
speed alone, but the
product of speed and                   0
quantity of matter
                                            0      1      2        3        4       5
(quantitas materiae). In
an anticipation of New-                           Log of Grain Size (nm)
ton’s first law of motion,
he maintained that once                With grain sizes averaging about 10 nanometers,
the initial impetus was            this nanophase copper metal has a hardness more than
supplied, motion contin-           three times the hardness of normal copper metal. Why?
ued indefinitely. The
spheres of heaven, for
instance, having been              120. Head of a Pin
put in motion by God,
                                   What is the smallest amount of charge that can sit on
continued so and
required no constantly             the head of a pin? Some people say that the smallest
working angels to keep             nonvanishing amount of charge should be +e or –e,
them moving.                       where e is the fundamental unit of electrical charge.
    —I SAAC A SIMOV, A SIMOV ’ S   What do you say?
   B IOGRAPHIC AL E NCYCLOPEDIA
  OF S CIENCE AND T ECHNOLOGY,
                  2 ND REV. ED.    121. Coulomb Blockade
                                   The tunnel junction is a conductor-insulator-conductor
                                   device. Suppose a very small tunnel junction is operated
                                   at very low temperatures so that thermal fluctuations
                                   do not contribute to electron tunneling across the junc-
                                   tion. Now connect the tunnel junction to a source of

56       Mad about Modern Physics
constant electrical charge. Will the flow of current
across the junction be steady?

                                                             Descartes regarded
122. Deterministic Competition                               the conservation of
Consider a simplified system, one that can be described       momentum (quantity of
by Nt objects at time t. For example, one could consider     motion) as divinely
the number of grasshoppers on the plains of Africa, or       ordained. He wrote:
on some small plot of land. Let there be competition         “[God] set in motion in
between the growth processes and the decay processes         many different ways
                                                             the parts of matter
so that the number of objects at time t + 1 is Nt + 1 = Nt
                                                             when He created them,
exp[r (1 – Nt)], an exponential growth relationship.
                                                             and since He main-
This equation is deterministic, for Nt determines Nt + 1
                                                             tained them with the
unambiguously. One can think of r as a measure of the        same behavior and with
ratio between growth and decay. Numerous mechani-            the same laws as He
cal, hydrodynamic, chemical, and electrical systems can      laid upon them in their
be approximately modeled by this relationship.               creation. He conserves
    How does the number of objects behave with               continually in this mat-
elapsed time? If Nt = 1, then N remains 1 forever. In the    ter an equal quantity of
general case, we can determine Nt as t → ∞ to find out        motion.”
whether N approaches the equilibrium value 1. For
instance, let r = 1 and begin with N0 = 0.5, and calcu-
late with a calculator or personal computer. Now try
different values for r. What behavior do you predict?
                                                             Earth is gradually slow-
123. Two Identical Chaotic                                   ing down; the day is
Systems                                                      about 16 milliseconds
                                                             longer now than it was
A chaotic system exhibits a sensitivity to initial condi-    1,000 years ago. This
tions and will evolve rapidly and deterministically          slowing is due largely to
toward different end states if begun in slightly different   frictional tidal effects
states. Although the chaos is unpredictable, each possible   of the Moon on Earth’s
outcome is deterministic—that is, an orderly behavior.       oceans.
    Consider two identical chaotic systems isolated
from each other. They will quickly fall out of step
because any slight difference between them would be
magnified. Assume that these systems have several

                                                                 Start Me Up       57
In the fall of 1915 it was          parts and that at least one of the parts is stable—that
widely expected that the            is, subjected to a perturbation, the part’s behavior
Nobel Prize in Physics              changes a little but settles back to its normal operation.
was to be jointly shared            Now drive both systems with the same chaotic signal
by Edison and Tesla.                applied to the same stable part. Can the two systems be
Then a Reuters dispatch             synchronized?
from Stockholm dropped
a bombshell. The Nobel
Committee announced
that the prize for                  124. Tilley’s Circuit
physics would in fact be
shared by William Henry             This electrical circuit near the permanent magnet has
Bragg and his son W. L.             two ideal switches and a galvanometer. When switch A
Bragg . . . . What had              is closed and switch B, on the right, is opened, there is
happened? The Nobel                 a large change in the magnetic flux in the galvanometer
Prize Foundation                    circuit. What do you predict the galvanometer response
declined to clarify. One            will be?
biographer reported
years later that the
Serbo-American had                                            Magnet
declined the honor,
stating that as a discov-                                               G
erer he could not share                           A       B
the prize with a mere
inventor. Yet another
biographer advanced the
theory that it was Edison
who objected to sharing             125. Thermal Energy Flow
the prize. . . . The Nobel          If two identical bodies at different temperatures are in
Foundation said simply,             contact, thermal energy will always flow from one to
“Any rumor that a person
                                    the other in such a direction as to increase the total
has not been given a
                                    entropy. In which direction will this flow be? That
Nobel Prize because he
has made known his
                                    depends on two factors, the amount of energy and
intention to refuse the             entropy the two bodies already contain. The second
award is ridiculous.”               law of thermodynamics implies that thermal energy
       —M ARGARET C HENEY,          must flow toward the region of lower temperature—
      T ESLA : M AN O UT OF T IME   that is, each unit of thermal energy acquires greater dis-
                                    order as it moves into the cooler region. Why?


58       Mad about Modern Physics
126. Cadmium Selenide                                          Is Galileo a beneficiary of
                                                               the Matthew effect? The
When atoms are arranged in nanometer-size clusters of          latter is a term introduced in
diameters from less than 100 nanometers to as large as         1968 by Robert K. Merton
700 nanometers, interesting optical properties can be          (1910–2003), a U.S. sociol-
demonstrated. For example, nanophase versions of               ogist of science, that refers
pure cadmium selenide can be made almost any color             to the disproportionately
in the spectrum simply by changing its cluster size.           great credit given to eminent
Indeed, some types of lipstick are made in many differ-        scientists for their contribu-
                                                               tions to science, while rela-
ent colors even though the predominant light-scattering
                                                               tively unknown ones tend to
molecule is the same in all color versions. What is the
                                                               get disproportionately little
physics here?                                                  for their occasionally com-
                                                               parable contributions. The
127. Optical Solitons                                          term derives, of course,
                                                               from the Gospel according
A light pulse is a continuum of optical carriers of dif-       to Matthew (13:12 and
ferent frequencies. Optical media are dispersive, so           25:29). In the New King
these carriers in the light pulse travel at different veloc-   James Version the passage
ities, causing the energy to spread over time and dis-         reads: “For whoever has, to
tance. In addition, there is the optical Kerr effect, which    him more will be given, and
“instantaneously” increases the refractive index of the        he will have abundance; but
                                                               whoever does not have, even
medium by an amount proportional to the optical
                                                               what he has will be taken
power. Can one use these two effects—dispersion and
                                                               away from him.” Recognition
the Kerr effect—to ensure that a light pulse retains its       tends to go to those who are
integrity while traveling thousands of kilometers              already famous. In Galileo’s
through an optical fiber?                                       case, both Philoponus in
                                                               the sixth century and the
                                                               Belgian–Dutch scientist
128. Ceramic Light Response                                    Simon Stevin in 1586 per-
Certain ceramic materials will change their shape upon         formed the key experiment
exposure to light. What is the physics here?                   of dropping two different
                                                               weights simultaneously and
                                                               observed that they struck
129. Random Movements                                          the ground at the same
                                                               time—the experiment that
Supposedly, research has revealed that random move-
                                                               today seems indissolubly, if
ments help explain how a tightrope walker stays aloft,         incorrectly, wedded to the
for instance. If understood, robotics engineers could          name of Galileo.

                                                                  Start Me Up        59
In Galileo’s time a                make their machines more stable by injecting a little
general impression                 noise into their systems. And persons having difficulty
prevailed that a falling           walking may be able to let some noisy vibrating shoe
body gained speed in               soles help them walk confidently again. What could be
proportion to the dis-             the physics here?
tance through which it
fell, and Galileo himself
held this opinion for a
time. It appears also              130. Gravitational Twins
that he abandoned this             Engineering physics involves the transport of people
idea, not because of               and materials in space as well as practical applications
contrary results of                here on Earth. So consider a pair of twins in free fall.
experiment, but
                                   Imagine that one twin is in circular orbit around a star
because deductive rea-
                                   and that her sister is shot out from this circular orbit
soning (not without a
                                   location on a radial orbit—that is, the traveling twin
flaw) had led him to the
impossible conclusion              will fall back to meet up with the stay-at-home sister in
that a body governed by            circular orbit. For simplicity, let them meet after an
this law would fall                integral number of revolutions around the circular
through a long distance            orbit for the one left behind.
in the same time as                    Any clock system in a gravitational potential, such
through a short one. He            as the clocks in the GPS system here on Earth, depends
therefore abandoned                on two relativistic effects on the clock rate: (1) a clock
the space-acceleration             ticks slower closer to a massive object than when far
idea and considered the            away, and (2) the faster-moving clock ticks slower than
possibility that the gain          the slower-moving clock.
in speed was propor-                   Initially, the clock rates of the twins are the same
tional to the time of fall.
                                   because they start out in the same circular orbit at the
               —PAUL R. H EYL ,
“TRANSCENDENTAL MECHANICS,”        same radial distance from the star. The traveling twin
 A MERIC AN J OURNAL OF P HYSICS   moves away from the star along the radial line, all the
                   (M AY 1941)     while slowing down and eventually coming to a momen-
                                   tary stop and returning with ever-increasing speed until
                                   rejoining her sister in orbit. So on average, the traveling
                                   twin experiences a smaller amount of gravitational time
                                   dilation and a smaller amount of speed time dilation
                                   than her stay-at-home sister. Therefore the traveling
                                   twin returns home older than her sister, because her
                                   clock ticked faster on average. What do you think?


60        Mad about Modern Physics
131. Photon Engine                                        Nikola Tesla (1856–1943),
                                                          the Serbian-born inventor
The ideal Carnot heat engine converts heat to             of the first practical
work without the engine itself being a source of any      alternating-current
work. The reversible closed Carnot cycle consists of      dynamo and power trans-
two isothermal (constant temperature) processes and       mission system, was
two adiabatic (no external exchange of thermal            known for unusual powers
energy) processes. No heat engine operating between       of visualization. He was
two temperatures can be more efficient than a Carnot       able to construct, modify,
cycle.                                                    and even operate his
                                                          imaginary devices, purely
                                                          by visualizing them. He
                              Hot atoms                   wrote in “My Inventions”
                                                          (Electrical Experimenter,
       Mirrors
                                                          1919), “It is absolutely
                                                          immaterial to me whether
 T1                                 Piston       T2       I run my turbine in
                                                          thought or test it in my
                                                          shop. There is no differ-
  Photon beam                                             ence whatever, the results
                                                          are the same. In this way I
                                                          am able to rapidly develop
                                                          and perfect a conception
                                                          without touching anything.
    But Carnot could be wrong. The challenger is the      When I have gone so far
new “quantum Carnot engine,” in which the radiation       as to embody in the
pressure from photons drives a piston in an optical       invention every possible
                                                          improvement I can think
cavity. The inward-facing surface of the piston is
                                                          of and see no fault any-
mirrored and the other cavity mirror is fixed in place
                                                          where, I put into concrete
while exchanging thermal energy with a heat sink at
                                                          form this final product of
temperature T1. A second heat bath at a higher            my brain. Invariably my
temperature, T2 , provides the source of thermal energy   device works as I con-
for the photons.                                          ceived that it should,
    This source of thermal energy is a stream of hot      and the experiment
atoms, which flows into the optical cavity and            comes out exactly as I
exchanges thermal energy with the photons through         planned it. In twenty
emission and absorption processes. These atoms exit       years there has not been
the cavity at a cooler temperature and are reheated to    a single exception.”


                                                              Start Me Up       61
The Earth is a somewhat          T2 in a second cavity, to be reinjected into the first
irregular clock. Some            cavity for the next cycle of the quantum Carnot engine.
years the length of the              Therefore, the quantum and classical Carnot
day is found to vary by
                                 engines operate in the same way as a closed cycle
as much as one part in
10 million, or three sec-        of two isothermal and two adiabatic processes. How-
onds in a year of 31.5           ever, in its simplest form, when each bath atom is
million seconds. In addi-        treated as a two-state system, the quantum Carnot
tion, there are also sea-        engine cannot extract work from a single heat bath.
sonal fluctuations of a          Why not? Will the engine work if each bath atom is a
few milliseconds per
year. In the winter the
                                 three-state system?
Earth slows down, and in
the summer it speeds
up. Think of the Earth
as a spinning skater.
During the winter in the
northern hemisphere,
water evaporates from
the ocean and accumu-
lates as ice and snow on
the high mountains. This
movement of water
from the oceans to the
mountaintops is similar
to the skater’s extend-
ing her arms. So the
Earth slows down in
winter; by the summer
the snow melts and runs
back to the seas, and
the Earth speeds up
again. This effect is not
compensated by the
opposite effect in the
southern hemisphere
because most of the
land mass is north of
the equator.
 —J AMES J ESPERSEN AND J ANE
      F ITZ -R ANDOLPH , F ROM
  S UNDIALS TO ATOMIC C LOCKS




62       Mad about Modern Physics
7 A Whole
  New World
  A      TOMIC PHYSICS BEGAN IN THE            1840 S   WITH

         the identification of the emission lines of hydrogen
  and of other atoms and ions in laboratory sources and in the
  solar spectrum. In the early 1900s, the Bohr-Sommerfeld
  model of the atom was the paradigm, but numerous prob-
  lems with its predictions existed that were finally resolved
  with the advent of quantum mechanics in 1925. The electron
  in the atom occupies particular quantized energy states of
  unequal energy spacing, and selection rules based on conser-
  vation of energy and angular momentum dictate which
  jumps between states to an available final state are allowed.
  In addition to a spontaneous electron jump to a lower energy
  level with the emission of a photon, external photons with
  the correct energy can stimulate the atomic absorption or
  emission of photons. The eventual application of quantum
  mechanics to the simple molecules proved very successful, if



                                                                 63
                        not challenging, and today faster computers continue
                        to calculate the properties of atoms, inorganic and
                        organic molecules, and very large biomolecules such as
                        DNA and proteins. Enormous progress has also been
                        made in understanding the fundamental properties of
                        condensed matter of fluids and solids such as crystals,
                        ionically doped materials, plastics, pseudocrystals, and
                        so on. Our lives are becoming more dependent on the
                        practical devices arising from this great endeavor called
                        molecular design. The challenges introduced in this
                        chapter are but a small sample of the wide range of
                        possible problems.




64   Mad about Modern Physics
132. Grain of Sand                                              Both Niels Bohr and
                                                                his wife had a similar
If the atoms in a grain of sand were laid out side by side      response to religion:
in a line, approximately how long would the line be?            Margrethe has written
                                                                about Niels’: “There
                                                                was a period of about a
133. Forensics                                                  year . . . [he was] 14 or
Historically, paintings could be verified with reasonable        15 . . . where he took it
assurance of authorship by experts who knew the                 all very seriously; he got
brushstrokes and color and paint choices of the artist as       taken by it. Then sud-
well as the overall style and character of the subjects.        denly it was all over. It
However, in some cases, fraudulent artworks have been           was nothing for him.”
                                                                About her own feelings,
successfully passed as genuine. New techniques for
                                                                Margrethe reported:
assessing all types of artwork are always needed, and
                                                                “You know it was often
the scientific community has been answering the call.
                                                                at that age . . . that
One scientific technique for checking the authenticity           one got very religious
of old paintings uses laser lights. How might this feat         and would listen to the
be accomplished?                                                minister about confir-
                                                                mation. Then it all
                                                                dissolved. And for me
134. Doppler Elimination?                                       it was exactly the
                                                                same; it disappeared
                                                                completely.”
 Intensity                   Doppler shifted
                                                                           —L ÉON ROSENFELD,
                             Natural width                      “B OHR , N IELS H ENRIK DAVID,”
                                                                      D ICTIONARY OF S CIENTIFIC
                                                                               B IOGRAPHY, VOL . 2

                             Doppler-broadened



              Frequency


When an atom emits or absorbs a photon, there is
always a recoil of the atom and a Doppler shift in the
photon frequency. Is it possible to have recoilless
atomic emission or absorption?


                                                             A Whole New World              65
FIELD THEORY IN THE            135. Light Tweezer
EIGHTEENTH CENTURY?
In many ways this major                       Laser light
advance had its origin in
1758 with the publication                                   Lens
of the immensely influen-
tial Theory of Natural
 Philosophy by Roger                      1                               2
Joseph Boscovich
(1711–1787), a theory of       F2
such importance that
                                                            f         1
nearly 150 years later
                                    F1
Lord Kelvin could describe
himself as a “true Boscov-                                      2
ichean.” Boscovich was
born Rudjer Josip              In science fiction movies we often see light beams shot
Bos      ´
    ˘kovic in the Republic     out from handheld light guns supplying a tremendous
of Dubrovnik. Today he is
                               impulse to knock over an enemy storm trooper
claimed with equal vigor by
                               approaching along the direction of the beam. By New-
the Croats, the Serbs, and
the Dalmatians. As Mar-
                               ton’s third law, the light gun itself should have experi-
garet Wertheim writes in       enced an equivalent recoil! We know that a ray of light
her Pythagoras’ Trousers,      has energy and linear momentum, so its impingence on
“His nationality is surely a   any surface will produce a slight backward movement
significant part of the        of that surface. However, we would like to know
reason that this visionary     whether a light beam can be used to physically move a
physicist isn’t more           tiny object, such as a small one-celled animal, in a
famous today—it is diffi-      direction perpendicular to the beam.
cult to imagine that any
Anglo-Saxon scientist of
such caliber would have        136. Fluorescent Lights
remained so outside the
spotlight.” Trained as a       The gas plasma inside a fluorescent tube emits mostly
Jesuit priest, he became a     ultraviolet radiation and very little visible radiation.
professor of mathematics       Electrons are captured by the ions and jump down to
in Rome, and over the          lower energies, emitting a characteristic UV photon for
course of his lifetime         each fluorescence jump. Why are fluorescent tubes so
published over a hundred       much more efficient in producing visible light than
books and papers, most of      incandescent lamps?

66      Mad about Modern Physics
    Why can the light from some fluorescent lights be              which still remain
dangerous to your health? Could it be that they emit              untranslated from the
some UV? Are there types of fluorescent tubes that are             Latin. To get rid of the
better for human working environments? Are they bet-              “spooky action at a dis-
ter because they do not emit in the UV?                           tance,” he introduced
                                                                  the proposition that
                                                                  atoms have no size;
137. Phase Conjugation Mirror                                     they are geometrical
                                                                  “points of force” that in
Can a light wave pass through a disturbing medium, be
                                                                  turn create fields of
distorted, reflect off a special mirror, and return to the
                                                                  force, an idea later
source undisturbed?                                               elaborated on by Fara-
                                                                  day. Moreover, he sug-
138. Stationary States                                            gested that all these
                                                                  atomic forces along with
In the Bohr model of the hydrogen atom, the angular               gravity, must be aspects
momentum for the orbital motion of the electron of                of one all-encompass-
mass m at distance r is quantized in integral units of            ing universal force, an
Planck’s constant h—that is, assuming the proton posi-            eighteenth century ver-
tion to be fixed, mvr = nh/2π, where n is an integer and           sion of the “theory of
v the electron velocity. Using mv = h/ λ, de Broglie was          everything!”
able to derive Bohr’s quantization rule and nλ = 2πr. If          —ADAPTED FROM LESLIE HOLLI -
                                                                   DAY, “E ARLY V IEWS ON F ORCES
f1 and f2 are the frequencies of the Bohr orbital motion            BETWEEN ATOMS ,” S CIENTIFIC
of the electron in energy states E1 and E2, then if an                     A MERIC AN (M AY 1970)
electron jumps down from state 2 to state 1, why isn’t
the energy of the emitted photon the difference energy
hf1 – hf2?


139. Angular Momentum
In classical calculations, the quantity that often appears
in the result is the square of the angular momentum J 2.
One can often guess at the correct quantum mechanical
formula by replacing J 2 by j (j + 1) h2/4π 2, where j is the
z-component of the angular momentum and h is
Planck’s constant. Why is the square of the angular
momentum in quantum mechanics proportional to
j ( j + 1) instead of just j 2?

                                                                A Whole New World           67
Five Venus rounds           140. Kinetic Laser
(synodic periods) equal
eight solar years within    A traditional laser involves the stimulated downward
about two days, and         electron transition in an atom in a background “sea of
equal also 99 lunar         photons,” with the emission of a characteristic photon
months with a discrep-      matching in fequency and momentum the stimulating
ancy of less than four      photons. This stimulated emission process was predicted
days. Specifically,         by Einstein. In 1951, J. Weber at the University of Mary-
5 × 583.9 = 2919.5,         land was the first to calculate the operating principles of
8 × 365.25 = 2922.0         the ammonium maser and laser. However, as the story
99 × 29.53 = 2923.5.        goes, upon asking for research monies to build the
For ancient astronomers
                            maser, a few hundred thousand dollars from the univer-
this was evidence of
                            sity, he lost out to the athletic department’s request for
profound unity and even
                            money to build up the Maryland football program.
preestablished harmony
within the cosmos.               The first operating ammonium maser was subse-
                            quently built by C. Townes in 1954, and the first oper-
                            ating device lasing in the optical part of the spectrum
                            was built in 1960 by T. H. Maiman. Laser action first
                            in the microwave region is no coincidence, for sponta-
                            neous emission is proportional to the cube of the tran-
We must be clear that,      sition frequency, and being extremely small in this part
when it comes to atoms,     of the spectrum, can be neglected compared to stimu-
language can be used        lated emission and absorption.
only as in poetry.               Among the more exotic lasers is the kinetic laser,
            —N IELS B OHR
                            which is an “exploding” material that emits light and
                            X-rays. In its simplest form, the material would be a
                            foil of a single element such as copper that is exploded
                            by focusing powerful laser pulses on it. How does this
                            type of laser produce coherent laser light?
If all the empty space
were squeezed out of a      141. Noninversion Laser
person, the amount of
solid matter remaining      For decades, lasers have been explained as the result of
would be no larger than     an inverted population of states with stimulated emission
a speck of dust.            of photons in a high-Q cavity. However, lasers can be
                            made without an inverted population. Can you explain
                            how this type of stimulated emission process works?


68      Mad about Modern Physics
142. X-ray Paradox
The index of refraction n gives the ratio c/v, the speed
of light in vacuum to the speed of the electromagnetic
wave in the material. Window glass, for example, can            Why do all FM radio
have an index of about n = 1.5 for visible light, with a        stations end in an odd
slight variation in the index with the color of the light.      number? FM radio sta-
A paradox arises with X-rays because they have an               tions all transmit in a
index of refraction value less than one in crystals! What       band between 88 MHz
does this behavior mean?                                        and 108 MHz. Inside
                                                                that band, each station
                                                                occupies a 200 kHz
143. Benzene Ring                                               slice, and all of the
                                                                slices start on odd num-
The benzene molecule is a
                                                                ber frequencies. This is
ring of six carbon atoms,
                                                                completely arbitrary. In
each C atom having one H                                        Europe, the FM stations
atom attached. There is a                                       are spaced 100 kHz
mystery about the energy                                        apart, and their fre-
contained in this molecule.                                     quencies can end on
The benzene ring can be                                         even or odd numbers.
broken up into pieces, and
chemists have measured
the energies associated
with the pieces and with                                        When Einstein regis-
                                                                tered for the draft in
the single bonds and the double bonds by studying
                                                                Switzerland at the age
ethylene and so on. The expected total energy can be            of 22, his height was
calculated from these data, but the actual total energy         recorded as five feet
of the benzene ring is much lower, telling us that the          seven and a half inches.
carbon atoms are much more tightly bound. Therefore,            His contemporaries
the bond picture would make the benzene ring easily             regarded him as tall. By
susceptible to chemical attack, yet the molecule is quite       way of comparison,
                                                                Isaac Newton is thought
resilient to breaking up.
                                                                to have been about five
    Using the Schrödinger equation by considering each          feet five inches tall.
carbon atom on this ring as the potential home for a                —A DAPTED FROM B ARRY
single electron, one can calculate the possible energy                PARKER , E INSTEIN : T HE
                                                                     PASSIONS OF A S CIENTIST
levels for the benzene ring. Why does this method of
calculation work?


                                                             A Whole New World           69
                                    144. Graphite
                                    Atoms in a crystal make a regular array if there are no
                                    dislocations. Most pure single-element crystals have a
                                    cubic or a diamond crystal structure, with all orthogonal
Niels Bohr discovered               directions showing the same structural spacing. Even for
his ideas in the act of             a pure element substance, however, the spacing may be
enunciating them, shap-             different in different directions. For example, take car-
ing thoughts as they                bon atoms, which probably are components of more
came out of his mouth.              than 75 percent of all known compounds. In diamond
Friends, colleagues,                they have the same structure in all orthogonal direc-
graduate students, all
                                    tions, but in graphite the third direction is definitely
had Bohr gently entice
                                    quite different than the other two directions, which
them into long walks in
                                    define a plane of hexagonal carbon rings. How can this
the countryside around
Copenhagen, the heavy               third direction be so different in an originally nonbiased
clouds scudding over-               environment?
head as Bohr thrust his
hands into his overcoat
pockets and settled into
an endless, hesitant,
recondite, barely audi-
ble monologue. While he
spoke, he watched his
listeners’ reactions,
eager to establish a
bond in a shared effort
to articulate.
      —ROBERT P. C REASE AND
       C HARLES C. M ANN , T HE
  S ECOND C REATION : M AKERS OF
             THE R EVOLUTION IN
         20 TH -C ENTURY P HYSICS   145. Ozone Layer
                                    We’ve heard so much in the past few decades about the
                                    ozone layer in the upper atmosphere and its possible
                                    demise. Yet ozone is only a minor greenhouse gas, far
                                    behind carbon dioxide, HOH vapor, and methane in
                                    overall importance. So why is there all this fuss over the
                                    ozone layer?


70       Mad about Modern Physics
146. Greenhouse Gases                                        The special theory of
                                                             relativity predicts that,
Why are the greenhouse gases carbon dioxide, HOH             for an observer moving
vapor, and methane important for human survival on           at the speed of light,
Earth? If they are good for our existence, shouldn’t         distance traveled
having more carbon dioxide, etc., in the atmosphere be       shrinks to zero while
encouraged?                                                  time slows to a stand-
                                                             still. Thus, as far as the
                                                             light itself is concerned,
147. LED vs. LCD                                             it does not travel any
An LED is a semiconductor device that emits visible          distance, and takes no
light when an electric current passes through it.            time to do so. As Gilbert
                                                             Lewis showed back in
The light is not particularly bright and usually
                                                             1926 (Nature, vol. 117),
monochromatic, occurring at a single wavelength. The
                                                             from light’s point of view
LED light output range is from infrared and red to
                                                             the Universe is so “bent”
blue-violet. The LCD is a type of display used in digital    that there is no separa-
watches and many portable computers that utilizes two        tion between the point of
sheets of polarizing material with a liquid crystal          emission of light and its
solution between them. An electric current passed            point of absorption. . .
through the liquid causes the crystals to align so that          If light does not expe-
light cannot pass through, each crystal acting like a        rience itself to have
shutter, either allowing light to pass through or block-     traveled any distance, it
ing the light.                                               does not need a vehicle
    What is the difference in energy requirements in         or mechanism by which
the operation of a light-emitting diode (LED) and a liq-     to travel. . . . It is only
uid crystal display (LCD)? After all, they both require      in our frame of refer-
                                                             ence—the frame of
energy to operate. And how is a plasma display differ-
                                                             observers with mass who
ent from both of them in its energy requirements?
                                                             move at sub-light
                                                             speeds—that light
148. Sonoluminescence                                        appears to travel
                                                             through space and time;
Sound energy is converted directly into light energy by      and only in that frame
a phenomenon called sonoluminescence. Discovered in          does the question of
the 1800s, the process lay dormant for more than 100         whether it is a wave, a
years, only to experience a revival in the 1990s. How        particle or both arise.
does one convert a small amount of sound energy into            —P ETER RUSSELL , “H ERE I S
a brief but brilliant flash of light?                             T HERE ,” “L ETTERS ,” N EW
                                                                                  S CIENTIST
                                                                   (N OVEMBER 23, 1991)

                                                            A Whole New World            71
Isidor Isaac Rabi, (Nobel             149. Siphoning Liquid Helium
Prize in Physics in 1944)
faced a crisis of faith
while only a child.
He decided to put the
tenets of Orthodox
Judaism to the test.                                                        Liquid helium
Jewish law prohibited
riding streetcars on the               Glass dewar
Sabbath, so one Sabbath
he climbed onto a street-
car, just to see what would
happen. Emboldened by
an uneventful ride, he
conducted another test:               Liquid helium can crawl up the wall of its container
“I remember being in the              without any additional help. How is this feat accom-
synagogue and the priests             plished?
. . . would stand up and
with their hands out-
stretched, they would
bless the congregation.
                                      150. Quantized Hall Effect
You were not supposed to              The Hall effect was discovered by Edwin Hall in 1879.
look at their hands, you              “. . . A charged particle moving in a magnetic field feels
might go blind if you did.            a ‘Lorentz’ force perpendicular to its direction of
Well I tried . . . with one           motion and the magnetic field. As a direct consequence
eye.” His sight left unim-            of this Lorentz force, charged particles will accumulate
paired, he dismissed                  to one side of a wire if you send current through it and
Jewish faith as supersti-
                                      hold it still in a [perpendicular] magnetic field. . . .”
tion but, to please family
                                      When the transverse voltage is measured at a fixed cur-
and friends, consented to
                                      rent, the Hall resistance is measured and increases lin-
a bar mitzvah, at which he
ended up lecturing them               early with an applied magnetic field.
in Yiddish on the workings                 The conduction electrons in a solid behave like a
of the electric light bulb.           gas of electrons. So the discovery of the quantized Hall
          —PAUL H OFFMAN , “T HE      effect in 1980 by von Klitzing and his research group
  H IGHER T RUTH OF P HYSICS ,” A     when he was investigating the conductance properties
  REVIEW OF R ABI : S CIENTIST AND
     C ITIZEN BY J OHN S. R IGDEN ,
                                      of two-dimensional electron gases at very low temper-
          N EW YORK T IMES B OOK      atures and high magnetic fields was a surprise. What is
          R EVIEW (M AY 10, 1987)     the physics behind this quantized Hall effect?


72      Mad about Modern Physics
151. Integrated Circuits                                       Planetary atomic mod-
                                                               els were already popular
As integrated circuits (ICs) become crowded with               a few years before
more semiconductor devices and internal connections,           Rutherford’s proposal.
one wonders how they will be connected to the                  The most elaborate
external world. We know also that cosmic rays and              attempt was that of
other particle radiation from the environment already          Hantaro Nagaoka,
disrupt some of the operations by random destruction,          whose “Saturnian”
and these effects will become worse as the scale dimin-        model was published in
ishes. However, neither connection to the external             1904. Nagaoka’s model
world via gold wires of any size nor the background            was astronomically
                                                               inspired, in the sense
particle radiation is the major problem today. What is?
                                                               that it closely relied on
                                                               Maxwell’s 1856 analysis
152. Atomic Computers?                                         of the stability of Sat-
                                                               urn’s rings. The Japan-
Atoms are busy collections of electrons and nuclear            ese physicist assumed
particles that are ever changing their positions in a ran-     that the electrons were
dom dance. In contrast, information storage requires           placed uniformly on
stable states over reasonable time intervals. Can infor-       rings moving around the
mation be stored on individual atoms in their restless         attractive center of a
world?                                                         positive nucleus.
                                                               Nagaoka’s calculations
                                                               led to suggestive spec-
153. X-ray Laser?                                              tral formulas and a
We know that there exist free electron X-ray lasers that       qualitative explanation
have an electron moving past a rippled surface and             of radioactivity. The
emitting X-rays, as well as X-ray laser sources based on       model was, however,
                                                               severely criticized, and
                                                               disappeared from the
                     Cu-W                                      scene only to reappear
                                                               in an entirely different
                                  Cu(111)                      dressing with Ruther-
                                                               ford’s nuclear theory.
                                                                  —H ELGE K RAGH , QUANTUM
                                                                    G ENERATIONS : A H ISTORY
                                                                           OF P HYSICS IN THE
                                                                        T WENTIETH C ENTURY
                                   Cu X-rays
        X-ray tube


                                                             A Whole New World          73
There was a time when          plasmas such as the kinetic laser. However, an X-ray
physics and philosophy         laser with a wavelength of about 1 Å or 0.1 nanometer
were allied disciplines.       or less that can be operated on a tabletop would be
However, Niels Bohr’s          convenient and would be able to resolve details down
three long historic papers     to nearly 1 wavelength. The uses in physics and medi-
on the structure of the        cine are expected to be many.
hydrogen atom were
                                   A very interesting tabletop device is the working
published in 1913 in
                               monochromatic X-ray source shown in the illustration
Philosophical Magazine.
                               that emits very intense, narrow beams at the Cu 1.54 Å
The journal, first pub-
lished in 1798, was at         characteristic emission line known as Kα1. There is a
that time accepting            special bimetal X-ray tube source of Cu-W that emits
articles from most             X-rays from both metals upon bombardment with
branches of science.           high-energy electrons in the standard way. These
This alliance began to         X-rays exit the tube and then Bragg scatter in an exter-
break up at the end of         nal Cu crystal to produce a very narrow, intense beam
the nineteenth century.        of Cu Kα1 X-rays. The first surprise is the enormous
Today, even though the         line intensity at a single wavelength, and the second
word “philosophical”           surprise is that no Cu Kα2 X-rays appear in the output
persists in its title, it is   from the external crystal. How does the external crys-
devoted primarily to           tal affect the X-ray beam? Is this device an X-ray laser
condensed-matter
                               or a super-radiant X-ray source?
physics.

                               154. Bose-Einstein Condensate
                               A Bose-Einstein condensate is a new form of matter
                               made at the coldest temperatures in the universe. Essen-
Things that cannot go on       tially the condensate is a collection of identical atoms
forever don’t.                 behaving as one entity. How do the individual atoms
               —A NONYMOUS     lose their self-identity?


                               155. Quantum Dots
                               Quantum dots are crystals containing only a few hun-
                               dred atoms and when illuminated with UV light, for
                               example, will fluoresce at only one specific wavelength
                               of light. Why does the dot emit only one wavelength of
                               light when excited?

74     Mad about Modern Physics
8 Chances
  Are
  Q          UANTUM MECHANICS ( QM ) ORIGINATED IN

             1925 as a theory to understand the internal
  behavior of the hydrogen atom. Since then, QM has
  evolved to encompass the behavior of practically everything.
  In its most rudimentary version, QM is based on three
  fundamental rules. The main idea of QM is not quantized
  energy and quantized angular momentum, for the classical
  physics of strings, tubes, drumheads, and so on, involve
  quantized states of energy and angular momentum.
     The heart of QM is the coherent superposition of states,
  as given in rule 2 below. From The Feynman Lectures on
  Physics, the three fundamental rules of QM are:

  1. The probability P of an event in an ideal experiment is
     given by the square of the absolute value of a complex
     number ψ, which is called the probability amplitude (or
     wave function):
                             P = |ψ| 2.

                                                                 75
                         2. When an event can occur in several alternative
                            ways, the probability amplitude ψ for the event is
                            the sum of the probability amplitudes ψ1, ψ2, ψ3
                            . . . , for each way considered separately; that is,
                            there is superposition and interference:
                                        ψ = ψ1 + ψ2 + ψ3 + . . .
                                        P = | ψ1 + ψ2 + ψ3 + . . . | 2

                         3. If an experiment is performed (or could be done)
                            that can determine whether one or another alterna-
                            tive is actually taken, the probability of the event is
                            the (classical) sum of the probabilities for each
                            alternative; that is, the interference is lost:
                                          P = P1 + P2 + P3 + . . .



                            We have no knowledge about a more basic mecha-
                         nism from which these rules can be deduced. Numer-
                         ous tests have verified their fundamental validity over
                         and over. You will need to apply them in the challenges.




76   Mad about Modern Physics
156. Schizophrenic Playing Card                                For centuries, Britain
                                                               and its colonies rang in
                                                               the New Year on March
                                                               25, Annunciation Day,
                                                               when according to the
                                                               biblical account the
                                                               angel Gabriel announced
                                                               to the Virgin Mary that
                                                               she would bear the child
                                                               of God. March 25 is
                                                               nine months before
                                                               Christmas.
                                                                 —D UNCAN S TEEL , M ARKING
An ideal playing card stands perfectly balanced on its                T IME : T HE E PIC QUEST
                                                                     TO I NVENT THE P ERFECT
edge. According to the rules of quantum mechanics,
                                                                                     C ALENDAR
this card will fall in both directions at once! That is, the
final state of the card is the superposition of the two         If you stood on the
alternative falling directions, with ψ1 for left and ψ2 for    moon’s near side, you
right. The card’s wave function changes smoothly and           would see the Earth
continuously from the balanced state to the mysterious         suspended against the
final state Ψ = ψ1 + ψ2 with two alternatives that seem         stars more or less in
to have the card in two places at once. Why haven’t we         the same direction
seen this happen in the everyday world around us?              with respect to your
                                                               horizon—never rising
                                                               or setting. But the Earth
157. Schrödinger’s Cat                                         as seen from the moon
In one version of the famous Schrödinger cat gedanken          would exhibit phases
                                                               over the course of a
experiment, a healthy cat is placed inside an ideal cat
                                                               month, just as the moon
playroom that is isolated from the rest of the world
                                                               does as seen from
whenever the door is closed. Inside is one deadly object
                                                               Earth.
left by mistake. The door is closed. After some time            —M ICHAEL Z EILIK AND J OHN
elapses, one wonders whether the cat is alive or dead,                 G AUSTAD, A STRONOMY :
the two classical possibilities. Rule 2 of QM tells us,              T HE C OSMIC P ERSPECTIVE

however, that the state of the cat is Ψ = ψ1 + ψ2 , where
ψ1 means alive and ψ2 means dead. So QM requires us
to consider the cat as being alive and dead simultane-
ously! However, you are curious. You push a button


                                                                    Chances Are           77
The bourgeois ambiva-               that opens the door just enough so you could look in to
lence of Werner Heisen-             determine the status of the cat. You could peek in but
berg’s childhood may                you decide not to. Now what does QM predict for Ψ?
have played a role in his
own adult ambivalence
toward the sweeping                 158. Wave Functions
claims of every system
of thought and belief,              Wave functions can be functions of many different
including science. At               physical parameters of the system of interest. For
middle age and again                example, one can define a wave function in coordinate
near the end of his life,
                                    space, in momentum space, in spin space, and so on as
Werner declared science
and religion to be “com-            long as the unit vectors of the space are orthogonal. For
plementary” aspects of              a single particle, the wave function ψ(x 1,y1 ,z1) is the
reality, each with its own          QM amplitude for finding the particle at the three-
language and symbolism              dimensional configuration space point (x 1,y1 ,z1), which
and each with its own               directly corresponds one-to-one to position space coor-
limited realm of validity.
                                    dinates x1, y1, and z1 for this one-particle system. For
Different religiously or
intuitively apprehended             the two-particle system, the wave function ψ(x 1,y1 ,z1 ;
truths should be viewed             x2 ,y2 ,z2) defines a six-dimensional configuration space.
as different sides of the           Is there a direct correspondence to three-dimensional
same truth, while                   position space coordinates for this two-particle wave
rational science—his                function as well? What about the multiparticle wave
own profession—should
                                    function?
be viewed as just one
among a variety of ways
of perceiving reality.
Shortly before his
                                    159. Wave Function Collapse?
death, Heisenberg                   Consider an electron in a box. Imagine partitioning the
remarked to a colleague,            box into N identical cubes and assume that the
“If someone were to say
                                    amplitude Ψ for finding the electron in the box is
that I had not been a
Christian, he would be              the superposition Ψ = ψ1 + ψ2 + ψ3 + . . . , that is, the
wrong. But if someone               sum over all N imagined identical cubes in the box.
were to say that I had              Now use a photon to observe where the electron is by
been a Christian, he                recording the scattered photon and so on. Suppose
would be saying too                 your incident probe photon passes right through the
much.”
                                    box and does not interact with the electron, which you
        —DAVID C. C ASSIDY,
        U NCERTAINTY : T HE L IFE   determine because the photon took a straight-line path
      AND S CIENCE OF W ERNER       to your detector. What happens to the electron wave
                  H EISENBERG
                                    function Ψ?

78     Mad about Modern Physics
160. Quantum Computer                                      QUESTION: WHAT IS “IT”?
                                                           Pascal did IT under
The new quantum computers rely on quantum coher-             pressure.
ence. That is, the quantum computer system contains N      Coulomb got all charged
identical quantum subsystems—for example, atoms, or          up about IT.
optical setups, or molecules, or resonant cavities. In     Hertz did IT frequently.
general, each quantum subsystem can be in many pos-        Boltzmann did IT in heat.
sible quantum states. Assume that the ψi for each quan-    Ampere let IT flow.
tum subsystem has only two states, which we label 1        Heisenberg was never
and 0. If N = 3, then Ψ = ψ1 + ψ2 + ψ3 is the QM state       sure whether he even
of the system. Therefore our quantum computer repre-         did IT.
sents all eight states simultaneously: 000, 001, 010,      Bohr did IT in an excited
011, 100, 101, 110, 111.                                     state.
                                                           Pauli did IT but excluded
                                                             his friends.
                                                           Hubble did IT in the
                                                             dark.
                                                           Theorists do IT on paper.
    That is, during calculations on Ψ all eight states     Astrophysicists do IT
participate in each calculation! If the quantum com-         with young starlets.
puter is actually a large molecule in a vacuum, then the   ANSWER: IT = science, of
molecule must be kept away from the walls of the con-        course!
tainer and away from other molecules. Why?                     —C OPYRIGHT © 2002 BY
                                                                    J UPITER S CIENTIFIC


161. Cup of Java Quantum
Computer                                                   It has been proved that
One day while looking into her cup of java, Laura real-    the 13th is more likely
ized that this slurry of caffeine molecules could be the   to fall on Friday than on
world’s natural quantum computer. How could this           any other day of the
inherent ability in coffee be possible?                    week. For a short proof
                                                           consult the reference
                                                           below.
162. Bragg Scattering of X-rays                                  —J OHN WAGNER AND
                                                                     ROBERT M C G INTY,
Bragg scattering of X-rays of wavelength λ in an ideal                “S UPERSTITIOUS ?”
crystal satisfies Bragg’s law: 2d sin θ = m λ, where d is       M ATHEMATICS T EACHER 65
the spacing between adjacent scattering planes and θ is               (1972): 503–505

the angle measured from the surface of the crystal, not

                                                              Chances Are          79
Pauli once referred to           the perpendicular. When this condition is met for vari-
quantum mechanics as             ous integer values of m, constructive interference from
Knabenphysik —boys’              the entire family of parallel planes occurs because the
physics—because so               path differences are integral multiples of the X-ray
many of the main con-            wavelength. One often reads that the Bragg scattering
tributors were still in          of X-rays from an ideal crystal is a coherent scattering
their twenties. For
                                 process—that is, all the Bragg-scattered X-rays arrive
example, in September
                                 in phase at the detector. Why is it not so?
1925 Heisenberg was
23 years old, Pauli 25,
Jordan 22, and Dirac
had just turned 22. In
                                 163. Beautiful Faces
1932 Friedrich von               Why can we see a person’s face in great detail in visible
Weizsäcker recalled,             light? Hint: think about coherent scattering versus non-
“The general attitude            coherent scattering of the light.
was one of immense                   Why is the image of a person’s face blurry in the
Hochmut, an immense              infrared (IR) and in the ultraviolet (UV)? For simplicity
feeling of superiority,          and idealization purposes, assume that we can see
as compared to old pro-          equally well in the IR, visible, and UV so that our phys-
fessors of theoretical
                                 iology is not the limiting factor.
physics, to every exper-
imental physicist, to
every philosopher, to            164. Gravitational Waves
politicians, and to what-
ever sorts of people you         In addition to telescopes for photons in the γ-ray, X-ray,
might find in the world,         UV, visible, IR, µ-wave, and radio parts of the electro-
because we had under-            magnetic spectrum, new windows to the universe are
stood the thing and they         opening up with neutrino and gravitational wave
didn’t know what we              observatories. Gravitational waves are expected to be
were speaking about.”            produced by a changing mass quadrupole—for exam-
   —H ELGE K RAGH , QUANTUM      ple, two masses revolving about their common
     G ENERATIONS : A H ISTORY
            OF P HYSICS IN THE   barycenter, such as the two stars in a binary star sys-
         T WENTIETH C ENTURY     tem. They would emit gravitational waves with wave-
                                 lengths of many kilometers that interact with all
                                 objects—that is, they exhibit most wave phenomena
                                 such as scattering, reflection, and transmission through
                                 objects in ways similar to other types of waves. The
                                 classical scattering cross section of gravitational waves


80       Mad about Modern Physics
by a mass pair in a detector was worked out by physi-         FROM AN INTERVIEW WITH
cist J. Weber about 50 years ago.                             THE AMERICAN PHYSICIST
    For simplicity, assume that each pair of identical        ISIDOR ISAAC RABI
atoms in a material is a mass pair quadrupole scatterer       “The nature of discover-
of gravitational waves. We would like to know whether         ies is so remarkable, so
gravitational waves can scatter coherently in the detec-      wonderful—if you want
tor—that is, whether a gravitational wave can simulta-        to think of the goal of
neously scatter from many mass pairs in the detector          the human race, there it
                                                              is. To learn more about
(such as an aluminum bar) or whether a gravitational
                                                              the Universe and our-
wave must scatter from a single mass pair at a time.
                                                              selves. In physics, the
What is the physics here?
                                                              newest discoveries like
                                                              relativity and the uncer-
165. Coherent Neutrino                                        tainty relation, uncover
                                                              new modes of thought.
Scattering                                                    They really open new
Another possible window or telescope for observing            perspectives.” A sudden
the universe is in the detection of neutrinos. The Super-     sad look passed over his
Kamiokande neutrino facility in Japan and the Sud-            face. “And I thought
                                                              that, say, fifty years
bury Neutrino Observatory (SNO) in Canada house
                                                              ago, that this would hap-
two of the largest neutrino detectors, containing thou-
                                                              pen, that these revolu-
sands of tons of water. Already they have determined
                                                              tions and advances in
that the flux of solar neutrinos from the Sun agrees           science would have an
with the standard solar model. In addition, research          effect on mankind—on
groups operating these neutrino detectors have verified        morals, on sociology,
neutrino oscillations in matter, the conversion of one        whatever. It hasn’t hap-
type of neutrino to another.                                  pened. We’re still up to
    The two neutrino detectors are enormous because           the same things, or,
neutrinos are notorious for their extremely small prob-       well, I think, regressed
ability to interact with matter. Billions of neutrinos pass   in values.”
through our bodies each second and do no harm! A sin-               —ROBERT P. C REASE AND
                                                                     C HARLES C. M ANN , T HE
gle electron neutrino would pass through solid lead             S ECOND C REATION : M AKERS OF
(Pb), filling space from Earth to Jupiter with only a                       THE R EVOLUTION IN

small chance of colliding with a Pb nucleus. However,                  20 TH -C ENTURY P HYSICS

in 1984 physicist J. Weber proposed that neutrinos of
all energies could be coherently scattered by the nuclei
in large defect-free single crystals of silicon, ruby, or


                                                                  Chances Are             81
                                   diamond, thereby enhancing the neutrino scattering
                                   probability by a factor of 1022. Therefore, in the ideal
                                   case, practically all incident neutrinos would scatter at
                                   least once from the carbon nuclei in a perfect diamond
                                   crystal within the first centimeter or less!
                                       Normally, one might expect only neutrinos of wave-
Compared to the theory
                                   lengths much greater than the spacings between the
of relativity, quantum
                                   nuclei in the crystal to have any chance at coherent
mechanics developed
                                   scattering, analogous to light scattering coherently from
rapidly, disseminated
very quickly, and met              a surface of atoms spaced much less than the wave-
almost no resistance.              length of the incident light. Otherwise, when the nuclei
Also contrary to relativ-          are treated as scattering potentials, the phases con-
ity, quantum mechanics             tributed by the scattering nuclei to the QM amplitude
attracted little public            are random, and the scattering probability will be pro-
interest. Eddington was            portional to N instead of N 2, like the result for X-rays
one of the few scien-              discussed in a previous problem. What assumption have
tists who wrote about              we made about the scatterers that Weber says leads to
the theory for a nonsci-           an incorrect conceptual argument against coherent scat-
entific readership.                tering for the shorter-wavelength neutrinos?
Although quantum
mechanics was no less
counterintuitive than
relativity, there was no           166. Magnetic Resonance
quantum counterpart to             Imaging (MRI)
the antirelativistic liter-
                                   Magnetic resonance imaging (MRI)
ature that flourished in                                                    Magnetic
the 1920s.                         is really the medical application         Field
   —H ELGE K RAGH , QUANTUM        of nuclear mag-
    G ENERATIONS : A H ISTORY OF   netic resonance,
     P HYSICS IN THE T WENTIETH    which physicists
                        C ENTURY
                                   have been doing
                                   since the 1940s.
                                   A sample of liv-
                                   ing tissue con-
                                   tains numerous hydrogen atoms bound in molecules.
                                   Each hydrogen nucleus has a spin with a magnetic
                                   moment that can be aligned by an applied magnetic
                                   field. The sample is placed in a very strong uniform

82        Mad about Modern Physics
magnetic field to align the spins of the hydogen nuclei.     The number of photons
A pulsed electromagnetic field is applied that would flip     is in general not con-
just one hydrogen spin, for example. What alternative       served in particle reac-
QM interpretation can one provide that treats the           tions and decays. I . . .
nuclei as a collective whole?                               would like to note here
                                                            an ironical twist of his-
                                                            tory. The term “photon”
167. Heisenberg Uncertainty                                 first appeared in the
                                                            title of a paper written
The Heisenberg uncertainty principle, also known as
                                                            in 1926. The title: “The
the indeterminancy principle worldwide, states ∆px∆ x
                                                            conservation of pho-
≥ h/4π , where ∆ x is the uncertainty in the x-position     tons.” The author: the
measurement, ∆ px is the uncertainty in the x-momen-        distinguished physical
tum measurement, and h is Planck’s constant. As some        chemist Gilbert Newton
people say, the uncertainty principle places a limit on     Lewis (1875–1946) from
the accuracy of knowing a particle’s position. What do      Berkeley. The subject: a
you think? Some people claim also that the Heisenberg       speculation that light
uncertainty principle is just an example of a more gen-     consists of “a new kind
eral uncertainty relationship for all waves, that the       of atom . . . uncreatable
position can be determined only at the expense of our       and indestructible [for
knowledge of its wavelength. Is this statement true?        which] I . . . propose the
    We also know that Niels Bohr, in his discussions        name photon.” This idea
                                                            was soon forgotten, but
with Albert Einstein over several decades on whether
                                                            the new name almost
quantum mechanics is a complete description of
                                                            immediately became
nature, would often invoke the uncertainty principle to
                                                            part of the language.
defend his point of view, known as the Copenhagen              —A BRAHAM PAIS , IN S OME
interpretation of QM. Bohr argued that if you pin                    S TRANGENESS IN THE
down the particle’s position more precisely for the              P ROPORTION , EDITED BY
                                                                          H ARRY WOOLF
famous double-slit experiment by observing with pho-
tons, their interaction with the particle disturbs its
momentum by giving it a random momentum kick.
That is, without looking, the particle exhibits an inter-
ference pattern on a distant screen behind the two slits.
However, if you look to see which way the particle
goes through the slits, the measurement disturbs the
system and there’s no interference pattern on the dis-
tant screen. You just see a classical two-hump distribu-
tion. What do you think about Bohr’s argument?

                                                               Chances Are        83
Quantum mechanics, in              168. Vacuum Energy?
matrix form, was born
when in July of 1925               Although the classical vacuum is a void, the quantum
Heisenberg, only                   vacuum is a virtual “soup” of particle-antiparticle pairs
twenty-three years of              that interact with real atoms to produce the Lamb shift
age, had a creative                (slight energy shift in atomic levels) and the Casimir
breakthrough on the                effect (attraction of two plates in a vacuum). Does the
fog-shrouded island of             quantum vacuum have energy content, or does the
Helgoland in the North             energy in the “soup” average out to zero?
Sea. He took as his
guiding principle the
proposition that a the-            169. Casimir Effect
ory should not traffic
                                   When two parallel uncharged metal sheets are placed in
with unverifiable
abstractions. He wanted            a perfect vacuum, they attract each other with a tiny
to deal only with meas-            force that is not gravitational. What is the source of
urable quantities. He              this effect?
later told Einstein that
“the idea of observable
quantities was actually            170. Squeezing Light
taken from his relativ-            Laser light can be described in many ways. If one con-
ity,” which had rejected           siders just the amplitude and the phase of one ray in a
such concepts as                   laser beam, there will always be shot noise—that is,
absolute speed for the             random variations caused by virtual particle interac-
same reason. In a letter
                                   tions in the vacuum with the beam. Yet we’ve heard
to Pauli, dated 9 July
                                   that there may be techniques to reduce the shot noise in
1925, Heiseberg wrote,
                                   the amplitude, for example. Then what happens to the
“My entire meager
efforts go toward killing          shot noise in the phase?
off and suitably replac-
ing the concept of the
orbital paths that one
                                   171. Electron Spin
cannot observe.”                   Does the vacuum affect the spin of a particle such as an
     —H ANS C HRISTIAN VON         electron?
B AEYER , TAMING THE ATOM : T HE
     E MERGENCE OF THE V ISIBLE
                  M ICROWORLD
                                   172. Superconductivity
                                   One quantum mechanical effect that shows itself on the
                                   macroscopic scale is superconductivity. Cooper-paired

84       Mad about Modern Physics
conduction electrons in superconductors have total          Creative thinkers seem
spin zero, that is, their paired spins are opposite, even   to possess the following
though their spatial separation can be enormous—cen-        characteristics in
timeters to meters, for example—because they have           common:
opposite momenta. These pairs can act like bosons of        • an acutely sensitive
spin zero, which obey Bose-Einstein statistics. Any            awareness of their
                                                               environment;
number of bosons can be in the same quantum state,
                                                            • the ability to generate
that is, have the same four-momentum (e.g., defined by
                                                               a large number of
the energy and three-momentum) and spin. Therefore,
                                                               ideas in response to a
all bosons in the same collective superconducting state        given problem;
have exactly the same energy. Yet this boson collective     • the ability to focus
state in a superconductor has a small energy width.            their faculties in sus-
Any thoughts about the cause of this energy width?             tained concentration;
                                                            • in most cases the cre-
                                                               ative individual’s work
173. Superfluidity                                             place is likely to be a
He-4 below the lambda transition temperature 2.7 K             cheerfully haphazard
can be analyzed as a two-fluid liquid composed of He            conglomeration of
atoms in the normal state and He atoms in the macro-           complete disorder;
scopic superfluid state. Superfluidity is a property of       • the majority of truly
He-4 in the liquid state because He-4 atoms obey Bose-         creative persons are
                                                               introverts;
Einstein statistics. Many He-4 atoms can be in the same
                                                            • they tend to be much
macroscopic quantum state—that is, the same momen-
                                                               less concerned with
tum states for these atoms moving in the superfluid. If
                                                               what others think of
so, then why can He-3 at low temperatures also                 them than most peo-
become a superfluid?                                            ple are; also, they are
                                                               often comparatively
                                                               indifferent to clothing
174. Gap Jumping                                               and appearance. Cre-
Tiny detectors on a person’s head have been used to            ative people do not
sense tiny fluctuations in the brain’s magnetic field.         seem to have a need
These SQUIDS, short for superconducting quantum                to present themselves
interference devices, are the most sensitive of any kind       in a favorable light to
and rely on the Josephson Effect, in which the Cooper          others.
pairs of electrons in a superconductor can sometimes                 OTTO H. T HEIMER , A
                                                                      G ENTLEMAN ’ S G UIDE
jump a physical spatial gap in the material to another                TO M ODERN P HYSICS
part of the superconductor. Manufactured SQUIDS

                                                               Chances Are          85
We all share a strange                                  Oxide layer
mental time lag, a phe-
nomenon first brought
to light in the 1970s by
neurophysiologist Ben-               Current in                            Current out
jamin Libet of the Uni-
versity of California at
San Francisco. In one
experiment, Libet docu-
mented a gap between
the time an individual
was conscious of the
                                                      Superconductor
decision to flex his fin-
ger (and recorded the
exact moment of that               have a thin film filling the gap. In fact, the direct cur-
consciousness) and the             rent (DC) SQUID used in laboratories worldwide today
time his brain waves               for sensing small magnetic fields is a superconducting
indicated that a flex was          ring with two gaps! The best DC SQUIDS have an
imminent. The brain                energy sensitivity capable of detecting a magnetic flux
activity occurred a third          change corresponding to about 10–34 joule in one sec-
of a second before the             ond, about the mechanical energy required to raise an
person consciously                 electron 10 centimeters in one second. Why do the
decided to move his                paired electrons jump the gap?
finger.
     —A NTONIO R. DAMASIO,
      “R EMEMBERING W HEN ,”
          S CIENTIFIC A MERIC AN
                                   175. Nuclear Decay
           (S EPTEMBER 2002)       In the nucleus of an atom, neutrons and protons are
                                   held by nuclear forces. Their total energy (ignoring the
                                   mc2 contributions) is less than the barrier height poten-
                                   tial energy. Yet some nuclear particles do escape. Any
                                   thoughts about the reason for an escape?


                                   176. Total Internal Reflection
                                   In total internal reflection of light—for example, at a
                                   glass-air interface or from a water-air surface—if the
                                   incident light is in the more dense medium, does the
                                   light penetrate into the air beyond the interface?

86        Mad about Modern Physics
                                                            The first wrist watches
         Light ray                      Air                 were decorative orna-
                                                            ments made for women,
                                                            and men consequently
                                                            shunned them as being
                                                            too effeminate. Again
                                                            war had its effect—the
                                                            first World War. A watch
                                                            that has to be fished
                                 Glass prism                out of a pocket in order
                                                            to be read is a lot less
                                                            convenient than one
177. Annihilation                                           strapped to the wrist,
                                                            particularly when you’re
We know that particles and their antiparticles annihi-      trying to fire a machine
late each other. For example, the electron and the          gun or charge over a hill
positron in positronium can annihilate into two pho-        at a particular moment.
tons or three photons in the final state, depending on       Realizing this, govern-
the total angular momentum of the positronium. Why          ments made wrist
would they do such a violent action?                        watches part of the
     Hint: why does any event occur in nature? We           standard equipment
know that the rate of any quantum mechanical event,         issued to their soldiers.
by Fermi’s Golden Rule, is proportional to the proba-       After the war, men sud-
                                                            denly felt that wrist
bility for the event times the density of final states. Is
                                                            watches were accept-
this statement all we need to say?
                                                            able for civilian wear, a
                                                            trend which clearly has
178. A Bouncing Ball                                        continued for both
                                                            sexes until the present
We see a kid bouncing a ball. According to quantum          day.
mechanics, which applies to everything that happens,          —J O E LLEN B ARNETT, T IME ’ S
why does the ball bounce?                                     P ENDULUM : F ROM S UNDIALS TO
                                                            ATOMIC C LOCKS , THE F ASCINATING
                                                                H ISTORY OF T IMEKEEPING AND
                                                                      H OW O UR D ISCOVERIES
179. The EPR Paradox                                                    C HANGED THE W ORLD
First of all, a short explanation. Although there are
other examples of the Einstein-Podolsky-Rosen (EPR)
paradox and violations of Bell’s inequalities, we
choose this version because we can provide you with

                                                                Chances Are             87
Einstein’s unease with             actual data to use in formulating your own solution to
quantum mechanics                  the paradox.
stemmed from a firm                    A source of two correlated identical particles of
belief in determinism. In          opposite spins sits on the straight line between two
a 1931 essay “The World            identical particle detectors. Each detector can measure
as I See It,” reprinted in         the polarization state of the entering particle, and each
Ideas and Opinions,
                                   detector has three polarization switch positions (1, 2,
he wrote, “I do not at all
                                   and 3) and two display lamps (green and red). Each
believe in human free-
                                   time the experimenter pushes the button, the two cor-
dom in the philosophical
sense. Everybody acts              related particles are shot out of the source in opposite
not only under external            directions into the detectors. The data show two pat-
compulsion but also in             terns: (1) For runs that have the same switch settings on
accordance with inner              the two detectors, the same color lights flash on them.
necessity. Schopen-                (2) For all runs, without regard for switch settings, the
hauer’s saying, ‘A man             pattern of flashing is completely random.
can do what he wants,                  This experiment gets to the heart of QM and the
but not want what he               application of its three rules for events. We can use
wants,’ has been a very            classical mechanics to explain the first pattern: let the
real inspiration to me             two particles carry the same instructions to be applied
since my youth; it has             at the detectors. For example, this instruction set might
been a continual conso-
                                   work: flash red at switch positions 1 and 3; flash green
lation in the face of
                                   at switch position 2. But this classical scheme with pre-
life’s hardships, my own
                                   determined instruction sets will not handle the second
and others’, and an
unfailing well-spring of           pattern. Why not? What is the surprising conclusion?
tolerance.”                            Reproduced here is a small part of a data set for the
            —A DAPTED FROM         experiment (from the Mermin reference in the answer).
             A LBERT E INSTEIN ,   Each entry shows the switch settings and the colors the
          I DEAS AND O PINIONS
                                   lights flashed for each run. The switch settings are ran-
                                   domly changed from run to run.

                                         31GR       13RG        31RR       33GG

                                         21RR       31RG        33GG       11GG

                                         22RR       12RG        31RG       13RG

                                         33GG       13GR        31RR       31RG


88      Mad about Modern Physics
      11GG       22GG       33RR       23GR

      23RR       12RG       32RG       31GR

      32GR       12GR       31RG       23RG

      12GR       22GG       11RR       22RR

      12RG       23GR       23GR       12GR                HEISENBERG’S
                                                           UNCERTAINTY PRINCIPLE
      11GG       33RR       12GG       32GR                SIMPLIFIED
                                                           “If you know where it is,
      12GR       23GG       21GR       12GG
                                                           you don’t know where
                                                           it’s going,” or, “If you
      22RR       23GG       13GR       31GG
                                                           know where it’s going,
                                                           you don’t know where it
      12GG       33RR       33GG       32RG
                                                           is.”
      33RR       23GR       11GG       21GR

      11RR       21GG       12RR       22GG

                                                           The ordinary adult never
180. Information and a Black Hole                          gives a thought to space-
                                                           time problems. . . . I, on
Classical information and quantum information are
                                                           the contrary, developed
not the same. Why? Because QM rule 2 tells us that in
                                                           so slowly that I did not
QM there can be a coherent superposition of quantum        begin to wonder about
states. No such state exists in classical physics. So      space and time until I
quantum information supersedes classical information.      was an adult. I then
    The classical and the quantum information content      delved more deeply into
in a system, such as a chair, can be determined or esti-   the problem than any
mated by standard techniques of classical and quantum      other adult or child would
information theory. Suppose the chair is tossed into a     have done.
black hole. The quantum information in the chair                  —A LBERT E INSTEIN ( TO
                                                                        N OBEL L AUREATE
seems to have gone with the chair into never-never              J AMES F RANCK ) IN A LICE
land. Why should we worry about this information               C ALAPRICE , T HE E XPANDED
loss?                                                                  QUOTABLE E INSTEIN




                                                              Chances Are           89
9 Can This
  Be Real?
  A          FTER QUANTUM MECHANICS EXPLAINED

             the internal behavior of the atom in the 1920s
  and the chemistry of atoms and molecules, physicists
  turned toward understanding the atomic nucleus in the
  1930s and 1940s. Rutherford in 1911 had determined that
  practically all the atomic mass was in the nucleus, and of
  course everyone knew that its positive protons balanced
  the electron negative charges in the neutral atom. But what
  held the nucleus of positive protons together? A nuclear
  strong force was eventually identified in the 1970s as the
  color interaction acting between quarks, and it is one of the
  four known fundamental forces in nature. The second
  nuclear force, the weak interaction, responsible for many
  nuclear decays, was identified completely in the 1960s. By
  the early 1980s three of the four fundamental interactions
  had been unified into the Standard Model (SM) of Leptons



                                                                  91
                        and Quarks. Only gravitation needs to be incorporated
                        into the unified model of nature. The selected chal-
                        lenges in this chapter range through the whole gamut
                        of nuclear and particle physics.




92   Mad about Modern Physics
181. Carbon-14 Dating                                        Roughly once a second,
                                                             a subatomic particle
Carbon-14 is produced when cosmic rays collide with          enters the earth’s
atoms in the atmosphere to create an energetic neutron       atmosphere carrying as
that then collides with a nitrogen-14 atom (seven pro-       much energy as a well-
tons, seven neutrons) to make a carbon-14 atom (six          thrown rock. Somewhere
protons, eight neutrons) and a hydrogen atom (one            in the universe, that
proton, zero neutrons). Carbon-14 is radioactive, with       fact implies, there are
a half-life of 5,730 years.                                  forces that can impart
    These C-14 atoms combine with oxygen to form             to a single proton 100
carbon dioxide, which plants absorb into plant cells         million times the energy
                                                             achievable by the most
through photosynthesis. Animals and people eat the
                                                             powerful earthbound
plants and take in the C-14 as well as the normal non-
                                                             accelerators.
radioactive isotope C-12. The ratio of C-14 to C-12 in
                                                              —J AMES W. C RONIN , T HOMAS
the air and in all living things at any given time is            K. G AISSER , AND S IMON P.
assumed constant; about 1 in 10 trillion carbon atoms          S WORDY, “C OSMIC R AYS AT
                                                                   THE E NERGY F RONTIER ,”
are C-14. The C-14 atoms are always decaying, so after                  S CIENTIFIC A MERIC AN
an organism dies, no new carbon atoms are taken in                          (J ANUARY 1997)
and this ratio of C-14 to C-12 atoms decreases.
    The carbon-14 radiocarbon dating of living and
once-living materials began with Willard Libby in the
1940s. Antiquities dated by C-14 agree with other date
records until they begin to disagree for dates more than
several thousand years ago. Why is there disagreement        [Pierre Curie] was
                                                             impressed by Marie’s
in the dates between C-14 dating and the written
                                                             courage and her amaz-
records?
                                                             ing love of work and
                                                             fascinated by her
182. Nuclear Energy Levels                                   lucidity, her challenging
                                                             questions, her reflective
In the 1930s and 1940s, physicists working on the            answers.
energy states of the nucleus of an atom concentrated on                 —J. A. DEL R EGATO,
various models, including a shell model using the                   R ADIOLOGIC AL P HYSICISTS
Schrödinger equation with an approximately constant
electrical potential inside the nucleus. Conceptually,
each nucleon is in a well-defined orbit within the
nucleus and moves in an averaged field produced by all
the other nucleons. However, even though quantum


                                                           Can This Be Real?            93
On August 6, 1945, an
atom bomb dubbed
                                     2p
“Little Boy” was dropped
from an American B-29                1f
bomber called the
Enola Gay on the city of
Hiroshima. It detonated
at 8:16 A.M. at a height
of 1,900 feet. Of                    2s
Hiroshima’s 330,000
                                     1d
inhabitants, approxi-
mately 70,000 were                                                       Energy
killed instantly. By the
end of 1945, the death               1p
toll had risen to
140,000. “Little Boy”
used the gun assembly                1s
design and uranium-
235 as the fissionable
                            states such as n = 1, with l = 0, 1, 2, 3, etc., are possi-
material. Because the
gun design was an inef-     ble in the shell model, the predicted energy levels did
ficient means of caus-      not fit the data. In fact, the actual energy levels were all
ing the chain reaction,     scrambled compared to the shell-model theoretical
about 50 kilograms of       predictions. Why?
89 percent U-235 and
14 kilograms of 50 per-
cent U-235 ended up         183. Nuclear Synthesis
being used. Of this it is
estimated that only         The championship of nuclear binding energy is often
about 2 percent actually    attributed to Fe-56, meaning that Fe-56 has the great-
fissioned. Three days       est binding energy per nucleon and therefore is the
later another atom          most stable nucleus. Most elements are synthesized in
bomb, dubbed “Fat           stars. Supposedly, elements higher on the periodic chart
Man,” was dropped on        than Fe cannot be synthesized in normal star burning
the city of Nagasaki.       cycles. Why not? Actually, the sequence of nuclear syn-
Approximately 40,000        thesis does not stop at iron, because Ni also is synthe-
were killed instantly.      sized. What happens to the Ni isotopes that are
                            synthesized?



94      Mad about Modern Physics
184. Heavy Element Synthesis                                    By the time she enrolled
                                                                at the Sorbonne in 1891,
If we are “truly the stuff of stars,” then where do all the     Maria Sklodowska was
heavier elements beyond iron come from if they are not          twenty-four years old. In
made in normal star burning cycles?                             1893 she passed the
                                                                 license in physics, com-
                                                                ing first in her class. In
                                                                that year Maria
185. Neutron Decay
                                                                Sklodowska met Pierre
A free neutron will decay with a half-life of about 14.8        Curie. The meeting
minutes, but it is stable if combined into a nucleus.           between the two was
Why would the neutron be stable in the nucleus?                 arranged for scientific
                                                                purposes, with little hint
                                                                of matchmaking. At the
                                                                time they met, both
186. Finely Tuned Carbon?                                       Maria and Pierre consid-
Eventually a star exhausts its supply of hydrogen in its        ered themselves des-
core, gravitational contraction occurs, the temperature         tined for single lives.
reaches about 108 K, and helium burning can occur via           After graduation Maria
the reaction 3He-4 → C-12 + 2 photons. In fact, the             intended to return to
                                                                Warsaw to look after her
nucleosynthesis of all the heavier elements essential for
                                                                aging father and teach
life relies on this reaction. However, the chance that
                                                                science. Pierre, mean-
three helium nuclei get together fast enough to form the        while, at age thirty-four
carbon nucleus is negligible. So this critical reaction         one of France’s leading
actually proceeds via an intermediate beryllium step            young physicists, was
given by 2He-4 + (99 ± 6) keV → Be-8 followed by                convinced that he would
                                                                never find a wife who
                                                                would tolerate his com-
                             7.70 MeV                           plete devotion to sci-
                             7.65 MeV                           ence. He was the first to
                                                                fall. Almost from the
                                                                beginning he realized
                                                                that in this severe Polish
                             7.40 MeV                           girl he had found the
                                                                woman of his dreams.
                                                                 —A DAPTED FROM M ARGARET
                                                                       W ERTHEIM , P YTHAGORAS ’
                                                                  T ROUSERS : G OD , P HYSICS , AND
                                                                     THE G ENDER W ARS ; S USAN
                             0.00 MeV                             Q UINN , M ARIE C URIE : A L IFE


                                                              Can This Be Real?             95
An ingenious “explana-                   Be-8 + He-4 → C-12 + 2 photons. Since the Be-8
tion” of the Michelson-                  lifetime of about 10–17 second is much longer than the
Morley null result was                   He-4 + He-4 collision time in a star, the beryllium will
found by George F.                       be around long enough for the reaction to occur.
FitzGerald of Dublin in                       The total energy of the Be-8 nucleus and a He-4
1889. He suggested                       nucleus at rest is 7.4 MeV above the energy of the nor-
that the lengths of bod-
                                         mal state of the C-12 nucleus. The radioactive state of
ies moving through the
                                         the C-12 is 7.65 MeV above the normal state. If the
ether at velocity v con-
                                         energy of the radioactive state were more than 7.7 MeV
tract in the direction of
their motion by a factor                 above the normal state, the formation of C-12 via Be-8
(1 – v 2 / c2)1/2—which                  plus He-4 would require the reactants to have at least
would just compensate                    0.3 MeV of total kinetic energy, which is extremely
for the ether drift in the               unlikely at the temperatures found in most stars.
Michelson-Morley                              The importance of this process is emphasized by
apparatus. A few years                   physicists who inject the Anthropic Principle, that cer-
later [Dutch physicist                   tain constants of nature have values that seem to have
Hendrik A. Lorentz]—                     been mysteriously fine-tuned to just the values that
apparently independ-                     allow for the possibility of life. Recently, others have
ently—made the same                      introduced a further extension that claims that this car-
hypothesis and incorpo-                  bon nucleus coincidence can be explained only by the
rated it into his ever
                                         intervention of a designer with some special concern
more comprehensive
                                         for life. Both groups cite the closeness of the required
ether theory. This
                                         energy to the actual limit, 7.7 MeV – 7.65 MeV = 0.05
“Lorentz-FitzGerald
contraction” then                        MeV, a quantity less than 1% of 7.65 MeV, as their evi-
quickly diffused into the                dence for the fine-tuning. Why is their reasoning sus-
literature.                              pect with regard to this carbon formation process?
       —WOLFGANG R INDLER ,
   R ELATIVITY : S PECIAL , G ENERAL ,
              AND C OSMOLOGIC AL
                                         187. Proton-Proton Cycle
                                         The thermonuclear reactions in the proton-proton
                                         cycle inside the Sun convert four protons into an alpha
                                         particle, two positrons, two electron neutrinos, and
                                         two photons with the release of 26.7 MeV of energy.
                                         First, two protons collide to form a deuteron H-2, then
                                         this deuteron collides with a proton to form He-3, then
                                         finally two He-3 nuclei must find each other to collide


96         Mad about Modern Physics
and form an He-4. The overall representation of this           Bertrand Russell would
proton-proton cycle is:                                        sometimes liken the
                                                               scientific method to the
              4H → He-4 + 2e+ + 2ν + 2γ.
                                                               following syllogism:
    The six photons ultimately produced, including the           Bread is made of
four 0.511 MeV photons from two positron-electron                  rock;
annihilations, take about a million years to reach the           Rock tastes good;
Sun’s surface to be emitted eventually as visible pho-           Therefore bread
tons, which then take about another eight minutes to               tastes good.
reach Earth. The two neutrinos carry away about 3
                                                                  In other words, you
percent of the energy to balance the energy conserva-          can never be sure that
tion equation and to conserve lepton family number.            correct conclusions
    Presented as the primary source of our Sun’s energy,       don’t follow from incor-
this method of burning hydrogen is not the primary             rect premises.
method for fusion energy in many stars. Why not?
What reaction sequence is the primary candidate?               The illumination pro-
                                                               vided at eye level in
                                                               artificially lighted rooms
188. Oklo Nuclear Reactor                                      is commonly from 50 to
In the 1970s, uranium samples from the Oklo uranium            100 footcandles, or less
mine in Gabon, Africa, were discovered to have abnor-          than 10 percent of the
mally high concentrations of the isotope U-235, as high        light normally available
as 3 percent, when only about 0.72 percent of the iso-         outdoors in the shade of
                                                               a tree on a sunny day.
tope was expected in a natural source. Supposedly the
                                                               As a result, the total
high concentration of U-235 is explained by realizing
                                                               amount of light to which
that the uranium deposits at Oklo acted as a natural
                                                               a resident of Boston,
nuclear reactor. Could this natural reactor have been a        say, is exposed in a
breeder reactor making its own Pu and U-235?                   conventionally lighted
                                                               indoor environment for
189. Human Radioactivity                                       16 hours a day is con-
                                                               siderably less than
Radiation doses are expressed in SI units as milliSievert      would impinge on him if
(mSv) effective doses. This unit takes into account            he spent a single hour
the type, the intensity and duration of radiation, the         each day outdoors.
amount and type of body tissues irradiated, and the            —R ICHARD J. W URTMAN , “T HE
                                                                   E FFECTS OF L IGHT ON THE
different radiation sensitivity of the irradiated tissues.
                                                                   H UMAN B ODY,” S CIENTIFIC
The average natural background dose rate in many                        A MERIC AN (J ULY 1975)


                                                             Can This Be Real?            97
In early 1940, Paul                 countries is 1–5 mSv a year. On average, medical expo-
Harteck, a German physi-            sures contribute about another 0.5–0.7 mSv a year. The
cal chemist, felt he’d need         current recommended limit for occupational exposure
up to 300 kilograms of              in many countries is about 20mSv effective dose per
uranium to test his idea of         year averaged over five consecutive years.
using carbon dioxide as a
                                        The typical human adult body has an inherent
moderator. He arranged to
                                    internal radiation dose from its natural amounts of
get the frozen carbon
dioxide (dry ice) from I. G.        radioactive elements, including its major contribution
Farben, and the necessary           of about 40 milligrams of radioactive potassium as the
uranium from Heisenberg.            isotope K-40, which has a half-life of about 1.3
But at the last moment,             Gigayears. This isotope is not the result of artificial
Farben declared they                radioactivity but remains from the formation of potas-
could only supply the dry           sium in the supernova that gave birth to our Solar Sys-
ice until early June; they’d        tem about 5 billion years ago. There has not been
need it after that for              enough time for all of the radioactive potassium to
keeping food fresh during           decay, so that is why there is so much in our bodies.
the hot summer months.
                                    Eileen wonders whether this inherent K-40 radioactive
Harteck scraped together
                                    source is exposing our bodies to more than the recom-
about 200 kilos of ura-
nium, but with that low             mended limit? Is the limit exceeded when several peo-
amount his results were             ple gather together in a small circle?
inconclusive; Germany did
not go ahead with the               190. Nuclear Surprises?
easy, dry ice reactor that
would almost certainly              Which of the following statements is true?
have given them plenty of           1. A typical coal burning power plant releases more
radioactive metal early on             radioactive materials into the air than a typical
in the war. Thus was the               nuclear reactor plant.
clear hot weather of that
summer—so often cursed              2. Spreading all the nuclear waste equally around the
by the Allies for letting              surface of the planet will hardly change the back-
Panzer armies advance                  ground radiation level at all.
into France—central to
forestalling this greater           191. Cold Fusion
evil.
      —M ARK WALKER , G ERMAN       Is cold fusion—that is, the fusion of two deuterium
     N ATIONAL S OCIALISM AND THE   nuclei at about room temperature—a possibility, or
      QUEST FOR N UCLEAR P OWER
                                    can this process be eliminated by theoretical arguments
                     1939–1949
                                    alone?

98       Mad about Modern Physics
192. Fission of U-235                                      Although we are quite
                                                           unaware of their pres-
                                                           ence, there are, on the
                                        U-235
                                                           average, some 400
                                        Fragments          microwave photons in
                                                           any cubic centimeter in
                                        Neutrons           the universe left over
                                                           from the big bang.


                                                           In A.D. 499 the Indian
                                                           astronomer Aryabhata
                                                           presented a treatise on
                                                           mathematics and astron-
During World War II the Germans and the Allies were        omy, the Aryabhatiya.
both working on projects related to nuclear weapons        The Aryabhatiya is a
development. One can calculate the minimum mass of         summary of Hindu math-
                                                           ematics up to his time,
U-235 required for a fission weapon from present-day
                                                           including astronomy,
nuclear physics data sheets. That value is the amount
                                                           spherical trigonometry,
required if the neutrons produced by the fission of
                                                           arithmetic, algebra, and
U-235 encounter stationary target nuclei. The problem      plane trigonometry. The
is much more difficult for two important reasons. Can       Aryabhatiya presented a
you identify them?                                         new treatment of the
                                                           position of the planets in
                                                           space. It proposed that
193. Minimal Nuclear Device                                the apparent rotation of
What is the minimum mass of pure U-235 or Pu-239           the heavens was due to
required in a device for a nuclear event? How would        the axial rotation of the
you estimate this value?                                   Earth. Moreover, Aryab-
                                                           hata conceptualized the
                                                           orbits of the planets as
194. Large Nuclei                                          ellipses, a thousand
Small nuclei that become excited and deformed lose         years before Kepler.
                                                                        —D ICK T ERESI , L OST
their energy by breaking up into smaller fragments. A       D ISCOVERIES : T HE A NCIENT R OOTS
larger nucleus, with 150 or more nucleons, stores           OF M ODERN S CIENCE — FROM THE

most of its excitation energy as rotational energy. As             B ABYLONIANS TO THE M AYA

they slow down and de-excite, these nuclei lose energy


                                                         Can This Be Real?              99
In the end we—that is,           and return to their unexcited shape. What do these
Bohr, Pauli and I—knew           nuclei emit, and how would you characterize the
that we could now be             energy spectrum?
sure of our ground, and
Einstein understood
that the new interpreta-         195. Human Hearing
tion of quantum                  The human eardrum is sensitive to displacements of
mechanics cannot be
                                 less than the diameter of an atomic nucleus. How have
refuted so simply. But
                                 such minute displacements been measured via nuclear
he still stood by his
                                 physics techniques?
watchword, which he
clothed in the words:
“God does not play at            196. 1908 Siberia Meteorite
dice.” To which Bohr
could only answer: “But          In an article by Andrew Chakin in Sky & Telescope in
still, it cannot be for us       January 1984, pages 18–24, the author states:
to tell God, how He is to           A grande dame of scientific mysteries—the Tun-
run the world.”
                                    guska event—turned 75 last summer, her charm
      —W ERNER H EISENBERG ,
    E NCOUNTERS WITH E INSTEIN      very much intact. She continues to seduce both
                                    scientist and charlatan alike, both hoping to
                                    explain what happened over a remote stretch of
                                    Siberian taiga on June 30, 1908. All that can be
When Rutherford was                 said from direct eyewitnesses is that a fireball
offered a position at               nearly as bright as the Sun streaked to Earth out of
Yale that required some             a cloudless morning sky. The bolide’s plunge was
teaching, he turned it              abruptly terminated by an explosion so great that
down, commenting,                   it registered on seismic stations across Eurasia.
“They act as if the uni-            The resulting shock wave circled the Earth twice.
versity was made for
students.”                           The article relates that in 1908 in Siberia a huge
  —A DAPTED FROM E. S EGR È,     meteorite is supposed to have crashed in the forest,
     F ROM X- RAYS TO QUARKS     causing huge fires and a crater many kilometers long,
                                 but no rocky debris was ever found.
                                     By radiocarbon dating tree rings from old trees that
                                 have been living since 1908, Willard Libby and Edward
                                 Teller in 1963 may have learned something very
                                 important about the constitution of the meteorite.
                                 What could the radiocarbon data have suggested?

100       Mad about Modern Physics
197. The Standard Model                                        The Schrödinger equa-
                                                               tion, published in March
                                                               of 1926, was designed to
  Electron neutrino             Up quark                       explain almost all
  Electron                      Down quark                     aspects of the behavior
                                                               of electrons in terms of
                                                               de Broglie waves, rather
  Muon neutrino                 Charm quark                    than of matrices. Physi-
  Muon                                                         cists now could visualize
                                Strange quark
                                                               the atom in terms of
                                                               continuous processes—
                                                               the ripple and flow of
 Tau neutrino                   Top quark
                                                               standing waves—
 Tau                            Bottom quark                   whereas with matrices
                                                               they had to deal with
                                                               Heisenberg’s assertion
                                                               that the nature of the
The Standard Model (SM) of Leptons and Quarks is
                                                               microworld was discon-
the most successful physics model of all in terms of test-
                                                               tinuous and impossible
ing its concepts. The model has six leptons in pairs in        to picture. Little wonder
three lepton families and six quarks in pairs in three         that many physicists
quark families, with the quarks in three different col-        threw away their matri-
ors. Aesthetically, the matching of three to three is          ces and started working
pleasing. Mathematically, this matching of numbers of          with Schrödinger’s
lepton and quark families cancels out infinities in quan-       methods. Even today,
tum field theory calculations, such as the infinities that       most physicists would
would arise from the famous triangle anomaly. How-             say that the Schrödinger
ever important this family matching may be, can you            equation, being nonrela-
provide a fundamental physics argument for the spe-            tivistic, has no right to
cific matching of the first lepton family to the first         be this good.
                                                                            —A DAPTED FROM
quark family, of the second lepton family to the second               ROBERT P. C REASE AND
quark family, and so on?                                                  C HARLES C. M ANN ,
                                                                       T HE S ECOND C REATION :
                                                                M AKERS OF THE R EVOLUTION IN
198. Spontaneous Symmetry                                              20 TH -C ENTURY P HYSICS

Breaking
Spontaneous symmetry breaking is a concept first intro-
duced by W. Heisenberg in describing ferromagnetic

                                                             Can This Be Real?           101
Maria Sklodowska was a       materials. A ferromagnet has a perfect geometric sym-
daughter of a Polish         metry until the Curie transition temperature is reached;
freethinker but reared       then the material becomes magnetized and one particu-
by a Catholic mother.        lar direction of magnetization is chosen. The theory is
She abandoned the            symmetrical still, but the actual material is not. One can
church before she was        summarize the process by stating that microscopic
twenty and her marriage
                             events can have macroscopic consequences. Near the
to Pierre Curie was a
                             critical point of a phase transition, small, random fluc-
purely civil ceremony
                             tuations can grow to make their presence felt through-
because she says in her
memoir of him, “Pierre       out the material. A few aligned spins can propagate
belonged to no religion      their influence throughout the whole crystal, and the
and I did not practice       symmetry is broken.
any.”                            Other examples are the Schrödinger equation and
                             Maxwell’s equations. As successful in helping to
                             describe nature as they have been, these equations
                             have more symmetry than the underlying phenomena
                             they describe. Interest in their symmetry-breaking
                             applications has led to significant new insights into new
                             connections between macroscopic and microscopic
                             phenomena.
                                 In particle physics, the spontaneous symmetry
                             breaking is achieved by the Higgs mechanism. The
One must make of life a
                             Standard Model of Leptons and Quarks relies upon
dream, and of that
dream a reality.             the Higgs particle to spontaneously break symmetry to
           —P IERRE C URIE   provide three of the electroweak bosons with mass
                             while leaving the photon massless. Simultaneously, all
                             the leptons and quarks get their mass values. Moreover,
                             the effect of the Higgs field is to provide a frame of ref-
                             erence in the vacuum for the isotopic spin directions
                             that distinguish the particles of each grouping—for
                             example, neutrons from protons.
                                 Is spontaneous symmetry breaking by the Higgs
                             mechanism the only way to go? Are there other ways to
                             spontaneously break symmetry to achieve the Standard
                             Model of Leptons and Quarks?




102     Mad about Modern Physics
199. Proton Mass                                              Contrary to the claim
                                                              found in some dictionar-
Kate sees that the chart of the fundamental leptons           ies, the word algebra
and quarks shows that the up and down quark masses            does not derive from an
are ~ 5 MeV/c2 each. Yet the proton, which is com-            Arabic expression for
posed of two up quarks and one down quark as the              bone setting but rather
combination uud, has an enormous mass of 938                  it means compulsion,
MeV/c2. She asks why there is such a large mass differ-       as in compelling the
ence between constituents and the final product.               unknown x to assume a
                                                              numerical value.

200. Right- and Left-Handed
Neutrinos?
                                                              The amount of ultra-
Neutrinos are lepton family partners to the electron,         violet radiation that
muon, and tau particles of the Standard Model of Lep-         penetrates the atmos-
tons and Quarks. Each neutrino is thought to be dis-          phere varies markedly
tinct, the electron neutrino being different from the         with the season: in the
muon neutrino, for example. We now know, however,             northern third of the
that each lepton family neutrino type has a very small        U.S. the total amount
mass and is actually a linear combination of three fun-       of erythemal (skin-
damental neutrino states: ν1, ν2, and ν3.                     inflaming) radiation that
                                                              reaches the ground in
    For the weak interaction, there is the left-handed
                                                              December is only about
doublet state | νL, eL> and the two right-handed singlet
                                                              a fifteenth of the
states | νR > and | eR >, with the consequence that the
                                                              amount present in June.
right-handed states interact with the Z 0 boson but do        —R ICHARD J. W URTMAN , “T HE
not participate in the weak interaction mediated by the           E FFECTS OF L IGHT ON THE
W + and W – bosons. The left-handed doublet interacts             H UMAN B ODY,” S CIENTIFIC
                                                                       A MERIC AN (J ULY 1975)
with all three weak bosons. Must one resort solely to
the explanation “that is how Nature behaves,” or is
there another fundamental reason for left-handed dou-
blet and right-handed singlet states?


201. Physics without Equations
John von Neumann and Stanislaw Ulam in the
1940s were among the first to consider attempting to


                                                           Can This Be Real?           103
In 1911 Marie
Sklodowska-Curie, by                 Cellular Automaton Rule 30 for 50 steps
then a double Nobel
Prize winner, was asked
to write a letter of rec-
ommendation for Albert
Einstein who was being
considered for a posi-
tion at the ETH, the
Zurich Polytechnic. She
wrote, “In Brussels,
where I attended a sci-                                                         Time steps
entific conference in
which M. Einstein also            understand natural phenomena via cellular automata
participated, I was able
                                  and computers. Cellular automata (CA) involve adja-
to admire the clarity of
                                  cent cells in a 1-D, 2-D, 3-D, and so on grid of cells (or
his intellect, the breadth
                                  nodes) that take on new numerical values at each tick
of his information, and
the profundity of his             of the clock according to given rules. The future state
knowledge. Considering            of each cell is determined only by the present state of its
that M. Einstein is still         local neighborhood. One can even remove the external
very young, one is justi-         clock and still maintain a progression of states within
fied in placing great             the CA grid to simulate the passage of time.
hopes in him and in                   Some people claim that all of nature will be simu-
regarding him as one of           lated eventually on computers using cellular automata.
the leading theoreti-             Certainly, fluid flows and other large-scale systems in
cians of the future.”             nature can be simulated to a reasonable degree by CA.
 —A LBRECHT F ÖLSING , A LBERT    But concerning the motion of electrons and other fun-
       E INSTEIN : A B IOGRAPHY
                                  damental particles, which involves quantum mechanics
                                  and the fundamental interactions, how will these parti-
                                  cles show their behavior with this CA technique?




104       Mad about Modern Physics
10 Over My
   Head
   U       NTIL THE    1920 S   NO ONE WAS SURE THAT WE

           were seeing stars outside our own Milky Way galaxy.
   Then, after Edwin Hubble established in 1927 that extra-
   galactic galaxies existed and had recession velocities pro-
   portional to their distance from us, the cosmology game was
   afoot. The rules of the game had been established already by
   Einstein in 1916 with his general theory of relativity (GTR).
   The verification of one of its major predictions by analyzing
   the deflection of starlight passing near the Sun during the
   1919 total solar eclipse told everyone that solid theoretical
   foundations were in place. But only in the 1990s did the vast
   accumulation of data on distant objects, by orbiting satellites
   such as the Hubble Space Telescope and the COBE micro-
   wave detector, and by a new generation of ground-based
   telescopes, transform a conjectural science into real testing
   of models of the universe. We present a sample of challenges
   from a vast range of possibilities.

                                                                     105
The possibility that a             202. Olbers’ Paradox
massive object could bend
light rays was discussed           While walking through the fields one night with his
by Newton as early as              dog, Jan looked up to see a remarkably clear night sky.
1704, and later by Henry           In an instant, a famous question flashed in his mind:
Cavendish. However, the            “Why is the sky dark at night?” With his engineering
first actual calculation           background, he determined that if the universe is uni-
 of the deflection angle           formly filled with stars, then their successive spherical
was published by a                 shells would contribute equal amounts, and the sky
Bavarian astronomer
                                   should be ablaze with light from all directions. Yet the
named Johann Georg
                                   night sky remains dark. What is the resolution of this
von Soldner in 1803.
Assuming that light was a          paradox?
corpuscle undergoing the
same gravitational attrac-         203. Headlight Effect
tion as a material particle,
Soldner determined how             We live in a universe in which very distant stars have
much bending would occur           enormous cosmological redshifts of their light. This
for a path that skimmed            fact is interpreted as a cosmological recession velocity
the surface of the Sun.            at nearly the speed of light. Unusual relativistic effects
The deviation, while small,        can be observed when looking at such fast-moving light
is calculable, and                 sources. We consider a more local version here.
Soldner’s value was
                                       Suppose you are standing next to a straight test
0.875 seconds of arc. In
                                   track that carries a vehicle with a light that shines in a
1911 Einstein, using the
principle of equivalence,          cone with an apex angle 45 degrees about the forward
obtained the same result.          direction. In the past, you have always seen the light as
Then in 1915, using the            the vehicle approached. One day the vehicle for the first
equations of the general           time is able to reach its highest speed ever: v = 0.9999
theory of relativity,              times the speed of light. But this time you do not see the
Einstein found that the            light as it approaches. Why not? Do you see the vehicle
deflection had to be 1.75          as it passes and then recedes into the distance? Suppose
seconds of arc, twice the          a distant star or galaxy is approaching you at this speed.
previous value. In recent          What would you see? And if receding?
decades the result as
been confirmed to a
precision better than 0.1          204. Incommunicado?
per cent.
         —C LIFFORD M. W ILL ,
                                   In later problems we will encounter the behavior of light
         W AS E INSTEIN R IGHT ?   near a black hole, particularly its inability to escape


106      Mad about Modern Physics
from a black hole. That is, if you are trapped inside a    At age twelve Einstein
black hole and are still alive, you cannot communicate     suddenly became com-
with your friends outside because nothing escapes.         pletely irreligious. Ironi-
    Meanwhile, consider a related problem in a normal      cally, this conversion was
space environment. Suppose you and your friend are in      the consequence of the
                                                           only religious custom his
separate rocketships that begin next to each other and
                                                           parents observed, namely
accelerate with respect to the stars in opposite direc-
                                                           to host a poor Jewish stu-
tions. You both maintain steady pulsed light communi-      dent for a weekly meal.
cation with each other via intense, nondiverging laser     The beneficiary was Max
beams. But your relative speed is increasing each sec-     Talmud (later “Talmey”),
ond as the separation distance grows ever faster. Will     a medical student from
there come a time when neither of you will receive the     Poland, ten years older
other’s light beam?                                        than Albert. Talmud
                                                           directed his attention to
                                                           popular science books as
205. Local Accelerations                                   well as to various books in
                                                           mathematics. Einstein
                                                           summed up the results of
                                   Test masses             Talmey’s influence:
                                                           “Through the reading of
                                                           popular scientific books I
                                                           soon reached the convic-
    Massive body                                           tion that much in the sto-
                                   Laboratory              ries of the Bible could not
                                                           be true. The consequence
                                                           was a positively fanatic
Einstein formulated the general theory of relativity       [orgy of] freethinking
(GTR) in 1915 based on his Equivalence Principle. In       coupled with the impres-
prerelativistic terms, a uniform gravitational field of     sion that youth is inten-
strength g may be exactly simulated inside a rigid lab-    tionally being deceived by
oratory in a completely gravity-free region of space by    the state through lies.
                                                           Suspicion against every
accelerating this laboratory with a constant accelera-
                                                           kind of authority grew out
tion g m/s2 relative to an inertial frame. By releasing
                                                           of this experience.”
two small test masses, their behavior reveals the physi-             —A DAPTED FROM M AX
cal environment.                                                      J AMMER , E INSTEIN AND
    Suppose an unseen massive body is near the rigid       R ELIGION —P HYSICS AND T HEOLOGY

laboratory. What behaviors of the two small test masses
will reveal its presence?

                                                            Over My Head             107
Are the planets “arranged”     206. Twin Paradox
so that the gravitational
perturbations between          The twins are five years old when one of then is sent off
them are smaller than          in a spaceship that travels nearly the speed of light and
would be expected from a       the other remains on spaceship Earth. After 50 years
random configuration, thus     Earth time the spaceship returns. The twins greet each
resulting in long-term sta-    other and compare their experiences. We know that
bility of the system? Some     the twin who experiences accelerations will age slower
argue that an example of       and return to Earth much younger than 55 years old.
this favorable arrangement     Precisely how does the general theory of relativity
is given by Neptune and        explain the aging of the twin during accelerations?
Pluto, whose orbits appear
to cross if we neglect their
inclinations. Since their      207. Twin Watches
orbital periods are in the     This problem came from Richard P. Feynman in the
ratio 3:2 Pluto never gets     1960s while one of us (F. P.), an undergraduate, was with
near to Neptune and actu-      him in his car on the way to Malibu, California, where
ally approaches more           he gave weekly physics lectures. The third person in the
closely to Uranus. It is
                               car, B. Winstein, then a graduate student in physics, con-
worth noting that the
                               tributed to a discussion that became quite involved!
orbital planes of the plan-
                                    Charlotte holds two identical ideal watches at the
ets are not coincident with
each other or with the         same height, one in each hand. She holds one steady in
equatorial plane of the sun,   her left hand and tosses the other into the air straight
which is inclined at about     up. At the instant the upward-moving watch is along-
7° to the ecliptic.            side the other at the same height above the ground, she
                               sees that the two watches are synchronized, with the
                               exact same readout value. Later, on its downward free-
                               fall path, she reads the time on both watches when they
                               are again alongside each other and at the same height
                               above the ground. Assuming that the moving watch is
                               always in free fall, what would you predict for the two
                               watch readings?

                               208. Global Positioning Satellites
                               The global positioning system (GPS) is a modern marvel,
                               with a constellation of at least 24 satellites, each in a 12-
                               hour orbit at an altitude of about 20,200 kilometers,

108      Mad about Modern Physics
whizzing around Earth at enormous speeds with respect           The Russian astrophysi-
to the ground beneath them. Each satellite knows its            cist George Gamow
own position and sends out signals with this informa-           decided early in life
tion. The GPS handheld receiver uses the signals from at        that traditional religion
least four different satellites to calculate its own position   could not be trusted.
to within a few meters or better when a local reference         After watching Commu-
                                                                nion in the Russian
signal is present. Yet within minutes the accuracy would
                                                                Orthodox Church, he
reduce to many kilometers of error if one of Einstein’s
                                                                decided to see for him-
discoveries were not an essential part of the calculations
                                                                self whether red wine
in the GPS system. What are we referring to?                    and bread could trans-
                                                                form into the blood and
209. Solar Redshift                                             flesh of Jesus. He held
                                                                a bit of the blessed
The light emitted from the Sun shows a redshift of the
                                                                bread and wine in his
spectral lines even though our distance to the Sun is
                                                                mouth, ran home from
fixed during the measurement process. Why so?                    church, and placed the
                                                                specimen under the
210. Orbiting Bodies                                            lens of his new toy
                                                                microscope. It looked
When a body such as a planet orbits around a more
                                                                identical to an ordinary
massive body such as the Sun, the orbit does not close
                                                                bread crumb that he
on itself, as expected from Newton’s universal law of
                                                                had prepared at home
gravitation and Kepler’s laws. The general theory of            earlier for comparison.
relativity (GTR) calculates the correct value for this          “I think this was the
precession of the orbital ellipse, determining that its         experiment which made
complicated equations reduce to an equation similar in          me a scientist,” he
form to that of the classical Kepler problem, with an           recalled. In the 1940s,
additional quadratic term that causes the precession.           Gamow and others pre-
Can you provide a conceptual argument in GTR for the            dicted the existence of
precession of the ellipse?                                      a cosmic background
                                                                radiation.
                                                                 —C OREY S. P OWELL , G OD IN
211. Gravitational Lensing                                       THE E QUATION : H OW E INSTEIN
                                                                  B EC AME THE P ROPHET OF THE
In examining the universe, astronomers utilize a tech-                     N EW R ELIGIOUS E RA
nique called gravitational lensing of the light from dis-
tance stars. Supposedly, space itself can act as a lens for
light rays. How can the emptiness of space—the vac-
uum itself—around stars and galaxies focus light?

                                                                Over My Head            109
In various writings, not
all published, Isaac New-
ton intimated that
comets were divine
agents destined to                  Earth                                             Quasar
reconstitute the entire
solar system, to prepare
sites for new creations,
and to usher in the                                          Galaxy lens
Millennium.
 —S ARA S CHECHNER G ENUTH ,
    C OMETS , P OPULAR C ULTURE ,   212. Cosmological Redshifts
    AND THE B IRTH OF M ODERN
                    C OSMOLOGY      The light from a distant galaxy can exhibit a significant
                                    cosmological redshift. If the cosmological redshift is
                                    not a velocity redshift, what is its origin? Can the two
In a letter to E. Büsching,
the author of There Is              effects be distinguished from each other by observing
No God, dated 25                    the spectrum of the galaxy or other light source?
October 1929, Einstein
declared that a belief in           213. Tired-Light Hypothesis
a personal God seems
“preferable to the lack of          Since the 1920s there has been a popular hypothesis
any transcendental out-             trying to explain the cosmological redshift as a so-
look of life.”                      called tired-light effect—that is, the light loses energy as
      —M AX J AMMER , E INSTEIN     its photons race through space, getting more tired with
       AND R ELIGION —P HYSICS
                                    distance, like a long-distance runner completing a race.
                AND T HEOLOGY
                                    What two specific pieces of evidence rule out this expla-
                                    nation for the cosmological redshift?
My comprehension of
God comes from the
deeply felt conviction of           214. Black Hole Entropy
a superior intelligence             A black hole has an entropy proportional to its surface
that reveals itself in the          area, so it must have a temperature above absolute
knowable world. In                  zero. What would be evidence for this temperature?
common terms, one
can describe it as
“pantheistic” (Spinoza).            215. Black Hole Collision
  —A LBERT E INSTEIN , 1923 IN      Two black holes collide head-on. Will they coalesce
        A LICE C ALAPRICE , T HE
  E XPANDED QUOTABLE E INSTEIN      into one black hole?


110      Mad about Modern Physics
216. Centrifugal Force Paradox                              RIEMANN ANTICIPATES
                                                            EINSTEIN
The general theory of relativity (GTR) predicts that in     Ever since Newton, scien-
certain circumstances the centrifugal force may be          tists had considered a
directed toward, not away from, the center of circular      force to be an instanta-
motion. In fact, if an astronaut could steer a spacecraft   neous interaction between
sufficiently near to a black hole, the astronaut would       two distant bodies. How-
feel a centrifugal force pushing inward, not outward,       ever, over the centuries,
contrary to everyday experience! What is the concep-        critics argued that this
tual explanation of the unusual result?                     action-at-a-distance was
                                                            unnatural, because it meant
                                                            that one body could change
217. Geodesics and Light Rays                               the direction of another
                                                            without even touching it.
In conventional geometry, the geodesic is the shortest
                                                                Georg Bernhard
curve between two points measured by counting how           Riemann (1826–1866)
many rulers fit along the curve. When learning relativ-      developed a radically new
ity theory, one often reads statements that conflict with    physical picture, banishing
intuition, such as the following: “In any space-time,       the action-at-a-distance
with or without a gravitational field, light always         principle. To Riemann,
moves along geodesics and traces out the geometry of         “force” was a consequence
space-time.” “In a space warped by a gravitational           of geometry. He con-
field, the light rays are curved and in general do not       cluded that electricity,
coincide with geodesics.” Why are these phrases, taken      magnetism, and gravity are
                                                            caused by the crumpling
from the general theory of relativity (GTR), not really
                                                            of our three-dimensional
in conflict with each other?
                                                            universe in the unseen
                                                            fourth dimension. Thus a
218. Galaxy Rotation                                        “force” has no independent
                                                            life of its own; it is only
                                                            the apparent effect
         Velocity
                                                            caused by the distortion
                                                            of geometry.
                                                                —M ICHIO K AKU , H YPERSPACE :
                                                                A S CIENTIFIC O DYSSEY THROUGH
                                                                           PARALLEL U NIVERSES ,
                                                                  T IME W ARPS , AND T HE T ENTH
                    Visible edge                                                     D IMENSION



                        Radius

                                                               Over My Head              111
Venus always presents        One of the great surprises in astronomy is the rota-
the same face to Earth       tional behavior of galaxies—that is, all the stars in the
when the two planets         galactic disk revolve at roughly the same tangential
are at their closest
                             speed! Two immediate conflicts with conventional
approach, suggesting
that its peculiar rotation   physics arise: (1) If Newtonian gravitation and Kepler’s
may be due in part to        laws apply, they would dictate a decrease in star veloc-
terrestrial action. A        ity with increasing radius from the galactic nucleus,
simple calculation will      like the planets of the Solar System do. (2) If the spiral
show, however, that          arms of a spiral galaxy are to retain their integrity and
solar tidal action will      persist for at least ten complete revolutions, as they
dominate that of Earth
                             have for the Milky Way, there must be something pre-
on Venus, and so it is
difficult to see how         venting them from wrapping numerous times. By
such a situation has         assuming that Newton’s universal law of gravitation
evolved.                     applies to these galactic problems, what general type of
                             mass/energy distribution must be proposed to explain
God is what mind             the rotational velocity curve? What further hypotheses
becomes when it has          might you propose?
passed beyond the
scale of our comprehen-
sion. God may be either      219. Cosmic Background Radiation
a world-soul or a collec-
tion of world souls. So I    Cosmic background radiation was first detected in the
am thinking that atoms       microwave region in the 1960s and exhibits a perfect
and humans and God           blackbody spectrum equivalent to the radiation from a
may have minds that          source at a temperature of 2.72 K. One would expect
differ in degree but not
in kind.
          —F REEMAN DYSON
                               Intensity (W/m3)

                              3
                                                         2.72 K
                              2

                               1


                                         0.005       0.001    0.0015         0.002
                                                       Wavelength (m)

112     Mad about Modern Physics
lots of remnant starlight all over the universe in all       NOT ALL STARS ARE
parts of the electromagnetic spectrum emitted for the        ROUND!
past 10 billion years or so. Yet this microwave back-        Many red giant stars,
ground radiation is not this light emitted from stars.       including the bright star
How do we exclude this starlight?                            Betelgeuse and the well-
                                                             known variable star Mira,
                                                             exhibit peculiar egglike
220. Planetary Spacings                                      shapes, presumably
For some people the orbital radii for the planets in the     because of the huge
                                                             convection currents roil-
Solar System seem to follow a regular pattern. The pat-
                                                             ing their filmy outer lay-
tern was originally called the Titius-Bode law before
                                                             ers. Astronomers also
Pluto was discovered. According to this numerical
                                                             found huge cocoons of
scheme, the semimajor axis of a planet’s orbit a = 0.4 +     hydrogen gas surround-
(0.3) 2n, where n is taken as negative infinity for Mer-      ing hot blue stars and
cury, zero for Venus, and has increasing integer value       clouds of titanium oxide
by one unit for each successive planet. Neptune does         billowing off red giants’
not fit in this scheme, and the scheme may not repre-         surfaces.
sent an underlying physics.                                  —“M OONBALL : A STRONOMERS
                                                                    B EAT A PATH TO H IGH
                                                                 R ESOLUTION ,” S CIENTIFIC
                       Jupiter                                      A MERIC AN (J ULY 1993)



 Actual
                                                             Many of the oldest
 Bode                                                        stars and star clusters
                                                             in the galactic halo of
 Nottale                                                     the Milky Way move in
                                                             retrograde orbits—that
                  Planet distances                           is, they revolve around
                                                             the galactic center in a
    In the 1990s, applying chaos theory to gravitation-      direction opposite to
ally bound systems, L. Nottale found that statistical fits    that of most other stars.
                                                                 —S IDNEY VAN DEN B ERGH
indicate that the planet orbital distances, including that
                                                               AND J AMES  E. H ESSER , “H OW
of Pluto, and the major satellites of the Jovian planets,        THE M ILKY WAY F ORMED,”
follow a numerical scheme with their orbital radii pro-                 S CIENTIFIC A MERIC AN
                                                                            (J ANUARY 1993)
portional to the squares of integers n2 extremely well!
The planets were divided into two groups, the inner


                                                              Over My Head              113
Experience may suggest                planets Mercury, Venus, Earth, and Mars being at
the appropriate mathe-                n = 3, 4, 5, and 6, respectively, and the outer planets,
matical concepts, but                 starting with Jupiter at n = 1. The two sets can be
they most certainly can-
                                      combined into one set with Mercury at n = 3, Jupiter at
not be deduced from it.
Experience remains, of                n = 10, and so on. The lack of planets at some integers
course, the sole criterion            can be attributed to the history of the Solar System and
of physical utility of a              does not indicate a failure of the prediction.
mathematical construc-                    Other researchers claim that the Nottale sets of
tion. But the creative                integers are not unique and that several alternative sets
principle resides in math-            of integers exhibit excellent fits, raising the question of
ematics. In a certain
                                      whether there is actually a unique pattern in the orbital
sense, therefore, I hold it
true that pure thought can            spacings. In addition, there are known orbital reso-
grasp reality, as the                 nances for the satellites of the Jovian planets that cause
ancients dreamed.                     some of the apparent patterns in the satellite spacings.
    —A LBERT E INSTEIN , 1933, IN         How would you determine whether any of the
 G ERALD H OLTON , T HEMATIC O RI -
                                      claimed patterns are physically significant or simply
      GINS OF S CIENTIFIC T HOUGHT
                                      numerology?

Popular books claim that
Fred Hoyle coined the                 221. Entropy in the Big Bang
term big bang to ridicule
                                      “The primordial fireball was a thermal state—a hot gas
the theory; but Hoyle dis-
puted that. “The BBC was              in expanding thermal equilibrium. But the term ‘ther-
all radio in those days,              mal equilibrium’ refers to a state of maximum entropy.
and on radio, you have no             However, the second law demands that in its initial
visual aids, so it’s essen-           state, the entropy of our universe was at some sort of
tial to arrest the attention          minimum, not a maximum!” How would you resolve
of the listener and to hold
                                      this paradox raised by R. Penrose?
his comprehension by
choosing striking words.
There was no way in which             222. Gravitational Wave Detectors
I coined the phrase to be
derogatory; I coined it to            Radio wave detectors are calibrated by sending out
be striking, so that people           radio waves from a transmitter several wavelengths
would know the difference             away and more. Why can’t builders of gravitational
between the steady state
                                      wave detectors do the same thing? After all, one could
model and the big bang
model.”                               put two large masses at opposite sides of a rotating
              —K EN C ROSWELL ,       platform and spin them around to have a gravitational
       T HE U NIVERSE AT M IDNIGH     wave source for detector calibration.

114       Mad about Modern Physics
223. Space Curvature                                         WHAT’S IN A NAME?
                                                             Planck happened upon
The general theory of relativity (GTR) has been              “Relativtheorie” when
checked and verified at local distance scales. We know        in 1906 he was groping
that the GTR may not explain the rotation curves of          for a name to distin-
galaxies without the introduction of “dark matter.” We       guish the Lorentz-
do not expect the GTR to work for extremely small dis-       Einstein theory (of the
tances, for extremely short time intervals, or for cos-      deformable electron)
mological distances—that is, whether the GTR                 from Abraham’s (theory
                                                             of the rigid, spherical
correctly explains the universe on a global scale. The
                                                             electron). It was in the
GTR, for example, allows for the overall curvature of        discussion following
space but does not predict its global value. In better       Planck’s 1906 lecture
words, the GTR does not fully predict the geometry of        on Kaufmann’s experi-
space, neither determining the global shape nor the          mental electron data
connectedness of space.                                      and their theoretical
    Suppose you were given the task of measuring the         interpretation that
overall curvature of space. One way might be to count        Planck’s term
the number of stars at each radial distance, say, and plot   Relativtheorie was
                                                             embellished by the
the number found versus distance. How does this
                                                             experimentalist
method determine the curvature of space? Does this           A. H. Bucherer to
technique work for both continuous and discrete spaces?      Relativitätstheorie.
                                                             Although Einstein
                                                             had in his published
224. The Total Energy                                        work from the very
The total energy in the observable universe can be           first referred to the
shown to be zero by adding the total mass energy in          Relativitätsprinzip, he
matter and radiation to the total gravitational potential    did not use the designa-
energy. That is: energy total = mass energy + gravita-       tion Relativitätstheorie
                                                             in print until 1907, as
tional energy. Does this result mean that the creation of
                                                             a variant of Planck’s
matter out of nothing contradicts no physical conser-        expression. Felix
vation law?                                                  Klein’s suggestion of
                                                             Invariantentheorie did
                                                             not catch on.
225. Different Universes?                                    —E RIC S HELDON , “R ELATIVITY
                                                              OR I NVARIANCE ?,” A MERIC AN
The present limited understanding of our universe
                                                                        J OURNAL OF P HYSICS
allows for much speculation with regard to whether                       (S EPTEMBER 1986)
we live in but one of many universes, possibly with


                                                              Over My Head             115
This world is indeed a               connections among them. These wild conjectures are
living being endowed                 permitted simply because we do not know enough
with a soul and intelli-             about the origins of the fundamental constants such as
gence . . . a single living          Planck’s constant, the gravitational constant, or the lep-
entity containing all                ton and quark masses, for example. Indeed, the other
other living entities,               possible universes could have different values for these
which by their nature
                                     constants. Suppose that the lepton and quark mass
are related.
                                     values are discovered to be determined by some funda-
                        —P LATO
                                     mental properties in mathematics such as the invariants
                                     of elliptic functions. How might this discovery end
                                     speculation about many universes?
Cosmic bombardment
of Mars by planetary
debris has sent hun-
dreds of tons of Martian
rocks falling to Earth.
Experts estimate that a
Mars rock lands as
often as every three
days, and that billions
have done so over
time—even though only
13 of them have been
identified so far. The
rain of debris from
Mars was hardest,
experts agree, in the
Earth’s early days. And
the reverse trip was far
less likely because the
Sun pulls Earth debris
away from Mars and
toward itself.
—W ILLIAM J. B ROAD, “WANNA
S EE A R EAL L IVE M ARTIAN ? T RY
   THE M IRROR ,” T HE N EW YORK
      T IMES (M ARCH 14, 1999)




116       Mad about Modern Physics
11 Crystal Blue
   Persuasion
   S       OME PROBLEMS THAT COULD HAVE BEEN PUT

           in the previous chapters are presented in this chap-
   ter. We have collected these special problems for this grand
   finale. Some of the previous problems will provide significant
   clues toward answering challenges in this hodgepodge of a
   collection, while many challenges here are new to most
   readers. We hope that you enjoy them as much as we have.




                                                                  117
HIMMEL’S THEORY OF                  226. Iodine Prophylaxis
ACADEMIC TYPES
The greater the cer-                Supposedly, in the event of a nuclear emergency, iodine
tainty of one’s results,            tablets offer protection from radioactive iodine. How
the less the concern                can this preventive measure work? Isn’t all the iodine,
with others’ opinions of
                                    tablets or not, exposed to the ambient radiation?
oneself.
   Thus at the end of
the spectrum occupied               227. Bicycle Tracks
by sociologists and pro-
fessors of literature,              If you came upon this set of
where there is uncer-               bicycle tracks meandering
tainty as to how to dis-            around in the mud,
cover the facts, the                could you determine
nature of the facts to
be discovered, and                  which way the bicycle
whether indeed there                was going simply by
are any facts at all, all           examining the tracks?
attention is focused on             Remember that the
one’s peers, whose                  front wheel and the rear wheel make separate tracks.
regard is the sole crite-
rion for professional
success. Great pains                228. Earth Warming
are taken in the devel-
opment of the impres-               Over the past several decades there has been consider-
sive persona, with                  able concern over the possible slow rise of a few tenths
excessive attention                 of a degree Celsius in the average temperature of
given to distinguished              Earth’s atmosphere and its surface. Most of the concern
appearance and fault-
less sentence structure.
                                    seems to be associated with the greenhouse effect on
   At the other end,                the radiation from the Sun. However, assuming that the
where, as the mathe-                average temperature of Earth can be defined unam-
maticians themselves                biguously so that the small rise in average temperature
are fond of pointing out,           is true, can there have been additional thermal energy
“a proof is a proof,” no
concern need be given               coming from within Earth to cause this rise?
to making oneself
acceptable to others;               229. Frequency Jamming
and as a rule none
whatsoever is given.                Suppose one desired a noisy emitter of electromagnetic
 —R EBECCA G OLDSTEIN , T HE        waves at all frequencies simultaneously. Such a device
 M IND -B ODY P ROBLEM : A N OVEL   might be useful to jam undesired cell phone signals, for
                                    example. How could you do this simply?


118      Mad about Modern Physics
230. Light Energy                                            If the universe of sci-
                                                             ence is not evident to
We know that the speed of light is the same for all          our ordinary senses but
observers in inertial frames. If so, are the momentum        is elaborated from cer-
and energy of light as measured by all observers the         tain key perceptions, it
same even when the light source is moving toward the         is equally the case that
observer?                                                    these perceptions
                                                             require their appropriate
                                                             instruments: micro-
231. Acid Rain                                               scopes, Palomar tele-
Chemists define the quanity pH = -Log [H + ], where           scopes, cloud chambers,
[H + ] is the aqueous concentration of H + ions. Pure        and the like. Again, is
                                                             there any reason why the
rainwater is a neutral solution that has a pH of 7 when
                                                             same should not hold for
the droplets form. Will these droplets falling through
                                                             religion? A few words by
clean, unpolluted air have a pH of 7 by the time they
                                                             that late, shrewd lay the-
hit the ground?                                              ologian, Aldous Huxley,
                                                             make the point well. “It is
232. Electrical Current                                      a fact, confirmed and
                                                             reconfirmed by two or
Upon turning on a lamp, Raymond wonders, “About              three thousand years of
how fast do the electrons in house wiring move as they       religious history,” he
provide electrical energy to the lamps and other elec-       wrote, “that Ultimate
tronic devices?”                                             Reality is not clearly and
                                                             immediately appre-
                                                             hended except by those
233. Earth’s Orbit                                           who have made them-
Earth’s elliptical orbit is not fixed in orientation with     selves loving, pure in
respect to the stars. Why not?                               heart, and poor in spirit.”
                                                              —H USTON S MITH , B EYOND THE
                                                                      P OST-M ODERN M IND



                                                             I swear to you that to
                                                             think too much is a
                                                             disease, a real actual
                                                             disease.
                                                              —F EODOR D OSTOEVSKY, N OTES
                                                                       FROM U NDERGROUND




                                                      Crystal Blue Persuasion         119
Even when all the possi-          234. Crystal Growth
ble scientific questions
have been answered, the           Many children grow crystals in solution as a science
problems of life remain           project. How does a crystal grow from a small “seed”
completely untouched.             to its final size? That is, exactly how do the atoms
    —L UDWIG W ITTGENSTEIN        know where to adhere to the growing structure without
                                  fouling up the precise cubic crystal structure develop-
                                  ment, for example? Do you see the dilemma? And the
The shooting stars are            surprise?
meteors usually the size
of a grain of sand.
                                  235. Ruby, Sapphire, and Emerald
                                  How are ruby, sapphire, and emerald crystals related?
General relativity is             How do they produce their colors?
actually less relativistic
than special relativity.
The complete feature-             236. Kordylewski Clouds
lessness of flat space-
                                  By Kepler’s laws any object orbiting the Sun in an orbit
time, its homogeneity
and isotropy, are what            smaller than Earth’s has a faster speed. So how can dust
ensure that positions             particles placed in solar orbit along the Earth-Sun
and velocities are                radial line but closer to the Sun have the same speed as
strictly relative. As soon        Earth?
as spacetime acquires
“bumps,” or local regions
of curvature, it becomes
                                  237. Twist
absolute because posi-            Scooter
tion and velocity can be
                                  There is a type of three-
specified with respect
                                  wheeled scooter with a
to the bumps. Space-
                                  handlebar extending ver-                Feet go here
time, instead of being
merely a featureless              tically upward from the
arena for physics, is             apex of a V-shaped hori-
itself endowed with               zontal metal frame that
physical properties.              has three wheels. The
        —B RYCE S. D E W ITT,     front wheel can rotate
       “Q UANTUM G RAVITY,”       about a vertical axis at
         S CIENTIFIC A MERIC AN                                     Flexible hinge
           (D ECEMBER 1983)       the foot of the vertical
                                  handlebar at the apex

120      Mad about Modern Physics
intersection, and the two rear wheels are at the ends of         A 1972 study of the polit-
the arms of the V. The two arms of the V are hinged at           ical views of academic
their intersection at the front of the scooter so they can       scientists shows that the
rotate about the vertical axis of the hinge—that is, they        high achievers in acade-
can form a wider or a narrower V angle within limits.            mia, particularly the
                                                                 physicists and mathe-
The rider places one foot on each arm of the V and
                                                                 maticians, are signifi-
swivels his or her body from side to side to tilt the verti-
                                                                 cantly more liberal than
cal handlebar from side to side to make the vehicle go           the rank and file of the
forward or backward. Can you explain the physics of its          academic scientists and
forward motion?                                                  engineers. The study
                                                                 ranks the percentage of
238. Unruh Radiation                                             scientists ranked liberal,
                                                                 from the highest to the
Physicist J. Bekenstein determined that a particle accel-        lowest, as follows: math-
erating in a vacuum experiences a blackbody radiation            ematics, physics, biologi-
bath around itself at a temperature directly propor-             cal sciences, chemistry,
tional to its acceleration. By the equivalence principle,        geology, medicine, and
would a particle at rest in a gravitational field also           engineering. The authors
experience this blackbody radiation bath?                        suggest that if the scien-
                                                                 tist’s work is character-
                                                                 ized by a high degree of
239. Star Diameters                                              intellectuality, creativity,
One can determine the diameter of a distant star even            and orientation toward
though the diameter cannot be measured by parallax.              basic research, the sci-
The process uses the intensity interference, not the             entist will very likely be
                                                                 near the liberal end of
            Star light                                           the scale, and will tend to
                                       Star light
                                                                 be critical of existing
                                                                 social institutions and
                                                                 practices. Conversely,
                                                                 if his work is character-
                                                                 ized by practicality,
                                                                 routine, and know-how,
                                                                 the scientist will tend to
                                                                 be conservative.
                                                                      —E. C. LADD J R. AND S. M.
                                       Multiplier                  LIPSET, “P OLITICS OF ACADEMIC
                                                                  NATURAL SCIENTISTS AND ENGI -
          Correlator                                              NEERS ,” S CIENCE (J UNE 9, 1972)




                                                          Crystal Blue Persuasion           121
The surest sign that            amplitude interference, between the light entering two
intelligent life exists         identical photodetectors (telescopes) from the left side
elsewhere in the uni-           of the star surface, and from the right side of the star
verse is that it has never      surface. This process is valid even though the star is
tried to contact us.            simply a point in the light-gathering optics of either
   —B ILL WATTERSON , C ALVIN
                                photodetector. Can you explain the physics?
    AND H OBBES COMIC STRIP



                                240. Glauber Effect
Among those who                 Does a standard incandescent lightbulb emit single
become famous, one              photons? Paired photons? Triplets?
specific category is
greatly overrepresented:        241. Bird Sounds
firstborn sons who lost
their fathers at a young        Practically all birds and other animals emit sounds that
age and were emotion-           have a fundamental frequency and several harmonics.
ally rejected by their          Some birds, however, can emit just the fundamental
mothers. The foremost           with no harmonics! Measurements inside the bird
example in physics:             reveal that the original sound has harmonics. So how
Isaac Newton.                   does the bird eliminate them before they escape into the
                                great outdoors?


Q: Why do chemists call         242. Spouting Alligator
helium, curium, and             Some alligators can submerge themselves slightly
barium the medical              underwater and vibrate their bodies so that numerous
elements?
A: Because if you can’t
helium or curium, you
barium!                                                               Water surface




                                                                Talented alligator

122      Mad about Modern Physics
individual water droplets are projected upward simulta-        When Glenn T. Seaborg
neously a foot or more directly above the backs of their       and his colleagues at the
heads. What is the physics here?                               Lawrence Berkeley
                                                               National Laboratory in
                                                               California were able to
243. Hair-Raiser Function                                      make a new element in
                                                               1940 with 94 protons in
One of us (F. P.) first heard about the hair-raiser func-       its huge nucleus, they
tion (HRF) from physicist Richard Feynman. We intro-           could not at first imag-
duce this function here as a curiosity to stimulate the        ine that anything more
mind. And if one pulls a hair on top of the head               massive would ever be
upright, this function is probably a good representation       obtained, and so they
of its fast vertical increase with horizontal distance.        called their new element
                                                               ultimium (later it would
                                                               be renamed plutonium).
1200                        2 × 1024


                                                               A typical person age fif-
                                                               teen to forty-five grows
                                                               about an inch overnight.
  0                               0                            When we lie down to go
       0   1    2    3                                         to sleep at night, the
                                      0    1   2 3    4
                                                               spaces in our spine
                    Hair-raiser function                       expand and we get taller.
                                                               In the morning when we
                                                               get up, gravity reasserts
    Most mathematical functions are easy to define,            its downward pull and
and so are their inverses. Physicists utilize a tremendous     very quickly we go down
variety of mathematical functions, the two most com-           a bit. Astronauts typi-
mon being the power of a quantity and the exponential          cally stretch more than
of a quantity. Physicists also use mathematical opera-         two inches in weightless-
tions that may not seem to qualify as a function, such         ness, but as soon as
as the Dirac delta function δ(r – r0). Powers are preva-       they return to Earth
lent in fundamental laws dictated by geometrical sym-          gravity, over a matter of
                                                               hours they shrink to
metries such as the universal law of gravitation and the
                                                               normal height.
Coulomb law, both having potentials proportional to
1/r and forces proportional to 1/r 2 for ideal point
sources. The exponential function increases more


                                                       Crystal Blue Persuasion     123
Cosmic radiation comes        rapidly than practically any other known function and
from outer space. The         is used whenever the change in a quantity is propor-
radiation dose from           tional to the quantity itself, such as in growths and
cosmic radiation              decays.
increases with altitude,          The hair-raiser function HRF(x) can be defined by
roughly doubling every        example on how it maps integers to integers. The
6,000 feet. Therefore,
                              HRF(1) is 1. The HRF(2) = 22. The HRF(3) = (33)3, and
a resident of Florida (at
                              so on. Notice the grouping with parentheses. One
sea level) on average
                              needs a calculator for most of the higher-integer HRFs.
receives about 26 mil-
lirem, one-half the dose      Certainly the HRF is a one-to-one mapping.
from cosmic radiation             How does one calculate the HRF of a non-integer?
as that received by a         Of a complex number? How does one determine the
resident of Denver,           inverse of the HRF? That is, given a number such as 42,
Colorado, and about           how does one determine what number is mapped by
one-fifth of that by a        the HRF into 42? And finally, of what potential use is
resident of Leadville,        the hair-raiser function?
Colorado (about two
miles above sea level). A
passenger in a jetliner       244. Space Crawler
traveling at 37,000 feet      In 1999, the U.S. Patent Office awarded a patent to a
would receive about 60        propulsion device that is a base frame with a sliding
times as much dose
                              carriage on the frame and two counterrotating masses
from cosmic radiation
                              that together couple and decouple to the base frame to
as would a person
                              move the carriage forward and backward in a compli-
standing at sea level for
the same length of time.      cated motion. An onboard battery provides the energy
                              for all internal movement. When the rotating masses
                              are not coupled to the frame, and when placed on a
I don’t think there is one    nearly frictionless air table, the whole device simply
unique real universe. . . .   oscillates forward and back repeatedly, as expected.
Even the laws of physics      When the coupling is allowed to occur at specific
themselves may be             phases in their rotational cycle, the whole device moves
somewhat observer-            only forward in a continual sequence of spurts that are
dependent.                    longer with less air table friction! Will this device oper-
        —S TEPHEN H AWKING    ate likewise in space?




124      Mad about Modern Physics
Answers
                                           inside the bottle will begin to cool just
       Chapter 1                           as the burning finishes. The egg seals
     The Heat Is On                        the opening, so the air pressure inside
                                           decreases as the cooling progresses.
                                           The total downward force will eventu-
1. Egg into a Bottle                       ally be greater than the total of the
Newton’s second law explains the           forces resisting the egg at the entrance,
result. When the egg is resting on the     and the net force downward will accel-
bottle, the ambient air in the room        erate the egg into the bottle. The
plus the gravitational force on the egg    movement continues until the egg
by the Earth together exert a total        drops to the bottom. Kerplop!
downward force on the egg that is
equal to the total upward force pro-
vided by the contact force of the bottle
                                           2. Egg out of a Bottle
plus the force of the air inside. By       The hard-boiled egg is in the bottle
Newton’s second law, a downward            and the goal is to remove this egg
acceleration begins when there is a net    without damage. When the total out-
force downward. To get the egg to          ward force acting on the egg exceeds
accelerate downward, you must              the total opposing force resisting its
reduce the upward force of the air         exit, then the egg will accelerate out of
inside the bottle. This action reduces     the bottle. Assume that the bottle is
the total upward force, allowing a net     held vertically upside down. Earth’s
force downward, with the resulting         gravitational force acting on the egg—
acceleration downward dictated by net      that is, its weight—and the force of the
force = mass times acceleration.           air inside the bottle both contribute to
    The correct timing requires one to     the total downward force acting on
wait until the paper dropped inside the    the egg. The upward force is the con-
bottle has stopped burning, then           tact force of the bottle (which includes
immediately and carefully place the        static friction) plus the force of the
egg on the opening. The warmed air         ambient air in the room.


                                                                               125
    To remove the egg, you need to          latticework with water molecules
create a pressure difference between        loosely connected, so the “holes” in
the air inside and the air outside the      this water lattice can accommodate a
bottle, with the greater pressure inside.   large number of other molecules. The
Hold the bottle with its mouth near         sugar molecules form temporary
your mouth and its bottom somewhat          hydrogen bonds to the water mole-
higher so that the egg is positioned far    cules, these bonds breaking and re-
forward, to the neck of the bottle, but     forming continually. Essentially the
not completely blocking the neck.           rule is “like dissolves like.” Of course,
Exhale a tremendous burst of breath         the sugar molecules are quite large,
into the bottle to suddenly increase the    and one cup of sugar molecules con-
inside air pressure. The Bernoulli          tains only about one twenty-fifth the
effect, the reduction in air pressure       number of water molecules in a cup.
perpendicular to the air stream flow         So there are many water molecules to
direction, caused by the rush of air        every sugar molecule in the solution.
flowing around the egg plus the              Wolke, R. L. What Einstein Told His Cook:
increased inside air pressure, will aid       Kitchen Science Explained. New York:
                                              W. W. Norton, 2002, pp. 21–22.
in pushing the egg through the neck
and mouth of the bottle. When the net
force outward occurs, the egg acceler-      4. Kneading Bread
ates outward. Sometimes the egg just
pops out and must be caught in the          Each successive kneading of the bread
mouth, and at other times the egg           dough distributes the CO2 gas released
must be gently removed at the open-         by the action of the yeast to make a
ing. The more vertical the bottle is, the   finer texture—that is, smaller holes
more help one has from the gravita-         more evenly distributed throughout
tional force of the Earth.                  the bread volume.
                                                Initially, the concentration of yeast
                                            is not uniform but has some non-
3. Sugar                                    uniform volume distribution in the
One can usually dissolve about five         bread dough. Where there is more
cups of sugar in one cup of water!          yeast, there will be more CO2 gas pro-
Very simply, sugar molecules can            duced by the yeast chemistry and usu-
squeeze into the empty spaces among         ally bigger bubbles in the region. At
the water molecules, so they are not        the molecular level, CO2 molecules
really occupying more space. The            released by the yeast will diffuse into
water forms somewhat of an open             the surrounding dough somewhat,

126     Answers
probably not moving very far in the             If ice chunks were used instead of
available time. Some of the gas bub-        butter, then the procedure is correct.
bles may even coalesce to form bigger       As a check, when floating ice melts,
bubbles. Without further kneading,          the water level does not change.
some places in the bread will possess       Therefore, the recommended proce-
many bubbles or large bubbles, and          dure is correct for measuring one-half
other places may have very tiny bub-        cup of ice. But the density of butter is
bles or none. We have all seen bread        not the same as that of the ice, and the
with a non-uniform distribution of          density of melted butter is not the
bubbles, or even with one large bubble      same as that of the water. Therefore,
somewhere. A more thorough knead-           the immersed volumes of ice and but-
ing would eliminate these oddities,         ter would be different.
unless they are planned.                        However, if the butter is held
                                            under the water surface when water is
                                            added, then the butter measurement is
5. Measuring Out                            correct when the water reaches the
                                            one-cup level. One is simply measur-
Butter
                                            ing the volume of the butter, and one
Butter floats on water because its den-      does not use Archimedes’ principle for
sity is less than the density of water.     this measurement.
The recommended procedure in many
cookbooks for measuring one-half cup
of butter is: put one-half cup of water
                                            6. Milk and Cream
into the measuring cup and then add         The milk is “heavier,” meaning more
pieces of butter until the water level is   dense. Cream floats to the top in a milk-
pushed up to the one-cup level. Often       cream mixture, so the cream is less
a reference to Archimedes’ principle is     dense. Quite often people confuse mass
given. (According to Archimedes’ prin-      density with liquid flow sluggishness.
ciple, a body wholly or partially           The two properties are unrelated. Many
immersed in a fluid will be buoyed up        of these people will think that the cream
by a force equal to the weight of the       is more dense. Anyone who has milked
fluid that it displaces.)                    a cow or a goat has seen the cream at
    However, this recommended meas-         the top. A pint of light cream weighs
uring procedure does not use                about 1.5 grams more than a pint of
Archimedes’ principle! And the meas-        heavy cream on average. Why? Because
ured amount of butter is not exactly        heavy cream has more fat per volume,
one-half cup!                               and fat is less dense than water.

                                                                    Answers      127
    Separation of materials by density    on the top or the bottom of the potato.
has been an important method for mil-     With a sudden thrust, drive the
lennia. Gold and platinum have been       pinched straw into the potato held in
separated from other elements by          the other hand. The straw goes right
dumping the ores into a hot bath of       through. Why? The trapped air upon
lead. The specific density of lead is     contact is compressed inside the straw
11.36, of gold 19.32, and of platinum     and helps the straw remain rigid. The
21.45, so gold and platinum atoms         paper or plastic is stretched taut and is
sink and practically all other elements   more difficult to bend significantly.
and compounds float. Of course, the        Alternately, one could pinch the straw
workers must not breathe in the lead      in a vertical position in a vise or a
atoms in the vapors.                      clamp and drop the potato onto the
                                          straw (or a collection of straws).

7. Straw and Potato
Simply placing the straw in one hand
                                          8. Blueberry Muffins
and trying to push the straw through      The downward drift of the blueberries
the hard potato leads to a frustrating    in the warm batter is caused by the
failure. The straw material—paper or      gravitational force of the Earth, so one
plastic—cannot take much compres-         must increase the friction between the
sion before bending sideways. So this     batter and the surface of the berry to
sideways bending must be prohibited       hinder this settling. One could make
if success is possible. We can use        the batter thicker, but this solution
air pressure to help make the straw       may not produce the desired muffin
more rigid.                               texture. Instead, before mixing the
     Pinch the straw between your         blueberries in the batter, dampen them
thumb and forefinger about two            slightly and shake them in a bag of
inches from the end that is farthest      flour. The flour attaches to the berry
from the potato and squeeze tightly.      surface and increases the static friction
Hold the potato carefully but securely    with the batter, keeping the berries
in the other hand placed in a horizon-    uniformly distributed in the muffin.
tal plane with thumb on one side and          The physics involves Newton’s
fingers on the other side of the potato.   second law, found in every high
Make sure that no part of the hand        school textbook, that is, net force =
will be in the path of the straw—that     mass × acceleration. The downward
is, avoid having any part of your hand    gravitational force on the blueberry


128     Answers
must be balanced by an upward force             9. Can of Soup
applied by the batter through friction
to produce zero net force in the verti-         Turn the can of soup upside down and
cal direction and therefore zero accel-         open the bottom. Then turn the can
eration downward from rest.                     over and watch the concentrate being
    In this case, the maximum value of          pushed out by the weight of the more-
the static frictional force (equal to the       liquid stuff, assuming that the
coefficient of friction times the force of       upward-acting static frictional force of
the batter perpendicular to the blue-           the wall with the concentrate balances
berry’s attempted movement) has not             the weight of the concentrate. The
been exceeded. Without the flour coat-           weight of the liquid thus provides the
ing, the static friction coefficient is too      nonzero net force downward to accel-
small. The maximum static frictional            erate the liquid downward by New-
force upward provided by the batter             ton’s second law.
alone is too small. The berry acceler-              If there is delay in this evacuation
ates downward until reaching the crit-          process, allowing some air into the liq-
ical velocity, for which the net force is       uid region behind the solid concentrate
again zero, so the blueberry drifts             might expedite the motion. Sometimes
downward with no acceleration. With             the air seal at the wall is very good, so
the flour coating, the coefficient of           that as the concentrate slips outward, a
static friction is large enough to not be       significant inward pressure difference
exceeded and the downward gravita-              can build up to slow down the extrac-
tional force is always balanced by the          tion process. In addition, the molecules
static frictional force upward.                 in the soup may interact more vigor-
    At the atomic level, friction involves      ously than expected via the electrical
electrical forces acting between atoms          force between the concentrate and the
and between molecules. Still an active          wall of the can (i.e., the viscous force,
research area, the influence of quantum          the surface tension, etc., may be large
mechanical effects is significant. Even          enough to make the extraction even
sound waves contribute good vibrations          more challenging).
in this field, called tribology!
 Krim, J. “Friction at the Atomic Scale.”       10. Salt and Sugar
  Scientific American 275, no. 4 (1996):
  74–80.                                        Salt and sugar do their work on bacte-
                                                ria by osmosis, dehydrating them so
 Miller, J. S. The Kitchen Professor. Sydney:
  Australian Broadcasting Commission,           that they die or are deactivated. A
  1972, pp. 22–24.                              bacterium in very salty water has a


                                                                        Answers     129
saltier environment outside its cell         12. Ice Cream Delight
membrane than inside. Water mole-
cules will move from its inside through      Good ice cream contains abundant air
its water-permeable membrane to the          bubbles to keep it light with very small
outside to balance the salt concentra-       ice crystals so that the texture is
tions, a process called osmosis. The         smooth. There are several good small
bacterium shrivels up and dies. Sugar        appliances for making ice cream and
works by the same process to preserve        sorbet. Some ice cream makers rely on
fruits and berries. In the markets today     human musclepower to turn a large
one can buy “cured” hams and other           spatula to control the ice cream tex-
pork products that use both salt and         ture; others are electric. But with an
sugar to enhance the flavor.                  abundant supply of liquid nitrogen at
 Wolke, R. L. What Einstein Told His Cook:
                                             –196°C and about an equal volume of
  Kitchen Science Explained. New York:       ice cream mixture, the freezing of the
  W. W. Norton, 2002, pp. 137–138.           ice cream occurs so fast that only small
                                             crystals have time to grow. As the
                                             nitrogen furiously boils, plenty of small
11. Defrosting Tray
                                             gas bubbles form in the mixture. All
The “miracle” defrosting tray is simply      these effects make for a delicious treat.
aluminum metal, and one could use a           Kurti, N., and H. This-Benckhard. “Chem-
thick aluminum frying pan or other             istry and Physics in the Kitchen.” Scientific
piece of aluminum or copper metal              American 270, no. 4 (1994): 66–71.

tray to do the thawing as quickly, as         Walker, J. “The Physics of Grandmother’s
                                               Peerless Homemade Ice Cream.” Scientific
long as there is no coating on the             American 250, no. 4 (1984): 150–153.
metal. Nonstick pans have coatings
that are poor heat conductors. Metals
are the best heat conductors because         13. Cooking a Roast
they have about 1023 conduction elec-
                                             The bone-in roast cooks faster because
trons per cubic centimeter available to
                                             the bone, even though porous, rapidly
transfer thermal energy from the hotter
                                             conducts thermal energy to the inside
source to cooler regions. For thawing
                                             faster than the meat itself does. So the
purposes, the metal tray will conduct
                                             bone-in roast cooks from both direc-
thermal energy from the room air into
                                             tions—outside in and inside out. There
the frozen food very efficiently.
                                             will be some minor speedup effect
 Wolke, R. L. What Einstein Told His Cook:
  Kitchen Science Explained. New York:       from the difference in specific heats of
  W. W. Norton, 2002, pp. 202–203.           bone and meat, and there will be


130     Answers
slightly less meat if both roasts weigh    chunks cook faster and therefore
the same, but in the simplified, ideal      require less fuel. The faster cooking
case we ignore these differences. One      occurs because (a) the inside of the
could use computer modeling with the       small cube is closer to the heat source
appropriate physics equations to           than for a thicker piece and (b) the meat
determine the temperature distribution     is tumbled during stir-frying, exposing
in various parts of the meat and bone      different small surfaces to the higher-
in the two cases, showing that some        temperature direction. The temperature
parts of the meat are cooked more          sensed by the meat tends to decrease
than others in both types of pieces.       with distance from the thermal energy
    Any general physics text discussing    source, in this case the pan bottom.
thermal conductivity and specific heat           The amount of cooking experi-
contains the pertinent information for     enced by a small volume of interior
analyzing this problem, but the actual     meat is proportional to the tempera-
temperature profile as a function of       ture experienced and the duration of
the elapsed time is difficult without      cooking at this temperature. Both of
some idealizations about the shape, the    these quantities are changing during
uniformity, and other things. We have      the cooking process. In addition, the
considered the idealized case above        thermal conductivity and the thermal
that ignored the change in bone prop-      heat capacity of the meat are changing
erties with temperature change, such       because the meat material itself is
as the specific heat of the bone and       changing. For example, if the outside
its thermal conductivity. Somehow          becomes charred, its thermal conduc-
Nature has figured out all these things     tivity is significantly reduced, so that
without special computer modeling!         the transmission of thermal energy
                                           decreases compared to its prior rate.
                                           Therefore hamburgers, which must be
14. Cooking Chinese                        cooked thoroughly inside to kill the
                                           bacteria on the surfaces of the ground-
Style                                      up meat, must never be charred on the
There are at least two good reasons for    outside because the inside will tend to
cutting up meats into small volumes: (1)   remain uncooked or partially cooked,
marinades and spices penetrate more        creating a dangerous eating condition.
thoroughly into the meat in a shorter           The physics here is found in any
time because the inner-volume elements     high school physics text, but the
are closer to the surface; (2) smaller     application to cooking was devised


                                                                    Answers     131
millennia ago by ancient chefs who          tension has exceeded its elastic limit.
desired a particular result with per-       During the rupture process, bean skin
haps a minimum of fuel expenditure.         molecules have been separated because
Certainly one can go to the other           the electrical force holding one mole-
extreme by slow-roasting a whole car-       cule to the next has been exceeded.
cass in a revolving spit or in a hot         Miller, J. S. The Kitchen Professor. Sydney:
ember-lined pit in the ground.                Australian Broadcasting Commission,
                                              1972, pp. 81–82.

15. Baked Beans
The hot beans have taken up water
                                            16. Ice Water
and swelled so that the skins are under     The ice at the top brings about faster
great tension. Blowing cool fast-mov-       cooling of the water in the pitcher. As
ing air across the beans from pursed        some ice melts, this cold water is more
lips (instead of an open mouth) lowers      dense than the surrounding water and
their surface temperature and reduces       sinks, cooling the water it passes
the ambient air pressure. The inside is     through. The warmer, less dense water
still hot, so the larger pressure differ-   at the bottom is buoyed upward into a
ence results in the hot, high-pressure      cooler region. This mixing helps the
vapor under the skin pushing outward        water cool faster than when the ice is
just a bit more. If the pressure gradient   held at the bottom, because the cold
is great enough, the skin will rupture.     water produced by the ice would
The slight cooling of the skin material     remain at the bottom. Thermal con-
increases the skin tension to reduce the    ductivity throughout the water would
time to rupture.                            eventually cool the water above, but
     We can simplify the physics to         the convection currents work faster.
applying Newton’s second law perpen-        Of course, vigorous stirring of the ice
dicular to the skin surface. Three          water eliminates any need for the pre-
forces acting on the skin are impor-        vious discussion!
tant: (1) the inward force of the ambi-          One could say that the discussion
ent air pressure, (2) the inward force      above is incomplete because in our ide-
of the tension in the skin, and (3) the     alization we have ignored the ice ther-
outward force produced by the hot,          mal interaction with the ambient air.
high-pressure gas within. Blowing the       This interaction can be important,
air reduces the ambient air pressure        especially on hotter days. The ice does
enough to create a net force outward,       its job when thermally interacting with
and skin rupture occurs when the skin       the water, not with the air! The ice held

132     Answers
in the water would be a more efficient      swollen. Cold water on the outer sur-
direct interaction procedure. So when      face of the hot beet causes the skin to
the ambient air temperature is great       shrink, but the innards remain hot and
enough, they could be competitive.         swollen. So the stretching skin bursts
    By the way, this cooling process is    in several places and becomes easier to
exactly the same as the sequence of        remove with a paring knife without
events that occurs when a pond freezes     being so messy.
over in winter. However, in that case,         The procedure is exactly opposite
the pond water is prevented from cool-     to placing an ice cube in hot water.
ing further and from freezing through      Now the outside of the cube tries to
until all the water reaches 4°C first.      expand, but the inside is still cold. One
This delay in freezing throughout          can hear the thermal stresses crack the
saves the lives of pond organisms          ice cube.
through the winter if spring comes
soon enough. Evaluated in a different
way, there would be no life surviving      18. Igniting a Sugar
the worst ice ages on Earth if water
                                           Cube
did not reach its maximum density at
about 4°C!                                 Very small particles tend to ignite
                                           more easily. The large surface-area-to-
                                           volume ratio for a collection of small
17. Peeling Vegetables                     particles aids ignition, providing a
When a tomato is held carefully over a     large combustion area for the chemical
flame and rotated, some of the thermal      interaction of their surface molecules
energy gained by the tomato vaporizes      with oxygen and also providing a
the water just under the skin to locally   nearby heat source for sustenance.
rupture it. A paring knife can remove      Therefore, rub the far corner of the
the skin easily after the tomato has       sugar cube in some cigarette ash or
cooled. Often, just pulling on the rup-    tiny ash particles from burned paper,
tured skin is enough. Very hot water       then light the ashen cube with the
can be used instead of a flame, but the     burning match. Ignition is now easy.
effects are not as dramatic, and the       Oxygen molecules react with mole-
peeling is a bit more difficult.            cules in the ash to produce thermal
    The boiling raises the temperature     energy and product molecules, includ-
of the beets to cook them and, simul-      ing water.
taneously, a small amount of hot               Historically, there have been many
water enters them, so they are slightly    examples of the spontaneous ignition

                                                                   Answers     133
of dust particles in the air, such as       process is more complicated than the
explosions in granaries where grains        simple, ideal version we have consid-
crops are stored and in mills that grind    ered here, involving nucleate boiling,
grain into smaller particles. A small       transition boiling, etc., in the real situ-
warm spot in the dusty air, perhaps         ation. However, the complete analysis
produced by sunlight, by a match, or        produces the same general argument.
by friction, can rapidly spread into a           One should ask whether different
full-scale explosion.                       results would occur with sea salt, a
    On a less violent scale, simply         mixture of KCl and NaCl and dead
lighting a campfire outdoors begins         organic matter in larger grain sizes
with kindling, very small sticks and        than normal table salt. The slow rate
shavings of wood, which have a very         of dissolving the larger sea salt grains
large surface-area-to-volume ratio.         may delay the reboiling longer than
The slightly larger sticks can be added     experienced for NaCl.
once the flame sustains itself. Finally, a        The relevant physical data are in
whole faggot of sticks can be placed in     most chemistry and physics hand-
the firepit to generate a lasting fire.       books as well as in some textbooks.
                                            The physics pertinent to the idealized
                                            case discussed above is part of tradi-
19. Water Boiling
                                            tional physics and chemistry courses,
The major thermal effect is the raising     but the more detailed complete analy-
of the boiling point (i.e., boiling tem-    sis can be found only in the technical
perature) by the added salt in solution     literature.
from 100°C (standard conditions at 1
atmosphere) to about 104°C (if the
salt is pure NaCl), a significant change     20. Put the Kettle On
that results in a small time lag before
boiling begins again if thermal energy                               Steam
input continues.
    In contrast, the actual amount of
thermal energy needed to raise the                                  Spout Gap
temperature of the sprinkled salt itself
is minuscule because the specific heat
of NaCl is much lower than for water,
and the amount of water by weight in
the pot is enormous compared to the
amount of salt. The actual boiling

134     Answers
No, you cannot see water vapor, that         level. So at higher elevations potatoes
is, water molecules in their gaseous         may not cook as quickly in the open
state. If you look closely at the orifice     pot, and the lukewarm water will not
of the spout, there is a clear region per-   make good tea or instant coffee. In the
haps up to one inch long. That’s where       mountains we would be wise to use a
the water vapor is before it condenses       pot with a lid so that the total pressure
into the steam you can see. The tem-         acting downward on the water surface
perature of the vapor in the clear           can be higher—vapor pressure plus
region is still too high for droplets of     atmosphere—and so that the water
steam to form—that is, collisions of         boils at a higher temperature than
water molecules are too violent to           without the lid, hopefully almost at
allow them to bind together to form          100°C.
droplets.                                        Suppose the cooking at sea level is
    In the clear region at the orifice of     done in a pot with a lid. Now the
the kettle, the water molecules are          action of lifting the lid to “watch the
moving so rapidly that when they do          pot” reduces the thermal energy in the
collide, the van der Waals attractive        air above the liquid surface as some
force—an induced dipole-dipole elec-         molecules escape. These escaping water
tromagnetic interaction—at close             molecules are among the most ener-
range cannot keep them together. As          getic in the vapor, so they can carry
the water vapor cools farther away           away much thermal energy. The pres-
from the end of the spout, these same        sure above the liquid is now lower
collisions produce droplets that grow        than before, the boiling occurs at a
in size.                                     lower temperature, and the cooking
                                             takes significantly longer. One should
                                             replace the lid and let the food cook
21. The Watched Pot
                                             undisturbed! Hence the expression “A
Put a pot of water on a flame atop a          watched pot never boils.” This state-
stove. The thermal energy from the           ment actually refers to the extended
flame raises the water temperature. If        cooking time and involves some good
the pot has no cover, soon the water         physics.
vapor pressure above the water sur-
face equals the pressure of the ambient
atmosphere. The water is now boiling.
                                             22. Ice in a Microwave
If we are high in the mountains, the         Yes! The water molecules in the liquid
boiling has occurred at a lower tem-         state rotate a bit in the microwaves
perature than when we are near sea           and transfer energy to the surrounding

                                                                     Answers     135
molecules to make them jiggle ran-         given above. At the water molecule’s
domly. The water molecules in ice are      resonant frequencies in the microwave
locked into crystals and are unable to     region of the electromagnetic spec-
rotate. (Note: The actual details of       trum, the applied field changes so
molecular bonding in the ice are more      rapidly that very little energy is trans-
complicated and show that a minus-         ferred to the nearby molecules.
cule amount of rotation is possible,       Microwave ovens actually operate at a
but an insignificant amount to change       frequency that is lower than the
the ice to water.) Using microwaves,       frequency at which the absorption is
therefore, one can boil water inside an    greatest. The food needs to be heated
ice block!                                 throughout, and by lowering the
    Boiling the water inside the ice       applied frequency a bit, more micro-
block is an example of selective energy    waves penetrate farther inside, past the
absorption. Numerous examples of           outer layer.
selective absorption occur in the natu-     Kurti N., and H. This-Benckhard. “Chem-
ral world. For example, the green            istry and Physics in the Kitchen.” Scientific
                                             American 270, no. 4 (1994): 66–71,
leaves of plants have chlorophyll A          120–123.
and B molecules that selectively            Walker, J. “The Secret of a Microwave
absorb bluish and greenish light for         Oven’s Rapid Cooking Is Disclosed.”
photosynthesis. At an even smaller           Scientific American 256, no. 2 (1987):
                                             134–138.
scale, nuclei are very selective in
absorbing gamma rays of specific ener-
gies. At the macroscale of meters, we
know that rooms can absorb and
                                           23. The Glycemic Index
amplify sound energy at selected reso-     The rate of conversion from one type
nance frequencies. Some materials are      of molecule to another is a chemical
even useful for just the opposite          process, with the ratio of surface area
behavior, such as window glass, which      to volume for particles in the food
has no selective absorption in the visi-   being converted being an important
ble part of the electromagnetic spec-      factor. Smaller spherelike particles
trum. You can take your pick, but the      have a higher ratio of surface area (SA)
game is played by the rules of nature.     to volume than larger ones. Conse-
    The selective absorption by water      quently, since the reactions occur on
molecules (and some other molecules)       the surface, material consisting of
in a microwave environment is a little     smaller-diameter spheres convert to
different from the other examples          sucrose faster than the same material


136     Answers
consisting of larger spheres. In fact, for   the glycemic index for numerous foods
the limiting case of a sphere, the ratio     and some of the recent results from
SA/Vol. = 3/R, where R is the radius of      nutrition and diabetes research world-
the sphere. Physical and chemical            wide. Excerpts from the book and
processes initiated by the environment       many other resources on the glycemic
occur first at the surface of the parti-      index and its comparison to the insulin
cle. In addition, for biological systems,    index can be found on the Internet.
larger particles must be reduced to
smaller ones before passing through
membranes. So a collection of small
                                             24. Electric Pickle
particles equal in total mass to one         Even though the electrical energy
large particle will be reduced to            source is provided through an AC cur-
acceptable size faster than the large        rent, the pickle glows predominantly
one because the same amount of               at one end with a yellowish color that
chemical solution acts upon a much           is determined by the pickling solution
larger total surface area.                   and the pickle type. Reliably predict-
     The smaller the particles of            ing which end will glow has not been
ingested food, the faster can be the         achieved. There is no actual symmetry
digestion of the molecules in the intes-     here in the shape or chemical compo-
tines because the surface-area-to-           sition of the pickle, so alternate glow-
volume ratio is higher, and the sooner       ing is less likely and never seen. The
is the uptake into the blood. The            conjecture is that the pickle is now act-
higher temperature used for baking           ing like an electrical diode, passing
the potato makes its particles smaller       current in one direction only!
than for the same potato when boiled              The authors listed below performed
at 100°C, so its glycemic index is           an experiment by taking a visible light
greater, reflecting its faster uptake into    spectrum of the glowing pickle, using a
the blood.                                   spectrometer with a diode array detec-
     Dates contain some maltose, a           tor. A fiber-optic probe was used to
sugar that is even faster than glucose       channel the yellow glow to the spectro-
in its basic conversion to sucrose in        graph, and a calibration spectrum was
the blood, so their glycemic index is        taken of a sodium chloride flame test.
above 100.                                   The emission spectra of the two are
     A popular book The New Glucose          nearly identical
Revolution by J. Brand-Miller is avail-           This pair of emission lines, at
able in many libraries and has tables of     589.00 nanometers (nm) and 589.59


                                                                     Answers      137
                                                 25. Space-Age
                               Pickle
                               NaCl              Cooking
                                                 Unlike electric cooktops, which gener-
                                                 ate thermal energy by the electrical
                                                 resistance of the burner coils, magnetic
                                                 induction cooktops generate thermal
                                                 energy by the magnetic resistance of
                                                 the metal cooking vessel itself. The 60
        587.5 590.0 592.5
                                                 Hz AC current flowing in the induc-
            Wavelength (nm)                      tion coil beneath the ceramic surface
                                                 produces an alternating magnetic field
nanometers, indicate a characteristic            than interacts with the Fe atoms—for
of sodium emission, called the sodium            example, in the iron frying pan—to
D line doublet. Josef Fraunhofer                 oscillate its magnetization 120 times
observed these lines in the emission             per second. The magnetization direc-
spectrum of the Sun, in about 1817.              tion changes have resistance, so much
We know now that these lines are due             energy goes into thermal energy in the
specifically to an electronic transition          metal of the pan. Iron and stainless
of sodium atoms in the gas phase.                steel pans will work, but aluminum,
    The pickle conducts electricity due          copper, glass, and ceramic pans and
to the vinegar (acetic acid) and sodium          pots will not. The advantages are no
chloride salt used to make it. Sodium            noise and no hot cooktop except
ions in the pickle liquid attach elec-           where the pan has been in contact.
trons from the flowing current. These                 Cooking with light is not done by
ions are neutralized electrically, form-         lasers! Light is meant in the broader
ing excited sodium atoms in two dif-             sense of the word, the infrared (IR)
ferent excited electronic states (hence          through the visible into the ultraviolet
the emission doublet). Because of the            (UV) part of the electromagnetic spec-
heat and sparks and general pandemo-             trum. Banks of 1500-watt halogen
nium around the electrodes stuck in              lamps in the oven walls put out about
the pickle, these sodium atoms are in            70 percent IR, 10 percent is visible
the gas phase. They emit yellow light            light, and the remaining 20 percent is
as they relax to the ground state.               simply heat. The IR is not thermal
 Appling, J. R., F. J. Yonke, R. A. Edgington,   energy, but when IR is absorbed by
  and S. L. Jacobs. “Sodium D Line Emission
  from Pickles.” The Journal of Chemical         molecules, their random motions can
  Education 70, no. 3 (1993): 250.               be increased. Thermal energy (“heat”

138      Answers
in the vernacular) is the random
kinetic energy of molecules and atoms.
These frequencies penetrate meat only
                                              now
about half an inch at most. Thermal                  future
conduction transfers some of this ther-
mal energy farther inside.                                         Sun
    However, these light ovens also          Earth
have a microwave source to penetrate
with microwaves to cook the interior
of the meat. So while the outside is
being browned by the light, the inside        North Pole will be alternating from
is cooking via microwaves. The overall        the extreme of being pointed toward
benefit is much faster cooking than is         the Sun and to being pointed away
possible in conventional ovens.               from the Sun in January. At present
 Wolke, R. L. What Einstein Told His Cook:    and for some years to come, the North
  Kitchen Science Explained. New York:
                                              Pole points away from the Sun when
  W. W. Norton, 2002, pp. 303–307.
                                              Earth is at the perihelion position in its
                                              orbit on about January 5 each year.
                                              Gradually over the next 12,900 years
     Chapter 2                                the North Pole will precess around to
                                              receive more and more radiant energy
   Does Anybody                               in January.
  Really Know What                                However, we need not wait nearly
                                              so long because the ellipse of Earth’s
     Time It Is?                              orbit is also precessing, so our summer
                                              will coincide with perihelion in only
                                              about 10,000 years!
26. January Summer
Yes, the Northern Hemisphere enjoys
summer in January quite often (in the          27. Proximity of Winter
cosmic scheme of things), repeating,
every 25,800 years, the period of
                                               Solstice and Perihelion
Earth’s precession. Just like a top with      The proximity of the two dates is an
its axis precessing, Earth experiences a      artifact of the particular century we
precession of its axis with respect to        live in. The date of perihelion does not
the stars with a 25,800-year period of        remain fixed, but slowly moves later
oscillation. So every 12,900 years the        into the year at the rate of about one

                                                                      Answers      139
full day every 58 years. It turns out          29. The Equinox
that the period from perihelion to
perihelion (the anomalistic year) is
                                               Displaced
about 25 minutes longer than the year          On the dates of the equinoxes, the day
defined from equinox to equinox (the            is about seven minutes longer than the
mean tropical year). The date of               night at latitudes up to about 25
perihelion thus moves completely               degrees, increasing to 10 minutes or
through the tropical year in about             more at latitude 50 degrees.
21,000 years. This slow change in the              The moment of the equinox occurs
date of perihelion may have a long-            when the geometric center of the Sun’s
term effect on Earth’s climate. At this        disk crosses the celestial Equator. But
time the temperature extremes are              the standard definition of sunrise is the
moderated somewhat in the Northern             time when the Sun’s upper limb is just
Hemisphere, but that will change as            breaking the horizon, and sunset when
the perihelion shifts in the direction         the Sun’s upper limb is just disappear-
of summer.                                     ing below the horizon. This adds one
                                               Sun semidiameter (about 16 arc min.)
                                               at both sunrise and sunset, extending
28. Earth’s Speed                              the duration of daylight by a little over
                                               two minutes.
Since perihelion occurs in early Janu-
                                                   The other factor is atmospheric
ary, Kepler’s second law implies that
                                               refraction, which causes the rays to
Earth is traveling faster during the
                                               bend around the horizon. As a result,
winter months. The time for Earth to
                                               we see the Sun about 34 arc minutes
travel from the autumnal to the vernal
                                               higher at both sunrise and sunset,
equinox, taken as a fraction of the
                                               adding roughly four minutes to the
year (T = 178.83/365.25), can be used
                                               time that the Sun is above the horizon.
to find an accurate value of the eccen-
                                                   In spring the days get longer as we
tricity of Earth’s orbit, ε = 0.5 π
                                               approach March 20, and the date of
(0.5 – T) = 0.01632, about 2 percent
                                               equal day and night occurs several
away from the precise value of ε =
                                               days before the March equinox, about
0.016713. A more accurate formula
                                               March 17 at latitude 40 degrees.
based on T is found in the reference
                                                   Conversely, in the fall it takes
below.
                                               several extra days for the time when
 Snyder, R. “Kepler’s Laws and Earth’s
  Eccentricity.” American Journal of Physics   the Sun is seen above the horizon
  57 (1989): 663–664.                          to shrink to 12 hours. The date falls


140      Answers
on about September 26 at latitude             north, the equation of time has the
40 degrees.                                   dominant influence over the changes
    On their website, the U.S. Naval          in sunrise and sunset times. Prior to
Observatory publishes excellent sun-          December 8, however, the declination
rise and sunset tables for any location.      effect is dominant, pulling the sunset
                                              to its earliest time on December 8.
                                              Then the equation of time takes over,
30. The Dark Days of                          and during the two weeks before win-
                                              ter solstice all the shortening of the
December                                      day comes from the later clock time of
There are two effects that, together,         sunrise. After winter solstice the days
determine the local times of sunrise and      lengthen, even as the sunrises continue
sunset. One is called the equation of         to get later until January 5.
time; the other is the Sun’s declination.      Steel, D. Marking Time: The Epic Quest to
     Earth’s orbit around the Sun is            Invent the Perfect Calendar. New York:
                                                John Wiley & Sons, 2000.
slightly elliptical. As a result, the speed
of the Sun’s apparent motion across
the sky is a bit faster in winter than in     31. Days of the Year
summer. Clocks, however, run at a             While the time interval to return to its
constant speed, so there is usually a         same point in the orbit is 365.2422
discrepancy—up to 16 minutes—                 days, Earth executes 366.2422 rota-
between clock time and the solar time         tions on its axis. One can demonstrate
shown by a sundial. We refer to this          this result by taking two coins, holding
discrepancy as the equation of time.          one in place on a table, and rolling the
     The Sun’s declination, its angular       second coin in contact with the fixed
distance above or below the celestial         coin without slipping. Therefore the
equator, determines the maximum               number of solar days is 365.2422, but
height of the Sun in the sky on any           the number of sidereal days (i.e., with
given day, thus causing our seasons. In       respect to the stars) is one more for
late December, the daily rate of change       one orbit of the Sun.
of the Sun’s declination is rather small.
It is, in fact, exactly zero at the Decem-
ber solstice (“solstice” means “sun sta-
                                              32. Leap Years
tionary”). Hence in late December, or         In years divisible by four, every four
more precisely from about December            years is a leap year except years divisi-
8 to January 5 at latitude 40 degrees         ble by 100. If the mean interval


                                                                       Answers      141
between vernal equinoxes, called the       time periods occurs because Earth is
tropical year, lasts 365.2422 days,        moving with respect to the stars, so the
then in 100 years we should experi-        Moon must travel slightly farther
ence 36,524.22 days. But there will be     around its Earth orbit to reach its full
24 leap years in a century normally, so    Moon position along the Sun-to-Earth
there will be 0.22 day left over. So       radial line.
every 400 years is declared to be a leap    U.S. Naval Observatory, Nautical Almanac
year with one extra day to approxi-          Office. The Astronomical Almanac for
                                             the Year 2000. Washington, D.C.: U.S.
mate the 0.88 day. The year 2000 was         Government Printing Office, 2000.
the first such leap year on a year divis-
ible by 100 since the modern calendar
began general use in the late 1600s.       34. Moon Time
By the time the British were ready         The person is at the desk at 12:20 dur-
to go along with the rest of Europe in     ing the noon hour, not at night. The
the 1700s, the old Julian calendar         Sun must be at the upper left because
required a correction of eleven days!      the Moon is illuminated from this
The Gregorian calendar was adopted         direction. This daytime Moon is sel-
in Britain in 1752, with Wednesday,        dom noticed because the sky is nor-
September 2, 1752, being followed          mally bright, but the daytime Moon
immediately by Thursday, September         is up as often and as long as the Moon
14, 1752.                                  at night.
    The famous physicist Isaac New-            Perhaps the easiest way to appreci-
ton was born on Christmas Day, 1642,       ate the appearances of the Moon at
on the Julian calendar but on January      night and during daylight is to use a
4, 1643, on the Gregorian calendar in      lamp for the Sun and two spheres, one
use today. Therefore, Newton was not       representing Earth and the other the
born in the year of Galileo’s death,       Moon. Fix the lamp position and the
1642!                                      Earth position, but move the Moon
                                           around Earth to observe its illumina-
                                           tion phases. Stop the motion at several
33. Full Moons                             points in the orbit of the Moon to
                                           observe its view from daytime loca-
No, the orbital period of the Moon is
                                           tions on Earth. You also might recon-
27.554 sidereal days, and the average
                                           struct the scene in the diagram.
interval between full moons was
                                            Pryor, M. J. “Phases, Models, and Car-
29.535 days for the twentieth century.       toons.” The Physics Teacher 3, no. 6
The difference between these two             (1965): 264, 288.


142     Answers
35. Lunar Calendar                          the seed parts develop) and may vary
                                            extremely from one variety to the next,
Modern farming methods tend to plant        from days to months. Photoperiod sen-
crops at approximately the same time        sitivity is a natural mechanism based
year after year, with minor adjustments     on the plant’s ability to distinguish pre-
for quirks in the weather. Hence, a par-    cise differences in the ratio of day
ticular crop is usually planted about       length to night length. The biological
365 days after its planting the previous    mechanism causing photoperiod sensi-
year, plus or minus about 10 days. One      tivity is quite complex and involves sev-
example is spring wheat, usually            eral genes. Essentially, some varieties
planted on about April 15 in the north-     should be planted only during certain
ern Plains states of the United States.     times of year to ensure that prevailing
    Rice is a different kind of organism    day length/night length conditions will
than wheat as far as its environmental      trigger panicle initiation when desired.
needs. Rice planted at about the same            Many varieties of rice are sensitive
date every year will sometimes permit       to bright moonlight, which can inter-
two good rice crops per year, but in        rupt their growth sequence. However,
most years the farmer will get only one     newer varieties have been bred and
good rice crop. The cause is the some-      others will be genetically modified to
times detrimental appearance at night       decrease their light sensitivity during
of the full Moon, which can interfere       critical photosensitivity times so that
with the growth cycles of the rice plant.   the moonlight will have a minimal
    By planting rice according to the       effect.
same date on the lunar calendar                  The use of the Moon for timing of
instead of the solar calendar, farmers      events is not restricted to rice farming
can often harvest two good rice crops       and its related festivals worldwide. For
every year. The young rice shoots are       Christians, Easter Sunday is the first
very sensitive to the light intensity at    Sunday after the first full Moon after
night during their photoperiod-sensi-       the vernal equinox!
tive stage, so the timing of the Moon’s      University of the Philippines College of Agri-
brightness is essential for a good crop.      culture in cooperation with the Interna-
Because the lunar calendar shifts with        tional Rice Research Institute (IRRI),
                                              comps. Rice Production Manual, rev. ed.
respect to the solar calendar dates each      Manila: compilers, 1970.
year, the solar calendar provides bad        Yano, M., et al. “Hd1, A Major Photoperiod
timing for planting rice.                     Sensitivity Quantitative Trait Locus in Rice,
                                              Is Closely Related to the Arabidopsis Flow-
    The photoperiod-sensitive stage           ering Time Gene Constans.” Plant Cell 12
occurs before panicle initiation (where       (2000): 2473–2484.


                                                                       Answers       143
                                             Jargocki, C. P. Science Braintwisters, Para-
36. The Sandglass                             doxes, and Fallacies. New York: Charles
The hourglass shape ensures that the          Scribner’s Sons, 1976, pp. 6, 70.

time scale on the glass is uniform, with
equal distances between scale divisions
corresponding to equal time intervals.
                                            37. Old Watch
If the sandglass didn’t taper, the top of   The old watch will run fast. The bal-
the sand column would descend at            ance wheel is the basic component
increasing speed. We can mathemati-         that oscillates “exactly” 300 times for
cally determine the proper shape. The       each minute on the watch face—that
speed, V, at which the sand is escaping     is, the wheel changes direction 10
from the opening is given approxi-          times each second! The moment of
mately by Torricelli’s formula V ~          inertia of the balance wheel depends
√
(2gy), where g is the local accelera-      on how much ambient air is dragged
tion of gravity and y is the height of      along during each oscillation. The
the sand column in the upper glass. Let     source of energy is a wound spring
A = π r 2 be the circular cross-sectional   that is essentially unaffected by the air
area of the upper glass at the top of the   because of its extremely small change
sand column and v the speed at which        in configuration.
this top is falling, then Av = aV, where        In the mountains, the viscosity and
a is the area of the opening, because       density of the air decrease slightly,
the sand is approximately incompress-       allowing the balance wheel to oscillate
ible. Substitution produces y ~ cr4,        faster. Newton’s second law applied to
with constant c = π 2v 2/(2ga 2). A plot    this rotational motion is required.
of y versus r will produce the familiar     From its momentary stop to change
hourglass shape.                            direction, the balance wheel must
     Sandglasses without time interval      accelerate to its maximum angular
markings have been used since before        velocity, then accelerate back to rest,
the fourteenth century to time speeches     etc. The net torque τ equals the
at town meetings and other events.          moment of inertia I times the angular
When the sand had run its course, the       acceleration α—that is, τ = Iα. The
speaker’s time was done. They are still     moment of inertia is determined by the
used today in some board games and          mass distribution with respect to the
as kitchen timers. Sandglasses with         rotation axis, and the air carried along
ruled markings haven’t been so popu-        with the balance wheel motion adds to
lar, being replaced early on by mechan-     the moment of inertia of the wheel
ical watches and clocks.                    alone. That is why the watch must be

144     Answers
recalibrated when the location of the           as, say, 3.45 seconds ± 0.005 second,
owner is at a different elevation than          which is an awkward notation,
the factory.                                    because the display goes only to hun-
    When the balance wheel drags                dredths of a second, yet the uncer-
along less air mass at a higher eleva-          tainty is smaller. Therefore, by
tion, the moment of inertia is less for         agreement, the elapsed time is given as
the same net torque, so the angular             3.45 seconds ± 0.01 second so that the
acceleration is greater. Less time is           number of decimal places is the same
needed to reach top angular speed,              for the value and the uncertainty—
and less time is needed to come to rest         that is, there is an uncertainty of plus
again.                                          or minus one digit in the smallest time
 Jargocki, C. P. Science Braintwisters, Para-   interval position of the display.
  doxes, and Fallacies. New York: Charles
  Scribner’s Sons, 1976, pp. 6, 71.
                                                39. Eternal Clocks?
38. Reading a Digital                           The laser clocks and the atomic clocks
                                                must maintain a vacuum within a rea-
Timer                                           sonably small range of parameters to
For a digital timer that displays the           function accurately. The temperature
elapsed time to one-hundredth of a              and pressure must be maintained
second, the minimum uncertainty in              within a certain tolerance because
the interval depends on the software            temperature fluctuations or pressure
and/or hardware method used to dis-             fluctuations could bring about inaccu-
play the last digit. Suppose that the           racies. Even the outgassing of atoms
hundredths digit fraction from 0.00             and molecules from the container
through 0.49 is displayed as zero hun-          walls can create severe problems for
dredths and from 0.50 through 0.99 is           some designs. Certainly, improvements
displayed as one hundredth. Likewise,           will be made to ensure longer lifetimes
1.00 through 1.49 is displayed as one           and more robust timepieces. But main-
hundredth, and 1.50 through 1.99 as             taining vacuums, low temperatures,
two hundredths. Seeing a 1 in the hun-          and so on, for decades and centuries
dredths place then corresponds to the           will be constant problems in these
range from 0.50 through 1.49, so the            sophisticated systems.
minimum uncertainty in the elapsed                  Whether a 10,000-year (or even a
time value is ± 0.50, or one-half of            1,000-year!) mechanical clock will
one-hundredth of a second. The                  ever exist and can stand the test of
reported elapsed time should be given           time and environment is doubtful. A

                                                                       Answers     145
group of engineers and futurists are         and then a decrease to zero after the
presently developing such a clock con-       light from the far-corner reflections is
taining a stack of rotating metal rings      received. The detailed intensity curve
connected to a torsion pendulum. Peri-       could be simulated on a computer.
odic winding will be required, perhaps            With the photodetector array, the
once a year or so.                           image shows the six nearest spots—the
    No special environment is needed,        centers of each equidistant flat sur-
although the assumption seems to be          face—which grow bigger and then
that a standard atmosphere with lim-         form rings of reflected light, then
ited pollutant content is a reasonable       many arcs of light until eight corners
expectation. We do know from a vari-         appear and disappear.
ety of scientific research projects that           When the flash length is extended
the oxygen concentration in the              to 1 microsecond, the light pulse is
atmosphere has been very nearly con-         300 meters long. There will be an ini-
stant at 21 percent for millions of          tial detector response rise and the rings
years, so the rusting rate of the            of light from the walls will be seen for
exposed metal can be predicted. But          a very small fraction of the total imag-
we do not know what future chemistry         ing, then flooded, then decreased.
will bring. Even a local environmental            We are not accustomed to light
disaster such as excess acidity in the       pulses lasting for milliseconds or less in
air from volcanic eruptions, a chemi-        daily life. But even nanosecond pulses
cal explosion, or careless disposal          are very long in some research fields.
could shorten the life span of the clock     For example, femtosecond (10–15 sec-
dramatically.                                ond) and shorter light pulses are used
 Gibbs, W. W. “Ultimate Clocks.” Scientific   in chemistry to watch molecular inter-
  American 287, no. 3 (2002): 86–93.         actions in progress. The present record
                                             for subdividing the second with a laser
40. Room Light                               strobe light is a few hundred attosec-
                                             onds, an attosecond being a billionth
For a nanosecond flash, the light pulse       of a billionth of a second!
length d = 30 cm is calculated with d =
ct, where c is the speed of light and t is
the time interval. When summing all          41. Right to Left Driving
the entering light, the photodetector
displays an initial rise from the light
                                             Switch
scattering from nearby walls with            Yes, as long as the accelerations and
increasing intensity until maximum,          decelerations required are within the

146      Answers
normal driving ranges, there should be        frames, in the laboratory frame, and in
no problems. One could check out              the rest frame of the clock. But the tick
each roadway with physical trials, or         rates will be different. In the frame
one could make an aerial video and            moving with the clock, the light flashes
play back the video in the reverse            follow the same path as before, reflect-
direction, essentially reversing time. If     ing perpendicularly to each mirror,
the car accelerations appear unusual          keeping the same tick rate.
in the reverse sequence, there will be
driving problems.                                Mirror
     Even nature at the most fundamen-
tal level is cognizant of left versus
right, a surprising discovery in the
1950s related to the weak interaction,           Mirror
one of the four fundamental interac-
tions, the other three being the gravita-
                                                  In the laboratory frame, the light
tional, the electromagnetic, and color
                                              continues to reflect off each mirror
(also called the strong interaction). For
                                              repeatedly, but during transit from one
the latter three, the interaction
                                              mirror to the other the path length is
strengths are the same for left-hand
                                              longer, being the diagonal of a right
spinning particles and right-hand spin-
                                              triangle. If the speed of light is the
ning particles. But for the weak inter-
                                              same value in both reference frames,
action, the evidence shows that nature
                                              then the time interval between reflec-
actually excludes any weak interaction
                                              tions (and clock ticks) will be longer in
behavior for a right-hand spinning par-
                                              the lab frame now than for the clock
ticle! The origin of this behavior biased
                                              at rest. Therefore, a moving clock ticks
toward left-hand spinning particles is
                                              slower than an identical clock at rest.
described by the mathematics in the
                                              And this phenomenon is true for all
standard model of leptons and quarks.
                                              clocks, no matter how they are made.
 Bartlett, A. A. “A Simple Problem from the
  Real World That Can Be Solved through           A more complicated situation
  Time Reversal.” American Journal of         occurs when the light clock is acceler-
  Physics 42 (1974): 416–417.                 ated parallel to the mirrors. We suggest
                                              that you think about this case when
                                              there is ample time in your schedule.
42. Light Clock                                Feynman, R. P., R. B. Leighton, and M.
                                                Sands. The Feynman Lectures on Physics.
Yes and no. The clock will continue to          Vol. 1. Reading, Mass.: Addison-Wesley,
keep accurate time in both reference            1963, pp. 15-5–15-7.


                                                                      Answers      147
43. Time Reversal                          a critical essential protein coding does
                                           not produce a viable organism even
Answer b: the acceleration is still        though a change in the nonessential
downward. The reversed motion is           DNA might—that is, changes in the
upward, but the object is decreasing its   DNA that do not affect the biochem-
speed because the acceleration is down-    istry critically will be tolerated. There
ward. A good example of this behavior      are vast regions of DNA where such
is the flight of a ball tossed upward. At   ineffective changes can occur, but any
all moments the acceleration is down-      change in the other regions pro-
ward, toward Earth’s center. Yet the       grammed for the production of essen-
ball moves upward with decreasing          tial biomolecules will be disastrous.
speed, turns around, and moves down-            If we assume the ideal case, that in
ward. Even at the turnaround point its     principle the changes would be equally
acceleration is downward.                  probable at any random location along
     Quite often people become con-        any DNA chain, and we assume that
fused between velocity and accelera-       the organism will grow and reproduce
tion. They are two different vector        the next generation, then we could have
quantities that should be separated        a molecular clock. However, as we
conceptually, but they are mathemati-      know, just as all genes are not equal in
cally related. Their directions can be     value at any given time, all DNA
the same or opposite along the line of     sequences are not equal in value. In par-
motion. Newton’s second law relates        ticular, some DNA sequences code for
forces and accelerations but says noth-    proteins that control the expression of
ing about velocities, for example. And     other DNA genes themselves, turning
we know that Aristotle was wrong           them on and off at appropriate times in
when he proposed that a force was          the development of cells in the organ-
required to keep an object moving.         ism. The Hox protein in insects, for
The real world operates with just the      example, will determine the structure of
opposite rule because no net force is      several different body parts, and slight
required to keep an object moving in a     changes in its amino acid sequence are
straight line at a constant speed!         major contributors to insect evolution.
                                           Therefore, both complementary DNA
                                           strands at the critical location need not
44. Molecular Clock                        be affected for the appearance of obvi-
The changes in the DNA during the          ous phenotype changes.
evolution of organisms do not occur at          However, the whole DNA mecha-
a common rate because any change in        nism and its subsequent biochemistry

148     Answers
in the cell are much more robust than         problems in their circadian rhythms
originally realized. The fact that many       because the bright artificial lighting
of the amino acids have several DNA           means that people stay awake long
base code triplets of nucleic acids for       past sunset, delaying and shifting the
their selection is a built-in resiliency      maximum in certain biochemical
that can produce a viable organism            cycles beyond their evolutionary time
even when the DNA has an error of             of day. In particular, greenish light
this kind. In addition, if the erroneous      from televisions and clock radios
amino acid substitutes at a location          passes into the eyes, even through
that is not critical for the 3-D shape        closed eyelids while asleep, to trigger
and the operation of the protein, once        the pineal gland to initiate some of the
again there is a built-in resiliency. Let’s   biochemical circadian rhythm shifts.
face the fact that Nature is much more         Wright, K. “Times of Our Lives.” Scientific
clever than we can ever hope to be!             American 287, no. 3 (2002): 59–66.
 Gibbs, W. W. “The Unseen Genome: Gems
  among the Junk.” Scientific American 289,
  no. 5 (2003): 46–53.                        46. Two Metronomes
 Ronshaugen, M., N. McGinnis, and W.
  McGinnis. “Hox Protein Mutation and         For the case of the periodic perturba-
  Macroevolution of the Insect Body Plan.”    tion of one metronome by the other,
  Nature 415 (2002): 914–917.
                                              the mode-locking occurs when the per-
                                              turbing frequency is sufficiently close
                                              to the unperturbed frequency of the
45. SAD                                       metronome. When a metronome is
Yes, if they suffer from SAD. At first         placed on the skateboard, the move-
one might think that the variation in         ment of the pendulum causes the
the length of day and night changes so        skateboard itself to move slightly, usu-
little from January to June that no one       ally in the opposite direction to the
living at the Equator would suffer from       pendulum swing, since the metronome
SAD. This reasoning is true if everyone       base is kept in place on the skateboard
went to bed at sunset and arose at sun-       by static friction or could be bolted
rise. The increasing light at sunrise         down. Some of the energy of the
would trigger the start of another cir-       metronome base motion is transferred
cadian rhythm that brings about bio-          to the skateboard, and this very small
chemical changes in our bodies.               amount of energy is further trans-
     But even people living near the          ferred along the skateboard in several
Equator are tuned no longer into the          directions, with some amount reaching
rise and setting of the Sun. There are        the other identical metronome.

                                                                       Answers      149
    If at first this energy arrives at some   of motion for each oscillator is mathe-
random phase point in the oscillation        matically equivalent to describing a lin-
of the second metronome, eventually          ear spring in a viscous medium with a
its regular energy delivery becomes          fluctuating driving force.
more and more effective in synchron-             According to the first reference
izing the pendulum oscillations. Of          below, the mode-locking can occur
course, the second metronome is acting       also for a wide range of aperiodically
on the first metronome in the same way        driven nonlinear oscillators in the
simultaneously. The synchronization is       physical and biological sciences, from
normally in-phase, but antiphase syn-        nonlinear electrical circuits to neural
chronization can occur in special con-       systems. As in the periodically driven
ditions. (See the second reference below     systems, the synchronization of ran-
for details.)                                domly driven nonlinear oscillators was
    The behavior can be represented          found to be structurally stable, which
by two equations for two harmoni-            means that even in the presence of
cally driven oscillators with a signifi-      small amounts of noise an approxi-
cant amount of dampening. If the             mate synchronization is achieved.
dampening were not significant, then           Jensen, R. V. “Synchronization of Driven
we would see two coupled pendulums              Nonlinear Oscillators.” American Journal
                                                of Physics 70 (2002): 607.
alternating their swing behavior out of
                                              Pantaleone, J. “Synchronization of Metro-
phase from maximum amplitude to                nomes.” American Journal of Physics 70
nearly zero amplitude. In the actual           (2002): 992–1000.
case, the pendulums simply synchro-
nize and keep nearly identical time.
    For the case of one of the pendu-
                                             47. Time Symmetry
lums being driven by a force random in       No, nature does not need to obey time
time, their fluctuating behavior can         symmetry at the most fundamental
converge to an identical response. Both      level. Equations sometimes have more
pendulums would exhibit the same             symmetry than the actual underlying
random fluctuations eventually. For          physics behavior. For example, even
both periodic and aperiodic driving          though the tensor equations of general
forces, asymptotic stability results for     relativity are time-symmetric, they
linear oscillators properly damped.          can be derived from a more funda-
That is, small changes in the parame-        mental type of mathematical entity
ters of the linear oscillator or the driv-   called a twistor. Twistor equations of
ing force result in only small changes in    general relativity are not time-symmet-
the asymptotic behavior. The equation        ric. In applications to a black hole, for

150      Answers
example, the tensor equations predict
time symmetry, but the twistor ones
do not. As a consequence, the forma-
tion of a black hole and the time-
reversed version cannot both represent
real physical behavior.
    One might think that the quantum
theory described by the Schrödinger
equation is time-asymmetric, the equa-
tion being first order in time. As Roger
Penrose points out in the reference
                                            and draw the straight-line path. Of
below, quantum theory and its equa-
                                            course, all parts of the path must be on
tions are indeed time-asymmetric. The
                                            the faces, and the appropriate faces
wave function can be used to calculate
                                            sharing a common edge must retain
the probability of a future state on the
                                            their relative positions and orientations.
basis of a known past state, but not
                                             Steinhaus, H. Mathematical Snapshots, 3rd
the other way—that is, one cannot cal-        ed. New York: Oxford University Press,
culate the probability of a past state on     1983, pp. 173–176.
the basis of a future state. You cannot
retrodict the past!
 Hilgevoord, J. “Time in Quantum Mechan-
                                            49. Moon Distance
  ics.” American Journal of Physics 70      The laser light pulse traveling from the
  (2002): 301–306.
                                            Earth to the Moon and back will
 Penrose, R. The Emperor’s New Mind.
  Oxford: Oxford University Press, 1989,    encounter the Earth’s atmosphere
  pp. 354–359.                              twice. The pulse will have an initial
                                            known rise time and decay time, but
                                            these times will be extended by pas-
                                            sage through the air medium. We
        Chapter 3                           assume ideal reflection at the Moon’s
       Crazy Circles                        corner reflector—that is, no pulse
                                            spreading in angle or in time.
                                                First, the ideal case. We assume
48. Spider and Fly                          that the laser source and the reflector
To find the shortest path between any        on the Moon are opposite each other
two points on a cube not on the same        on the line connecting the centers of
face, one convenient method is to lay       Earth and Moon and that Earth’s
out the six faces of the cube on a plane    atmosphere does not affect the transit

                                                                     Answers      151
time. The major source of uncertainty       table in a gridwork. Imagine placing
will be the ability of the detection sys-   the ball in appropriate adjacent tables
tem to locate the half-height point on      at mirror image positions; then draw
the rise time of the outgoing pulse and     the straight line to the pocket to find
the same point on the incoming pulse.       the collision points on the cushions.
If the system is good to about a
picosecond in detecting this point,
then a transit time of 2.56 seconds for
the 3.84 × 108 meter distance corre-
sponds to a timing uncertainty of bet-
ter than one part in 100 billion, with
an uncertainty in distance of less than          Normal billiards and pool tables
4 millimeters. That is, with the pro-       are marked around the perimeter to
posed laser system, one can measure         accomplish precision bank shots. Using
the distance to the Moon to almost the      these markers takes practice. These real
same distance uncertainty as one can        tables with their markings are not very
measure the length of a table with a        similar to the ideal table discussed
meterstick!                                 above. Moreover, the rolling ball before
    Of course, the atmosphere will          the collision with the cushion may have
foul things up a bit. The index of          additional spin—“English”—about an
refraction and the change of this index     axis not parallel to the table and per-
with altitude will both alter the light     pendicular to the travel direction. The
pulse speed and spread out the pulse        professional player uses all these quan-
rise time and rise shape. Sophisticated     tities in the particular shot, but we
signal processing techniques can elim-      amateurs simply enjoy the results as we
inate most of these atmospheric             practice more of the many possible
effects. So the final uncertainty will be    improvements to our games.
determined in the electronics creating       Steinhaus, H. Mathematical Snapshots, 3rd
the laser pulse and detecting the             ed. New York: Oxford University Press,
arrival of the pulse’s leading edge.          1983, pp. 61–64.



50. Ideal Billiards Table                   51. Wallpaper Geometry
On this ideal billiards table for which     Standing inside the cube, to your right
the incident and reflection angles at the    you see your left side of your person in
cushion are equal, one simply consid-       the cube to the right. To your front
ers adjacent mirror image copies of the     you see your back. To the top you see

152     Answers
the soles of your shoes. You see a 3-D           52. Space-Filling
array of yourself from many different
views, at many different distances, at
                                                 Geometry
many apparent sizes, and at many dif-            First consider a two-dimensional flat
ferent image intensities. This view is           space. A plane tesselation (or two-
not like being inside a cube with                dimensional honeycomb) is an infinite
reflecting mirrors on all sides because           set of polygons fitting together to
no image is reversed.                            cover the whole plane once, with every
     Cosmologists are trying to deter-           side of each polygon belonging to just
mine whether our 3-D space is mathe-             one other polygon. A regular tessela-
matically and physically discrete—that           tion has regular polygons. There are
is, compartmentalized into large cubes           three regular tesselations of the plane:
or regular dodecahedrons, each being             equilateral triangles, squares, and reg-
perhaps as large as 10 billion light-            ular hexagons. There are additional
years or bigger. If so, seeing a galaxy in       plane tesselations with two or more
one direction could be complemented              convex polygon shapes. One also can
by seeing the same galaxy in the oppo-           cover the plane with Penrose tiles,
site direction. Of course, several prob-         polygon pairs with at least one poly-
lems exist, such as the distance being           gon not being convex.
greater in one direction than in the                  Now consider an additional spatial
other, with the consequence that the             dimension. A three-dimensional honey-
galaxy is being seen not only from               comb (or solid tesselation) is an infinite
the other side but also at a different           set of polyhedrons fitting together to fill
time in its evolution of structure.              all space once, so that every face of each
There may be multiple copies of the              polyhedron belongs to one other poly-
galaxies to confound things. There               hedron. If we require all the polyhe-
might even be multiple copies of each            drons to be identical, then the only
of us! Any positive results will bring           regular honeycomb is the one filled with
about a revolution in our thinking               cubes, eight at each vertex. If we allow
about space and time in the universe.            two different regular polyhedrons, one
 Levin, J. How the Universe Got Its Spots:       can fill the space with eight regular
  Diary of a Finite Time in a Finite Space.      tetrahedrons and six regular octahe-
  Princeton, N.J.: Princeton University Press,   drons surrounding each vertex. These
  2003, pp. 132–155.
                                                 space fillings and others determine
 Thurston, W. P., and J. R. Weeks. “The          many of the natural crystal structures.
  Mathematics of Three-Dimensional Mani-
  folds.” Scientific American 251, no. 1              From the apparent simplicity of a
  (1984):108–120.                                3-D space filled with cubes, one may

                                                                         Answers      153
think that this solid tesselation would          headstone. Archimedes was killed in
be the most likely mathematically if real        212 B.C.E. during the capture of Syra-
space is discrete instead of continuous.         cuse by the Romans in the Second
However, mathematicians can show                 Punic War after all his efforts to keep
that the most likely and interesting 3-D         the Romans at bay with his machines
discrete space is the non-Euclidean tes-         of war had failed. Plutarch recounts
selation by dodecahedrons, of which              three versions of the story of his killing
there are two kinds, depending on the            that had come down to him:
angle of twist in relating one dodecahe-         1. “Archimedes was, as fate would
dron to the adjacent one. For further               have it, intent upon working out
information see the Thurston and                    some problem by a diagram, and
Weeks reference below.                              having fixed his mind alike and his
 Coxeter, H. S. M. Regular Polytopes. New           eyes upon the subject of his specula-
  York: Dover, 1973, pp. 58–74.                     tion, he never noticed the incursion
 Levin, J. How the Universe Got Its Spots:          of the Romans, nor that the city was
  Diary of a Finite Time in a Finite Space.
  Princeton, N.J.: Princeton University Press,      taken. In this transport of study and
  2003, pp. 132–155.                                contemplation, a soldier, unexpect-
 Thurston, W. P. and J. R. Weeks. “The Math-        edly coming up to him, commanded
  ematics of Three-Dimensional Manifolds.”
                                                    him to follow to Marcellus; which
  Scientific American 251, no. 1 (1984):
  108–120.                                          he declining to do before he had
                                                    worked out his problem to a demon-
                                                    stration, the soldier, enraged, drew
53. Archimedes’                                     his sword and ran him through.”
                                                 2. “A Roman soldier, running upon
Gravestone                                          him with a drawn sword, offered
Archimedes (287?–212 B.C.E.), perhaps               to kill him; and that Archimedes,
the greatest mathematician of ancient               looking back, earnestly besought
times, was the first to calculate the vol-           him to hold his hand a little while,
ume ratio of the sphere inside the cylin-           that he might not leave what he was
der. With a sphere and a cone inside                then at work upon inconclusive and
the cylinder touching top, bottom, and              imperfect; but the soldier, nothing
sides, Archimedes determined that                   moved by his entreaty, instantly
their volumes are in the ratios 1:2:3!              killed him.”
     The Roman general, Marcellus,               3. “As Archimedes was carrying to
tells of how he searched for and found              Marcellus mathematical instru-
Archimedes’ gravesite with this                     ments, dials, spheres, and angles, by


154      Answers
   which the magnitude of the Sun           Therefore, a model with 106 neurons
   might be measured to the sight,          will not work as a useful scale model
   some soldiers seeing him, and think-     of the real brain.
   ing that he carried gold in a vessel,        Of course, one could simply take a
   slew him.”                               small volume of the brain containing 1
    Archimedes was buried in Syra-          million neurons and ignore connec-
cuse, where he was born, where he           tions to other parts. Or one could arti-
grew up, where he worked, and where         ficially modify the unusable computer
he died. On his grave there is an           model above by ensuring a few con-
inscription of π, his most famous dis-      nections or more to each neuron.
covery. Also placed on his tombstone        Whether the behavior that ensues is
is the figure of a sphere inscribed         realistic must be determined. The more
inside a cylinder and the 2:3 ratio of      practical approach is to model a small
the volumes between them, the solu-         section of the brain—perhaps tens of
tion to the problem he considered his       thousands of neurons and all their
greatest achievement.                       interconnections—in a focused study
    His nicknames were, “the Wise           and simulation. A grid of computers,
One,” “the Master,” and “the Great          each representing one small section,
Geometer.”                                  could then be used to simulate a larger
 Plutarch. Lives of Noble Grecians and      portion of the brain. Hopefully, when
  Romans. Translated by A. H. Clough. New
                                            quantum computers become a reality,
  York: Random House, Modern Library,
  1992, p. 517.                             they will be able to simulate the whole
                                            brain. Whether the brain behaves
                                            quantum mechanically and requires
54. Brain Connections                       quantum superposition for its opera-
A million neuron model of the brain is      tions is presently unknown.
still quite a formidable programming            There is the remarkable problem
task for a computer simulation, but         of information storage in the brain—
there would be no information trans-        that is, where exactly is information
fers from neuron to neuron. Why not?        stored? If each neuron stores only one
Because any neuron in this model of         bit of information, then the human
the human brain would have on aver-         brain is not large enough by many fac-
age nearly zero inputs. One calculates      tors of ten! In 1989 Roger Penrose
as follows: if 1011 neurons have 1,000      suggested that each neuron must be
connections each, say, then the average     capable of storing many bits of infor-
is 1 connection per 108 neurons.            mation, in contrast to the prevailing


                                                                   Answers     155
ideas. Subsequently, the numerous           calculate the extent of the arm to
microtubules in each neuron were            determine its end point distance, thus
found to participate in the information     requiring three more numbers, the
storage game. There still remains the       lengths of the three parts. The space of
question of what each stored bit of         operation is nine-dimensional and is
information represents.                     called a 9-D configuration space to
 Penrose, R. Shadows of the Mind. Oxford:   distinguish between physical space and
  Oxford University Press, 1994, pp.        coordinate space. Of course, one could
  358–377.
                                            have determined this result by realiz-
                                            ing that each rod end point requires
55. Configuration                           three coordinate values to be specified.
                                                The movement of the arm to touch
Space                                       the point is the next challenge. If feed-
There are many ways to approach this        back exists in the robot, such as visual
problem of describing the arm position      feedback of the hand position and the
in physical 3-D space. We consider one      desired point location, the movement
approach only. In all approaches, the       algorithm can use a correction proce-
end of the rodlike hand must touch the      dure that becomes finer and finer as the
specified point, so three numbers           fingertip approaches the point, as we
define the end point of the hand.            humans tend to operate. If there is no
    Let’s start at the fixed shoulder       continual feedback mechanism, then
position. Two numbers will describe         the algorithm must move the arm to
the upper arm position, the angle in        the point directly, somehow knowing
the vertical plane measured from a          where the fingertip is at all times. A
fixed vertical axis through the shoul-       systematic error cannot correct itself if
der, and an angle about this vertical       no feedback exists. Many robotic arms
axis. Two more numbers describe the         operate in both modes, first without
forearm position, an angle in the verti-    feedback for rapid deployment and
cal plane measured from a vertical axis     then with feedback for fine adjustment.
through the end of the upper arm, and           As humans, we learn to perform
an angle about this vertical axis. Like-    many tasks and do many of them sev-
wise, two more angles are needed for        eral times daily. As a result, we often
the hand.                                   forget how we learned a particular
    At least six numbers are necessary      procedure and how much practice was
for the robot to locate the particular      required. To relive that learning expe-
point in the room. The program will         rience, try using the “other hand” to


156     Answers
punch in data in a calculator, or some      57. A Spooky
similar task. The learning curve is
sometimes very steep!
                                            Refrigerator
                                            Yes. Just as you could remove a dot
                                            from a piece of paper with an eraser
                                            brought in from the third spatial
56. Farmer Chasing a                        dimension, a 4-D being could enter the
Goose                                       refrigerator without needing to open
                                            the door and remove a piece of food.
If the farmer is restricted to chasing
                                            That is, 3-D objects are open in the
the goose along the instantaneous
                                            direction of the fourth dimension.
line of sight to the goose, the farmer
                                                Conceptually, visualizing a 4-D
will never be able to catch the goose
                                            object is difficult in our 3-D world.
unless there is a head-on encounter.
                                            Some mathematicians suggest letting
The best strategy for the goose is to
                                            the fourth coordinate direction be rep-
run in a straight line, for then the
                                            resented by flashing color, such as the
farmer’s velocity is eventually in the
                                            sequence of colors in the visible spec-
same direction as the goose’s direction,
                                            trum from red to indigo. Take any 3-D
and the relative distance remains
                                            object—a sphere, for example. As the
constant. Note that even when the
                                            sphere moves in the fourth coordinate
goose changes direction often, the
                                            direction its flashing color changes
farmer cannot close the gap com-
                                            from red to orange to yellow, etc. A
pletely because the closer they are, the
                                            2-D sheet of paper moving in the
more toward being parallel are the
                                            fourth dimension would be changing
velocities!
                                            its flashing color also to indicate its
    In the real open-field experience
                                            fourth coordinate value. The inherent
without the restriction, one strategy
                                            color of the object does not change, of
that might work if the goose is inexpe-
                                            course.
rienced is for the farmer to anticipate
                                                Descriptions of 4-D objects inter-
the position of the goose and to get
                                            secting our 3-D world are quite fasci-
there at the same time the goose
                                            nating. For example, a 4-D sphere
arrives. However, most geese “read”
                                            intersecting our 3-D world would first
the game plan and change course in
                                            appear as a point, then an increasing
midflight.
                                            3-D sphere, then a decreasing 3-D
 Behroozi, F., and R. Gagnon. “The Goose
  Chase.” American Journal of Physics 47,   sphere, then a point, then gone! The
  no. 3 (1979): 237–238.                    analog in fewer dimensions would be a


                                                                   Answers     157
3-D sphere intersecting a 2-D sheet of         eight cubes. In each case we obtain the
paper, being first a dot, then a widen-         number 2 raised to an integer power.
ing circle followed by a narrowing cir-        We can make a table and generalize to d
cle, then a dot again, and then gone.          arbitrary dimensions.
    Although most people would
expect there to be more mathematical           Figure            Dimension   No. of Copies
difficulty and complications in even           Line segment          1          2 = 21
higher dimensions than four, this              Square                2          4 = 22
expectation is false. The mathematics          Cube                  3          8 = 23
                                               Doubling figure       d          n = 2d
actually simplifies with five dimensions
and more! Much geometry remains to
                                                   We can now determine the dimen-
be worked out in a 4-D space, whereas
                                               sion of an interesting but strange geo-
the mathematics is better understood
                                               metrical object, the Sierpinski triangle,
in the higher dimensions.
                                               named after the Polish mathematician
 Gardner, M. The Colossal Book of Mathe-
  matics: Classic Puzzles, Paradoxes, and      who originally thought it up in 1916,
  Problems. New York: W. W. Norton, 2001,      shown here with its holes being gray.
  pp. 137–149.
                                               Double the length of the sides, and you
 Peterson, I. The Mathematical Tourist. New
  York: W. H. Freeman, 1988, pp. 82–107.
                                               get another Sierpinski triangle, similar
 Pickover, C. A. Surfing through Hyperspace:
                                               to the first. For example, if the first
  Understanding Higher Universes in Six Easy   Sierpinski triangle has one-inch sides,
  Lessons. Oxford: Oxford University Press,    the doubled one has two-inch sides.
  1999, pp. 44–70.
                                               How many copies of the original tri-
                                               angle do you have? Remember that the
58. Fractional                                 gray triangles are holes, so we can’t
                                               count them.
Dimensions?
                                                   Ignoring the hole in the center of
Yes. Noninteger dimensions are known           the double-sized Sierpinski triangle,
as fractal dimensions. A pathway
toward understanding fractal dimen-
sions begins by considering duplications
of well-known objects. A line segment
can be duplicated to produce two line
segments. A square can be duplicated
in each direction to produce four
squares. A cube can be duplicated in
each of its three directions to produce


158      Answers
we learn that doubling the sides of the    their applications to the familiar phys-
original gives us three copies, so 3 =     ical world. One interesting question is
2d, where d = the dimension according      whether two or more objects with the
to the scheme in the table. Using a        same fractal dimension must be
calculator, one finds that its dimension    related in some fundamental way,
d = 1.585 . . . , a noninteger!            either mathematically or physically.
    In general, the mathematical            Peterson, I. The Mathematical Tourist. New
expression for the dimension of the          York: W. H. Freeman, 1988, pp. 114–142.
figure is given by the ratio of two
logarithms:                                59. Platonic Solids
  dimension = logarithm (number of
     self-similar pieces)/logarithm
         (magnification factor).

    For simplicity:
1. A dimension between 0 and 1 is
   supposed to correspond to the
   capacity of a set of points to partly
   fill a line without achieving it com-    One must turn the second identical reg-
   pletely, out of having the whole        ular tetrahedron upside down and
   value 1 that is needed.                 rotate by 30 degrees about the vertical
2. A dimension between 1 and 2 is          axis before mathematically pushing
   supposed to correspond to the           them through each other to form a six-
   capacity of a line to partly fill a     vertex regular solid. Obviously these six
   plane, without achieving it com-        vertices correspond to the vertices of a
   pletely, out of having the whole        regular octahedron if the new object
   value 2 that is needed.                 were placed inside one. However, the
3. A dimension between 2 and 3 is          sides are not convex and flat. Our bi-
   supposed to correspond to the           tetrahedron does have twofold symme-
   capacity of a surface to partly fill a   try axes even though the two tetra-
   volume without achieving it com-        hedrons are rotated with respect to each
   pletely, out of having the whole        other.
   value 3 that is needed.                     We might have asked you to place
    There is a whole world of mathe-       the two regular tetrahedrons face-to-
matics to be learned with fractal          face in congruence so that the com-
dimensions and fractal geometry and        bined object has five vertices defining


                                                                    Answers      159
a triangular bipyramid. There are now        that the Greeks were absolutely
three twofold rotational symmetry            right. That is, what we will be saying
axes, each one being through an edge         in a very profound way, the finite
of the joining faces. However, this          groups of symmetries in 3-space
bipyramid is not a regular polyhedron.       “see” the simple Lie groups (and
    If we were to extend the discussion      hence literally Lie theory) in all
to four spatial dimensions, there are        dimensions.
six regular solid convex objects analo-
gous to the five Platonic solids in three       One of us (F. P.) has proposed that
dimensions. The number of regular          the fundamental building blocks of
convex solids, called regular poly-        matter, the leptons and quarks of the
topes, does not increase with more         Standard Model of particle physics,
spatial dimensions. Instead, all dimen-    are described mathematically by the
sions beyond four have only three reg-     specific rotational symmetries of the
ular convex solids.                        3-D Platonic solids for the leptons and
    The Platonic solids are special in     by their 4-D analogs for the quarks.
many ways, but perhaps their most          The key arguments are that the
important mathematical property was        mathematical symmetry groups for
pointed out by mathematician B.            these regular solids are subgroups of
Kostant:                                   the Standard Model symmetry group
                                           and that the lepton and quark mass
  The ancient Greeks, especially the       ratios can be directly related to the
  school of Plato, had great reverence     ratios 1:108:1728 of the invariants of
  for the regular polygons in the plane    these subgroups. Whether the natural
  and regular solids in 3-space. The       world mimics this fundamental math-
  latter—the tetrahedron, cube, octa-      ematical behavior has yet to be deter-
  hedron, dodecahedron, and the            mined by experiments at particle
  icosahedron—are often referred to as     colliders.
  the Platonic solids. The Greeks           Coxeter, H. S. M. Regular Polytopes. New
                                             York: Dover, 1973, pp. 41–57, 126–144.
  believed that these regular figures
                                            Kostant, B. “Asterisque.” In Proceedings of
  were fundamental in the structure of
                                             the Conference “Homage to Elie Cartan,”
  the universe. If symmetry or its           Lyons, July 1984, p. 13.
  mathematical companion—group              Potter, F. “Geometrical Basis for the
  theory—is fundamental in the struc-        Standard Model.” International Journal of
                                             Theoretical Physics 33 (1994): 279–305.
  ture of the world, then one of the
  points of our lecture is the statement



160     Answers
                                             Potter, F. “Geometrical Basis for the
60. Intersecting                              Standard Model.” International Journal of
Spheres                                       Theoretical Physics 33 (1994): 279–305.

Two identical three-spheres can inter-
sect in a point, a circle, a sphere (two-   61. Arm Contortions
sphere), and a three-sphere. Now
bring in a third identical three-sphere
to intersect with the former two in
appropriate combinations of points,
circles, spheres, and a three-sphere,
the latter when all three are coinci-
dent. With three intersecting identical
three-spheres, a resulting single two-
sphere can be obtained only when the
three three-spheres form a symmetrical
configuration.
    If the leptons and quarks of the
Standard Model of particle physics are
physical manifestations of the finite
rotational symmetries of the 3-D Pla-
tonic solids and their 4-D analogs as
proposed in a model by F. Potter (see
the reference below), then the intersec-
tions of three-spheres will become
important in fundamental physics. A
quark would be defined in a 4-D
space, and its mathematical behavior        Yes, if one allows the arm to move over-
would depend on the properties of           head. This second rotation untwists the
three-spheres. The proton, for exam-        arm and brings the orientation of
ple, is a real particle composed of three   the book back to the initial one again.
quarks in our 3-D world—that is,            One can say that the arm-object pair
three 4-D entities according to the pro-    requires two 360-degree rotations to
posed model. So three three-spheres         return to the initial orientation. Such an
(representing the quarks) must inter-       entity is said to mathematically corre-
sect to form a two-sphere that “lives”      spond to a spin 1⁄2 system—that is,
in our 3-D space.                           related to the continuous symmetry


                                                                      Answers      161
group SU(2). Any lepton or quark               Coxeter, H. S. M. Regular Polytopes. New
                                                York: Dover, 1973.
wave function, such as the electron
                                               Rieflin, E. “Some Mechanisms Related to
wave function, behaves in this way              Dirac Strings.” American Journal of Physics
with respect to rotations and angular           47 (1979): 379–381.
momentum.
     Spheres, cubes, and other objects
with spatial symmetry also can be clas-       62. The Rotating
sified as spin 1⁄2—that is, their rota-       Cup
tions are described by symmetry
                                              Done properly, you would see the
groups that are subgroups of SU(2)
                                              same sequence in both cases. The cup
and SU′ (2) = SU(2) × Ci , where Ci is
                                              appears to rotate with changing rota-
the two-element inversion group. For
                                              tion rates as time passes. We have here
the Platonic solids, the rotational sym-
                                              an example of Galilean relativity for
metry groups are discrete instead of
                                              uniform motion. At these familiar
being continuous, and some of these
                                              slow speeds compared to the speed of
symmetry groups are subgroups of
                                              light, the behavior of the rotating cup
both SU(2) and SU′ (2) because among
                                              produces no surprises. We can walk
all the elements of both can be found
                                              past the cup, or the cup can move past
the elements of finite order for the dis-
                                              our stationary location.
crete subgroups.
                                                  In a later chapter, where we intro-
     As you know, our practical experi-
                                              duce the special theory of relativity
ence is mostly with spin 1 entities—
                                              (STR), we examine an object such as a
that is, those needing a 360-degree
                                              cube moving past a stationary
rotation to return to the initial orien-
                                              observer at enormous speed, and we
tation. Mathematically, these spin 1
                                              could consider the opposite case, of
properties can be constructed from the
                                              the observer moving past the station-
spin 1⁄2 symmetry properties. Mathe-
                                              ary object. Of course, one sees the
maticians know that even more funda-
                                              same behavior in both cases, just like
mental are the reflection groups from
                                              the symmetry we observe in Galilean
which all spin 1⁄2 properties can be
                                              relativity. However, in STR an effect,
derived as two reflections in perpendi-
                                              now called the Terrell effect, explains
cular planes. The two books listed
                                              why a cup approaching and passing by
below discuss these hierarchical rela-
                                              at near-light speeds appears addition-
tionships and many more.
                                              ally rotated, the observer being able to
 Altmann, S. L. Rotations, Quaternions, and
  Double Groups. Oxford: Clarendon Press,     see the back side of the cup as it
  1986.                                       approaches!


162      Answers
63. Space and Time                          describe the dynamics of motion in
                                            three-space. Spinors are equivalent
Together                                    mathematical entities used in quantum
                                            mechanical wave functions to describe
Using three real spatial coordinates
                                            the electron and other fermions in
and one imaginary time coordinate for
                                            (3 + 1)-D space.
calculations works correctly when
                                                Quaternions were first “discov-
calculating the squares of space-time
                                            ered” by W. R. Hamilton in the 1800s,
coordinates and their sums and differ-
                                            and the quaternion q has one real com-
ences. The important relationship is
                                            ponent and three imaginary compo-
the space-time interval τ defined by τ2
                                            nents. Just as complex numbers are
= c2 ∆t 2 – ∆ x 2 – ∆y2 – ∆z 2, where the
                                            formed from pairs of real numbers,
∆x’s are the four “distances.” How-
                                            quaternions are formed from pairs
ever, physics textbooks that use an
                                            of complex numbers. Thus one should
interval τ defined by τ2 = + ∆x2 + ∆y2 +
                                            assign the time coordinate to the
∆z2 – c2 ∆t2 are making a mathemati-
                                            real component and the three space
cal faux pas in choosing three real
                                            coordinates to the three imaginary
space coordinates and one imaginary
                                            components of a quaternion. Hence,
time coordinate—that is, the set (x, y,
                                            mathematically, we live in a quaternion
z, ict) with i being the imaginary and c
                                            world with an imaginary 3-D physical
being the speed of light—instead of
                                            space and a 1-D real-time clock!
vice versa. Fortunately, this fundamen-
                                             Altmann, S. L. Rotations, Quaternions, and
tal error does not affect the calcula-        Double Groups. Oxford: Clarendon Press,
tions of time intervals and spatial           1986.
separations because these calculations       Pickover, C. A. Surfing through Hyperspace:
involve the differences of squared            Understanding Higher Universes in Six Easy
                                              Lessons. Oxford: Oxford University Press,
quantities. To be mathematically cor-         1999, appendix D.
rect in the (3 + 1)-dimensional space-
time, one must use quaternions, which
are numbers in the form q = a + bi + cj     64. Space > 3-D?
                               √–
+ dk, with i, j, and k being  1 and a,    There are several arguments for why
b, c, and d being ordinary real num-        space is not larger than three dimen-
bers, because they are the numbers in       sions. Planetary orbits are not stable
four dimensions that properly handle        when n > 3, except for a circular orbit
rotations, translations, and Lorentz        for n = 4, because the attractive force
transformations. Today, quaternions         and the centripetal force both do not
are used everywhere in science to           have the correct dependence on radial


                                                                     Answers       163
distance. In 1917 P. Ehrenfest showed          The considerations could be
that one needs to consider the Poisson     extended to a universe with more than
equation for arbitrary dimensions to       one time dimension! However, this
determine orbit stability. When the        matter and others we leave for future
n ≥ 4 circular orbit for a body around     challenges.
a central mass becomes slightly per-        Büchel, W. “Why Is Space Three-Dimen-
turbed, one can show that the com-           sional?” Translated by I. M. Freeman from
                                             Physikalische Blätter 19, no. 12 (1963):
parison of the central force to the          547–549. American Journal of Physics 37
centripetal force for the orbit depends      (1969): 1222–1224.
on the perihelion value r1 and the          Ehrenfest, P. Annalen der Physik 61 (1920):
                                             440.
aphelion value r2 according to [1/2 –
                                            Pickover, C. A. Surfing through Hyperspace:
(n – 2)–1]/r12 < [1/2 – (n – 2)–1]/r22,
                                             Understanding Higher Universes in Six Easy
which cannot be true for n = 4 and           Lessons. Oxford: Oxford University Press,
larger. In a 4-D space, a satellite          1999, pp. 202–205.
launched from Earth toward the Sun
would either fly away to infinity or spi-
ral into the Sun.
    The hydrogen atom is not stable
                                                 Chapter 4
when n > 3 because there is no energy       Fly Me To The Moon
minimum for n ≥ 5, which is shown
using the indeterminancy principle—
that is, the Heisenberg uncertainty
                                           65. Gunfight
principle. For the case n = 4, the rela-   One could do good classical physics
tivistic energy equation must be exam-     here, but the filmmakers have turned
ined to show that no energy minimum        the scene into Hollywood exaggera-
is available and the atom is not stable.   tion. The physics is determined by the
    Several other physical phenomena       conservation of linear momentum.
would be unusual for n ≥ 4 dimensions.     Assume that the victim of the shooting
There is no satisfactory propagation of    is initially at rest, so the total momen-
sound waves or electromagnetic waves       tum initially is all in the bullet (or
free of distortion and reverberation in    buckshot) of mass m and speed V
spaces other than n = 1 and n = 3.         before hitting the victim, at mV. After
Also, axial vectors such as the mag-       the collision, the final total momentum
netic field and the angular momentum        is in the backward “flying” person
vectors do not exist in even-dimen-        plus bullet. If the victim has mass M
sional spaces.                             and the combined victim-plus-bullet


164     Answers
object has speed v, the total final         backward. Falling backward, perhaps,
momentum is (M + m)v. For all inter-        but not “flying”!
actions, by the law of conservation of          There is the story of a famous
linear momentum, the final momen-           physicist back in the 1950s who loved
tum equals the initial momentum.            to watch gunfights in Western movies.
     In the simplest case (assuming no      The bad guy always draws first, he
frictional drag at the feet and ignoring    noticed, but the good guy wins the gun-
transfer of momentum away to the            fight. How could this outcome happen?
earth, etc.), its application yields (M +   His hypothesis was that psychology
m)v = mV. Solving for the velocity v of     played an important role, slightly hin-
the victim afterward produces v =           dering the man who had to make the
mV/(M + m). Substituting reasonable         conscious decision to draw first. The
values of M = 80 kg, V = 400 m/s, and       second man simply had to react.
m = 0.03 kg yields a “blow-back”                Even today, the psychology of
velocity maximum of v = 0.15 m/s.           choosing a physical action is an impor-
Most people can walk about 2 m/s            tant factor, particularly in sports.
(i.e., about 4 mph). So we can con-         There are tennis coaches (and coaches
clude that any shooting victim              in other sports) who preach the psy-
depicted as being blown backward by         chology of playing tennis, saying that
the impact of the bullet (or shotgun        when you think too much on the court
blast of pellets) is ridiculous and         instead of simply reacting, as you learn
belongs in the fantasy world only!          to do in practice, then you are in trou-
     A physicist wouldn’t actually need     ble. You are letting self no. 1 (your
to calculate the velocity backward          mind) control self no. 2 (your body),
using linear momentum conservation          and your tennis game will suffer. We
explicitly. Simply watching the behav-      wonder whether the concepts hold
ior and movement of the shooter hold-       true for playing the physics game, too!
ing the gun before and after the shot
reveals approximately how much
momentum is available by using New-
                                            66. Body Cushion
ton’s third law. If the shooter isn’t       We doubt whether landing on top of
blown backwards by the recoil force         another body after such a long fall
of the shot, the victim won’t be either.    provides much cushion! The impor-
Of course, someone will suggest that        tant parameter here is the extent of the
involuntary muscle contraction in the       collision time ∆t—that is, how long
stunned victim causes the “flying”          the collision of the hero’s body with


                                                                   Answers     165
the object actually lasts. The longer      because the change in linear momen-
the ∆t, the better. We also need to        tum will be almost double, even
know the acceleration a versus time        though the collision duration may be
profile. In better words, what is the      increased slightly. Automobile colli-
maximum acceleration to be experi-         sions provide plenty of evidence about
enced by the hero’s body? By defini-        the damages done to internal organs
tion, the average a = ∆v/∆t, where ∆v      by sudden collisions with very short
is the velocity change during the time     collision times. We doubt whether our
interval ∆t. Shorter ∆t’s make the         hero will be able to walk away from
experience more painful.                   the “body cushion.” In fact, our hero
     Stunt professionals are often seen    will be lucky to survive!
leaping off buildings or falling through
windows in movies, but their colli-
sions are with huge air-filled balloons
                                           67. Cartoon Free Fall
that effectively extend the total colli-   When the cartoon character steps off
sion time to half a second or more. We     the cliff, the fall should begin immedi-
do not see their collision with the bal-   ately, of course. The natural path is
loon in the program because the edit-      essentially a parabola, with approxi-
ing process substitutes the desired        mately free-fall acceleration downward
body lying dead on the concrete.           and a constant velocity horizontally.
     Back to the hero landing on the       Even a cartoon character must have
other body. The collision time here        some mass; otherwise the character
will be less than one-tenth of a second,   could not exert a force on anything,
producing dangerous accelerations.         including the ground being walked on.
For example, if the body falls from the    Unless the upward buoyant force of the
top of a two-story building, its speed     air balances exactly the gravitational
will be approximately 11 m/sec just        force downward, no cartoon character
before collision. The acceleration dur-    stepping off the cliff would remain in
ing the collision will be greater than     suspension at the height of the cliff.
110 m/s2, very dangerous. Even if          Even if the buoyant force was suffi-
bones are not broken in bringing the       cient, why would its upward push dis-
hero’s body to rest, the internal organs   appear suddenly to allow the character
will continue to move until they suffer    to free fall?
a collision inside the hero’s body. And        We should see the character accel-
if the hero bounces back upward, the       erating downward with ever-increasing
acceleration can be even worse,            speed unless the terminal velocity is


166     Answers
reached or the buoyant force balances        increases rapidly in time as the ball
the weight. The collision at the bottom      and wall both bend a little during the
is also subject to analysis. To prevent      collision. The initial kinetic energy of
this sudden collision, sometimes             the ball just before the collision
another character is able to run down        becomes distributed in the distortions
from the clifftop just in time to catch      of the ball and the wall. The interac-
the falling character. And sometimes         tions between the molecules of the
we even see another person falling with      wall material change as some of the
a greater acceleration downward to           available energy from the collision
arrive in time to catch the victim! If the   spreads from the immediate impact
fall is nearly at the free-fall accelera-    area. If the total collision time is
tion, the runner must be mighty swift!       extremely short, the energy distribu-
There are measured examples of skiers        tion will be quite limited in distance,
going down Mount Fuji with accelera-         and much of the energy is available for
tions greater than the free-fall accelera-   ripping. Otherwise, if the collision
tion, but no runner has achieved this        time is much longer, a big portion of
feat yet. And yes, an anvil always has a     the wall will respond by deforming
greater acceleration downward than           just a little, and ripping may not occur.
any other object (sure!)!                         A bullet going through the paper
                                             at a practice target makes a fairly clean
                                             hole for two reasons: (1) the contact
68. Silhouette of                            time is extremely short; (2) the bullet
                                             offers a nearly round profile, so there
Passage                                      is symmetry about an axis perpendicu-
The condensed matter physicist knows         lar to the hole. Even then, close exam-
a lot about the physical properties of       ination of the bullet hole reveals an
liquids and solids, so he or she proba-      irregular surface and additional tear-
bly would say, “Wow! How was that            ing beyond the actual round hole.
done?” The only realistic possibility is          You can check out the advantage of
that the wall cut should be quite messy      a symmetrical object. Now, with
and the cartoonist cleaned the edges         appropriate safety precautions, try to
for heightened dramatic effect!              rapidly push any shaped profile other
    We can estimate how difficult the         than round through a sheet of target
cookie cutter hole would be to achieve       paper. The lack of cylindrical symme-
by considering a ball thrown at              try perpendicular to the surface usually
the wall. The impact surface area            creates enormous problems for the


                                                                     Answers      167
material because slightly more energy        resulting centrifugal force is called a
goes into some directions. Moreover,         pseudo-force because the actual force
tearing is required to occur at different    is acting radially inward toward the
distances along a noncircular profile,        axis of rotation to accelerate the object
so there may be points where the trans-      from its inertial straight-line motion.
ferred energy density is significantly       We must assume that the structural
higher or lower than the surrounding         integrity of the space station remains
paper regions. All these factors and         intact—that is, the station was
several more act against a very clean        designed for the rotation and for the
cut through the material. The cut            allowed distributions of mass on
through a thicker piece of paper or a        board.
wall that has significant depth would             Using the relation for the angular
be even messier.                             velocity ω = v/r, we can express the cen-
    Of course, the cartoon character         trifugal force as Fc = mrω2. The whole
has several other options for getting        spaceship has the same angular velocity
through the wall when time permits:          about the rotation axis, so an object’s
(1) simply paint an exit onto the wall       radial acceleration increases linearly
through which only he can pass, or (2)       with distance r from the rotation axis.
the character can carry around a hole        An astronaut at one end of the dumb-
to be affixed where needed!                   bell must climb—that is, walk up a lad-
                                             der and then down a ladder—from one
                                             end to the other through the middle,
69. Artificial Gravity                       where the radial acceleration is zero.
Space stations and spaceships have           The muscular effort required changes
been created by authors and screen-          throughout the climb, so the sensations
writers in a vast array of shapes and        must be wonderful!
sizes. A rotating dumbbell shape has
appeared in many space adventures,
the rotation providing a pseudo-
gravity force for Earth creatures. The
                                             70. Small Wings
rotation about the center of mass per-       The small wings on such alien beings
pendicular to the long axis provides a       are probably much too small for a 20
pseudo-acceleration acting radially          kg body. One could argue that the
outward called the centrifugal acceler-      planet’s gravitational force at its sur-
ation ac = v2/r, where v is the tangential   face is much less than the value here
velocity value and r is the radial dis-      on Earth, so the alien being’s weight is
tance from the axis of rotation. The         much less also. That may be so, and

168      Answers
the proposition is not unreasonable.        the weight F = g′ m of the alien being
However, we still require a sufficient       with small wings, or 200 N if g′ =
air density for the wings to do their       10 m/s2.
work and a breathing atmosphere for              We assume that a very strong 20
our Earthling on the foreign planet’s       kg individual can stretch out his arms
surface. (After all, this example has the   horizontally to the side and push
human standing there breathing with-        upward against two supports with
out any special oxygen supply.)             about 200 N downward force to lift
     We need to determine the required      the body. However, this same individ-
density of the atmosphere of this alien     ual will not be able to use small wings
planet, assuming an adequate supply         of the same length and perhaps just a
of oxygen molecules for breathing by        little wider than the arms to beat
our visiting human. Earth’s atmos-          against the air with equal effect. If you
phere at sea level has a total density of   doubt this hypothesis, put some arm-
about 1.4 kg/m3, of which O2 com-           length wings on a strong person and
prises about 20 percent by molecular        observe how easily he or she can lift
composition. That is, 1 cubic meter of      off and hover a few centimeters above
air weighs 1.4 kilograms. The remain-       the ground!
der, of about 80 percent, is N2 , which          If the alien being were hollow
has a molecular weight of 28, com-          inside so that its total mass is signifi-
pared to 32 for O2. For simplicity, we      cantly less than expected for the body
assume they have the same molecular         size, there may be no problem with
weight, so we require an alien atmos-       hovering.
phere to have about 0.3 kg/m3 of oxy-
gen available for breathing.
     The gravitational acceleration g′ at
its surface determines the air density at   71. Shrunken People
the surface for a given molecular com-
position and air temperature profile.        We assume that the proposed shrinking
Most planets will have an acceleration      to one-hundredth scale can be accom-
not much different from the value of        plished. Unfortunately, your weight
9.8 m/s2 here on Earth, as one can          would remain the same (unless you get
check out for the planets in the Solar      rid of molecules somehow), and your
System, for example. So the wings           density would increase a millionfold!
must be capable of exerting an upward       The area of contact of your feet would
force at least as great as the downward     be 10,000 times smaller, so the pres-
gravitational force—in our example,         sure at your soles would be 10,000

                                                                    Answers     169
times greater, rising to about 20,000        forces, but the gravitational force
psi. Every step would break the con-         always acts toward the center of Earth,
crete, or you would sink into the            sometimes being a help and sometimes
ground until the upward normal force         being a hindrance. The major problem
could balance you. Among other               is the enormous energy requirement in
changes, your metabolism must change         getting from the surface of Earth to a
enormously, for your high ratio of sur-      reasonable distance away. Once the
face area to volume will mean that the       spaceship is more than a few Earth
rate of heat loss has increased at least     diameters away, its nuclear engine
100 times. Of course, we choose to           operation can be reasonably efficient
ignore any consequences inside the           in accelerating the vehicle. However, to
body for simplicity.                         get off the surface requires a tremen-
    Notice that if the opposite happens      dous amount of energy, and its rocket
and you grow bigger and increase your        engines must throw out a lot of
size by a factor of 100 in all directions,   momentum in the exhaust gases at
without adding molecules, your den-          high speeds to achieve “escape veloc-
sity decreases a millionfold. You            ity.” Newton’s third law dictates this
would be blown away by practically           momentum requirement. The particles
any breeze! But your greater problem         ejected backward act on the rocket in a
would be that your density is now            force pair, the rocket pushing particles
much less than the density of air, so the    backward while the particles are push-
buoyant force upward would be                ing the rocket forward.
greater than your weight. You are now             To reach outer space from Earth,
a giant balloon being pushed upward          the vehicle must provide a large supply
toward the upper atmosphere! Also,           of energy and be able to eject a large
your metabolism would change dra-            amount of momentum, usually by
matically, but again, we ignore any          having a large supply of mass to eject.
consequences inside your body. You           The energy needs can be accommo-
could probably make the journey              dated by a variety of engineering
around the world in 80 days without          designs. However, the physics is quite
the hot air!                                 demanding on the amount of mass
                                             ejected per second. The fuel mass used
                                             for this propulsion is not consumed
                                             instantly, so this fuel mass adds to the
72. Spaceship Designs                        mass of the vehicle at launch time.
Landing a spaceship on Earth and then        Consequently, even more propulsion
taking off for space involve the same        fuel mass and energy are required for a

170     Answers
launch than simply accounting for the      73. Warp Speed
payload itself. The mass of the fuel
supply soon becomes many times             Spaceships with a warp drive to accel-
larger than the actual payload             erate beyond the local speed of light
launched into space.                       cannot be ruled out just yet! One
     So when the spaceship leaves its      example is the expansion of the uni-
Earth spaceport and doesn’t eject a lot    verse, which carries everything along,
of stuff backward out of its rocket        and speeds can exceed the speed of
engines, the film is expressing a mode      light. Distant quasars with recession
of operation that is not achievable        velocities greater than c are a reality.
with present technology. But perhaps       The analog on Earth may be to have a
the propulsion will be different in the    100-meter race on a stretchable track
future, say the disbelievers. So let’s     that can change length during the race.
now go to the extreme propulsion               However, if the spaceship has the
limit. The most efficient process would     technology to be able to distort space-
be particle-antiparticle annihilation,     time itself in its local vicinity, then
conversion of fuel and antifuel com-       there is no need for enormous speeds.
pletely to energy in the form of high-     Simply contract the space in the front
energy photons according to Einstein’s     to bring distant points closer, warping
famous E0 = mc2. If we ignore many         space-time directly. That starbase that
problems such as a source for antipar-     formerly was light-years ahead of you
ticles, radiation exposure, and so on,     is now close to you, reachable by nor-
and also assume that all the photons       mal propulsion in minutes!
                                            Krauss, L. The Physics of Star Trek. New
are eventually directed rearward, each       York: HarperPerennial, 1996, pp. 56–58.
kilogram of fuel could provide 3 ×
1016 joules of energy and 3 × 108 kg-
m/sec of linear momentum. To acceler-
ate upward at about 10 m/s2, a
                                           74. North Pole Ice Melt
kilogram of this fuel can provide a mil-   There would be no change in the sea
lion-kilogram spaceship with 30 sec-       level if all the ice at the North Pole
onds of thrust. If one requires 3,000      melted. Why? Because this ice is float-
seconds of thrust, simply use 100 kg of    ing on water. Upon melting, the water
matter-antimatter fuel. We look for-       molecules in the ice simply occupy the
ward to the future of space travel with    space of the liquid displaced by the ice
antimatter engines, but for the present    originally. Of course, we need to be
we can enjoy the entertainment pro-        more careful on defining the extent of
vided by space travelers in films.          the North Pole. If we include ice on

                                                                    Answers      171
some landmass, then this ice will add       kilometers inland from the sea, and the
extra water molecules to the liquid         arid climate made living off the land
seas and slightly raise the sea level. In   very difficult. So they migrated along
contrast, most of the ice at the South      the eastern coast of Africa all the way
Pole is several kilometers thick and is     through the Middle East and India to
predominantly on the Antarctica land-       Australia. The Aborigines in Australia
mass, so its melting could significantly     are descended directly from these peo-
raise the sea level. Some films depict       ples from Namibia and South Africa
oceans that have risen hundreds of          and together with the Bushmen are the
meters from the worldwide ice melt.         oldest civilizations on Earth.
Simple estimates easily reveal that this
conjectured amount of sea level
change is ridiculous.                       75. Lightning and
    Another concern might be the lin-
ear expansion of water when its tem-
                                            Thunder
perature is above 4°C of about 70 ×         Let’s consider identical explosions on
10–6/°C. Even if the water temperature      the battlefield at two different dis-
rose by 10°C throughout the first 10         tances from the observer (i.e., the cam-
kilometers of ocean depth, the              era) but seen simultaneously. The
expected water level rise for a column      sound intensity emanating from the
of water would be no more than 7            farther explosion should be less loud
meters if the surface area remained         compared to the closer one, and the
constant. But the surface area will         sound of the farther explosion should
expand, so the actual water level will      be delayed more with respect to its
rise about 2 meters of so. For a 1°C        light flash than for the closer explo-
increase in temperature, the ocean          sion. The extra distance affects both
level rise will be several centimeters.     the sound intensity and its time delay.
    Sea level changes played major          Depending on the distance and the
roles in the migrations of our human        temperature gradient in the air, there
ancestors. Some Bushmen left their          could be additional effects, such as dif-
homeland and their cave dwellings           ferent frequencies having slightly dif-
about 40,000 years ago, seeking better      ferent speeds and/or paths en route.
climates and less arid lands. A mini ice        For example, if the closer explo-
age had developed, so seawater              sion is one-fifth of a mile away and the
became locked into the ice at the poles.    farther one is two-fifths of a mile
Their caves, which were originally near     away, the differences in arrival times
the sea, were now several hundred           should be clearly heard. The sound

172     Answers
from the closer one begins to be heard         As a dramatic example, in 1987
one whole second after the light flash      Supernova 1987A in the Large Magel-
and the weaker sound from the farther      lanic Cloud, bound to our Milky Way
explosion should be heard one second       galaxy, blew apart, dumping practi-
later, two seconds after the apparently    cally all its energy into neutrinos, but
simultaneous light flashes. Count these     there was still enough energy to pro-
time delays aloud and you will imme-       duce a lot of photons for a bright flash
diately realize that most battlefield      of visible light that was first seen by
scenes do a false portrayal of the audio   amateur astronomers in Japan. Today
timing. But who really cares? The dra-     that light flash continues to expand
matic license enhances our viewing         with decreasing intensity, and the gas
experience of the fantasy world of the     cloud of particles also continues to
cinema! But experienced military peo-      stream outward, impacting molecules
ple know the difference.                   and other stuff in various directions to
    Many more audio violations can be      help “paint” the region in beautiful
recalled from the movies. We simulta-      colors. Data seem to indicate that the
neously hear and see airplane crashes in   remains at the origin consist of two
the distance, cars falling and crashing    small, massive objects orbiting a com-
into a chasm far below, jet airplanes      mon barycenter.
passing overhead with no sound delay,          The real problem with hearing an
etc., all for the benefit of a moviegoing   explosion in space is that there is no
public who should know the difference.     medium to carry the sound waves, so
                                           no sound from the original explosion
                                           should be heard by the spaceship crew
76. Explosions in Outer                    or by the audience in the theater. The
                                           light travels at nearly 3 × 108 m/s, and
Space                                      the sound would transit at the snail’s
The colors of the explosion are proba-     pace of 3 × 104 m/s or slower. The light
bly okay, and perhaps one can appre-       flash comes before the sound for any
ciate the beautiful outward-going          safe distance. Of course, any debris
streamers isotropically distributed.       from the explosion hitting the space-
However, very seldom does a real           craft would make an impact sound
explosion, even in a vacuum, distrib-      carried by the walls of the ship and by
ute the stuff isotropically. One also      the internal artificial atmosphere.
would expect bits and chunks of vastly     Then there is the noise of the other
different sizes, with a large chunk or     spaceship going past, but that is
two left over near the explosion origin.   another story.

                                                                   Answers     173
77. Space Wars                             scattering of the light to your eyes by
                                           the molecules in the air. Spots where
Don’t try to learn your physics from       the beams strike walls, mirrors, or any
space battles. Practically everything is   objects could be seen, but their
incorrect, except the ability to cause     straight paths to the spots are invisi-
something to explode if your weapon        ble. Therefore you have two ways to
can dump enough total energy into the      make the laser beams visible so your
target in a short time!                    audience can see the sequence of
     The laser beams, no matter how        maneuvers required to succeed in the
powerful they are, would not be seen       heist: (1) put something in the air itself
en route. Seeing them requires that a      to scatter laser light such as chalk dust,
reasonable amount of light be scat-        smoke, or a fog of liquid nitrogen
tered back to you along their paths.       droplets or water mist, or (2) artifi-
But there is practically nothing in the    cially put in the light beams during
vacuum of space to scatter off. A few      editing, after the scene is done.
hydrogen atoms per cubic meter are             We suppose that some real-world
all there is out there.                    locations may use an arrangement of
     The explosion would not be heard      crisscrossed laser beams for security
at all. Sound requires a medium for its    aids, but we do not know of any. Cer-
transport, and the vacuum of space is      tainly, infrared lasers are much
not a material medium able to conduct      cheaper than visible lasers, and they
sound waves. You will feel something,      would not be seen by injecting a fog
the impact of the bits of the exploding    into the space. Nor would their spots
debris from the enemy battlecruiser        on the wall be seen.
because they travel unimpeded from             Some heist scenes show the substi-
the explosion volume to your space-        tution of a mirror to fool the security
ship. Their speeds may be enormous,        system while the protagonist steals
so they could do significant damage if      the item. Indeed, the mirror substitu-
your shields are not in place.             tion may reflect the beam in the cor-
                                           rect direction, but the tenths of a
                                           second disturbance in the original
78. Security Lasers                        beam would be very easy to detect by
As the director of the theft scene, you    the security system electronics, which
would know that the laser light would      can sense disturbance spikes quickly
not be visible in the air in a normal      and sound an alert. Of course, the
room because there is not enough           human security operator may choose


174    Answers
to ignore the alert, which so often hap-    80. Internet Gaming
pens in movies!
                                            The Internet is usually not the culprit,
                                            being extremely fast compared to the
79. Bullet Fireworks                        hand-eye coordination of the player.
Normal bullets are copper-clad lead         The actual travel time between the
and do not spark upon impact with           major switching stations along the
steel or any other surface. Just go to a    Internet is extremely short because
grinder to check out the properties of      the data packages travel at nearly the
copper versus other metals in the pro-      speed of light in a vacuum. So a data
duction of visible sparks. The grinding     package can go 20,000 kilometers in
of steel produces sparks everywhere,        about 70 milliseconds with no delays.
seen even in sunlight. The many small,      Delays at the switching stations along
hot particles of steel are actually burn-   the Internet are typically in hundreds
ing. Now grind a piece of copper tub-       of milliseconds, with your local Inter-
ing. No sparks. You might see an            net service provider (ISP) contributing
occasional spark due to contamination       most of the delay time, on the order of
on the grinding wheel or in the copper.     300 milliseconds or so.
The copper bits ground off do react             In contrast, your local computer
with oxygen, but they do not get very       and cable modem, for example, usu-
warm. Please do not grind lead because      ally will be slower in responding to
toxic particles will be released into the   your input and getting the data out to
air and, besides, lead is known not to      your ISP and onto the Internet. The
produce sparks. So the conclusion is        faster the equipment at your end, up to
that practically no bullets produce a       a certain speediness, the quicker your
brilliant flash of light on impact.          response will appear in the game. If
    In support of the film depictions is     your equipment delay time becomes
the fact that the military does have        less than the Internet delays, there’s
machine gun bullets containing white        nothing further to be improved by
phosphorus so that the point of impact      spending more money on a faster com-
can be seen by the gunner. These bul-       puter system.
lets are used also to ignite fuel tanks
and other possible containers of explo-
sives by producing sparks to ignite the     81. Cartoon Stretching
vapors. But phosphorous bullets are         There are many ways to approach the
very rare outside the military.             problem of determining the speed of


                                                                    Answers     175
sound in the cartoon character’s body.       slow speed of sound indeed when com-
We consider one approach only. Start         pared to most materials, which have a
with the application of an external          speed of sound of about 300 meters per
force to the surface of the body at          second. So we conclude that cartoon
some location—the character’s foot,          characters are made of very unusual
say. We see the stretch region progress      materials. Perhaps designer materials in
up the leg over a period of one to two       the future will be able to mimic a car-
seconds.                                     toon material. According to the world’s
    “Wait just one minute!” you              greatest detective, the game is afoot!
exclaim. How is the speed of sound in
the material related to its speed of
stretching in response to an applied
                                             82. Infrared Images
force? The answer is that both               The actual converted infrared image
processes require communication from         would be a blurred black-and-white
one molecule to the next molecule out-       image, not a sharp one. The physics
ward, from the applied force region to       places a limit on the resolution of
the far reaches. Usually the much            images we see in the infrared com-
smaller energy in the sound application      pared to the visible. When we look
produces a very tiny stretch followed        through binoculars or any lens system
by a relaxation and overshoot, then          in the visible, these systems have a res-
another stretch, etc., repeatedly at         olution limit that depends on the qual-
some frequency above 14 Hz or so.            ity of the optic elements. No matter
The stretch produced by the tug of a         how much the image is enlarged or
much larger applied force shows a            enhanced by dithering, etc., the origi-
much larger displacement of the mole-        nal resolution is not improved even
cules that may or may not relax when         though the image looks cleaner. But
the applied force is lessened. The larger    the physics is even more restrictive
displacement between molecules for           when comparing an infrared image to
the stretch process may require slightly     a visible image because the visible light
more time if the process cannot be           has been coherently scattered from the
modeled by a collection of linear har-       object, whereas the infrared light has
monic oscillators, but the speed of          not, as explained below. Of course,
stretch will be very close to the value of   one cannot exceed the Rayleigh crite-
the speed of sound for most materials.       rion for resolution of approximately
    The stretched cartoon character’s        one wavelength of the light, except by
body exhibits a communication speed          using interference techniques, which
of about a meter per second, a very          we do not consider here.

176     Answers
    In the visible part of the electro-     will be able to take an infrared image
magnetic spectrum, atoms absorb and         and make a sharpened black-and-
emit the photons of light in a two-step     white image true to the original object.
process, usually absorbing and emit-            In the ultraviolet, the scattering
ting in about 10–16 second. During this     time is very short, but so is the wave-
time interval, the molecule holding         length, so the extent of the coherent
that atom moves very little. Nearby         scattering area also is very short; thus
atoms also scattering this impinging        the image is blurred compared to the
beam of numerous photons tend to            visible one. Nature has given us a
remain in place during their scatter-       vision range in the visible that ensures
ings. In effect, during the scattering of   the best resolution of detail.
each photon there always will be a
fixed phase relationship among all the
atoms scattering light from the object
                                            83. Light Sabers
to your light sensors. Hence, photons       Yes, the light sabers would pass
scattering from different areas on the      through one another as if nothing were
object’s surface carry detailed phase       there! The photons of the light are
information with fixed phases to            bosons, which do not repel each other.
achieve nearly maximum resolution. If       The clashing of the light sabers is pure
the phases actually varied randomly         artifact and far beyond artistic license.
from one location to another, the visi-     It’s an outright lie to the public!
ble image would become blurred.                  There are two general categories of
    In the infrared, the image is blurred   particles in the universe: fermions and
because most of the infrared is             bosons. Two identical fermions (e.g.,
absorbed and emitted by molecular           think electrons, or any other funda-
vibrations and rotations that have ran-     mental spin 1/2 particle) cannot exist
dom phases over the object’s surface.       in the same quantum state defined by
This scattering process takes much          its 4-momentum and spin. The exis-
longer—about 10–12 second—sufficient         tence of all matter, including us
time for the molecule to move consid-       humans, critically depends on the
erably during the scattering, so there      inability of two identical fermions to
will not be a fixed phase relationship       get together in the same state, so mat-
among neighboring molecules on the          ter occupies a volume. That is, objects
surface of the object. The same surface     can be bigger than a point!
that appeared well resolved in the visi-         In the case of bosons, one can not
ble will now be quite blurred in the        only put as many identical bosons
infrared. No magical digital techniques     (think photons or any integer spin

                                                                     Answers     177
fundamental particles) into the same       the battlefield scatters enough of the
quantum state, they also prefer to be      light. But when the laser beam origi-
in one, with the probability to do so      nates in space where there is no air (or
enhanced by the number N of bosons         other scatterers in significant density),
already in the state. There is no repul-   one can still see the laser beams. Is
sion experienced. So two light sabers      there no correct laser physics in these
intersecting at an angle would pass        movies? Is there any correct physics
right through one another with no          at all?
change in either. If the light beams
were powerful enough, though, they
could cut material objects made out of     85. Cold Silence of
fermions—that is, the ordinary stuff
all around us—whenever this material
                                           Space
stuff intercepts the beams because suf-    Yes and no. The very good vacuum
ficient energy could be absorbed to        between Earth and Venus, for exam-
change the physical state of the mate-     ple, certainly does not conduct sound,
rial and result in vaporization.           so the space is silent. But what is the
                                           temperature of outer space? This ques-
                                           tion is an improper question, for we
84. Force Fields                           must instead ask: “What temperature
We don’t know why we can see the           would be recorded by a thermometer
good guys through the battlefield force     placed in space between the Venus and
field via visible light at the same time    Earth orbital distances?” Whatever
that the visible laser beams cannot get    kind of thermometer we use, if the sen-
through! This result might be due to       sor is not rotated, one side will be
some dramatic nonlinear natural effect     exposed always to the direct rays from
not yet experienced in research labs, or   the Sun and the other side will be in
the phenomenon is a pure artistic          shadow. At thermal equilibrium, the
falsehood. We would bet on the latter.     amounts of outgoing radiation energy
Obviously, the light transmission          and incoming radiation energy in all
properties of a transparent material       directions will balance.
can change dramatically on absorbing           The equilibrium temperature T
energy, but the usual effect is hole-      for an object at Earth’s average dis-
burning, not reflection.                    tance from the Sun is about 280 K, or
    Then there is the problem of why       +7°C, the actual value being lower
the laser beams can be seen along their    because there will be some energy
paths, unless the dust in the air above    reflected away and not absorbed by

178     Answers
the thermometer. One calculates T            effective in limiting the initial spread
from this equation: flux absorbed =          of the debris from the thermal explo-
flux emitted. If we assume for simplic-       sion. No nuclear explosion would
ity that the sensor is a sphere of radius    occur, or else everything around would
R, then the equation becomes S (1 – A)       be vaporized by the energy released, in
πR2 = σT4 (4πR2), where S = 1.4 kW           which case the depth in the water
s–1 is the solar constant at Earth dis-      would help very little in preventing the
tance and σ = 5.67 × 10–8 W m–2 K–4.         spread of energy in many forms. The
The parameter A is the reflectivity,         thermal explosion in a nuclear reactor
which we take as zero in the ideal case      in the sub releases the fuel and
of a perfect absorber and radiator.          coolant, so radioactive particles and
Real materials will have an A value          debris will be sent out in all directions.
between 0 and 1.                             Some of this stuff would be slowed
     If you are orbiting at the Earth dis-   effectively by the water, and some
tance from the Sun, you may desire to        probably would make the surface and
rotate your spacecraft slowly in space       escape into the air.
so that all sides cook evenly! This pas-         When the Chernobyl nuclear reac-
sive heating can be augmented by             tor broke its containment vessel in the
active heating from within to maintain       1980s, its nuclear particles were
a cozy environment.                          detected around the world within
     If you are orbiting closer to the       hours to days. At the University of
Sun than Earth, the equilibrium tem-         California at Irvine, the air filtering
perature will be higher, the solar flux       system at the local nuclear reactor
increasing as the ratio of the squared       recorded the radioactive cesium and
distances. Near the orbit of Mercury         iodine particles in parts per billion
you may be too hot! If you are farther       from the Chernobyl incident 10 days
away, the temperature decreases, so          after the chemical explosion.
you may need artificial heating. Some
rotation may produce a system that
requires less fuel for heating, but the      87. Plutonium vs.
details need to be worked out.
                                             Uranium
                                             A plutonium bomb would be much
86. Nuclear Submarine                        safer to handle. Weapons-grade pluto-
Submariners love to dive in their sub-       nium (Pu-239) emits primarily alpha
marines. A dive to several hundred           particles and low-energy gamma rays,
meters under the surface may be              both being easy to shield. The trace

                                                                      Answers      179
amounts of even-numbered Pu iso-           88. Nuclear
topes have spontaneous fission reac-
tions that emit neutrons. However,
                                           Detonation
neutron detectors would need to be         We know of no nuclear explosive
within a few meters to detect these        device that does not require at least
neutrons above background. So a lost       very good spherical symmetry to be
plutonium bomb could be very diffi-         detonated. The simpler atomic devices
cult to find.                               have either two hemispheres that must
    The plutonium usually is coated        be rapidly moved together into a
with beryllium or another appropriate      sphere, or two spherical sections held
sealant because the exposed element        apart until a slab of nuclear material is
will react chemically with the oxygen      shot into the gap to start the fission
in the air or in water and increase its    reaction. The more complicated hydro-
temperature considerably. The person       gen devices require a strong, spheri-
holding a plutonium bomb would             cally symmetrical implosion from the
probably feel that the protective casing   perimeter shell to initiate the fusion
is warm, because the alpha particles       reaction.
deposit their kinetic energy in the cas-       Dropping the weapon from any
ing material.                              height would damage the casing asym-
    Of course, inhaled or ingested Pu is   metrically. Even shooting pellets or
one of the worst carcinogens known.        bullets, etc., through the casing into
Any explosion releasing Pu into the air    the warhead would create an asym-
creates a hazard for all life that would   metrical result but no explosion. Initi-
remain for a long time.                    ating the nuclear reaction and keeping
    In contrast, weapons-grade U-235       the reaction going are not easy. The
releases gamma rays at several ener-       process is certainly not worth worry-
gies, the most intense at 186 KeV. So      ing about to the extent portrayed in
the detection of a device containing U-    the movies. One should be more con-
235 is much easier than trying to find      cerned about whether the proper
plutonium. Some films portray the dif-      safety precautions are being practiced
ferences correctly, while other films      against accidentally dropping the thing
dramatize any nuclear device and its       on one’s toes.
possible dangers with remarkably ado-          By the way, the smallest practical
lescent scare techniques.                  nuclear weapon tips the scales at only
                                           about 9 kilograms, about 20 pounds,
                                           and is small enough to fit into a bulky


180     Answers
attaché case. For the physics details         that is, time passes at its normal rate
and an estimate of the smallest device,       for all particles and collections of par-
see the reference below.                      ticles. One possible explanation for all
 Bernstein, J. “Heisenberg and the Critical   particles behaving the same with
  Mass.” American Journal of Physics 70       regard to the passage of time relies on
  (2002): 911–916.
                                              the direction of time being built into
                                              the quantum state definition of a fun-
89. Fabric of                                 damental particle, with the opposite
                                              direction of time for its antiparticle.
Space-time                                         The phrase “speed of gravity” is
Space-time is not a piece of cloth, nor       meaningless in the simplest interpreta-
does space-time behave like a real mate-      tion, being an error in stating the
rial. The metaphor “fabric of space-          acceleration of gravity. Most films
time” allows one to visually model            confuse the concepts of speed and
space-time with its coordinates of space      acceleration in the same manner that
and time in a way similar to the simpler      most people do. Unfortunately for
two-dimensional construction of woven         society, the scriptwriters did not learn
cloth. There is no way for space-time to      their beginning physics, and we all
rip or tear, although mathematically          continue to suffer from their intuitive
there can be singularities of various         mistakes in describing the behavior of
dimensions and other mathematical             nature. Or perhaps a more positive
properties that some people have              view should be taken in that the enter-
stretched into real physical properties       tainment value is improved by ignor-
in the name of dramatic license.              ing the laws of physics!
    As far as being able to go back in
time as a time traveler, the time dimen-
sion of space-time is not like the spatial
dimensions of space-time. Mathemati-
                                                      Chapter 5
cally, the passage of time operator in               Go Ask Alice
quantum mechanics is antiunitary,
while the spatial displacement operator
is unitary. In addition, no one has
                                              90. Spotlight
shown convincingly that a fundamen-           Yes, a spot of light from the lighthouse
tal particle can progress either forward      beacon when moving across your field
or backward in time differently than          of vision can move faster than c. But
we now experience and understand—             the actual light itself (i.e., the photons


                                                                       Answers      181
in the beam) moves from the source             91. Quasar Velocity
to the reflecting spot in the sky at c,
no faster.                                     The special theory of relativity says
     A good example from astrophysics          that information cannot be transmit-
is the radio wave beam from the pulsar         ted faster than light. The photons
in the Crab Nebula, which sweeps               always travel at light speed in the local
across our Earth observatory thirty            reference frame, but the space in which
times a second from a distance of a            the photons travel may be expanding.
few thousand light-years. The electro-         An analogy would be a 100-meter race
magnetic waves coming to us from the           with the track lengthening during the
distant source travel at light speed, but      race. The elapsed time to reach the
the sweep across our planet moves              100-meter tape depends on the model
faster than c.                                 for the expansion rate for the runner
     Occasionally one encounters other         and for the photons from that distant
suggested examples of spots traveling          quasar. Under these expansion condi-
faster than light speed, such as the           tions, there can be recessional veloci-
intersection edge in a very long pair of       ties greater than c!
                                                Chown, M. “All You Ever Wanted to Know
scissors progressing outward when                about the Big Bang.” New Scientist (April
closing. Unfortunately, this intersec-           17, 1993): 32–33.
tion edge’s speed is limited by the             Peebles, P. J. E., D. N. Schramm, E. L.
speed of sound in the metal of the scis-         Turner, and R. G. Kron. “The Evolution of
                                                 the Universe.” Scientific American 271, no.
sors, which is quite slow compared to            4 (1994): 52–57.
c. However, the spot of light on an             ———. “Out with the Bang.” Scientific
oscilloscope trace can move across a             American 272, no. 3 (1995): 10.
screen faster than c even though this           Stuckey, W. M. “Can Galaxies Exist within
spot is produced by the slower-moving            Our Particle Horizon with Hubble Reces-
                                                 sional Velocities Greater than c?” Ameri-
electrons striking a phosphor.                   can Journal of Physics 60 (1992): 142–146.
 Bergmann, P. G. “Can a Spot of Light Move
  Faster than c?” The Physics Teacher 19
  (1981): 127.
 Rothman, M. A. “Things That Go Faster         92. Spaceship
  than Light.” Scientific American 203, no. 1
  (1960): 142–152.
                                               Approach
 ———. “Not So Fast.” Scientific American        The observer sees the highly relativistic
  269, no. 6 (1993): 10.                       object approaching back side first!
                                               Therefore, the spaceship seems to be
                                               approaching tail first! What is often
                                               referred to as a contraction in the

182      Answers
direction of motion for a rela-
tivistically approaching object is
actually a rotation known as the
Terrell effect.
    We need to discuss some aspects
of the Terrell effect to explain the
behavior of the approaching space-
ship. Consider a solid, opaque cube
approaching. At low speeds, the light
rays emitted off the back side of the     actually a rotation for a real three-
approaching cube cannot pass through      dimensional object. What we have
the box to reach the observer. At         described above is a snapshot of the
higher speeds nearing the speed of        spaceship (and cube)—that is, what
light, however, enough of the box         photons from different parts of the
moves out of the way for light emitted    object would imprint on a camera
from part of the back side to reach the   sensor simultaneously. E. Sheldon (see
observer. When this behavior happens,     the reference below) discusses the
the observer will not see all of the      stereoscopic appearance of a three-
front side because some of the light      dimensional object that involves shear-
rays from the front are intercepted by    ing and other distortions in addition to
the extremely fast moving box. The        rotation, all these effects first discussed
box appears rotated, with the away        by J. Terrell.
side of the front hidden, and the near     Sheldon, E. “The Twists and Turns of the
                                            Terrell Effect.” American Journal of Physics
side of the back visible. The rotation      56 (1988): 199–200.
angle increases with increasing speed,     Terrell, J. “Invisibility of the Lorentz Con-
nearing c and with proximity to the         traction.” Bulletin of the American Physical
trajectory. Additional complications        Society 5 (1960): 272.
also occur, such as nonrigidity, which     Weisskopf, V. F. “The Visual Appearance of
                                            Rapidly Moving Objects.” Physics Today
we ignore in this simple explanation.       13 (1960): 24–27.
    So the spaceship approaching at
near light speed will appear rotated so
that the back end is almost totally
                                          93. Mass and Energy
visible and the front end is almost       The answer to both questions is equa-
totally hidden from view. J. Terrell in   tion 1, although the majority of physi-
1959 was the first to recognize that      cists seem to prefer equations 2 or 3!
what physicists had been calculating      Their choices probably are caused by
as a Lorentz-Fitzgerald contraction is    the confusing terminology widely used

                                                                    Answers       183
in the physics literature that says that          the measurement of the length compo-
a body at rest has a “proper mass”                nent along my direction of motion of
or “rest mass” m0 , and a body in                 the metal bar that appears to be
motion has a “relativistic mass” m =              rotated. The atoms do not move closer
     √(1 – 
m0 / v2/c2) .                                to one another, so the strain gauge
    There is only one mass in physics,            remains at zero.
m, which does not depend on the ref-                  The apparent rotation is called the
erence frame. This mass m is the rela-            Terrell effect: if a snapshot is taken of
tivistic invariant quantity in E2 – p2c2          a moving object, the object does not
= m2c4, whereas the energy is different           appear contracted, but rather rotated.
in different reference systems. There is          A snapshot is understood to be a two-
no need to place the index 0 with the             dimensional, nonstereoscopic photo-
mass. However, the total energy E                 graph. The stereoscopic appearance of
needs the 0 index if the particle has no          a three-dimensional object is more
momentum in that reference frame—                 complicated because shearing and
that is, E0 = mc2.                                other distortions can be present. In
    For a complete and stimulating                fact, there is no such thing as a rigid
discussion of these ideas and their               object in relativity!
history see the L. V. Okun reference               DeCampli, W. M. “A Gedanken Experiment
below.                                              to Demonstrate Lorentz Contraction.” The
                                                    Physics Teacher 13 (1975): 420–422.
 Einstein, A. “Zur Elektrodynamik bewegter
  Körper.” Annalen der Physik 17 (1905): 891.      Terrell, J. “Invisibility of a Lorentz Contrac-
                                                    tion.” Bulletin of the American Physical
 ———. “Ist die Trägheit eines Körpers von
                                                    Society 4 (1959): 294.
  seinem Energieinhalt abhängig?” Annalen
  der Physik 18 (1905): 639. Translated by         ———. “Invisibility of a Lorentz Contrac-
  W. Perrett and G. B. Jeffery in The Principle     tion.” Physical Review 116 (1959):
  of Relativity. New York: Dover, 1923, p.          1041–1045.
  71.                                              ———. “The Terrell Effect.” American
 ———. “Zur Theorie der Brownschen                   Journal of Physics 57 (1989): 9–10.
  Bewegung.” Annalen der Physik 19 (1906):
  371.
 Okun, L. V. “The Concept of Mass.” Physics
  Today 42, no. 6 (1989): 31–36.
                                                  95. Mass/Energy
                                                  Mass is energy. There is no distinction
                                                  to be made. There is no conversion! In
94. Strain Gauge                                  1905, Einstein states explicitly: “The
The strain gauge continues to show a              mass of a body is a measure of its
zero value. What I interpret as a length          energy content . . . ”. What Einstein
contraction when I run past is really             was stating is that mass and energy are


184      Answers
equivalent, that they are possibly two            the amount equal to the total kinetic
different aspects of the same physical            energy of all the particles as seen in the
quantity, only their units have been              frame in which the total momentum is
chosen to be different. One does not              zero. The exception “yes” occurs when
convert one to the other if they are              all the particles move in the same direc-
equivalent.                                       tion with the same speed—that is, have
    We can imagine a conversation                 the same velocity.
with a student: “Does a photon have                    The value of defining the mass in
mass?” asks the student. “Yes, because            this relativistic fashion means that M
the photon has energy.” The student               determines the system’s inertia, its
counters, “But for a photon E = pc, so            resistance to acceleration by a force
the relation E2 – p2c2 = m2c4 becomes             that acts on the system as a whole. A
E2 – p2c2 = 0. Therefore m = 0 for                box with a hot gas of particles has
the photon.” Can you complete this                more mass than the same box after the
dialogue?                                         gas has cooled. Also, the box of hot
 Einstein, A. “Ist die Trägheit eines Körpers     gas exerts a greater gravitational pull
  von seinem Energieinhalt abhängig?”             on a test particle. In addition, a box of
  Annalen der Physik 18 (1905): 639. Trans-
  lated by W. Perrett and G. B. Jeffery, in The   photons exerts a gravitational pull on
  Principle of Relativity. New York: Dover,       a test particle, and vice versa.
  1923, p. 71.
                                                   Taylor, E. F., and J. A. Wheeler. Spacetime
 ———. Relativity: The Special and General           Physics: Introduction to Special Relativity.
  Theory. New York: Crown, 1961, p. 47.             San Francisco: W. H. Freeman, 1992, p.
                                                    135.

96. System of
Particles                                         97. Light Propagation
No and yes! Except in the special cir-            According to the special theory of rel-
cumstance described below, the answer             ativity (STR), (1) no object can move
is no. Energy and momentum are addi-              at the speed of light, (2) the speed of
tive, but not mass. Mass is a measure             light is the same for all observers, and
of the magnitude of the energy-                   (3) the space-time interval τ between
momentum 4-vector. From the total                 two events defined by τ2 = c2 ∆t 2 – ∆x2
energy E and the total momentum P                 – ∆y2 – ∆z2 is the same for all
can be determined the mass M of the               observers, but the ∆t and ∆x may be
system: M2c4 = E2 – P2c2. Therefore the           different, for example.
mass M of the system is greater than                  For one-dimensional motion τ2 =
the sum of the masses of its particles by         c2 ∆t2 – ∆x2. The driver has ∆x ≠ 0, so


                                                                            Answers       185
her ∆t must be greater for the two           One also could use a regular n-gon of
events than the elapsed time for the         flat mirrors to reflect the light around
observer on the ground. Therefore the        the Equator and then take the limit as
driver measures the longer time inter-       n becomes infinite. The light leaves
val between events A and B.                  from point P on the Equator of the
    Suppose the car makes a second           rotating Earth and returns to point P
run at a great speed. The nearer to the      in time T. The light going eastward has
speed of light the car goes, the smaller     traveled the distance 2πR + ωRT in the
is ∆t for the ground observer, and           inertial system, where ω is the angular
τ = c ∆t is smaller in this case also. But   frequency of rotation with respect to
again, as expected, the time interval is     the inertial reference frame. The point
longer for the driver. The nearer to the     P has traveled ωRT. The ratio of point
speed of light the car goes in the           speed to light speed is ωR/c =
ground frame, the difference will be         ωRT/(2πR + ωRT), from which T =
the difference in arrival times as           2πR/(c – ωR). For the system at rest,
observed in the two frames.                  T = 2πR/c. Hence, when ω ≠ 0, define
 Taylor, E. F. “Light Propagation.” The      δT = T – 2πR/c as the extra time
  Physics Teacher 25 (1987): 252.            required. Substitution for T gives δT =
                                             2πωR2/[c (c – ωR)]. Upon returning to
                                             point P on the Equator after one cir-
98. Sagnac Effect                            cuit, the clocks will differ by 2δT for
No, they do not tick at the same rates.      the measured elapsed times.
Their tick rates are different because        Schlegel, R. “Comments on the Hafele-
Earth is rotating with respect to an           Keating Experiment.” American Journal of
                                               Physics 42 (1974): 183–187.
inertial reference frame such as the dis-
tant stars. The clock moving eastward
has a higher velocity with respect to
the inertial frame than the clock mov-
                                             99. Light Flashes
ing westward at all moments. Accord-         The observer on planet A sees the
ing to the STR, the higher the velocity,     flashes 20 minutes apart. From the
the slower the clock ticks. That is, a       STR postulate, we know that no
clock ticks fastest when at rest in an       observations of the light flashes only
STR inertial reference frame.                can discern which inertial frame is at
    The difference in the elapsed time       rest. If the flashes sent out by the
for the two clocks can be calculated by      spaceship at 10-minute intervals are
considering a light clock following a        seen at planet B separated by 5 min-
circular light path around the Equator.      utes, then flashes sent out from B at

186      Answers
10-minute intervals will be seen on the            To understand why the object
spaceship at 5-minute intervals.               slows in the x-direction when the
    One also realizes that if there were a     applied contact force is in the y-direc-
light flash every 5 minutes from planet         tion, we begin with the space-time
A, the observer on planet B would see          interval: (interval)2 = c2 ∆t2 – ∆x2 – ∆y2
them at 5-minute intervals. What is the        – ∆z2. For real objects traveling at
interval for these flashes from A as seen       speeds less than c, the time term is
by the spaceship observer? The answer          much larger than the spatial terms,
is every 10 minutes, by invoking the           and the interval is called the proper
STR postulate above. So the spaceship          time τ. The linear momentum px in the
sees planet A’s flashes to be spaced twice      x-direction in Newtonian physics is
as much apart as the interval at the           defined as px = m dx/dt (for an object
source on A; likewise, the 10-minute           that is not changing its mass, i.e.,
flash intervals from the spaceship must         excluding objects such as a leaking
be twice the interval at planet A, or 20       bucket of water). The correct STR
minutes apart.                                 expression simply substitutes proper
 Hewitt, P. G. Conceptual Physics, 6th ed.     time τ for Newtonian time t so that px
  Glenview, Ill.: Scott, Foresman, 1989, pp.   = m dx/dτ. For a low-velocity object,
  650–656.
                                               dτ ~ dt. But the actual relationship
                                               between τ and t depends on the mag-
100. Forces and                                nitude of the object’s total velocity, a
                                               vector quantity, not just the speed
Accelerations                                  component in the x-direction. There-
No. In the STR, all contact forces will        fore, as the object speeds up in the y-
produce an acceleration in a direction         direction, its speed in the x-direction
not parallel to the applied force! For         must decrease to maintain a constant
example, a rigid sphere is moving              total velocity magnitude, otherwise its
along the plus x-direction of an iner-         x-component of linear momentum
tial reference frame. Now let an               would change, forbidden by the law of
applied contact force act in the plus          conservation of linear momentum.
y-direction to increase the speed of           The pertinent relationship is dt/dτ =
the sphere in the y-direction. What               √(1 – 
                                               1/ v2/c2) . Remember, the mass is
happens to the speed in the x-direc-           fixed in value.
tion? The x-component of the speed                 One could write down the relativis-
decreases—the object slows down in                                           √(1 – 
                                               tic momentum px = mvx//  v2/c2)
its original direction, corresponding to       and argue that since m is constant, the
a negative acceleration!                       component of velocity in the original

                                                                        Answers     187
direction must decrease to keep the             lab frame, let’s review the simpler prob-
momentum component constant.                    lem of how velocities are added in rela-
 Ficken, G. W. Jr. “A Relativity Paradox: The   tivistic frames. If an object moves
  Negative Acceleration Component.” Amer-       forward with the velocity V ′ in the
  ican Journal of Physics 44 (1976):
  1136–1137.                                    spaceship frame, then the object’s veloc-
 González-Díaz, P. F. “Relativistic Negative    ity V in the lab frame is determined by
  Acceleration Components.” American Jour-      the law of addition of velocities V/c =
  nal of Physics 46 (1978): 932–934.
                                                (V ′/c + Vs /c)/(1 + V ′Vs /c2), where Vs is
 Tolman, R. C. Philosophical Magazine 22
                                                the uniform velocity of the spaceship in
  (1911): 458.
                                                the lab frame. One can check the limit-
                                                ing case for low velocities, when
101. Uniform                                    V ′Vs /c2 is very small, to verify agree-
                                                ment with Galilean relativity—that is,
Acceleration                                    the two velocities simply add.
In the STR, the velocity in the lab                  To relate the acceleration of the
frame is no longer V = a ′t for a uni-          object as seen by both observers, the
form acceleration a ′ in the moving             addition of velocities expression is
frame. However, in the moving frame             differentiated with respect to the time
at each instant the expression V ′ = a ′t ′     in the lab reference frame to obtain
continues to be true. To convert from                                       Vs
                                                                          2/c
                                                a = a′ /{(1 + V′Vs/c2) √(1 –  2))3 }, a
the moving frame to the lab frame, we           messy expression. The velocity of the
must essentially convert the clock              accelerating object in the lab frame is
readings and time interval using dt/dτ          found by substituting V ′ = a′ τ. There-
     √(1 – 
= 1/ v2/c2) . Here, τ is the proper        fore a ≠ a′ and V < c.
time—that is, the clock reading on a                 An alternative mathematical tech-
wristwatch worn by an observer on               nique using a velocity parameter
the spaceship, say, and dτ is the proper        defined in terms of hyperbolic func-
time interval between two events at             tions is given in the Taylor and
the same location. In the example, τ is         Wheeler reference below.
the elapsed time on the wristwatch of            Taylor, E. F., and J. A. Wheeler. Spacetime
the person on the moving frame.                   Physics: Introduction to Special Relativity.
                                                  San Francisco: W. H. Freeman, 1966, pp.
Hence, on the moving spaceship                    47–58.
frame, V ′ = a′τ.                                Tipler, P. A. Modern Physics. New York:
    Before we determine the answer                Worth, 1978, p. 27.
for the velocity of the object in the




188      Answers
102. Long Space                              time dτ. In the same astronaut time the
                                             velocity parameter of the spaceship
Journey                                      with respect to the lab frame changes
The 7,000 light-year journey with 40-        from θ to θ + dθ. But dθ = a dτ/c. That
year aging is possible in STR physics        is, each time interval dτ on the astro-
but not in Newtonian physics!                naut’s wristwatch is accompanied by
     Define V/c = tanh θ, where tanh is       an additional increase dθ = a dτ/c in
the hyperbolic tangent. Substitute into      the velocity parameter of the space-
the law of addition of velocities to         ship. Since the spaceship starts from
obtain tanh θ = (tanh θ ′ + tanh θs)/(1 +    rest, we get θ = a τ/c, telling us the
tanh θ ′ tanh θs). Some checking of the      velocity parameter θ of the spaceship
mathematics of hyperbolic functions          in the lab frame at any time τ in the
will reveal that the θs are additive, just   astronaut’s frame.
as velocities are additive in Newtonian           Our solution is V = c tanh (aτ/c).
physics with Galilean relativity. That       There is no limit to the product aτ,
is, θ = θ ′ + θs. Some people call θ the     which can be much greater than c, but
velocity parameter.                          tanh ≤ 1, so the lab velocity V only
                                             approaches c after a long wristwatch
                                             elapsed time. The distance traveled in
                                             the lab frame is d x = tanh (aτ/c) cdt. In
                                             the lab frame, the astronaut’s wrist-
co-moving frame      co-moving frame         watch seems to be ticking slower than
                                             the lab clock, so dt = cosh θ dτ, with θ
Astronaut time τ    Astronaut time τ + dτ    = a τ/c. Therefore dx = c sinh (aτ/c) dτ,
                                             which can be integrated from zero
    Back to the problem at hand: How         astronaut time to the final time T to
much velocity V in the lab frame does        produce the distance traveled x = (cosh
the accelerating spaceship have after a      (aT/c) – 1) c2/a.
given time? We need three frames of               The journey would be done by
reference: the lab frame, the spaceship      accelerating to the halfway distance at
frame, and an instantaneously comov-         3,500 light-years, then decelerating to
ing inertial frame that for an instant       the 7,000-light-year distance. Substi-
has the same velocity as the spaceship.      tuting 3,500 light-years in units of
With respect to the instantaneously          meters, g as 9.8 m s–2, and the speed of
comoving frame, the velocity parame-         light, one calculates a journey dura-
ter changes from 0 to dθ in wristwatch       tion for the space traveler’s wristwatch


                                                                     Answers      189
of T ~ 8.62 years. The round-trip                and, to a first approximation, the time
would require about 34.5 years. So               interval between ticks differs by (δr/r)
the space travelers would age less than          GM/rc2 ∆T, where δr is the altitude dif-
40 years!                                        ference, M is Earth’s mass, r is the
    Are there any plans to make this             radial distance from the center of
journey? Assuming human volunteers               Earth, G is the gravitational constant, c
are available who want to achieve this           is the speed of light, and ∆T is the time
feat, other factors, such as a reliable          interval between ticks on the reference
food supply, sufficient health care, and          clock. Substituting r = 6.37 × 106 m
an energy source for the constant 1-g            and dr ~ 1.5 m produces a value of 1.6
acceleration for 40 years would be dif-          × 10–16 ∆T, an incredibly small change
ficult to provide with present technol-           in rate. Over a lifetime of about 80
ogy. And, of course, more than 14,000            years, the head becomes about 0.4
years would have passed for civiliza-            microsecond older than the toes.
tion here on Earth. Who or what                       To understand the effect of gravi-
would be here to greet them on their             tation on the clock rate, we can utilize
return?                                          the equivalence between an accelerat-
 Taylor, E. F., and J. A. Wheeler. Spacetime     ing rocket frame and being in a uni-
  Physics: Introduction to Special Relativity.   form gravitational field. Consider two
  San Francisco: W. H. Freeman, 1966, pp.
  47–58.
                                                 light flashes sent from the bottom of
                                                 the accelerating rocket to its top, as
                                                 shown in the animation diagram from
                                                 the view of our inertial reference frame
103. Head to Toe                                 with respect to the stars. The two light
Yes, your feet and toes age slower than          flashes are one second apart in our
your head. That is, whenever you are             frame but arrive at the top of the
standing or sitting, a clock at the alti-        rocket three seconds apart. Why?
tude of your head will tick faster than          Because the top has moved away
an identical clock at the altitude of            from the approaching light flash with
your toes. The ambient gravitational             the appropriate acceleration value.
field affects the tick rate of all clocks in      Therefore the frequency of arrival is
the same way. A clock will tick fastest          lower than the starting frequency. In a
at rest in an inertial reference frame.          stroke of genius, Einstein realized that
The difference between clock rates in            the only reason for different flash fre-
different gravitational environments is          quencies would be if the clock at the
normally minuscule but measurable                top ticked at a different rate than the


190      Answers
                                                 photon traveling at light speed, the
                                                 clock would not tick. As a photon
                                                 traverses the universe, no time elapses
                                                 in its reference system. The photon can
                                                 be absorbed by an atom and disap-
                                                 pear, but the photon cannot change
                                                 directly into another photon. Like-
                                                 wise, if all three neutrino types did not
                                                 have any mass, none could oscillate
                                                 into another neutrino type because
                                                 they do not experience the passage of
                                                 time. Therefore, for neutrino oscilla-
identical clock at the bottom. There-
                                                 tions to occur, at least two neutrino
fore, gravitation makes time run slow.
                                                 types must have mass. The data indi-
    Is there a place where one can put
                                                 cate that the sum of the three neutrino
a clock so that the time interval
                                                 masses cannot exceed about 1 eV/c2,
between ticks becomes infinite? Yes,
                                                 very much smaller than the 0.511
near a black hole, at the event horizon.
                                                 MeV/c2 mass of an electron.
 Hewitt, P. G. Conceptual Physics, 6th ed.
  Glenview, Ill.: Scott, Foresman, 1989, pp.
  671–678.
                                                 105. Spaceship
 Taylor, E. F., and J. A. Wheeler. Spacetime
  Physics: Introduction to Special Relativity.   Collision
  San Francisco: W. H. Freeman, 1966, p.
  154.                                           The method of determining position
                                                 and clock reading for the three events
                                                 first before answering the question is
104. Neutrino Mass                               a good one. However, the values
For a change in a system to occur—               inserted already are not all correct for
such as the change of a muon neutrino            the observer. Simultaneous measure-
to an electron neutrino, for example—            ments at both the origin X1 = 0 and at
time must elapse. That is, the reference         X2 = L cannot be made by the method
clock must tick in the rest frame of the         assumed since they are not equidis-
muon neutrino. We know that the                  tant. Therefore, if the notation (X, T)
greater the velocity of a real clock in          is correctly (0, 0) for event 1, then
our laboratory reference system, the             event 2 is labeled by (L, –L/c) because
slower is its ticking rate. In the speed         the light from event 2 takes L/c sec-
limit of a massless particle such as a           onds to travel the distance L to the


                                                                          Answers     191
observer. Event 3 is not at position L/2                  result in less aging than for his twin
between the two spaceships at T = 0                       brother, who has remained at home on
because spaceship B has already trav-                     Earth. Even if the acceleration was
eled for L/c seconds. Therefore the                       simply an immediate turnaround at
distance between the two spaceships is                    the farthest distance, the spaceship
L – VL/c. Thus T3 = L(1 – V/c)/2V. We                     velocity vector reversed direction from
can summarize the events as:                              +V to –V, a change of 2V, in a time
Event 1:    X1 = 0              T1 = 0
                                                          interval T. Peter felt the acceleration.
Event 2:    X2 = L              T2 = – L/c
                                                          Therefore, all observers will agree that
Event 3:    X3 = L(1 – V/c)/2   T3 = L(1 – V/c)/2V
                                                          Peter was the traveler and that his
                                                          clocks ran slow, so he ages less than
    These same events can be specified
                                                          his stay-at-home twin.
in the inertial frame (primed) of space-
                                                           Feynman, R. P., R. B. Leighton, and M.
ship A as:                                                  Sands. The Feynman Lectures on Physics.
                                                            Vol. I. Reading, Mass.: Addison-Wesley,
Event 1′:   X1′ = 0             T1′ = 0
                                                            1966, pp. 16-3 to 16-4.
Event 2′:   X2′ = γL(1 + V/c)   T2′ = – γL(1 + V/c)/c
Event 3′:   X3′ = 0             T3′ = γ–1 L(1 – V/c)/2V

                             (1 – 
     We have defined γ = √V 2/c 2
and have used the normal Lorentz
                                                                   Chapter 6
transformations x′ = γ (x – Vt) and t′ =                          Start Me Up
γ (t – Vx/c2) of the STR.
     Now, finally, we can determine the
clock reading—that is, the elapsed                        107. Air-Driven
time—for the observer who sees the                        Automobile Engine
collision a distance L(1 – V/c)/2 away
                                                          Yes. Many companies worldwide have
as T = L(1 – V/c)/2V + L(1 – V/c)/2,
                                                          been operating compressed-air-driven
which reduces to T = L(1 – V2/c2)/2V.
                                                          cars using a standard gasoline four-
The observer on spaceship A has an
                                                          cylinder engine but replacing the gaso-
elapsed time of γ –1 L(1 – V/c)/2V.
                                                          line fuel input with compressed air
 Chai, A.-T. “Some Pitfalls in Special Relativ-
  ity.” American Journal of Physics 41
                                                          from a tank. Of course, there is no
  (1973): 192–195.                                        combustion, so the electrical supply
                                                          for the spark plugs is not needed, nor
                                                          will there be any need to change the oil
106. Twin Paradox                                         very often. The compressed-air tank is
Peter experiences actual accelerations                    stored in the trunk.
during his spaceship journey that will                        The piston upstroke compresses


192         Answers
and heats the atmospheric air in the         the characteristics of randomness do
cylinder chamber until just about top        not contain long runs, you should be
dead center, when cool compressed air        able to distinguish them reliably.
is injected to drive the piston down             The actual estimate of the number
and turn the crankshaft. The process         of runs with 6 or more heads or tails is
repeats itself until the compressed air      4, meaning that you should be able to
is depleted. The exhaust is just cool air.   find about 4 of these long runs. For a
The horsepower rating is about 35            run of at least k heads in n tosses,
horsepower for some models, but the          where k ≥ 1, the mean number of runs
value will increase to more practical        is ~ n/2(k+1); thus 2 (256/27) = 4. The
values with further development.             following table contains actual data
Using traditional electricity sources to     for 256 coin tosses, with a 1 repre-
compress the air, there will be some         senting heads. You can count the num-
carbon dioxide air pollution for the         bers of the different run lengths.
overall process, but only about a fifth
or less that of conventional autos.          10111110101101101011010010001100
    Perhaps the best-known air-pow-          11011110100001001001110010100100
ered car is that designed by French          11001100111110001000001011111000
inventor and engineer Guy Nègre for          10110010001111100110111001110010
Motor Development International              11111000011011100000001011111000
(MDI) in France. The car has a maxi-         11110110110000001010000010111110
mum speed of about 110 kilometers            11111100111011001011100010111110
per hour and can travel about 300            01110110111100001111111000001100
kilometers at a cost of less than a cent
per kilometer. (Details can be found on       Silverman, M. P. A Universe of Atoms, an
                                               Atom in the Universe. New York: Springer-
the Internet.)                                 Verlag, 2002, pp. 284–291.


108. Coin Tosses
                                             109. More Coin Tosses
You should be able to pick out the
experimentally obtained sequences            Most people would expect to return to
with about 98 percent accuracy! In a         the lamppost quite often—20 or more
random sequence of 256 fair coin             times during the 1,000 tosses. How-
tosses, you would expect to find at least     ever, returning more than 2 times is
1 run of 6 heads or 6 tails with a prob-     unlikely! There will be a long drift
ability of 98.2 percent. If the sequences    away from the lamppost for most of
imagined by students unfamiliar with         the coin-tossing time.


                                                                      Answers      193
40                                                      if the switching time t is large enough
30                                                      to assure that the particle can adjust in
20                                                      the trapping minimum (adiabatic
 10                                                     adjustment time) and also is small
 0
      0   200   400      600        800   1000   1200
                                                        enough to fulfill the above require-
-10                                                     ment for the variance. Roughly, one
-20                                                     can say that a net flux to the left
-30                                                     always occurs when thermal energy is
-40                                                     significantly smaller than the potential
                      Toss Number
                                                        maximum, the external force chosen is
     One can do the actual coin tossing                 not too big, and the driving frequency
 to experience the drift away from the                  matches the adiabatic adjustment time
 origin for long time periods, or one                   needed for the particle to move into a
 can run a computer simulation. The                     potential minimum.
 expected distance after N tosses will                       Where does the energy come from
     N
 be √ times the unit step distance,                    leading to a drift against the external
 the random walk distance in one                        force? The energy does not come from
 dimension.                                             the heat bath but from the ratchet
                                                        potential when it is switched on. At
                                                        that moment the potential energy of
 110. Brownian Motor                                    the particle will suddenly be increased.
 As long as the ratchet potential is off,               In a simulation, this can be seen by a
 there can be no net movement to the                    sudden increase of the energy. But
 right or the left because the particles                most of the energy pushed into the sys-
 will move diffusively according to a                   tem will just be dissipated into the heat
 (biased) random walk, leading to a                     bath due to the relaxation of the parti-
                              √
 variance in position of δx = (2Dτ) and                cle into a potential minimum. Only a
 a mean position of <x> = f τ/γ, where                  tiny portion will be used for doing
 D = kT/γ is the diffusion constant.                    work. Thus a Brownian motor does
 When the ratchet potential is switched                 not violate any law of thermodynam-
 on, one or more particles get trapped                  ics because it only turns one type of
 in one of the potential minima. If αL ≥                work into another one. Nevertheless,
 δx ≥ (1 – α)L for the variance holds,                  the fluctuating force due to the heat
 the particle on average gets trapped                   bath is essential for a Brownian motor.
 into the minimum left of the starting                       For more details and possible
 point. The maximum flux is obtained                     applications in biology and chemistry


 194       Answers
read the following review articles. For        which probably would be costly. The
a simulation, there are Java applets on        alternative would be to have a small
the Internet.                                  closed loop of ferrofluid in contact
 Astumian, R. D. “Thermodynamics and           with a large loop of piping containing
  Kinetics of a Brownian Motor.” Science 276   the water to be heated in a heat
  (1997): 917 –922.
                                               exchange device. The advantage over
 ———. “Making Molecules into Motors.”
  Scientific American 285, no. 1 (2001):       typical systems would be no moving
  57–64.                                       mechanical parts in the solar heating
                                               system.
111. Magnetocaloric                             Rosensweig, R. E. “Magnetic Fluids.”
                                                 Scientific American 247, no. 4 (1982):
Engine                                           136–145.

The ferrofluid is cycled around the
loop by the stationary permanent
magnet. A small volume of ferromag-            112. Magnetorheological
netic material has less energy where           Fluid
the magnetic field density is greater,
                                               The flow properties of the fluid change
just like iron filings are pulled to the
                                               so radically that the fluid becomes gel-
poles of a magnet. So the ferrofluid
                                               like and can be pushed to one side of
approaching the magnet becomes
                                               the beaker where no relaxation may
magnetized and drawn into the loop
                                               occur. The degree of solidification
volume between the magnetic poles.
                                               depends on the inherent properties of
But the heat source nearby warms the
                                               the fluid and the strength of the mag-
ferrofluid to partially randomize the
                                               netic field. Of course, its solidification
magnetic dipoles in the ferrofluid, so
                                               may vary within the gel itself because
the energy of the system can be low-
                                               the magnetic field may vary with posi-
ered again by drawing in some more of
                                               tion in the beaker. Practical applica-
the cooler magnetized ferrofluid,
                                               tions of these materials with their
which pushes out the warmed fer-
                                               unusual properties are being devised
rofluid. The heat put in by the heat
                                               and tested. Perhaps automobile brak-
source is deposited at the heat sink,
                                               ing systems may someday use these
and the cycle repeats.
                                               types of fluids to replace solid materi-
    To use this engine for the solar
                                               als that wear away.
heating of buildings, two avenues of
                                                Klingenberg, D. J. “Making Fluids into
operation are possible. One could                Solids with Magnets.” Scientific American
have all the piping contain ferrofluid,           269, no. 4 (1993): 112–113.



                                                                       Answers      195
113. Binary Fluids                         low T, changing the entropy has a min-
                                           imal effect because the product TS
Both phase diagrams can represent          may be small. But at high T, the prod-
actual binary fluids, although the dia-     uct can be large. So systems at high T
gram to the right is quite rare. To        tend to maximize their entropy, that
understand these phase diagrams, both      is, their randomness or disorder.
energy and entropy must be consid-              We now have a good argument for
ered. The energy part involves the van     ruling out the diagram to the right,
der Waals interaction between adjacent     with its reappearing miscible phase at
molecules, an induced dipole-dipole        low temperatures. Not so! For some
electromagnetic interaction. In general,   molecules, hydrogen bonding occurs
this attractive force between unlike       with its very small angular spread,
molecules is much weaker than the          locking two molecules together. This
attractive force between like molecules.   hydrogen bonding occurs primarily
The stronger the force of attraction       at lower temperatures because of
holding the molecules together, the        the orientation dependence, with
lower the energy of the system. Hence,     “orientation” entropy lost in forming
when most of the molecule’s neighbors      the hydrogen bond being greater than
are of the same chemical species, the      the “compositional” entropy gained.
system energy is lowest and immiscibil-    Therefore both energy and entropy are
ity is favored. Even the increased ran-    lowered, and the lowered energy of the
dom tumbling about of the molecules        hydrogen bond has a large effect on
at higher temperatures doesn’t disrupt     the free energy. Water and butyl alco-
this clustering of like molecules.         hol is one example of a binary liquid
    However, energy considerations         with the rare phase diagram.
alone do not explain the behavior of        Walker, J. S., and C. A. Vause. “Lattice
binary liquids. Why are they miscible        Theory of Binary Fluid Mixtures: Phase
at all? The miscibility occurs at lower      Diagrams with Upper and Lower Critical
                                             Solution Points from a Renormalization-
temperatures because the system tends        Group Calculation.” Journal of Chemical
to minimize not its energy but rather        Physics 15 (1983): 2660–2676.
its free energy, Efree = Esys – TS. The     ———. “Reappearing Phases.” Scientific
free energy is the energy of the system      American 256, no. 5 (1987): 98–106.
minus the product of the temperature
T and the system’s entropy S. At a
given T, the free energy can be            114. Baseball Bats
decreased by decreasing the system’s       The main source of drag on the swing
energy or by increasing its entropy. At    of a baseball bat is not air friction but

196     Answers
   the retarding force produced by the            115. Old Glass
   pressure difference across the bat from
   front to back. As the bat carves its           Many people have suggested that the
   swath, the air in front gets separated         glass experiences some flow downward
   into two boundary layers that pass             in response to the gravitational pull of
   around the bat and recombine behind            Earth. Contrary to popular conjecture,
   the bat. In the wake of the bat,               there is no evidence that any of this old
   between the two separated boundary             glass could flow enough during the
   layers, the “lack of air” means a lower        time interval of centuries to create the
   pressure immediately in back of the            difference from top to bottom.
   bat, with a resulting backward force               Another factor against the flow
   due to the pressure difference. There-         hypothesis is the actual profile, which
   fore, some of the energy of the swing          is essentially a linear relationship of
   does work against this backward                thickness to vertical distance. As a
   force.                                         simple model, assume that the proper-
                                                  ties of the glass are identical at each
                                                  vertical position along the pane. If a
              Boundary layer         Direction    fixed amount of glass material flows
                                     of bat       from position 10, say, the same
                                                  amount would replace this amount
                                                  from position 11, slightly higher up
                Bat                               the glass. The major changes over a
                                                  long time interval would be a thick
                                        Airflow
                                                  buildup at the very bottom and a
Low-pressure High-pressure
                                                  depletion at the very top, with practi-
region       region
                                                  cally no thickness change between, in
                                                  contrast to the linear dependence of
        A dimpled bat sends the boundary          glass thickness to height.
   layers tumbling in turbulent eddies                In the old days, window glass pro-
   into the space behind the bat, reducing        duction made panes that varied
   the pressure difference and cutting the        slightly in thickness from one end to
   drag. More swing energy is now avail-          the other because the flat support sur-
   able to accelerate the bat and to trans-       face had a slight tilt. The installers
   fer to the ball, so the ball’s exit velocity   simply put the thicker end on the bot-
   will be increased.                             tom. Quality control must have been
    Gibbs, W. W. “To Fenway, with Love.”          marginal in some areas of the world,
     Scientific American 271, no. 1 (1994): 98.    because we have seen some large

                                                                          Answers      197
differences in glass thickness between       substances the dipoles are far enough
the two ends!                                apart to behave approximately inde-
    Glass is normally elastic at temper-     pendently, and when no applied field is
atures below about 1000 K, and glass         present, these dipoles have random
may break but never deform perma-            orientations. Each dipole is affected by
nently because the solid is crystalline.     the applied magnetic field but not by
Delicate telescope and camera lenses         its neighbors. The applied magnetic
would reveal such creep by changing          field competes with the random ther-
their optical characteristics in obvious     mal motion to cause a net magnetiza-
ways.                                        tion that increases nearly linearly with
 Pasachoff, J. M. “Comment on ‘Magnetic      the strength of the applied field, the
  Fluids.’” American Journal of Physics 66   ratio being known as the magnetic
  (1998): 1021.
                                             susceptibility.
 Zanotto, E. D. “Do Cathedral Glasses
  Flow?” American Journal of Physics 66           When the density of magnetic
  (1998): 392–395.                           dipoles becomes high enough for
                                             neighbors to affect each other, only
                                             neighbors in the head-to-tail configu-
116. Ferromagnetism                          ration will tend to align one another.
                                             The side-by-side neighbors will be
                                             oppositely aligned because all the
                                             fields from its neighbors are opposite
                                             at its location. Thus every other dipole
                                             in each layer of the crystal will be
                                             aligned and form one sublattice, like
                                             the white squares on a checkerboard,
                                             and the remaining dipoles (on the
                                             black squares) will form a second sub-
                                             lattice of dipoles pointing in the oppo-
Many atoms and molecules have an             site direction. The two sublattices
inherent magnetic dipole moment.             interact strongly or ferromagnetically,
When we assume that each dipole              but they cancel each other’s magneti-
behaves independently of its neighbors       zation. Therefore, when magnetic
except for its alignment, the magnetic       dipole moments are crowded together,
field direction next to the dipole is        they are more likely to disalign their
opposite to the direction in which the       nearest neighbors than to align them.
dipole itself points. In paramagnetic        So ferromagnetism is rare.


198     Answers
    Then how can ferromagnetic sub-           that accounts for the change in angular
stances exist at all? Via a cooperative       momentum.
effect when the dipoles are very close            If the pulleys are the same size, this
and no longer behave independently.           additional torque does not exist unless
In these conditions a state of lower          the belts are crossed.
energy can form if groups of dipoles
align each other into magnetic
domains that themselves point in ran-
                                              118. Superconductor
dom directions. With an applied field,         Suspension
these domains will change their sizes         The demonstrated superconductor
to find the lowest energy state. Of           suspension does not illustrate the
course, domain formation cannot               Meissner effect. Instead, this demon-
form above a certain temperature              stration depends on the persistent
called the Curie temperature, because         eddy current in the zero resistivity
the thermal agitation interferes with         superconducting material induced by
the dipole interactions. Above the            the magnet. The eddy current direc-
Curie temperature the substance               tion is determined by Lenz’s law to
becomes paramagnetic.                         produce a magnetic field that ends up
 Kolm, H. H. “Why Are So Few Substances       causing a repulsion between the super-
  Ferromagnetic?” The Physics Teacher 20
  (1982): 183–185.                            conductor and the permanent magnet.
                                                  To show the Meissner effect, the
                                              sequence of events must be different.
117. Coupled Flywheels                        Place the superconductor on the mag-
                                              net at room temperature first, and then
The overall angular momentum of the           cool the superconductor below its crit-
system must be conserved, so including        ical temperature Tc. Then the magnetic
just the change in angular momenta of         flux will be “expelled” by the Meiss-
the flywheels leads to an incomplete cal-      ner effect and the superconductor will
culation. The tension is different in the     become suspended above the magnet.
two sides of the belt, so the belt exerts a    Wake, M. “Floating Magnet Demonstration.”
downward force on pulley 2 and an               The Physics Teacher 28 (1990): 395–397.
upward force on pulley 1. These forces
are counteracted by reactions at the
bearings, in addition to the reactions to
                                              119. Nanophase Copper
the weight of the components. These           With smaller grain sizes, one would
additional reactions produce a torque         expect there to be many more grain


                                                                      Answers      199
boundaries in the nanophase copper              fraction of the charge of a single elec-
metal than in normal copper. The                tron. This “transferred charge” is pro-
extra grain boundaries would stop or            portional to the sum of the shifts of all
impede any moving dislocation,                  the electrons with respect to the lattice
thereby making the nanophase copper             of nuclei. These electrons in the con-
much harder. However, the surprise              ductor can be shifted as little or as
turned out to be that nanophase cop-            much as desired, so the sum can
per is mostly dislocation-free! Lacking         change continuously, and therefore so
large numbers of moving dislocations,           can the “transferred charge.” The pin-
these nanophase metals are much                 head can have any charge value, not
stronger.                                       just integer multiples of the fundamen-
 Siegel, R. W. “Creating Nanophase Materi-      tal charge.
  als.” Scientific American 275, no. 6 (1996):    Likharev, K. K., and T. Claeson. “Single Elec-
  74–79.                                          tronics.” Scientific American 266, no. 6
                                                  (1992): 80–85.


120. Head of a Pin
Experiments show surprising results.
                                                121. Coulomb Blockade
Any fraction of the fundamental                 No, the current across the junction
charge, such as +0.5 e or –0.1 e, can           will not be a steady current. There will
exist on the head of a pin! The con-            be single electron tunneling (SET),
ceptual argument goes as follows. The           with the voltage across the junction
metal pin is an electrical conductor. In        changing periodically with a frequency
general, an electric current flows in the        equal to the current divided by the
conductor because some free electrons           fundamental unit of charge e.
move through the lattice of atomic                  The tunnel junction is a conductor-
nuclei. Any particular volume of the            insulator-conductor device, so trans-
conductor has virtually no charge               ferred charge flows through the
because the negative charges are bal-           conductor to accumulate on the sur-
anced by the positive charges of the            face of the electrode against the insu-
nuclei.                                         lating layer of the junction. An
    So the important physical quantity          opposite surface charge of equal
is not the electric charge in any given         amount accumulates on the other elec-
volume but instead how much charge              trode across the junction. The actual
has been carried through the conduc-            amount of surface charge has a con-
tor—that is, the “transferred charge,”          tinuous change in value as the charge
which can have any value, even a                accumulates, including fractional

200       Answers
values such as +0.8642 e, because the      electrode is changed by only half the
electrons near this surface can adjust     charge of an electron!
their positions slightly.                   Likharev, K. K., and T. Claeson. “Single Elec-
    However, only discrete amounts of        tronics.” Scientific American 266, no. 6
                                             (1992): 80–85.
charge can tunnel through the insulat-
ing layer—that is, each electron tun-
neling through changes the surface         122. Deterministic
charge by +e or –e, depending on the       Competition
direction of tunneling. The tunneling
process is energy-dependent. If the        The time evolution here depends on
charge at the junction is greater than     the value of r. One finds that Nt = 1 is
+e/2, an electron can tunnel through to    a stable equilibrium only when r lies
reduce the surface charge by e, thus       between 0 and 2. If r = 2.3 with N0 =
reducing the electrostatic energy of the   0.5, then successive Nt will oscillate
system. And if the surface charge is       between about 1.59 and about 0.40 as
less than –e/2, an electron can tunnel     a stable 2 cycle. For r > 3.102, no cycle
in the opposite direction to decrease      is stable, all cycles are possible, etc.
the energy. But if the surface charge           In the chaotic regimes, the equa-
value is greater than –e/2 or less than    tion results are deterministic, but the
+e/2, tunneling would not occur            time evolution is indistinguishable
because the system energy would            from that governed by probability
increase. This tunneling suppression is    laws. One really needs to see the cal-
known as the Coulomb blockade, first        culations proceed to appreciate the
studied in the 1950s.                      amazing behavior of this simple-look-
    The tunnel junction connected to a     ing equation.
constant current source begins in the       Gleick, J. Chaos: Making a New Science.
                                             New York: Penguin, 1987, pp. 166–186.
Coulomb blockade condition, then
reaches tunneling for the one-electron
condition, then back to the Coulomb        123. Two Identical
blockade, then one-electron tunneling,
etc. The analogue may be a dripping
                                           Chaotic Systems
faucet.                                    Yes, the two identical chaotic systems
    Many electronic devices are being      described can be synchronized.
made with SET operation. For exam-         Chaotic systems are very useful for sev-
ple, an SET transistor can switch on or    eral reasons: (1) Chaotic systems are a
off the flow of billions of electrons per   collection of many regular, ordinary
second when the charge on the middle       behaviors, none of which dominate.

                                                                      Answers       201
(2) The proper perturbation can               the rate of change of magnetic flux Φ.
encourage the chaotic system to follow        But work must be done for V to be
one of its many regular behaviors. (3)        generated because the change in the
Chaotic systems are very flexible             work dWork = V dt.
because they can rapidly switch among          Nussbaum, A. “Faraday’s Law Paradoxes.”
different behaviors. (4) Chaotic sys-           Physics Education 7 (May 1972): 231–232.
tems are deterministic and, although
no one can say which output will              125. Thermal Energy
result, two identical chaotic systems of
the appropriate type will produce the
                                              Flow
same output in response to the same           The classical flow of thermal energy
signal input.                                 toward the cooler region occurs
     To synchronize two identical             because the free energy of the com-
chaotic systems each with the stable          bined system Efree = Esys – TS becomes
subpart behavior, one can apply the           less, where T is the temperature and S
appropriate pseudoperiodic signal             is the entropy. If the free energy is the
(one type is called a Rössler signal) to      same at two temperatures, one can see
coax them into step. For the reasons          that for a given amount of system
listed above, the outputs will be the         energy there is more disorder at the
same. The details can be learned in the       lower temperature. Assuming that the
reference below, where the chaotic            two-block system initially simply
attractor and the Poincaré section are        transfers thermal energy from the
discussed. Applications to secure com-        warmer to the cooler block, with no
munications and to biological systems         other energy transfers, then a cooler
are included also.                            system is preferred.
 Ditto, W. L., and L. M. Pecora. “Mastering    Dyson, F. J. “What Is Heat?” Scientific
  Chaos.” Scientific American 269, no. 2        American 191, no. 3 (1954): 58–64.
  (1993): 78–84.

                                              126. Cadmium Selenide
124. Tilley’s Circuit                         The wavelength of visible light is com-
The galvanometer does nothing! There          parable to nanophase cluster sizes. For
is no induced potential because no            example, greenish light has a wave-
work was done (assuming frictionless          length of about 580 nanometers, five
switches). This result appears to vio-        to ten times the nanophase cluster
late Faraday’s law V = d Φ/dt, where V        sizes. Clusters behaving as particles
is the potential difference induced by        ranging from about 1 nanometer to 50

202       Answers
nanometers in diameter are too small        These pulses conserving their shape
to have any significant scattering of vis-   and integrity are exhibiting soliton
ible light, so these materials are effec-   behavior. Optical solitons were first
tively transparent. Clusters of sizes       observed in fibers in 1980 and are now
comparable to particular wavelength         fundamental components in optical
ranges of visible light are subject to      transmission systems.
quantum confinement restrictions.             Desurvire, E. “The Golden Age of Optical
    Quantum mechanics predicts the            Fiber Amplifiers.” Physics Today 47 (1994):
                                              20–27.
correct behavior at the small cluster
sizes. The smaller the nanophase clus-
ter size becomes, the greater are the       128. Ceramic Light
energy spacings for the electron states.    Response
Which colors of light are absorbed and
emitted are determined by these energy      Certain ceramic materials will change
spacings. If the energy spacings are too    their shape upon exposure to light
great, the incoming light will not be       because some molecules in the mate-
absorbed, and light of that wavelength      rial have changed their shape upon
and longer will not be scattered. For       absorption of particular frequencies of
example, a typical semiconductor is         light. If the responses of many mole-
cadmium selenide. When the size of the      cules are coordinated, the overall
cluster is 1.5 nanometers, the cadmium      effect can be a macroscopic shape
selenide appears yellow, but when the       change. Called the photostrictive
size is 4 nanometers, it will appear red.   effect, research began in the 1990s,
And larger clusters appear black.           and some practical devices are begin-
Therefore, the observed color of the        ning to be developed, such as direct
clusters in the nanophase depends on        conversion of light to mechanical dis-
their actual sizes.                         placement for speakers instead of con-
                                            version to an electrical signal first. A
                                            telephone speaker could be one of the
127. Optical Solitons                       first products.
Under the right conditions, the two             These ceramics are examples of a
effects—dispersion and the Kerr effect      new type of “smart” material. The
—can be made to cancel exactly. The         four most widely used classes of smart
nonlinearity of the Kerr effect can         materials are piezoelectrics, electro-
delay the “fast” carriers relative to       strictors, magnetostrictors, and shape-
the “slow” carriers, bringing them          memory alloys. The resulting changes
together to counter the dispersion.         in the shapes of these materials are

                                                                    Answers      203
large enough to make them useful as             more of them there are. But the typical
actuators. A sensor receives a stimulus         human reaction time for such balanc-
and responds with a signal; an actua-           ing acts is about 100 milliseconds, so
tor produces a useful motion or                 most of the wobbles are faster than
action. By definition, smart materials           humans can react. Mathematical mod-
are both sensors and actuators,                 eling of human balancing acts match
because they perform both functions.            the measured fluctuations only when
    Photostrictive materials such as            the person or object is on the verge of
PLZT—a combination of lead, lan-                falling. Then the random fluctuations
thanum, zirconium, and titanium—                cancel each other out and the object
someday may be used to control                  remains upright.
robots and machines. Engineers at                    Related research has found that
Pennsylvania State University, for              elderly people and others with balance
example, are exploring applications             problems showed signs of better bal-
for devices that move when light                ance when they stood on a pair of bat-
shines on them and have created a               tery-operated, randomly vibrating
two-legged stand that walks very                insoles. The idea is that these vibra-
slowly when illuminated.                        tions amplify balance-related signals
 Dogan, A., et al. “Photostriction of Sol–Gel   from the feet to the brain and vice
  Processed PLZT Ceramics.” Journal of          versa that may have become reduced
  Electroceramics. 1, no. 1 (1997): 105–111.
                                                by age or illness. When people walk,
 Newnham, R. E., and A. Amin. “Smart Sys-
  tems: Microphones, Fish Farming, and          then turn or reach, they are most vul-
  Beyond.” ChemTech 29, no. 12 (1999):          nerable to a fall. When a person leans
  38–46.
                                                or sways to one side, the pressure on
                                                the sole of that side increases, and the
                                                nervous system senses the change in
129. Random                                     pressure and sends a message to the
                                                brain so that the posture can be
Movements
                                                adjusted. In many people, those mes-
Wobbles in any system can be fol-               sages can be altered by age, stroke, or
lowed with fast cameras. For most               conditions such as diabetes. Further
human actions, from balancing a stick           testing is under way to optimize these
vertically on a finger to balancing on a         helpful insoles.
tightrope, wobbles occur that last
                                                 Cabrera, J. L., and J. G. Milton. “On-Off
from seconds to tens of milliseconds.             Intermittency in a Human Balancing Task.”
Usually the shorter the fluctuation, the           Physical Review Letters 89 (2002): 158702.



204       Answers
 Chow, C. C., and J. J. Collins. “Pinned          Then the Carnot engine efficiency
  Polymer Model of Posture Control.”
  Physical Review E 52 (1995): 907–912.
                                                  η = (Qin – Qout)/Qin.
 Priplata, A. A., et al. “Vibrating Insoles and
                                                      If the bath atoms are assumed to be
  Balance Control in Elderly People.” Lancet      two-state systems that absorb and emit
  362, no. 9390 (2003): 1123–1124.                radiation at the same photon
                                                  frequency, then we need the thermo-
130. Gravitational Twins                          dynamic properties of a photon gas in
                                                  order to determine the theoretical effi-
The traveling twin actually returns               ciency of this photon engine. Assuming
much younger than her stay-at-home                thermal equilibrium for the photon
sister. The argument given was cor-               gas, the average number of photons n2
rectly stated but incomplete. The local           with energy ε coming in from the heat
gravitational tidal effects are not the           bath at temperature T2 is given by n2 =
same for the twins—that is, the rate of           1/(exp[ε/kT2] – 1), while the average
change of gravitational potential expe-           number of photons n1 leaving at tem-
rienced was different. These tidal                perature T1 is n1 = 1/(exp[ε/kT1] – 1).
effects contribute to the clock rates             Since Qin ∝ n2 ε and Qout ∝ n1 ε, the
and, when included in the calcula-                efficiency of the quantum Carnot
tions, contribute enough to change the            engine is η = 1 – T1/T2, exactly the
result so that the stay-at-home twin              same as for the classical Carnot engine.
ages faster and is older upon return of           When there is only one heat bath, with
her sister. For a calculation, see the ref-       T1 = T2, no work can be done.
erence below.                                         A different quantum engine occurs
 Bradley, M., and J. Higbie. Physics Teacher      when the bath atoms have three states
  22, no. 1 (1984) 34–35.
                                                  instead of two, bringing in quantum
                                                  behavior called quantum coherence,
                                                  with a nonvanishing phase difference
131. Photon Engine                                between the two lowest atomic states
We can analyze the operation of the               induced by a microwave field. One can
quantum Carnot engine in the same                 eliminate the photon absorption
manner in which we would analyze a                process (analogous to laser operation
classical Carnot engine. Let Qin be the           without a population inversion). The
energy absorbed from the bath atoms               temperature T2 becomes altered to a
during the isothermal expansion and               different effective temperature, Tφ. The
Qout be the energy given to the heat sink         efficiency ηφ = (Tφ – T1)/T1 can exceed
during the isothermal compression.                the efficiency of the classical Carnot


                                                                        Answers     205
engine. This quantum engine can
                                                       Original       Copy
extract work from a single heat bath,
even when T1 = T2! For the details of
the three-state quantum engine’s oper-
ation, see the reference below.
 Scully, M. O., et al. “Extracting Work from a
  Single Heat Bath via Vanishing Quantum
  Coherence.” Science 299 (2003): 862–864.


                                                    400                          600
                                                            Wavelength (nm)
     Chapter 7                                        Different atoms absorb and emit
 A Whole New World                               their unique characteristic frequencies
                                                 of light in the visible and the ultravio-
                                                 let. The types of atoms present and the
132. Grain of Sand                               intensity of the characteristic spectrum
If one assumes that the grain of sand            from each atom type will create a
has a diameter that is a reasonable              “spectral fingerprint” for each artist.
fraction of 1 millimeter, then the line          As you know, some artists simply laid
of atoms would be about 1010 meters              out the design of the painting, for
long, about thirty times the distance to         example, and lesser painters filled in
the Moon!                                        the regions, with the master artist
                                                 completing the final touches. Even
                                                 these paintings have their own finger-
133. Forensics                                   print of spectral colors.
Until the mass production of paints                   With a tunable laser capable of
became available in the late 1800s and           scanning from the infrared frequencies
early 1900s, each paint used by an               to the ultraviolet frequencies, the
artist is known to contain atoms in              “spectral fingerprint” of any region of
particular characteristic amounts,               the painting can be recorded and
depending on the source. Paints were             compared to other paintings by the
originally made from natural materi-             same artist or even other artists,
als, so when an artist mixed his or her          including fraudulent painters. This
paints, there was usually a unique               laser approach is normally combined
mixture of atoms and molecules for               with other approaches to achieve the
each color and color combination.                comprehensive evaluation.


206       Answers
    The laser technique also permits       possible energies for the two photons
the identification and removal of envi-     emitted in this quadrupole emission
ronmental coatings on top of the paint     process. A very small fraction of these
beneath, such as dust and grime, and       two-photon emissions will spit out
ensures that no harm to the painting       two photons of the same energy, go off
occurs. Famous paintings such as           in opposite directions, and produce no
Rembrandt’s 1642 De Nachtwacht             recoil of the atom. The two-photon
(The Nightwatch) in Amsterdam’s            emission from hydrogen was the first
Rijksmuseum have had the soot and          atom to be measured and the first to be
grime safely cleaned off to reveal a       calculated by quantum electrodynamics
marvelously brighter background of         (QED) in the 1940s. Two-photon
faces when compared to the somewhat        emissions after laser excitations have
obscure dull background that had           become commonplace for many uses in
existed for centuries.                     today’s optics research.
                                               Likewise, simultaneous two-photon
                                           absorption is possible. A container of
134. Doppler
                                           single atoms is placed between two
Elimination?                               counterpropagating laser sources,
Yes. First consider the emission           shining two identical frequency laser
process. Normally, a typical electric      beams on an atom so that energy and
dipole emission occurs with a single       angular momentum will be conserved
photon exiting the atom as a result of     and recoilless absorption can occur.
an allowed transition within the atom      First achieved in the 1970s, the precise
that conserves energy and angular          energy-spacing values within atoms
momentum—that is, the angular              have been determined. Today, two-
momentum of the atom changes by ±1         photon absorption with nonidentical
unit of Planck’s constant h/2π. The        energies plays a critical role in the
probability for all other emission         upconversion of laser light to higher
processes is lower by a factor of 1/137,   frequencies to achieve coherent beams
or by a higher power of this factor.       in the UV and for providing light
    A two-photon electric quadrupole       sources of precise frequencies.
emission process is possible between           At the nuclear level, recoilless
two atomic states with angular             gamma-ray emission and absorption
momentum quantum numbers differ-           are possible if the whole crystal recoils
ing by zero or two units of h/2π. There    simultaneously with the photon emis-
is a broad continuous spectrum of          sion or absorption. This Mössbauer


                                                                  Answers      207
Effect transition, discovered in the           greater than the wavelength of the inci-
1950s, relies on the inability in princi-      dent light but smaller in size than the
ple of identifying the single nucleus          diameter of the incident light beam. Let
involved and includes an exponential           the light source be a parallel beam of
factor proportional to the negative            light rays all of the same frequency,
ratio of the temperature of the crystal        such as in a laser beam focused to the
to its Debye temperature.                      point f by a symmetrical lens. The
     As an interesting historical note,        object tends to focus the light rays
Albert Einstein in 1917 was among the          somewhat, changing the direction of
first to recognize that classical electro-      the light rays. The sideways recoil of
magnetism cannot explain sponta-               the object occurs to simply conserve the
neous emission of light from atoms. In         linear momentum. If the light beam has
particular, he inferred that an atom           an intensity gradient, brighter in the
must recoil upon spontaneous emis-             center than near the edge, the object
sion, in conflict with the symmetric-          will receive a net push back toward the
field distributions produced by                optical axis in the center. There must
electromagnetic theory based on                also be a recoil of the object in the
Maxwell’s equations. According to              direction of the original light beam,
Einstein, “. . . outgoing radiation in the     which usually is taken up by the appa-
form of spherical waves does not exist         ratus and Earth because the object is on
. . .” for if an atom radiated a classical     a horizontal platform. A one-celled
spherical wave it could not recoil.            paramecium remains well trapped in a
 Einstein, A. “Zur Quantentheorie der          microscope via this light tweezer tech-
  Strahlung.” Physika Zeitschrift 18 (1917):   nique, begun at Bell Labs in the 1970s.
  121–128.
                                                   When the object is smaller than the
                                               wavelength of the incident light, a
135. Light Tweezer                             more detailed analysis is required to
Yes. A focused laser beam can exert a          understand the 3-D trapping and the
trapping force perpendicular to the            quantum interference effects.
beam direction of 2 × 10–12 Newtons                Optical tweezers have been widely
or more to keep cells confined in a            used for several decades in applications
microscope at the optical axis. The            as diverse as experiments on molecular
intensity gradient across the light            motors in biology and the movement of
beam is the source of the force.               Bose-Einstein condensates in physics.
    In the simplest geometry, consider a       The capabilities of single optical tweez-
semitransparent object with a diameter         ers have been greatly improved and


208       Answers
extended by the development of                This powder absorbs the UV light and
tailored beams and by schemes for gen-        fluoresces in the visible. Very little
erating large numbers of trapping sites       heating of the fluorescent lamp occurs,
and shapes simultaneously.                    so the efficiency occurs before the pro-
 Block, S. M. “Making Light Work with Opti-   duction of the visible light, with very
  cal Tweezers.” Nature 360 (1992):           little electrical energy being converted
  493–495.
                                              into thermal energy. The conversion
 Chu, S. “Light Trapping of Neutral Parti-
  cles.” Scientific American 266, no. 2       process in the powder makes the tube
  (1992): 70–76.                              useful for room lighting.
 MacDonald, M. P., et al. “Creation and            So why is the incandescent lamp so
  Manipulation of Three-Dimensional Opti-
                                              inefficient, converting only about 4
  cally Trapped Structures.” Science 296
  (2002): 1101–1103.                          percent to 12 percent of the electrical
 Ulanowski, Z., and I. K. Ludlow. “Compact    energy to visible light? The incandes-
  Optical Trapping Using a Diode Laser.”      cent lamp is simply a resistor whose
  Measurement Science and Technology 11
  (2000): 1778–1785.
                                              filament temperature rises until it gets
                                              rid of thermal energy at the same rate
                                              that thermal energy is being generated
136. Fluorescent Lights                       in the filament. In a standard 100-
Today, artificial illumination requires        watt, 120-volt bulb, the filament tem-
more than 25 percent of the electricity       perature is roughly 2550°C, about
generated worldwide. There are two            4600°F, so that the thermal radiation
trends in “energy saving” technolo-           from the filament includes a significant
gies. The first trend is using improved        amount of visible light.
lamps, such as fluorescent, mercury,               The output is 17.5 lumens per
sodium, metal halide, and halogen             watt, compared to a maximum of 240
lamps. The second trend is improving          lumens per watt if all the energy could
the electronic circuit design for such        be converted to visible light. The rea-
lamps.                                        son for this poor efficiency is the fact
    Although fluorescent lights are           that tungsten filaments radiate mostly
four to six times more efficient than          infrared radiation at any temperature
incandescent lamps, there now exist           that they can withstand. An ideal ther-
many other types of light sources that        mal radiator produces visible light
are even more efficient. For the fluo-          most efficiently at temperatures of
rescent lamp, its efficient production         about 6300°C (about 6600 K or
of the UV is extended into the visible        11,500°F). Even at this high tempera-
by a powder coating inside the tube.          ture, a lot of the radiation is either


                                                                    Answers     209
    infrared or ultraviolet, and the theo-       vitamin D production in the skin can
    retical luminous efficiency is 95            lead to rickets and other problems
    lumens per watt.                             associated with the calcium and inor-
        Most fluorescent lights predomi-         ganic phosphate metabolism. Eskimos
    nantly emit light in the visible part of     and other indigenous peoples obtain
    the spectrum and they do emit some           plenty of vitamin D from the fish oils
    UV light, but only in a narrow range         in their diets.
    of the UV spectrum. Unfortunately,            Porter, J. P., ed. How Things Work in Your
    their UV emission range does not over-         Home (and What to Do When They Don’t).
                                                   New York: Henry Holt, 1985, p. 158.
    lap the two small ranges of UV light
    needed by humans for the best func-
    tioning of certain internal organs,
    which receive some of the UV light           137. Phase Conjugation
    that passes through the skin, as well as     Mirror
    vitamin D production from 7-dehy-
                                                 Yes, the light can return undisturbed if
    drocholesterol in the skin.
                                                 the light wave retraces it original path
                                                 as its time-reversed twin and the
Intensity                                        medium retains its previous integrity.
                                                 The phase conjugate of a wave pos-
                                                 sesses exactly the same spatial proper-
                                                 ties as the original wave, but it is said
                                                 to be reversed in time. This means that
                                                 a phase conjugate wave exactly
                   Sunlight                      retraces the path of the original beam.
                   Standard Fluorescence         This method has the useful property
                   Special Fluorescence
                                                 that if a light beam propagates
    400       Wavelength (nm)              700
                                                 through a distorting medium, then the
                                                 phase conjugate is produced and
        Special fluorescent lights more          exactly retraces the path through the
    benevolent to human needs are avail-         distorting medium, enabling the unfa-
    able and mimic sunlight to produce a         vorable effects of the distorting media
    UV spectrum better matching the              to be reduced or eliminated. Phase
    needs of these internal organs. Indeed,      conjugation is the general term for a
    the lack of the required UV parts of         process in which both the direction of
    the ambient light spectrum can lead to       propagation and the overall phase fac-
    certain illnesses. Of course, the lack of    tor of a wave function are reversed.


    210     Answers
Phase conjugate mirror                         states that twice the kinetic energy
                                               plus the potential energy add to
                                               zero, so mv2 = ke2 r, from which the
                                               electron’s frequency of orbit is f =
                                               n3h3/(4π2me4). The actual Bohr energy
Conventional mirror                            E = –2π2me4/(n2h2) is clearly a differ-
                                               ent quantity, and for an electron jump
                                               between two energy states, E2 – E1 ≠
                                               hf2 – hf1.
                                                Spielberg, N., and B. D. Anderson. Seven
                                                 Ideas That Shook the Universe, 2nd ed.
                                                 New York: John Wiley & Sons, 1995, p.
                                                 248.
    Some laser sources come with opti-
cal phaser conjugators to remove dis-
tortion in the laser beam. Optical             139. Angular Momentum
phase conjugation occurs also when             We take the space quantization of
there are four waves mixing with all           angular momentum as given, so there
four waves of the same frequency.              will be (2j + 1) components in the
Another useful application of a phase          z-direction from j to –j, decreasing by
conjugate mirror might be to put one           an integer each step. Since there is no
in one reflecting path of an interfer-          preferred direction, J2 = Jx2 + Jy2 + Jz2,
ometer as a reference for detecting            that is, J2 = 3 < Jz 2>avg , where avg rep-
changes in the other path.                     resents the average value given by [ j 2 +
 Blaauboer, M., D. Lenstra, and A. Lodder.     (j – 1)2 + . . . + (–j + 1)2 + (–j)2] h2/(4π2
  “Giant Phase-Conjugate Reflection with a
                                               [2j + 1]). Using a math table or finding
  Normal Mirror in Front of an Optical
  Phase-Conjugator.” Superlattices and         the sum of the series of squared inte-
  Microstructures 23 (1998): 937.              gers directly, one can verify that J 2 =
 Brignon, A., and J.-P. Huignard. Phase Con-   j (j + 1) h2/4π.
  jugate Laser Optics. New York: John Wiley
  & Sons, 2004, chap. 1.                        Feynman, R. P., R. B. Leighton, and M.
                                                 Sands. The Feynman Lectures on Physics.
                                                 Vol. II, Reading, Mass.: Addison-Wesley,
                                                 1965, p. 34-11.
138. Stationary States
In the Bohr model of the hydrogen
atom, one would calculate the fre-
                                               140. Kinetic Laser
quency f = 2πr/v of the electron’s             The explosion of the lasing material
orbital motion. The virial theorem             creates many free electrons, some of


                                                                          Answers       211
which have been blown out of low-          and that could be used for applications
lying atomic states, creating the          such as biological imaging, nonlinear
needed population inversion for possi-     optics, holography, and so on, a pre-
ble lasing action. Practically any mate-   pulse technique has been developed.
rial can be used. During an extremely      This technique has been used success-
short time interval after the explo-       fully to produce lasing in many lower-
sion—on the order of nanoseconds—          Z neonlike ions such as titanium (Z =
stimulated emission may occur as           22), chromium (Z = 24), iron (Z = 26),
photons from the exploding material        and nickel (Z = 28). The use of this
exit the expanding blast volume.           prepulse technique has opened up a
These photons pass through regions of      new class of neonlike X-ray lasers for
the expanding cloud of ionized debris      investigation.
and can stimulate the emission of           Chapline, G., and L. Wood. “X-ray Lasers.”
many more photons into the same              Physics Today 6 (1975): 40.
                                            Dunn, J., et al. “Demonstration of X-ray
quantum state at the same wavelength.
                                             Amplification in Transient Gain Nickel-like
The resulting coherent radiation at          Palladium Scheme.” Physical Review Let-
many frequencies, including the soft         ters 80, no. 13 (1998): 2825–2828.
X-ray region, will show intensity           Nilsen, J. “Reminiscing about the Early Years
                                             of the X-ray Laser.” Quantum Electronics
spikes in particular directions.             33, no. 1 (2003):1–2.
    Some of the first kinetic laser
explosions were done at Livermore
National Laboratory in the 1970s and       141. Noninversion Laser
1980s with exploding foils and the         Yes, lasing without inversion (LWI)
Nova laser system. Since the first         can occur whenever absorption can-
demonstration of soft X-ray lasing—        cellation is established. Light amplifi-
emissions at about 10 nanometers or        cation is then possible even when the
more—using the collisional excitation      upper-level population is less than the
mechanism in neonlike selenium,            lower-level population. This cancella-
many other neonlike ions, ranging          tion can be set up in a three-level sys-
from copper (Z = 29) to silver (Z =        tem in an atom in which the two
47), have been made to lase. However,      absorption transitions to the same
attempts to produce lower-Z neonlike       final state interfere and cancel, making
X-ray lasers have been unsuccessful.       the absorption probability zero.
    In the effort to develop a tabletop        In the diagram, upper-level state
X-ray laser that would require smaller     | a > is connected to lower levels | b >
high-energy laser drivers than Nova        and | c >. Use incident photons of the


212     Answers
                       State a                   142. X-ray Paradox
                                                 The index of refraction n for a mate-
      E1         E2 State b                      rial is normally stated with regard to
                                                 the phase velocity, unless indicated
                       State c
                                                 otherwise. The phase velocity is vph =
            3-state system                       c/n(k), where the index is a function of
                                                 the wave number k. If n(k)<1, then the
                                                 phase velocity is greater than the speed
appropriate energies E1 and E2, which            of the light in the crystal. There is no
correspond to transitions | a > to | b >         alarm that the energy is being trans-
and | a > to | c >, respectively. The            ported faster than c, for the group
uncertainty in these atomic transitions          velocity is still less than c.
leads to interference, since the transi-              Essentially, travelling harmonic
tions end in the same final state. There          waves in all physical examples require
is no way to determine which absorp-             wave packets or groups because of the
tion transition to the final state actu-          non-infinite extent of space and/or
ally occurred, so like the Young                 time. There are two velocities associ-
double-slit experiment, one must have            ated with these wave packets or
the interference. There is no interfer-          groups: the phase velocity and the
ence between the emission paths, since           group velocity. Harmonic waves or
they have different final states. By             harmonic components have a phase
arranging the phases of the two                  velocity vph = ω/k, where ω = 2πf and f
incoming light rays properly, one can            is the frequency. This phase velocity is
make the interference completely                 the velocity at which the wave fronts
destructive for absorption. Then stim-           travel. A group of harmonic waves or
ulated emission is the only process left.        wave packet has a group velocity vg =
For details of the probability calcula-          dω/dk, the velocity at which the packet
tions, see the references below.                 shape or envelope travels—that is, the
 Narducci, L. M., H. M. Doss, P. Ru, M. O.       velocity at which information or
  Scully, S. Y. Zhu, and C. Keitel. “ A Simple
  Model of a Laser without Inversion.”           energy is transported.
  Optics Communications 81 (1999): 379.               On the atomic level, the slowing of
 Scully, M. O., and M.S. Zubairy. Quantum        light passing through a material may
  Optics. Cambridge, Eng.: Cambridge             be considered as a continuous process
  University Press, 1997.
                                                 of absorption and emission of photons
                                                 as they interact with the atoms of the



                                                                        Answers     213
material. One assumes that between
each atom, the photons travel at c, as
in a vacuum. As they impinge on the
atoms, they are absorbed and nearly
instantly re-emitted, creating a slight
delay at each atom, which (on a large
enough scale) seems to be an overall
reduction in the speed of the photons.
                                                  Both states should have the same
Quantum mechanically, the scattering
                                              energy, and they do. Therefore we
is a two-step process of absorbing the
                                              really have a two-state system, analo-
incident photon and emitting a new
                                              gous to the hydrogen molecular ion or
photon.
                                              the ammonia molecule, so the analysis
    Experiments in other ranges of the
                                              should be for a two-state system. There
electromagnetic spectrum, particularly
                                              will be the possibility that configura-
in the visible, have shown that by stor-
                                              tion A changes into configuration B. As
ing the phase information of the inci-
                                              a result, quantum mechanics will
dent light beam in a gas vapor
                                              reveal that two new stationary states
temporarily, one can even claim that
                                              will occur, one state (the new ground
the light pulse can be brought to rest!
                                              state) with energy below the ground
 Addinall, E. “The Refractive Index of X-
  rays.” Physics Education 6 (1971): 77–78.   (lowest) state determined before, and
                                              one state with higher energy. The new
                                              ground state will be neither of the two
143. Benzene Ring                             configuration states shown but will be
The benzene ring has six-fold rota-           a linear combination of these two con-
tional symmetry about an axis perpen-         figuration states. Only this state is
dicular to the plane of the ring. One         involved in the chemistry of benzene at
simply requires a wave function solu-         room temperatures.
tion of the Schrödinger wave equation             Understanding benzene was one of
that has this six-fold symmetry, and          the first verifications of the linear super-
such a solution is easy to find. One          position of states that is at the heart of
would expect that knowing this solu-          quantum mechanics and also indicated
tion would allow one to calculate the         that quantum mechanics will be suc-
energy levels.                                cessful at larger scales than atomic.
    However, we are not done! There            Feynman, R. P., R. B. Leighton, and M.
                                                Sands. The Feynman Lectures on Physics,
are two possible configuration base             Vol III, Reading, Mass.: Addison-Wesley,
states, as shown in the diagram.                1965, pp. 10-10 to 10-12.


214      Answers
144. Graphite                             formation of intermediate structures,
                                          and so on. Fullerenes tend to form by
Place identical atoms into a diamond      “rolling up” a graphite sheet and
crystal structure. First, one would       adding carbon pentagons to achieve
mathematically find a wave function       curvature. If you just roll the sheet into
for the four bonding electrons using      a cylinder and cap off the ends with
the Schrödinger equation, resulting in    pentagon-curved hemispheres, you
what are called sp3 orbitals. Then one    make a carbon nanotube. These nan-
would represent the periodic symmetry     otubes are quite different from the tra-
in the crystal. Each carbon atom will     ditional fullerene-type materials (i.e.,
make four orthogonal bonds with           roundish cages), so they have quite dif-
tetrahedral symmetry if it can to its     ferent properties.
nearest neighbors. This diamond struc-     Collins, P. G., and P. Avouris. “Nanotubes
ture is one way to do this bonding.         for Electronics.” Scientific American 283,
    Another way to have four carbon         no. 6 (2000): 62–69.
bonds is for six carbon atoms to form      Pauling, L. General Chemistry. New York:
                                            Dover, 1988, pp. 168–170, 207–210.
a regular hexagonal ring with two
bonds in the ring for each carbon, and
the other two bonds extending per-
pendicular to the ring, one upward        145. Ozone Layer
and the other downward. Upon calcu-       Ozone plays two important roles with
lating the energy states for the four     regard to the energy balance for Earth.
carbon binding states, one learns that    As a minor greenhouse gas in all parts
the two perpendicular binding states      of the atmosphere, including near the
are held less securely than the ones in   surface, ozone helps maintain Earth’s
the ring that form a plane. The struc-    average temperature at about 13°C
ture makes graphite, a layered crystal    instead of a frigid –17°C. The concen-
that slips easily between the planes.     tration of ozone in the upper atmos-
Pencil writing surfaces have been         phere, however, regulates the UV
made from graphite for several thou-      intensity in sunlight reaching the sur-
sand years.                               face. All organisms need some UV
    Carbon in the fullerene structure     light to maintain a healthy existence,
is even more interesting. The structure   but any reduction of ozone in the
of 60 carbon atoms that results           upper atmosphere might allow dan-
depends on many factors, including        gerously large amounts of UV to reach
the velocity distribution of the free     the surface.
carbon atoms before collision, the            The two polar regions are extremely

                                                                   Answers      215
  susceptible to ozone depletion, particu-               equilibrium temperature of about
  larly by chlorofluorocarbon (CFC) mol-                  13°C. Without the greenhouse effect
  ecules and others, because the ice                     in our atmosphere, Earth’s average
  crystals in the air provide these fluoro-               surface temperature would be about
  carbons with platforms for rapid ozone                 256 K, or about –17°C, much too cold
  dissociation. Already, as a result of                  for many life forms. The greenhouse
  ozone depletions in the upper atmos-                   effect involves the influx of sunlight,
  phere above the polar regions, particu-                its absorption by the atoms and mole-
  larly above the South Polar region,                    cules of the stuff on Earth, and the
  there has been an increase in eye prob-                attempted emission of light and
  lems in land animals such as sheep in                  infrared energy back into space.
  the southern parts of South America                         Although carbon dioxide receives
  and in Australia and New Zealand.                      the most attention in the press, HOH
    Allègre, C. J., and S. H. Schneider. “The Evo-       vapor is the most important green-
     lution of the Earth.” Scientific American           house gas because the HOH molecule
     271, no. 4 (1994): 66–75.
                                                         absorbs energy over practically the
    Newchurch, M. J., et al. “Evidence for Slow-
     down in Stratospheric Ozone Loss: First             whole range of visible and infrared fre-
     Stage of Ozone Recovery.” Journal of Geo-           quencies, while carbon dioxide
     physical Research 108, no. D16 (2003):
                                                         absorbs in a small region of the near
     4507.
                                                         infrared only. Water vapor controls
                                                         about 60 percent of the greenhouse
  146. Greenhouse Gases                                  effect, carbon dioxide about 20 per-
                                                         cent, and the other trace gases in the
                   CO2
                         HOH                             atmosphere the remainder.
                                             HOH              Additional greenhouse gas concen-
                                O3                       trations added to the atmosphere
                                                         would be expected to trap even more
                                       280 K blackbody
Radiance                                                 infrared radiation and probably raise
                                                         the temperature further. However, a
           Absorption                                    convincing comprehensive model of
           curve
                                                         this process has not been achieved.
                                                         There are many complications to any
           400                                 1600      model of Earth, including the trans-
                         Wavelength (cm-1)
                                                         mission and reflection of light from
  The greenhouse gases trap most of the                  clouds, the movements of the ocean
  infrared, and this additional energy                   currents, the effects of human-made
  helps heat Earth to its present average                sources and sinks, the perturbations

  216            Answers
by vegetation, land animals, and sea               LCD is provided by the ambient light,
organisms such as plankton, the ther-              whereas all the energy for the LED and
mal energy input from additional heat              plasma displays must be provided by
sources such as mantle transport of                the electronic power source itself, such
thermal energy from the interior of                as a battery or the AC supply. In addi-
Earth, and the effects of the bombard-             tion, considerable thermal energy can
ment by cosmic rays from the galaxy                be produced in a plasma display, an
and beyond.                                        energy requirement beyond simply
    Many natural temperature records               producing a picture on the screen. Of
have been mined in the past few                    course, there are LCD displays that
decades that provide the history of                must provide their own ambient light
temperature changes, so fluctuations in             if they are to be used in a dark envi-
average temperature, a vaguely defined              ronment, so these displays have addi-
concept, are not new. The claim seems              tional energy requirements when
to be that the rate of increase of the             operated in this manner.
average temperature is among the                       So LCDs consume much less
greatest ever experienced on Earth.                power than LED and gas-display mod-
Whether this hypothesis is verified in              els because LCDs work on the princi-
the near future will take better models,           ple of blocking light rather than
meaning greater computing capability               emitting it.
and more included physical and
chemical processes and/or a definitive,
unambiguous example.
                                                   148. Sonoluminescence
 Gillett, N. P., F. W. Zwiers, A. J. Weaver, and   The light produced by sonolumines-
  P. A. Stott. “Detection of Human Influence        cence must originate in atomic transi-
  on Sea-Level Pressure.” Nature 422 (2003):
  292–294.                                         tions, electrons in excited states in
 Herzog, H., B. Eliasson, and O. Kaarstad.         atoms jumping down to lower energy
  “Capturing Greenhouse Gases.” Scientific          levels and emitting photons to con-
  American 282, no. 2 (2000): 72–79.
                                                   serve energy and angular momentum.
                                                   The apparatus consists of distilled
                                                   water with an admixture of a little
147. LED vs. LCD                                   helium or other inert gas in a spherical
We assume that they all have the same              flask surrounded by a piezoelectric
resolution, and we know that all three             crystal or two to send in sound waves
types of display—LED, LCD, and                     at practically any frequency. The
plasma—require energy to operate.                  details of the apparatus can be found
But the majority of the energy for the             at many sites on the Internet.

                                                                           Answers     217
     The sound energy creates bubbles      Baghdassarian, O., H.-C. Chu, B. Tabbert,
                                            and G. A. Williams. “Spectrum of Lumi-
in the water that rapidly collapse and      nescence from Laser-Created Bubbles in
emit a flash of light from their central     Water.” Physical Review Letters 86 (2001):
region. Instead of sound waves, a           4934.
powerful laser pulse also can create
the bubbles for the pulse of light. The   149. Siphoning Liquid
spectrum of the emitted sonolumines-
cent light pulse is similar to a black-   Helium
body spectrum of an object at about       At temperatures near absolute zero,
8,000 K, hotter than the Sun’s surface    normal liquid He I becomes superfluid
temperature of about 6000 K! And the      liquid He II by undergoing a second-
pulse of light lasts for picoseconds,     order phase transition. Its He atoms
with such an intensity that it can be     can move without viscosity in the
seen by the unaided human eye.            superfluid. Superfluidity is a quantum
     The reference below provides         mechanical phenomenon, with a
experimental results that support the     macroscopic volume (centimeter dimen-
popular theory that a plasma inside       sions) of liquid acting like a single
the bubble causes sonoluminescence.       macroscopic particle and described by a
The research team fitted their pulses’     single-particle Schrödinger equation.
spectra to a blackbody radiation curve        Immediately, superfluid He II in an
and found the correspondence to           open beaker will form a film that
plasma temperatures at about 8000 K.      crawls up the walls, over the top, and
The gas in the bubble becomes a           down the sides until the beaker is emp-
partially ionized plasma, and the radi-   tied. Normal fluids also can be
ation is emitted by an energy cascade     siphoned out of containers, but only if
from ions to electrons and finally to      their motion is started externally! The
photons.                                  solid surfaces in contact with He II are
     More details will be understood      covered with a film 50 to 100 atoms
eventually as faster optical response     thick along which frictionless flow of
systems become available to better fol-   the liquid occurs. Supposedly, mass
low the time development of the light     transport flow in the He II film takes
emission process. In fact, how quickly    place at a constant rate that depends
a state-of-the-art photodetector system   only on temperature.
operates is measured against what ini-        As the atoms of liquid He II move
tial parts of the sonoluminescent pulse   up the wall, they gain potential energy.
of light can be discerned!                What process provides the energy?


218     Answers
The answer lies in the ability of helium     150. Quantized Hall
atoms to wet any surface—that is, nor-
mal liquid He I atoms cling to the
                                             Effect
wall. The helium-helium force is the
                                              3
weakest force in nature because the K             RH (h/e2)
shell of electrons is complete and the
helium zero-point motion is signifi-          2
cant, so the helium-anything force is
stronger. Hence helium atoms would
                                              1
rather be next to anything other than
another helium atom. So He atoms
quickly form a film when presented
with the wall of the container because            0            10             20       30
                                                              Magnetic Field (Tesla)
the helium–anything attraction lowers
the potential energy and so on, while        In a two-dimensional metal or semi-
they gain gravitational potential            conductor, the standard Hall effect is
energy. These He atoms clinging to the       observed, but at low temperatures, a
wall are no longer in the superfluid         series of steps appear in the Hall resist-
phase because their flow velocities are       ance as a function of magnetic field
now lower than a critical velocity           instead of showing the typical monoto-
value.                                       nic increase. By confining the electron
    The thickness of the film is usually      system in the third dimension to con-
limited to a few hundred atomic diam-        fine the electron gas to two dimen-
eters because at some thickness the          sions, only specific electron wave
advantage of being near to the wall is       functions meet the boundary condi-
canceled by the increase in gravita-         tions, so only certain quantized energy
tional potential energy. Then, while         levels are available for the electrons.
the normal fluid is clamped to the           These steps in the Hall resistance occur
wall, the superfluid He II flows freely        at incredibly precise values of resist-
as the He atoms on the wall act as a         ance, which are the same no matter
siphon.                                      what sample is investigated—that is,
 Goodstein, D. L. States of Matter. Engle-   the resistance is quantized in units of
  wood Cliffs, N.J.: Prentice-Hall, 1975,
  p. 327.                                    h/e2 divided by an integer. This amaz-
                                             ing result is the quantized Hall effect.
                                                 Recall that electrons have a spin



                                                                            Answers    219
1/2 and obey the Pauli exclusion prin-          When the Fermi energy lies in a
ciple. As electrons are added to an         gap between Landau levels, there are
energy band, they fill the available        no available states to scatter into, so
energy band states, just as water fills a    there is no scattering, and the electrical
bucket. The states with the lowest          resistance falls to zero. The Hall resist-
energy are filled first, followed by the      ance for the Hall current cannot
next higher ones. At absolute zero          change from the quantized value
temperature T = 0 K, the energy levels      whenever the Fermi energy is in a gap
are all filled up to a maximum energy        between Landau levels, so one meas-
called the Fermi level. At higher tem-      ures a plateau. Only when the Fermi
peratures one finds that the transition      energy is in the Landau level can the
region between completely filled states      Hall voltage change and a finite resist-
and completely empty states is gradual      ance value appear.
rather than abrupt and described by          Kivelson, S., D.-H. Lee, and S.-C. Zhang.
the Fermi function, which has a value         “Electrons in Flatland.” Scientific American
                                              274, no. 3 (1996): 86–91.
of 1 for energies that are more than a
few times kT below the Fermi energy,
equals 1/2 if the energy equals the
Fermi energy, and decreases exponen-
                                            151. Integrated Circuits
tially for energies that are a few times    Heat dissipation is the biggest prob-
kT larger than the Fermi energy.            lem in ICs. Simple, old-fashioned ther-
     Consider the ideal case of a fixed      mal energy limits the density of
Fermi energy and a changing applied         electronic components. The ability to
magnetic field. In the presence of the       miniaturize continues to improve, but
magnetic field, the density of electron      unless thermal energy production per
energy states in 2-D is no longer con-      volume is decreased or new geometri-
stant as a function of energy and           cal paths for thermal energy transport
bunches into discrete energy levels,        away from the sources are devised, the
called Landau levels, of finite width       game is lost. At present, 3-D ICs offer
separated by the cyclotron energy,          a temporary reprieve, but even they
with energy regions between the             will find their limit.
Landau levels where there are no                Some short time scale solutions
allowed electron states. As the mag-        may be possible. The best thermal con-
netic field is swept to higher values, the   ductor among crystalline materials is
Landau levels move relative to the          diamond, so going to a diamond sub-
Fermi energy.                               strate may be a solution. However, the


220       Answers
technology of diamond is not yet com-     152. Atomic
petitive with silicon technology. Also,
components on these substrates that
                                          Computers?
require significantly less energy to      Yes. One can use electron spin direc-
function as a gate would delay the        tions as binary holders, for example.
overwhelming impact of thermal prob-      Even the nuclear spins can join in the
lems. Optical information transfer        game. Quantum computers already
between components would eliminate        use nuclear spins for storage. On a big-
electrical currents and their thermal     ger scale, DNA molecules are being
effects, but silicon does not have the    used for a DNA computer.
right optical properties; hence the           Several difficulties in making
active research into doping silicon to    atomic computers exist, but all of the
make the desirable optical properties.    difficulties can be overcome by clever
    At the longer time scale of several   techniques. Putting information in and
decades, perhaps the silicon and semi-    reading information out of these
conductor technology will simply fade     atomic systems have been done in the
away in favor of some other technol-      laboratory already. Maintaining their
ogy on the time horizon that seems        fixed states is a different kind of prob-
unachievable today but that would         lem that depends on which type of sys-
become viable then. Or the newer          tem is being used. Nuclear spin systems
technology hasn’t even been dreamed       have been used quite successfully since
about yet!                                the 1940s with the development of
    For any solid or liquid material,     nuclear magnetic resonance (NMR).
quantum disturbances from cosmic          Electron spin systems also are con-
rays may decide the ultimate limit in     trolled quite nicely in labs. If isolation
electronic component density unless       of the system is required, then vacuum
redundancy can solve this problem.        chambers work well for long enough
For optical systems based on light        time periods of particle isolation.
interference, and so on, who knows            At the other extreme are proposed
what is possible? Whatever wins in the    quantum computers utilizing the caf-
decades ahead will be numerous            feine molecules in a cup of java. They
orders of magnitude smaller and           are being bombarded constantly by the
faster, as well as more robust than       other molecules in the liquid, so the
what we have today.                       liquid environment brings about a
                                          rapid decoherence of the system. How-
                                          ever, there are an awful lot of caffeine


                                                                  Answers      221
molecules in the cup, at least 1020. The     X-ray line has been used to analyze
quantum computer requires probably           materials in minutes that formerly
only a million or so to retain their iso-    required hours to days to accumulate
lation for the duration of the computa-      enough data.
tion time—microseconds, perhaps—so               Whether the population inversion
the numbers may win out.                     for the 2p–1s transition in the external
     Like the limits to integrated circuit   Cu atoms actually occurs is unknown.
component density caused by thermal          The emission X-ray line is uncharac-
effects and by cosmic ray bombard-           teristically narrow and intense, and the
ment, atomic computers also may face         absence of the other competing line
similar limits. The type of atomic com-      indicates that whatever the selection
puter devised will determine how hos-        process is doing must be very efficient.
tile the environment can be.                 Other element sources such as nickel,
                                             based on the same mechanism, also
                                             have been made.
153. X-ray Laser?                             DasGupta, K. “CuKa1 X-ray Laser.” Physics
The mechanism for the intense X-ray            Letters A 189 (1994): 91–93.
source appears to be the following
according to K. DasGupta, the origi-         154. Bose-Einstein
nator of this unique X-ray source. The
W X-rays from the Cu-W X-ray tube
                                             Condensate
knock out K shell electrons and others       A Bose-Einstein condensate is formed
in the Cu atoms in the external Cu           at the coldest temperatures, which
crystal to produce a temporary (about        means that the atoms have been
10–15 second) population inversion,          slowed in their motion to be almost
and the Cu X-rays coming simultane-          stationary. By the de Broglie relation,
ously from the same tube then stimu-         each atom of mass m has a de Broglie
late transitions in these Cu atoms to        wavelength λ = h/p, where p is its
produce the Cu Kα 1 line at the Bragg        momentum mv and h is Planck’s con-
angle to the Cu(111) atomic planes.          stant. As the velocity v is further
This mechanism is very selective, the        reduced to cool the atoms, the de
line being so narrow and intense and         Broglie wavelength increases accord-
the process being so efficient that one       ingly. Eventually temperatures are
does not detect any of the competing         reached for which the wavelengths of
Cu Kα 2 emission to the available 1s         adjacent and nearby atoms begin to
state. The single-frequency intense          overlap in space considerably. Further


222       Answers
cooling places all the atoms in intimate        how many electrons end up in each
contact in one collective quantum               quantum dot.
state. Individual atoms can no longer               An electron in a 3-D box is con-
be discerned because they act like one          strained to have a quantum mechanical
big “atom.”                                     wave function that matches the bound-
    The first Bose-Einstein condensate           ary conditions for the Schrödinger
was achieved in 1995, even though the           wave equation, producing discrete
physics principles have been known              energy levels that are inversely pro-
since Einstein and Bose proposed them           portional to the square of the wave-
in the 1920s. About 2000 rubidium               length. As the box is made smaller, the
atoms in the gas were cooled to 170             energy levels become farther apart. If
nanoK when they formed a Bose-Ein-              the quantum dot diameter—that is,
stein condensate less than 100 micro-           box diameter—is made small enough
meters across. The condensate lasted            in fabrication, only a few energy levels
for about 15 seconds and was cooled             will exist inside for the electron. Hence
further, all the way down to 20 nanoK.          one can make quantum dots small
 Anderson, M. H., et al. “Observations of       enough to allow only one fluorescence
  Bose-Einstein Condensation in a Dilute        transition possible in the visible part of
  Atomic Vapor.” Science 269 (1995): 198.
                                                the spectrum.
 Castin, Y., R. Dum, and A. Sinatra. “Bose
  Condensate Make Quantum Leaps and                 The data from the first quantum dot
  Bounds.” Physics World (August 1999): 37.     spectrum showed a rich harmonic series
 Cornell, E. A., and C. E. Wieman. “The         of transitions between electron energy
  Bose-Einstein Condensate.” Scientific
                                                levels. Subsequent tweaking of the elec-
  American 278, no. 3 (1998): 40–45.
                                                trostatic potential was shown to reduce
 Townsend, C., W. Ketterle, and S. Stringari.
  “Bose-Einstein Condensation.” Physics         the dot size and increase the energy
  World (March 1997): 29–34.                    spacings. Later researchers have been
                                                able to magnetically link together quan-
                                                tum dots with the hope of making
155. Quantum Dots                               arrays of them for quantum computing.
Quantum dots are crystals, essentially           Flügge, S. “Particle Enclosed in a Sphere.” In
metal or semiconductor boxes, con-                Practical Quantum Mechanics. Vol. I. New
                                                  York: Springer-Verlag, 1974, pp. 155–159.
taining only a few hundred atoms
                                                 Reed, M. A. “Quantum Dots.” Scientific
and a well-defined number of elec-                American 268, no. 1 (1993): 118–123.
trons. The number of electrons can
                                                 Whitesides, G. M., and J. C. Love. “The Art
be controlled by the electrostatic                of Building Small.” Scientific American
environment. The trick is to adjust               285, no. 3 (2001): 38–47.



                                                                          Answers       223
                                              by rule 3. The cat is now either alive or
         Chapter 8                            dead, not both. The two QM alterna-
        Chances Are                           tives reduce to just one possibility.
                                                  Note that this example with the cat
                                              brings the connection between the non-
156. Schizophrenic                            intuitive behavior of Nature on the
Playing Card                                  microscopic scale up to the macroscopic
                                              scale of our everyday experiences.
According to the rules of QM, the final
                                              There has been an enormous amount of
state should be the superposition of
                                              controversy over this example and its
the two alternative falling directions,
                                              interpretation. Some of the issues are
with equal amplitudes ψ1 for left and
                                              discussed in the references listed below.
ψ2 for right. But we never see a card
                                               Albert, D. Z. “Bohm’s Alternative to Quan-
fall both ways simultaneously. Any air          tum Mechanics.” Scientific American 277,
molecule colliding with the card is             no. 5 (1994): 58–67.
equivalent to an observation, a meas-          Loeser, J. G. “Three Perspectives on
urement process, so QM rule 3 applies           Schrödinger’s Cat.” American Journal of
                                                Physics 52 (1984): 1089–1093.
and the outcome reduces to the classi-
                                               Wick, D. The Infamous Boundary: Seven
cal one, with equal probabilities P1 to         Decades of Heresy in Quantum Physics.
fall to the left side and P2 to fall to the     New York: Copericus Books, 1996, pp.
right side.                                     149–152.

     The term describing this reduction        Yam, P. “Bringing Schrödinger’s Cat to Life.”
                                                Scientific American 276, no. 6 (1997):
of the wave function to the classical           124–129.
probabilities that have no QM inter-
ference is often called decoherence.
The Schrödinger equation, which is            158. Wave Functions
deterministic, controls the entire            No. Beyond three dimensions there is
process.                                      no direct one-to-one correspondence
 Tegmark, M., and J. A. Wheeler. “100 Years
  of Quantum Mysteries.” Scientific Ameri-
                                              between many-dimensional configura-
  can 284, no. 2 (2001): 68–75.               tion space coordinates and the three-
                                              dimensional coordinates of position
                                              space.
157. Schrödinger’s Cat                            The misconception referred to here
In QM, it is irrelevant whether you           shows up in discussing the wave func-
actually peek or not. If in principle you     tion for two-particle systems, espe-
could have determined the status of the       cially when the discussion refers to the
cat, QM reduces to the classical result       two-particle wave function reducing to

224       Answers
the classical result. One often encoun-           Dicke, R. H. “Interaction-Free Quantum
                                                   Measurements: A Paradox?” American
ters questions about how the wave                  Journal of Physics 49 (1981): 925–930.
function can reduce instantaneously to            Hilgevoord, J. “Time in Quantum Mechan-
the result, as if there has been some              ics.” American Journal of Physics 70 (2002):
faster-than-light information transfer.            301–306.
Fortunately, the two-particle wave
function reduces in configuration                160. Quantum
space, not in position space!
                                                 Computer
 Hilgevoord, J. “Time in Quantum Mechan-
  ics.” American Journal of Physics 70 (2002):   A quantum computer relies on main-
  301–306.
                                                 taining its linear superposition of
 Mermin, N. D. “Is the Moon There When
  Nobody Looks? Reality and the Quantum
                                                 quantum states—that is, Ψ = ψ1 + ψ2 +
  Theory.” Physics Today 38, no. 4 (1985):       ψ3, its coherence during the calcula-
  38–47.                                         tions—so that all the states participate
 Styer, D. F. “Common Misconceptions             in the calculation. Quantum decoher-
  Regarding Quantum Mechanics.” Ameri-
  can Journal of Physics 64 (1996): 31–34.       ence is a bad thing for a quantum
 Wick, D. The Infamous Boundary: Seven           computer. A collision with the wall of
  Decades of Heresy in Quantum Physics.          the chamber or with another molecule
  New York: Copericus Books, 1996, pp.           will ruin the coherence because an
  162–166.
                                                 observation has been made. By QM
                                                 rule 3, we no longer sum over the
159. Wave Function                               amplitudes ψi . This decoherence then
                                                 ruins the quantum computation
Collapse?                                        because only one state will be partici-
The original wave function Ψ = ψ1 +              pating in the computations.
ψ2 + ψ3 + . . . will change. The probe               Maintaining coherence in a real
photon did not scatter off the electron          physical system has been progressing
in particular imaginary boxes, so we             slowly for the past decade, with coher-
know immediately that the wave func-             ence times of tens of nanoseconds for
tion should not include their ampli-             three identical subsystems working as
tudes. One could say that there has              a quantum computer. No one knows
been a partial collapse of the wave              what type of physical system will com-
function even though there has been              pose the first 18-subsystem quantum
no interaction! We believe that this             computer in the future, but this com-
gedanken experiment was discussed                puter probably will outdo all the other
first by physicist Robert H. Dicke in             classical computers combined in com-
the reference below.                             puting speed.

                                                                          Answers       225
 Awschalom, D. D., M. E. Flatté, and N.         flip a down spin to an up spin for a
  Samarth. “Spintronics.” Scientific American
  286, no. 6 (2002): 67–73.
                                                stimulated absorption transition or
 Lloyd, S. “Quantum-Mechanical Comput-
                                                can cause a stimulated emission of a
  ers.” Scientific American 273, no. 4 (1995):   photon by a spin flip from up to down.
  140–145.                                          Now for some coffee. The liquid
 Nielsen, M. A. “Rules for a Complex Quan-      contains about 1020 caffeine molecules.
  tum World.” Scientific American 287, no. 5
  (2002): 67–75.                                Even if we assume that all of them par-
                                                ticipate initially in bunches as coherent
                                                states of many quantum computers in
161. Cup of Java                                the cup just before the calculation,
                                                most bunches will experience colli-
Quantum Computer                                sions during the calculation time of a
Coffee contains caffeine molecules,             nanosecond, say, and drop out from
which may be useful as quantum sub-             the collection of coherent states of the
systems for a quantum computer                  system. However, a significant number
because they contain two rings in a             of bunches of coherent states may be
plane with many attached hydrogen               participating still when the calculations
atoms. The nuclear spin states of the H         are done, and these will provide a
atoms attached to the rings can be              strong signal above the background
used for information storage à la               noise. At least that’s the hope!
NMR. That is, a nuclear magnetic res-            Gershenfeld, N., and I. Chuang. “Bulk Spin
onance (NMR) system is a collection               Resonance Quantum Computation.” Sci-
                                                  ence 275 (January 17, 1997): 350.
of nuclear spin states in an external
magnetic field that tend to align the
spins. In the simplest ideal case at tem-       162. Bragg Scattering
perature T, the external magnetic field
B is uniform and there are two spin
                                                of X-rays
states, up and down. Let’s say that B           Bragg scattering requires λ < d; there-
aligns most of the spins to the up state,       fore there will not be any collective
with the ratio of down to up spins              scattering from a group of scatterers at
being determined by the exponential             different atoms within one wave-
factor Exp(– µB/kT), where µ is the             length. The actual scatterers of the
nuclear magnetic moment and k is the            X-rays are the electrons at each atom
Boltzmann constant. An external                 in these planes of the crystal. Coherent
radiofrequency pulse of the proper              scattering requires fixed phase rela-
frequency v and energy hv = 2 µB can            tionships, but there is no fixed phase


226       Answers
relationship between electrons at dif-       total amount is proportional to √ N
ferent atoms nor between the electrons                                   √N
                                             instead of N. Therefore Ψ =  ψ1, so
doing the scattering at any moment.          P = N | ψ1| 2. The Bragg scattering of
Therefore, the X-rays scattered into         X-rays is not a coherent scattering
the Bragg angle have a multitude of          process.
random phases and not fixed-phase
relationships. The scattering probabil-
ity is proportional to N, the number of
                                             163. Beautiful Faces
scatterers, and not N 2 , as it would be     Coherent scattering of light by the
for coherent scattering.                     atoms in the skin is the reason for our
     Here is the QM argument mathe-          ability to see details of a face. The
matically. Let ψi represent the proba-       ambient incident light is scattered by
bility amplitude to scatter an X-ray at      the molecules of the skin. Two factors
the i th atom. We know from QM rule 2        are significant for this two-step scat-
that Ψ = ψ1 + ψ2 + ψ3 + . . . , for alter-   tering process: the time interval
native ways to go from the X-ray             required and the number of coherent
source to the crystal to the X-ray detec-    scatterers. In the visible region of the
tor. Each ψi represents one atom, and        electromagnetic spectrum, this scatter-
we assume single scatterings on the          ing process occurs in atoms in less
way to the detector for simplicity. Each     than 10–8 second over an area of the
ψi = exp[iδ] φi , which includes a phase     skin involving about a million atoms
part exp[iδ] and the identical scattering    within a circle with a radius of about
amplitude φi at the identical atoms in       one wavelength of the light. The wave-
the crystal. If the phase part at each       length of greenish light is about 500
scattering atom is identical, then we        nanometers.
would have Ψ = N ψ1 and the proba-               Consider scattering one incident
bility P = N 2 | ψ1| 2, giving us coherent   photon at a time. During the scattering
scattering proportional to N 2.              time of a single photon by these one
     However, there is no correlated         million alternative paths there is
motion between electrons on different        almost no movement of the scattering
atoms, so their phases are random. If        atoms in the molecules, so alternative
the phase differences between scatter-       paths have essentially fixed phase rela-
ers—that is, the electrons on different      tionships. By QM rule 2, ψ = ψ1 + ψ2 +
atoms—are not fixed differences, then         ψ3 + . . . , and ψ = N ψ1 with probabil-
the sum is over random phases and,           ity P = N 2 |ψ1| 2, giving us coherent
like the random walk problem, the            scattering proportional to N 2. With


                                                                    Answers     227
incoherent scattering we would not see      164. Gravitational
much detail.
    In the UV, both factors are smaller
                                            Waves
than for light in the visible spectrum—     Yes, the coherent scattering of gravita-
the scattering occurs in less time, and     tional waves is expected to occur, with
the area for each scattering is less and    the scatterers being mass quadrupoles—
involves fewer atoms because the            that is, mass pairs in the antenna.
wavelength is much less. The face seen      J. Weber, the same physicist who first
in the UV would appear grainier with        calculated the classical cross section for
less detail because the adjacent coher-     gravitational wave scattering in 1959,
ent scattering areas are smaller and        proposed in 1981 that the coherent
the shorter time interval means that        scattering of gravitational waves would
they will have some effects of almost-      enhance the scattering cross section for
random phases.                              certain detectors by a factor of 106 or
    In the IR, most of the scattering       more. The larger cross section might
involves molecular transitions, which       explain the large responses of his two
are relatively slow processes, so the       independent one-ton cylindrical alu-
scattering process involves a much          minum bar gravitational wave detectors
longer time interval. But each mole-        every time either end faced the nucleus
cule itself is completely involved in the   of the Milky Way galaxy, approxi-
scattering. So even though the wave-        mately twice per day. If his proposal for
length is large, involving many more        a coherent scattering response is cor-
scattering centers, the molecular scat-     rect, then solid bar antennas would be
terers move significantly during the IR      much more sensitive to gravitational
scattering process, producing random        waves than large interferometers with
phases everywhere and a smearing of         their small masses at the mirrors such as
the image.                                  LIGO and VIRGO.
    Organisms of many different types           The QM calculation can be out-
see in the UV and/or in the IR to find       lined as follows. With wavelengths in
their nourishment, as well as in the vis-   the kilometer range being much longer
ible. However, we humans evolved            than the size of the Al bar antenna in
without being able to see either the UV     the lab, all the mass pair quadrupoles
or the IR, our vision being confined to      in the antenna are within this one
the visible part of the electromagnetic     wavelength. Hence, their responses are
spectrum. Why our eye-brain system          approximately in phase, and each mass
evolved in this way is not known.           pair offers an equivalent alternative


228       Answers
scattering path. By QM rule 2, Ψ = ψ1         when the original classical response
+ ψ2 + ψ3 + . . . , and Ψ ~ N ψ1 with         cross section is used. The rate of con-
probability P = N 2 | ψ1|2, giving us         version of mass to energy at the galac-
coherent scattering proportional to           tic nucleus should have devoured the
N 2, where N is the total number of           whole galaxy by now! I suppose that
mass pairs in the bar, about 1024.            we must wait for LIGO and VIRGO
However, the bar is actually composed         to detect and calibrate gravitational
of many microcrystallites, so one really      waves before we truly know whether
sums the QM amplitudes over the               gravitational waves can scatter coher-
number of mass pairs within each              ently in Weber bar antennas.
microcrystallite, then sums the proba-         Gibbs, W. W. “Ripples in Spacetime.” Scien-
bilities over all the microcrystallites.        tific American 286, no. 4 (2002): 62–71.
The coherent scattering probability is         Preparata, G. “Superradiance Effect in a
                                                Gravitational Antenna.” Modern Physics
still more than 10 million times larger         Letters 5 (1990): 1–5.
(after accounting for the crystalline          Weber, J. “Gravitons, Neutrinos, and Anti-
defects) than the classical non-coher-          neutrinos.” Foundations of Physics 14
                                                (1984): 1185–1209.
ent scattering response that Weber first
calculated in 1959.
     Whether any bar antenna for grav-        165. Coherent Neutrino
itational waves behaves as a coherent
scatterer has not been unambiguously
                                              Scattering
demonstrated. Instead of the classical        In 1984, so the story goes, J. Weber
result with the bar oscillating at its res-   proposed to build a detector for the
onant frequency and its harmonics             coherent scattering of neutrinos in a
when hit by a pulse of gravitational          proposal for research monies. The
waves, the coherent scattering bar            proposal review committee challenged
would essentially have an almost equal        him to write up the neutrino coherent
response to a wide range of frequen-          scattering idea and publish the paper
cies. The actual experimental bar             in a reputable physics journal. In
responses are complicated and require         December 1984 he submitted the
elaborate methods to find gravita-            paper, “Method for Observation of
tional wave scattering signals buried         Neutrinos and Antineutrinos,” to
in background noise.                          Physical Review C, and the paper was
     If the Weber bars were really            accepted by a referee within eight days
detecting gravitational waves from the        of the December 12 reception date!
galactic nucleus, there is an enigma              The paper triggered an enormous


                                                                      Answers      229
response in parts of the physics com-       of the neutrino took place. Hence their
munity. Numerous rebuttals of his           responses are in phase and offer equiv-
arguments appeared in the physics lit-      alent alternative scattering paths. One
erature within months after this publi-     must sum the amplitudes over all pos-
cation, but all of these rebuttals can be   sible paths—that is, all nuclei—to
refuted. Every paper erroneously            obtain the total amplitude for the
assumes that the nuclear scatterers act     neutrino scattering.
as potentials. Wrong! Weber shows in             By QM rule 2, Ψ = ψ1 + ψ2 + ψ3 +
the first section of the paper that such     . . . , and Ψ = N ψ1 with probability
an assumption cannot lead to coherent       P = N2 | ψ1| 2, giving us coherent scat-
scattering for neutrino wavelengths         tering proportional to N 2 , where N is
less than the spacing between nuclei.       the total number of nuclei in the bar,
However, everyone seems to ignore the       about 1023. One gains the enormous
details presented by Weber, who cor-        factor of 1023 for neutrino scattering
rectly explains why the nonrelativistic     over the noncoherent cross section!
calculation does not predict coherent       The only remaining contention is
neutrino scattering for neutrino wave-      whether all the phase relationships are
lengths less than the atomic spacing.       properly accounted for in this rela-
The QM argument is essentially              tivistic calculation.
dependent on the fact that the scatter-          Weber (now deceased) actually
ing phases among the nuclei will be         conducted several experiments to
random, leading to a scattering proba-      check his relativistic calculations for a
bility proportional to N instead of N 2.    long defect-free single crystal detector.
     In later parts of the paper Weber      He claims to have verified the turning
does the relativistic QM scattering cal-    on and the turning off of a nuclear
culations to show that coherent scat-       reactor in blind tests, the leaking of tri-
tering for all energies occurs—that is,     tium from a highly radioactive tritium
neutrinos of all energies will suffer       source, and the twice-daily passing of
coherent scattering. Included in these      the Sun though the long axis of his
calculations are terms involving the        crystal detector. In 1995 he deter-
stiffness of the defect-free crystal, and   mined that the total measured solar
so on. The conceptual idea is that          flux of neutrinos—all three types,
when the crystal as a whole recoils,        because the detector did not distin-
like a Mössbauer Effect scattering,         guish among them—was equal to the
then one cannot determine (even in          total neutrino flux expected by the
principle) where the nuclear scattering     standard solar model. This predicted


230       Answers
result agrees with the 2002 results            to allow different regions of the living
reported by the heavy water detector           tissue to be “seen” separately. A com-
at the Sudbury Neutrino Observatory            puter algorithm analyzes the data from
(SNO).                                         numerous RF detectors surrounding
 Ho, T. H. “Comments on the ‘Method for        the body and constructs an artificial
  Observation of Neutrinos and Antineutri-     image on a display screen. A dynamic
  nos.’” Physics Letters 168B (1986): 295.
                                               MRI instrument has a fast response
 Weber, J. “Method for Observation of Neu-
  trinos and Antineutrinos.” Physical Review   time to show changes occurring in
  C 31 (1985): 1468–1475.                      the microscopic environment in sec-
                                               onds or less, such as muscle action or
                                               heart contractions.
166. Magnetic
Resonance Imaging                              167. Heisenberg
(MRI)                                          Uncertainty
Nuclear magnetic resonance experi-             The uncertainty principle places no
ments began in the 1940s, and they             limit to the accuracy of measuring the
continue to be very useful today. Their        particle’s position. The uncertainty
alternative QM behavior is described           principle ∆px∆x ≥ h/4π forbids the
as a collection of spins acting together.      simultaneous measurement of both
Initially, the spin collection has a total     position and momentum in the same
spin S in a collective quantum state Ψ         direction to arbitrary accuracy, not an
= ψ1 + ψ2 + ψ3 + . . . and then the            individual measurement. Of course,
pulsed magnetic field rotates them all          practical design limitations exist that
so slightly to S – α with respect to the       probably limit the measurement, but
original direction—that is, they act           conceptually there is no limit. The
collectively and coherently. No one            same argument applies separately to
spin behavior is isolated from the oth-        the momentum.
ers in the same microscopic atomic                 An application of the Heisenberg
environment. All hydrogen nuclei in            uncertainty principle to the hydrogen
the same environment respond the               atom is an insightful example. The
same, while those in a different envi-         hydrogen atom is usually solved in
ronment respond slightly differently.          spherical polar coordinates instead of
    The MRI instrument for magnetic            Cartesian coordinates. In spherical
resonance imaging uses the differences         polar coordinates, the uncertainty
in the microscopic atomic environment          relations are a bit more complicated


                                                                      Answers     231
and the consequences can be some-           together many single-frequency waves
what bizarre. For example, since the        of different frequencies with properly
hydrogen wave function for the elec-        chosen amplitudes and phrases, we
tron about the z-axis—that is, in the φ     can build up a lump in a narrow
direction—is known precisely for the        region of space of approximate length
1s atomic state, and hence the angular      ∆ x. The range of wavelengths ∆λ
momentum has no uncertainty, the            needed can be represented by the cor-
uncertainty in φ is maximum. There-         responding range of wavenumbers ∆k.
fore, in the φ direction, one finds an       The approximate mathematical rela-
equal probability at all angles, produc-    tionship ∆ x ∆k ~ 1 can be established
ing the smeared-out probability distri-     by considering several examples, as
bution in φ.                                seen in the Krane reference below.
                                                Bohr’s famous measurement dis-
Two waves added                             turbance argument is faulty. For half a
                                            century physicists have regurgitated
                                            this argument of how the uncertainty
                                            principle acts to defend quantum the-
Twenty waves added
                                            ory. In experiments that first refuted
                                            Bohr’s argument, a beam of cold
                                            rubidium atoms is split to travel along
                                            two different paths; call them A and B.
                                            The beams still overlap and combine
    Many other uses for the uncer-          at the end of their journeys to create
tainty relation exist because it lies at    an interference pattern. Now the
the very heart of quantum mechanics.        researchers looked to see which path
However, one can see that any descrip-      the atoms followed by tweaking those
tion of a phenomenon using waves of         on path B into a higher energy state by
any kind will require an uncertainty        a pulse of microwaves. These atoms in
relation. Engineers are familiar with
the fact that about a one-MHz band-
width is required to reproduce a one-              Collimated atoms       Slit plate

microsecond pulse: ∆f∆t ~ 1, for           Atom wave
                                                                      A
example. Suppose there is a single-fre-
quency wave defined by y = y1 sin k1x.
This wave extends from – ∞ to + ∞,                                    B
and the question “where is the wave         Collimator
located?” has no answer. By adding                        µ waves                      Screen


232      Answers
their internal states kept a record of            Krane, K. Modern Physics, 2nd ed. New
                                                   York: John Wiley & Sons, 1995, pp.
which path they had taken. The                     93–106.
microwave pulse absorbed by an                    Styer, D. F. “Common Misconceptions
atom is insignificant by a factor of               Regarding Quantum Mechanics.” Ameri-
about 10,000 and can cause little                  can Journal of Physics 64 (1996): 31–34.
change to the atom’s momentum, not                Wick, D. The Infamous Boundary: Seven
                                                   Decades of Heresy in Quantum Physics.
enough to smear the interference pat-              New York: Copericus Books, 1996, pp.
tern. Yes, QM worked still. With the               152–156.
microwaves off, interference fringes
appear. Turn them on so you can tell
which path was taken, and the inter-             168. Vacuum Energy?
ference pattern vanishes. The uncer-
tainty principle is correct still, but the       There is always the zero-point energy
argument that “measurement disturbs              in the vacuum. Whatever QM model
the system” to explain the double slit           for the vacuum is considered, all can
experiment is wrong.                             be reduced in a first approximation to
     So what may be the deeper mecha-            a large number of harmonic oscilla-
nism at work in the double slit experi-          tors, which have a zero-point energy
ment, for example? Perhaps quantum               value that is non-zero. At present, QM
entanglement, in which every particle            calculations of the energy density of
is linked to every other particle it has         the vacuum seem to be too large by at
interacted with. Two-particle wave               least 30 orders of magnitude! The vac-
functions are linked in a six-dimen-             uum energy density should be about
sional configuration space with no               10–11 J m–3 if this vacuum energy is the
one-to-one correspondence to physical            source of the accelerated expansion of
3-D space, so the entanglement of N              the universe determined by the Type
particles will be described by a wave            1a Supernova measurements in 1998.
function in 3N-dimensional configura-                 One can do an energy estimate
tion space with no one-to-one corre-             using the Heisenberg uncertainty prin-
spondence to 3-D physical space. And             ciple. Or, if the vacuum has an effec-
now the mathematics becomes messier!             tive potential for a scalar field, the
 Dürr, S., and G. Rempe. “Can Wave-Particle
                                                 product of the visible matter density
  Duality Be Based on the Uncertainty Rela-      and the potential will give the energy
  tion?” American Journal of Physics 68          density for an assumed radius of the
  (2000): 1021–1024.
                                                 universe. In either case, the assump-
 Englert, B.-G., M. O. Scully, and H. Walther.
  “The Duality in Matter and Light.” Scien-      tions necessary to estimate this energy
  tific American 271, no. 6 (1994): 86–92.        density would take us too far astray.

                                                                        Answers      233
    However, we can determine                  is a soup of virtual particle-antiparticle
whether an electrically neutral particle       pairs that interact with the real atoms
of mass ∆m popping into existence for          in the metal plates, these pairs being
a time interval ∆t can be detected by its      created and annihilated in extremely
gravitational field. We use the uncer-          short time intervals in accordance with
tainty relation ∆E∆t ≥ h/4π in the form        the Heisenberg uncertainty principle.
c2 ∆m∆t > h/4π. Suppose we have the            That is, the more the total energy ∆E
most sensitive detector, a free particle       in the pair, the less time duration ∆t is
of mass M initially at a distance R            its existence so that ∆E∆t ≥ h/4π. This
away from ∆m; then in the Newtonian            vacuum pair “soup” pushes inward at
approximation the detector will                both plates when the plates are very
receive a pulse P = F∆t. Substituting          close to each other because certain
F = GM∆m/R2 into the uncertainty               particle-antiparticle pairs are practi-
relation produces GM∆m∆t/R2 ≥                  cally forbidden from momentarily
GMh/(4πR2c2).The initial state of the          appearing between them. Essentially,
detector also obeys the uncertainty            if their deBroglie wavelength exceeds
relation ∆P∆X ≥ h/4π, so that ∆m to            the plate spacing, these pairs have a
be noticeable requires the impulse P to        much lower probability to be between
be greater than about 2∆P, or ∆X ≥             the plates. But these same pairs appear
4R (R/rg), where the Schwarzschild             outside the plates and provide the
radius of the detector rg = GM/c2. For         additional forces, whence the net
objects ranging in size from protons to        inward force. Known as the Casimir
planets, rg lies within the object itself.     effect, it was first measured in 1958.
So the momentum transferred by the                  The Casimir force is too small to be
impulse will not be detected!                  observed for plates that are not within
 Haroche, S., and J.-M. Raimond. “Cavity       microns of each other. Two mirrors
  Quantum Electrodynamics.” Scientific         with an area of 1 cm2 separated by a
  American 268, no. 4 (1993): 54–62.
                                               distance of about 1 µm have an attrac-
 Ostriker, J. P., and P. J. Steinhardt. “The
  Quintessential Universe.” Scientific Ameri-   tive Casimir force of about 10–7 N.
  can 284, no. 1 (2001): 46–53.                Although this force seems very small, at
 Stefanski, B. Jr., and D. Bedford. “Vacuum    distances of less than a micrometer the
  Gravity.” American Journal of Physics 62
  (1994): 638–639.                             Casimir force becomes the strongest
                                               force between two neutral objects! At
                                               separations of 10 nanometer—roughly
169. Casimir Effect                            100 times the size of an atom—the
Although the classical vacuum is a             Casimir effect produces a force that is
void, the quantum mechanical vacuum            the equivalent of 1 atmosphere of pres-

234       Answers
sure. The resurgence of interest in the      that the photons are not spread out
Casimir force is because micromechan-        evenly in time nor in spatial extent.
ical devices on the scale of tens of nano-   Heisenberg’s uncertainty relation dic-
meters must accommodate its effects!         tates this behavior. The QM operators
 Haroche, S., and J.-M. Raimond. “Cavity     of phase- and amplitude-quadrature
  Quantum Electrodynamics.” Scientific       (i.e., for the perpendicular components
  American 268, no. 4 (1993): 54–62.
                                             of the E field) of the electromagnetic
 Kleppner, D. “With Apologies to Casimir.”
  Physics Today 43, no. 10 (1990): 9–11.     field do not commute, similar to posi-
                                             tion and momentum of a particle. The
                                             product of phase- and amplitude-
170. Squeezing Light                         uncertainty has a fixed lower limit.
Classically, a ray of light is an electro-   The more precisely the phase of a light
magnetic wave having an amplitude            wave is measured, the less determined
and a phase, both being expressed in         becomes its amplitude and vice versa.
terms of the electric field components        States of the light with the smallest
Ex and Ey. Quantum mechanically, the         possible amount of overall quantum
normal modes of the electromagnetic          noise are minimum uncertainty states.
field are quantized and treated as an              The reduction in quantum noise in
ensemble of harmonic oscillators, one        one observable of the light (e.g., the
harmonic oscillator per normal mode.         phase) at the expense of enhancing it
The number of photons in each har-           in the complementary observable (i.e.,
monic oscillator is the energy in the        the amplitude) can be done by para-
corresponding oscillator. An harmonic        metric amplification procedures. The
oscillator obeys the Heisenberg uncer-       resulting states of the light are called
tainty principle, so one expects the elec-   squeezed states, since the quantum
tromagnetic field to behave likewise.         noise got squeezed at a particular
    As the electric field in a light ray is   phase angle. Their wave packets oscil-
reduced, even a ray from a laser             late in time and get wider and nar-
source, the fixed amount of intrinsic         rower—that is, they breathe.
quantum noise in the light intensity              Alternately, the uncertainty in the
becomes more obvious. This quantum           amplitude of a laser beam can be
noise in an electrical field is ever pres-    reduced to a level below that normally
ent. If you shine any light on a pho-        allowed by the Heisenberg uncertainty
todectector such as a photodiode,            principle, a level known as the zero-
there will be fluctuations in the diode       point quantum noise level. However,
current corresponding to the individ-        this increased knowledge comes at the
ual photons being detected. One sees         expense of greater uncertainty in the

                                                                   Answers     235
frequency of the light. Essentially, one        increases the magnetic moment of the
is using an uncertainty relation of the         electron or muon to g = 2 (1 + a). The
form ∆Ex ∆Ey ≥ V, where V is a con-             small correction of about 0.12 percent
stant. Reducing the uncertainty in Ex           is called the anomalous moment but is
to gEx means that the uncertainty in Ey         often referred to as “g-2.” Its meas-
becomes Ey /g to keep their product             urement with gradually increasing
the same.                                       accuracy presents spectacular agree-
    Experiments with squeezed light             ment with calculation to better than
promise to enhance our understandings           24 parts per billion.
of quantum mechanics at the individual              The muon is 206 times heavier
atom and photon levels. Recently, a             than the electron, so the muon’s mag-
new type of ultraprecise laser pointer          netic moment is 206 times smaller, but
made by “squeezing” a beam in two               the virtual particles in the quantum
directions was able to position the             soup can be more massive. As a result,
beam with a precision of 1.6 Å, about           the anomalous moment is 40,000
1.5 times better than the theoretical           times more sensitive to undiscovered
limit for a conventional laser.                 particles and new physics at short dis-
 Treps, N., et al. “A Quantum Laser Pointer.”   tances. There is agreement to 4 parts
  Science 301 (2003): 940–943.                  per million that must be regarded as
                                                the best test of the theory, but there is
                                                also a small discrepancy that needs to
171. Electron Spin                              be explained, a difference in mean val-
Yes. Although the vacuum influence              ues of the experiments and the theory
on the electron spin is extremely small,        by 2.6 standard deviations.
the same effect of the vacuum on the                The muon g-2 result cannot at
muon’s spin has been measured at                present be explained by the established
Brookhaven National Laboratory. The             SM. Recalculations of the predicted
interaction magnitude is predicted by           theoretical value continue, and correc-
the Standard Model (SM) of Leptons              tions have been done. Moreover, the
and Quarks and their interactions. All          g-2 calculation involves three of
fundamental particle-antiparticle pairs         the four fundamental interactions—
momentarily appear in the vacuum                weak, electromagnetic, and color—so
and disappear sporadically, so the              there are many Feynman diagrams
electron (and muon) see them all, if            that contribute.
only for a fleeting moment. This vac-                Perhaps this unresolved g-2 differ-
uum “soup” is slightly magnetic, so it          ence is the harbinger of new physics


236       Answers
beyond the SM, such as new quarks,             2.7 millikelvins. The He-3 pairs form
or supersymmetric particles, or a sur-         one momentum macrostate. Because
prise in the vacuum.                           the component He-3 atoms are not
 Bennett, G. W., et. al. “Measurement of the   bosons, there should be some small
  Positive Muon Anomalous Magnetic             width to the macrostate momentum in
  Moment to 0.7 ppm.” Physical Review
  Letters 89 (2002): 101804.                   addition to the small width because the
                                               He atoms are composed of fermions.
                                                   The pairs of atoms are magnetic,
172. Superconductivity                         so the He-3 superfluid is more com-
The paired electrons in superconduc-           plex than its He-4 counterpart. In fact,
tors that are in the superconducting           superfluid He-3 exists in three differ-
state show Bose-Einstein condensation          ent phases related to different mag-
to a single macrostate. There is some          netic or temperature conditions. In the
small energy width to this macrostate          A phase, for example, the superfluid is
because the pairs are composed of spin         highly anisotropic—that is, directional
1/2 particles, and they are showing            like a liquid crystal.
remnant Fermi-Dirac behavior: no                Scientific American. Special briefing on the
                                                 Nobel Prizes in Science, “A New Super-
two identical fermions can ever be in            fluid.” Scientific American 276, no. 1
the same state as defined by their four-          (1997): 15–16.
momenta and spins no matter how
they behave collectively.
                                               174. Gap Jumping
                                               This Josephson effect is really quan-
173. Superfluidity
                                               tum mechanical tunneling across the
The odd number of constituents in              physical gap because the wave func-
He-3 (two protons, one neutron, and            tion for the superconducting pair
two electrons) classifies it as a fermion       extends beyond the end of the material
that obeys Fermi-Dirac statistics. So no       into the gap and to the other side. If
two He-3 atoms can share the same              the superconducting material is actu-
quantum state defined by the four-             ally in the form of a ring, then match-
momentum (energy and three-momen-              ing of the wave function for the pair
tum) and spin. The surprise in the early       around the ring must be made,
1970s was that He-3 can magnetically           restricting their angular momentum
couple with another He-3 to form a             quantization to multiples of h/2π.
boson and become a superfluid liquid             Clarke, J. “SQUIDS.” Scientific American
at the extremely low temperature of              271, no. 2 (1994): 46–53.



                                                                       Answers      237
175. Nuclear Decay                           nuclear decay rate may be influenced
                                             by its environment, see the Peres refer-
The wave function extends through            ence below.
the potential barrier to the outside          Halliday, D., and R. Resnick. Fundamentals
world. Therefore the probability to be         of Physics. New York: John Wiley & Sons,
outside the nucleus is not zero. So why        1988, pp. 1009–1010.
does the wave function itself extend          Peres, A. “Zeno Paradox in Quantum The-
                                               ory.” American Journal of Physics 48
into the barrier? All confinement prob-         (1980): 931–932.
lems, classical and quantum, have
solutions with functions that extend
into the barrier, usually decreasing         176. Total Internal
exponentially to almost zero within a
few wavelengths. Atomic particles
                                             Reflection
have relatively long wavelengths com-        Yes, the light goes a little beyond the
pared to the barrier thickness. So why       interface. One can treat this behavior
does the wave function itself not end        either classically or quantum mechan-
in the barrier? Because the effective        ically. In QM the wave function for
barrier height decreases with radial         the photon extends beyond the glass-
distance.                                    air interface into the air.
    The probability to tunnel through            You can see this behavior in the
the barrier is proportional to Exp           following manner. Fill a drinking glass
      (U
[–Ar√(r)–E)], where E is the energy      partially full with water. Tilt the glass
of the incident particle, U(r) is the bar-   and look down into it at the side wall
rier potential as a function of distance     at such an angle that the light entering
r, and A is a constant that includes         your eye has been totally internally
Planck’s constant h. Some closely            reflected from the wall. The wall will
related problems to be treated as tun-       look silvery when this condition holds.
neling through a barrier are:                Then press your moistened thumb
1. Bare copper wire is cut and the two       against the outside of the glass. You
   ends are twisted together. In spite of    will see the ridges of your fingerprint
   the fact that the copper is coated        because, at those points, you will have
   with copper oxide, the twisted ends       interfered with the total reflection
   still conduct electricity readily.        process. The valleys between the
                                             ridges are still far enough away from
2. Tunnel diode operation.
                                             the glass that the reflection here
3. Scanning tunneling microscope.            remains total and you simply see a sil-
   For a discussion about how a              very whorl.


238       Answers
177. Annihilation                            action. The compression of the ball
                                             (and the concrete being struck) sends
Fermi’s Golden Rule hints that we            phonons (quantum sound waves) run-
should consider the phase space avail-       ning around telling the material that
able for the final particles, and this       compression is occurring and that the
phase space is related to the entropy of     increased energy density in parts of the
the final particles. If the entropy of the    ball can be reduced by expanding back
final state is greater than the entropy       to its normal size. Of course, the
for the initial state, the process occurs.   expansion overshoots and the ball
In the simpler case, when an electron        “rings” as it leaves the concrete, each
at rest and its antiparticle, the positron   extended state also increasing the
at rest, annihilate each other, two pho-     energy density in parts of the ball. One
tons are produced to conserve quan-          can model much of this behavior by
tum numbers as well as energy and            assuming that the atoms and mole-
momentum. The entropy of the prod-           cules are in a potential well somewhat
ucts is greater than the reactants.          similar to the parabolic well of the
Why? Because there is much freedom           harmonic oscillator. However, instead
in the direction of the photon polar-        of a potential energy for an atom ver-
izations. The interacting particle and       sus the atomic separation distance
antiparticle begin with their spins          being proportional to r2 only, there
opposite but along a specific direction,      must be additional terms proportional
thereby having a total spin of zero. In      to r3, etc., where r is the distance from
the final state with two identical pho-       the equilibrium location.
tons emerging in opposite directions in          Eventually the phonons help the
order to conserve energy and linear          ball get back into its normal shape, but
momentum, the photon spins are               the atoms and molecules never quite
opposite—that is, both spin +1 or both       make their initial relative positions
spin –1 with respect to their momen-         again, there being some residual dis-
tum directions—but the polarization          tortion. Even the concrete being struck
vectors can be in any direction in the       by the ball never quite recovers. Wit-
plane perpendicular to the momentum          ness the eventual wear of a concrete
directions.                                  highway by cars and trucks compress-
                                             ing the road, a more vigorous process
178. A Bouncing Ball                         but conceptually the same.

We present a simplified version of the
complexities of this bouncing ball


                                                                    Answers     239
                                                 Silverman, M. P. A Universe of Atoms, an
179. The EPR Paradox                              Atom in the Universe. New York: Springer-
There seems to be no classical thinking           Verlag, 2002, pp. 92–102.

that would reproduce the data set. A             von Baeyer, H. C. Taming the Atom: The
                                                  Emergence of the Visible Microworld.
predetermined instruction set would               Mineola, N.Y.: Dover, 1992, pp. 210–211.
be akin to an algorithm for generating
random numbers—but no such sets of
numbers are truly random. One must              180. Information and a
accept the conclusion that Nature is            Black Hole
quantum mechanical and therefore
classical physics is only an approxima-         For certain, one should worry about
tion. The rules of QM agree with the            quantum information loss, especially
results, but the details are too compli-        if quantum mechanics is to provide a
cated mathematically to present                 complete explanation for everything in
herein. The references provide the              the world. Does the black hole infor-
extended discussion.                            mation increase with the inclusion of
    Even more surprising is the sugges-         the chair? Let’s see. A black hole has
tion that locality is violated. That is,        mass, spin, and possibly electric
information from the first detector             charge, weak charge, or color charge.
passes to the second detector without           That’s all! We cannot determine the
passing through imaginary spherical             information content of the black hole
surfaces surrounding each, as if more           from these quantities only. That is a
dimensions exist in our world! Some-            problem. The most likely solution that
one, someday, will determine a funda-           would prevent quantum information
mental reason for this behavior of              loss is that the surrounding space
nature.                                         just outside the event horizon of the
 Eberly, J. H. “Bell Inequalities and Quantum
                                                black hole takes care of the informa-
  Mechanics.” American Journal of Physics       tion equation to make everything
  70 (2002): 276–279.                           correct, emitting particles to compen-
 Einstein, A., B. Podolsky, and N. Rosen.       sate correctly.
  “Can Quantum-Mechanical Description of
  Physical Reality Be Considered Complete?”         The actual physics calculation of
  Physical Review 47 (1935): 777–780.           information change in the gravita-
 Mermin, H. D. “Is the Moon There When          tional field of a black hole is much
  Nobody Looks? Reality and the Quantum         more complex and difficult. Among
  Theory.” Physics Today 38 (1985): 38–47.
                                                the necessary concerns is the fact that
 Shimony, A. “The Reality of the Quantum
  World.” Scientific American 258, no. 1        the black hole has performed a non-
  (1988): 46–53.                                unitary transformation on the state of


240       Answers
system when it devoured the chair. A           also vary. As the variations in the
non-unitary evolution is excluded in a         cosmic rays are determined by other
quantum theory because it fails to pre-        independent methods, they can be
serve probability—that is, after a non-        incorporated into the C-14 dating as
unitary evolution, the sum of the              adjustments.
probabilities of all possible outcomes             According to research literature,
of an experiment may be greater or             tree ring counts indicate that C-14 dat-
less than 1. Quantum mechanics could           ing has fluctuations of the C-14 con-
not survive. Perhaps the QM of a               centration in the atmosphere between
black hole will eventually be done and         1400 and 1700 B.C.E. Furthermore, a
quantum gravity will save us from this         comparison of radiocarbon-deter-
catastrophe!                                   mined ages with ages of archaeological
 Bekenstein, J. D. “Information in the Holo-   materials accurately established by
  graphic Universe.” Scientific American 289,   other methods reveals that for the
  no. 2 (2003): 58–65.
                                               period from 100 B.C.E. to 1400, radio-
                                               carbon dating gives values that are too
                                               large, and that prior to 100 B.C.E. the
                                               radiocarbon values are too small.
     Chapter 9                                     At about 1600 B.C.E., the C-14
  Can This Be Real?                            date values are about 175 years (5
                                               percent) too small, increasing to about
                                               300 years (6 percent) at 3000 B.C.E.
181. Carbon-14 Dating                          The discrepancy appears to be a result
The ratio of C-14 to C-12 in living            of slight variations in Earth’s magnetic
organisms will depend on many fac-             field over the years, which would alter
tors, including the local climate and          the cosmic ray intensities and hence
the amounts of C-14 in the atmos-              C-14 production in the atmosphere.
phere, factors that can vary on time           These corrections allow C-14 dates to
scales as short as tens of years. The          be corrected, and even for 100,000
radiocarbon dating process assumes             years ago the radiocarbon dates are
in its zeroeth order approximation             good to within 5 percent.
no variation in these factors over              Staff of McGraw-Hill, eds. McGraw-Hill
                                                 Encyclopedia of Science & Technology. Vol.
hundreds and thousands of years. But             15. New York: McGraw-Hill, 2002, pp.
the cosmic ray intensity reaching the            136–144.
atmosphere may vary considerably,
so the amount of C-14 produced will


                                                                        Answers       241
182. Nuclear Energy                          are higher in energy than the corre-
                                             sponding neutron levels because there
Levels                                       is the added Coulomb repulsion. Any
Even the shell model, often called the       extra proton or neutron can be added,
independent particle model of the            but this additional particle must
nucleus, fails to correctly predict many     occupy a higher energy state, usually
of the energy level spacings unless the      leading to an unstable nucleus.
spin-orbit LS interactions are included.      Jolie, J. “Uncovering Supersymmetry.”
That is, the proton and neutron mag-           Scientific American 287, no. 1 (2002):
                                               70–77.
netic moments interact with magnetic
                                              Serway, R. A. Physics for Scientists &
fields produced by their orbital               Engineers with Modern Physics, 3rd
motions. These LS interactions add             ed. Philadelphia: Saunders, 1990, pp.
terms to the approximate constant              1352–1354.

potential of the shell model to domi-         Tipler, P. A. Modern Physics. New York:
                                               Worth, 1978, pp. 427–432.
nate the quantum state sequence inside
the nucleus. As a result, many energy
levels change their relative positions
on the energy scale, with levels from        183. Nuclear Synthesis
different principle quantum numbers          Look at the binding energy curve for
becoming interchanged! Once the LS           the elements and you will see that at
interaction was properly accounted           least one isotope of Ni is well bound.
for, all its predictions were shown to       Unfortunately, this isotope has a rapid
agree with the empirical data.               decay mode. In fact, all the Ni isotopes
    This model of the nucleus also           from Ni-49 to Ni-57 have half-lives of
explained why nuclei containing an           only milliseconds to at most 10 days.
even number of protons and neutrons
are more stable than others. Like the
energy levels for the electrons in quan-
                                           8.80
tum states outside the nucleus, the                                  Fe-58
                                                             Fe-56              Ni-62
Fermi exclusion principle allows two
identical particles per quantum state
                                           8.78      Cr-52
only. The nuclear quantum states for                                    Ni-60
                                                             Cr-54
the protons are separate from the
nuclear quantum states for the neu-
trons, and any particular state is filled   8.76
when there are two identical particles              50        55        60         65
with opposite spins. The proton levels                        Mass number A


242      Answers
    Although the championship of              and thus they occur in the normal stel-
nuclear binding energy is often attrib-       lar burning cycles. But since the “iron
uted to Fe-56, this isotope actually          group”—those elements with isotope
comes in third. The most tightly bound        mass number of about A = 60—is
of the nuclei is Ni-62. The binding           at the peak of the binding energy
energies are 8.790 MeV/nucleon for Fe-        curve, the fusion of elements above Fe
56 and 8.795 MeV/nucleon for Ni-62.           requires energy, with the exception of
The binding-energy curve shows those          the most tightly bound isotope—
nuclides that are close to the peak.          Ni-62, for example.
    The most tightly bound nuclides               The elements beyond Fe are
are all even-even nuclei. Fe-56 is about      expected to be formed in the cata-
a factor of ten more abundant in stars        clysmic explosions known as super-
than Ni-62. The Fewell reference              novae in which a large flux of
below indicates that the reason lies          energetic neutrons build up mass
with the greater photodisintegration          approximately one unit at a time to
rate for Ni-62 in stellar interiors. Oth-     produce the heavy nuclei. Following
ers have suggested that the very low          neutron capture, some isotopes beta
rate of multistep production of Ni-62         decay to change a neutron into a pro-
from Co-59 is the culprit.                    ton plus an electron and an electron
 Fewell, M. P. “The Atomic Nuclide with the   antineutrino, increasing the atomic
  Highest Mean Binding Energy.” American      number by one unit. Some sample
  Journal of Physics 63 (1995): 653–658.
                                              sequences are:
 Shurtleff, R., and E. Derringh. “The Most
  Tightly Bound Nucleus.” American Journal       Fe-56 + n → Fe-57 (stable)
  of Physics 57 (1989): 552.
                                                 Fe-57 + n → Fe-58 (stable)
                                                 Fe-58 + n → Fe-59 → Co-59 by
184. Heavy Element                                  beta decay
Synthesis                                        Co-59 + n → Co-60 → Ni-60 by
The synthesis of the heavier elements               beta decay
beyond Fe is done during supernova            In principle this process could con-
explosions, in a few days or less, and        tinue indefinitely, but the elements
the atomic debris are spewed out into         beyond uranium (Z = 92) are all
space to later collect into new stars         radioactive.
and planets and be there for incorpo-             The layers of the star containing
ration into life forms.                       the buildup of heavy elements may be
    The fusion process for elements up        blown off by the supernova explosion
to Fe in the periodic table yields energy,    to provide the raw material of heavy

                                                                   Answers     243
elements within expanding hydrogen          contributing to the energy of the final
clouds that much later can condense to      state. Therefore, energy conservation
form new stars, planets, and the stuff      dictates that the proton will be in a
of life.                                    proton energy state that is lower in
 Krane, K. S. Modern Physics, 2nd ed. New   energy than the initial energy of the
  York: John Wiley & Sons, 1995, pp.        neutron. In many nuclei all available
  290–291.
                                            proton states—that is, those that are
                                            not occupied by protons—have higher
185. Neutron Decay                          energies than the energy of the initial
                                            neutron state, so the decay cannot
The failure of the neutron in a nucleus
                                            occur.
to decay is a quantum mechanical
                                                The equivalent energy levels of the
effect. According to quantum mechan-
                                            protons in nuclei are higher than for
ics, the rate of decay is dictated by
                                            the neutrons because their energies
Fermi’s Golden Rule, which states that
                                            include the Coulomb repulsion
the rate is proportional to the proba-
                                            between two protons and other prop-
bility of decay (i.e., the absolute value
                                            erties of the nuclear force, especially
of the square of the matrix element
                                            the spin dependence. Obviously the
connecting the initial and final states)
                                            stable nuclei will include those for
times the density of final states.
                                            which neutron and proton decays do
Because the free neutron decays to a
                                            not occur!
proton plus electron plus electron anti-
                                             Asimov, I. Understanding Physics. New York:
neutrino, we know that the probabil-          Hippocrene, 1988, p. 245.
ity for this beta decay process is not
zero and that there are available final
states for the three product particles.     186. Finely Tuned
Energy conservation dictates that the
total final state energy equals the total
                                            Carbon?
initial energy of the free neutron.         The crucial energy comparison to
     Inside a nucleus, the decay of a       make is not simply the radioactive
neutron is a transition from an initial     state energy of 7.65 MeV to the prac-
energy state, the particular bound neu-     tical limit value of 7.7 MeV, but one
tron state that the neutron occupies, to    must include the comparison of the
a final state consisting of a proton in      radioactive state energy 7.65 MeV to
some final proton energy state plus a        the energy 7.4 MeV of the reactants at
free electron and a free electron anti-     rest. This energy of 0.25 MeV misses
neutrino, the latter two particles          being too high for the production of


244       Answers
carbon by the fractional amount of                Called the carbon cycle, this sequence
0.05 MeV/0.25 MeV, or 20 percent,                 of reactions occurs much more rapidly
which is not so critical after all.               than the proton-proton cycle sequence
 Barrow, J. D., and F. J. Tipler. The Anthropic   because the C-12 acts as a catalyst,
  Cosmological Principle. Oxford: Oxford          neither being produced nor consumed
  University Press, 1986, pp. 252–253.
                                                  by the totality of reactions. The net
 Livio, M., D. Hollowell, A. Weiss, and J. W.
  Truran. “The Anthropic Significance of the       process is still the same: 4 protons →
  Existence of an Excited State of C-12.”         He-4, and the net energy produced is
  Nature 340, no. 6231 (1989): 281–284.
                                                  the same, but the rate of energy pro-
 Weinberg, S. Facing Up: Science and Its
                                                  duction is much higher.
  Cultural Adversaries. Cambridge, Mass.:
  Harvard University Press, 2001, pp.                 The carbon cycle occurs at a higher
  235–237.                                        temperature than the proton-proton
                                                  cycle because the C and H Coulomb
                                                  repulsion is greater than the H and H
                                                  repulsion, so the Sun, with its internal
187. Proton-Proton                                temperature of about 15 × 106 K, is
Cycle                                             too cool to activate the carbon cycle,
The other stars are using the carbon              which requires about 20 × 106 K.
cycle for their fusion energy. The com-            Krane, K. S. Modern Physics, 2nd ed. New
                                                    York: John Wiley & Sons, 1995, pp.
mon proton-proton cycle reaction is                 282–285.
not the source of fusion energy in
many stars burning hydrogen because
the first reaction in this sequence has            188. Oklo Nuclear
two protons combining to form a                   Reactor
deuteron H-2, a very unlikely event
                                                  The reaction sequence shows how to
that occurs slowly. A more likely
                                                  breed Pu from local U-238, which is
sequence of reactions involves having
                                                  the most abundant naturally occurring
C-12 be a catalyst:
                                                  uranium isotope. Initially, neutrons
    C-12 + p → N-13 + γ                           come from the fission of U-235. How-
                                                  ever, the very high abundance of
        N-13 → C-13 + e+ + ν
                                                  U-238 means that this isotope will
    C-13 + p → N-14 + γ                           absorb some of the neutrons to
    N-14 + p → O-15 + γ                           become U-239, decay by beta decay to
                                                  neptunium 239, and then decay to Pu-
        O-15 → N-15 + e+ + ν
                                                  239. Some of the resulting Pu-239
    N-15 + p → C-12 + He-4                        undergoes fission.


                                                                        Answers      245
   U-238 + n → U-239 → Np-239 +              proximity would have been emitting
      e– + anti-ν                            enough gamma radiation to exceed the
   Np-239 → Pu-239 + e– + anti-ν
                                             recommended limit.
                                                 We can estimate the exposure
However, because the natural reactors        amount and compare its value to the
at Oklo probably operated for such a         recommended limit today. There are
long time, the Pu-239 had time to            approximately 105 decays of K-40 iso-
decay by alpha decay to U-235. Thus          topes per second in your body, but the
the Oklo natural reactors were true          decay chart tells us that only about 11
breeder reactors, fissioning more            percent yield a gamma ray, producing
U-235 than originally existed in the         about 1100 self-inflicted gamma rays
reactors. The evidence for the breeder       per second, amounting to about 0.36
process remains in the reactor as more       mSv per year, well below the recom-
of the fission products than could pos-       mended limit today. Even a group of 10
sibly be produced by the amount of U-        people closely packed would not pro-
235 that has been lost from each of the      vide a radiation exposure more than 3.6
reactor sites.                               mSv per year. So we are not radiation
    A second piece of evidence for Pu        dangers to ourselves nor to our friends!
fission is the isotopic composition of         Cohen, B. L. “Catalogue of Risks Extended
the fission products in the mass range          and Updated.” Health Physics 61 (1991):
100 to 110. To breed Pu and addi-              317–335.

tional U-235, the reactors must have
operated for periods significantly
greater than the half-life of Pu 239,        190. Nuclear Surprises?
about 24,360 years.                          Both are true statements.
 Cowan, G. A. “A Natural Fission Reactor.”       1. The only emissions from a
  Scientific American, no. 7 (1976): 36–47.
                                             nuclear power plant are (a) water
                                             vapor from its cooling towers, (b)
189. Human                                   thermal energy in the external cooling
                                             water, (c) any stray gamma rays not
Radioactivity                                shielded (unlikely to be above normal
At one time in the history of radiation      background), (d) any radioactive iso-
safety, before extensive and long-term       topes created in the external cooling
measurements, the recommended radi-          water (unlikely to be above normal
ation limit was much less than the           background), and (e) electrical energy.
limit today. During those times, in              The emissions and safety proce-
the mid-1900s, two people in close           dures at a coal-burning power plant

246       Answers
are not as strict and, because all coal      Th-232, Ra-226, etc.) and cosmic
naturally contains radioactive material      radiation (photons, muons, etc.) are
with many isotopes, some of these            fairly constant over the world in the
radioactive isotopes escape into the air     range of 8–15 µrads per hour. Assum-
when the coal is piled in storage, when      ing maximum damage to human tis-
the coal is burned, and so on. Measure-      sue, this present background radiation
ments at coal-burning plants verify that     level corresponds to about 1.8 mSv per
radioactive atoms and molecules are          year.
released.                                        If one spreads all the human-pro-
    Scientific researchers in the            duced artificial radioactive materials
McBride et al. reference below have          equally around the surface of Earth,
concluded from measurements “that            the local increase in radioactivity is
Americans living near coal-fired power        expected to be minuscule compared to
plants are exposed to higher radiation       this indigenous natural radioactive
doses than those living near nuclear         background. Suppose we had a million
power plants that meet government            metric tonnes of human-made radioac-
regulations. . . . The fact that coal-fired   tive material to be dispersed over
power plants throughout the world are        Earth of approximately 5 × 1014 m2.
the major sources of radioactive mate-       Each square meter would acquire an
rials released to the environment has        additional 0.2 × 10–5 kg of radioactive
several implications. It suggests that       material, compared to the natural
coal combustion is more hazardous to         amount of radioactive material in the
health than nuclear power and that it        top 10 centimeters of about 2 × 10–2
adds to the background radiation bur-        kg, producing an insignificant amount
den even more than does nuclear              of local radiation unless the half-lives
power. It also suggests that if radiation    were short, on the order of minutes to
emissions from coal plants were regu-        days. The additional amount adds
lated, their capital and operating costs     only 1 part in 10,000 when dispersed
would increase, making coal-fired            around the globe.
power less economically competitive.”         Aubrecht, G. J. Energy, 2nd ed. Upper Saddle
    G. J. Aubrecht, in the reference           River, N.J.: Prentice Hall, 1994.
below, states that the radioactivity          Cohen, B. L. “Catalogue of Risks Extended
danger from each coal-burning electri-         and Updated.” Health Physics 61 (1991):
                                               317–335.
cal plant is at least 100 times the dan-
ger from each nuclear plant.                  Eisenbud, M. Environmental Radioactivity:
                                               From Natural, Industrial, and Military
    2. Background radiation levels             Sources, 4th ed. San Diego, Calif.: Acade-
combining terrestrial (from K-40,              mic Press, 1987.


                                                                      Answers       247
 McBride, J. P., R. E. Moore, J. P. Wither-    room temperature, H nuclei in neigh-
  spoon, and R. E. Blanco. “Radiological
  Impact of Airborne Effluents of Coal and
                                               boring lattice sites experience enor-
  Nuclear Plants.” Science 202 (1978): 1045.   mous accelerations, as large as 1014
                                               m/s2 in random directions. Sometimes
                                               these accelerations are toward each
191. Cold Fusion                               other, so the two protons can approach
Cold fusion at room temperature is a           very close and perhaps fuse into a He
real but unlikely possibility. The key         nucleus. However, the actual calcula-
idea is the quantum mechanical over-           tion reveals the rarity of this event.
lap of the wave functions of two                   Despite the extreme improbability
nearby H-2 nuclei, for example. Their          of deuteron fusion at room tempera-
wave functions always overlap, no              tures, so-called cold fusion, research
matter how far apart they are. How-            groups worldwide continue its pursuit,
ever, the bigger the value of the wave         as revealed in the references below.
function overlap, the more probable             Iwamura, Y., T. Itoh, M. Sakano, and S.
will be the possibility of the fusion            Sakai. “Observations of Low-Energy
                                                 Nuclear Reactions Induced by D2Gas Per-
process to make a He-4 nucleus.                  meation through Pd Complexes.” Infinite
    Of course, there is a Coulomb bar-           Energy 47 (January–February 2003):
                                                 14–18.
rier to be overcome. In the 1940s came
                                                Mallove, E. F. “The Triumph of Alchemy:
the proposal that muonic atoms—a                 Professor John Bockris and the Transmuta-
proton with a muon replacing the elec-           tion Crisis at Texas A&M.” Infinite Energy
tron—might allow fusion because the              32 (July–August 2000): 9–24.
muonic atom ground state puts the               Miles, M. H., B. F. Bush, and J. J. Lagowski.
                                                 “Anomalous Effects Involving Excess
muon so close to the nucleus on aver-            Power, Radiation, and Helium Production
age that the muonic atom appears neu-            during D2O Electrolysis Using Palladium
tral to the approaching proton.                  Cathodes.” Fusion Technology 25 (1994):
                                                 478–486.
However, calculations have shown
that the produced He isotope decays
too quickly for this scheme to succeed
in fusion energy production.                   192. Fission of U-235
    In gaseous form at room tempera-           There are two major problems to be
ture, two colliding H-2 nuclei do not          overcome in designing a fission device.
get close enough for a large wave func-        The neutron distribution in a pure
tion overlap, being strongly repelled by       U-235 solid would decrease as the
electrical forces acting between two           inverse distance squared from each
positive nuclei. In a solid, however, at       nuclear decay source, and the target


248       Answers
nuclei would be moving away during             chain reaction, but pure Pu-239 can-
the expansion, so one has a diffusion          not start itself because the loss rate of
problem complicated by moving tar-             neutrons exceeds the production rate.
gets. The moving targets contribute at         On the average, each U-235 fission
least two difficulties: the density of         produces 2.5 neutrons for every inci-
targets is rapidly decreasing, and the         dent neutron. At the critical mass of
neutron-capture cross section is a             fissile material the chain reaction will
function of neutron kinetic energy as          be sustained. For U-235 this critical
seen from the reference frame riding           mass is about 7 kilograms for ideal
with each U-235 nucleus. Without the           behavior, requiring a sphere about the
proper neutron capture rate by the             diameter of a baseball of pure U-235.
receding U-235 nuclei, the chain reac-         Surely this baseball would be too hot
tion fizzles out.                               to handle!
     Of course, the nuclear device can-            Diffusion problems of an expand-
not be expected to be pure U-235               ing material would require a neutron-
because the isolation of enough quan-          reflecting strong tamper material
tities of U-235 from U-238 is too diffi-        surrounding the U-235 sphere to delay
cult and too costly. Therefore, there is       the expansion for a few microseconds
mostly U-238 in the expanding solid            to achieve additional fissions before
with some U-235, so we have all the            exploding. At 100 percent efficiency
previously listed problems to solve but        the explosion would be equivalent to
also must account for the nuclear              about 120 kilotonnes of TNT. How-
properties of the U-238 as well as the         ever, no nuclear device is that efficient.
U-235.                                             Declassified records indicate that
     Apparently the Germans during             about 60 kilograms of highly enriched
World War II did not solve these diffu-        uranium was used in the nuclear device
sion problems satisfactorily.                  that was released over Hiroshima,
 Bernstein, J. “Heisenberg and the Critical    Japan, in 1945. The explosive charge
  Mass.” American Journal of Physics 70, no.   for the device detonated over Nagasaki
  9 (2002): 887–976.
                                               three days later was provided by about
                                               8 kilograms of plutonium-239 (>90
193. Minimal Nuclear                           percent Pu-239).
                                                Bernstein, J. “Heisenberg and the Critical
Device                                           Mass.” American Journal of Physics 70, no.
                                                 9 (2002): 887–976.
Pure U-235 can be accumulated into a            Pochin, E. Nuclear Radiation: Risks and
critical mass for a sustained nuclear            Benefits. Oxford: Clarendon Press, 1983.



                                                                       Answers      249
194. Large Nuclei                           shapes. Upon excitation, the nucleus
                                            first deforms into a shape like a rugby
Small nuclei that become excited and        football, with a length-to-height ratio
deformed prefer to lose their energy by     of about two to one. Mg-24 appears to
breaking up into helium nuclei (alpha       behave as if two C-12 nuclei are its
particles) or C-12 nuclei whenever          major components and seems to behave
possible. In fact, researchers often talk   as a superdeformed nucleus in this
about “nuclear molecules” composed          rugby football shape. The next state
of these two entities.                      would have an elongated hyperde-
    The larger nuclei, with more than       formed shape as a result of perhaps six
150 nucleons, usually spin faster when      alpha particles lined up along the long
energy is added, and the result of a        axis. This nucleus is highly unstable,
higher angular momentum state is a          and this nuclear sausage would produce
nucleus that is more deformed. As they      an unmistakable debris pattern.
de-excite, up to about 40 gamma rays            Recent detailed investigations of
are emitted by descending an “excita-       several Pb isotopes have yielded sur-
tion ladder,” producing a characteris-      prises. The angular distribution and
tic gamma ray emission spectrum.            polarization of the gamma rays show
From this spectrum one can determine        that they were not electric quadrupole
the nuclear angular momentum states         (E2) transitions but magnetic dipole
and the nucleus’s deformation shape.        (M1). Classically, M1 radiation is pic-
Superdeformed nuclei were discovered        tured as being emitted from a rotating
in this way.                                current loop, with the field oscillating
    Rotational motion of quantum            at the same frequency as the frequency
objects such as atoms and molecules         of rotation. Similar gamma-ray emis-
has a long and distinguished physics        sion bands have recently been identi-
history. Quantized rotational motion        fied in other nuclei in the mass region
of molecules was first recognized from       around 110, where the nuclei also are
the absorption spectra of infrared light    nearly spherical. These spectra have a
in 1912. The occurrence of rotational       pattern that is typical of transitions
motion of atomic nuclei first became a       between rotation states, which poses
topic of interest in the late 1930s in an   an awkward problem: how can we
effort to explain observed nuclear          explain these regular patterns of M1
excitation spectra by physicists            gamma rays? Apparently there is
Edward Teller and John Wheeler in           much more to understand.
about 1938.                                  Clark, R. M., et al. “Evidence for ‘Magnetic
    Quantum mechanics dictates the            Rotation’ in Nuclei: Lifetimes of States in


250       Answers
  the M1 Bands of Pb-198, Pb-199.” Physical       about 10–8 eV, so that the Fe-57 recoil
  Review Letters 78 (1997): 1868.
                                                  energy of about 0.002 eV produces a
 Macchiavelli, A. O., et al. “Semiclassical
  Description of the Shears Mechanism and         Doppler shift so large that no absorp-
  the Role of Effective Interactions.” Physical   tion in the cooled, thin film normally
  Review C. 57 (1998): R1073.                     occurs. One can cancel this Doppler
 Nolan, P. J., and P. J. Twin. “Superdeformed     shift with a moving absorber or emit-
  Shapes at High Angular Momentum.”
  Annual Review of Nuclear and Particle           ter of only 0.0002 m s–1. Therefore,
  Science. 38 (1988): 533.                        when the eardrum moves forward
                                                  toward the stationary cooled thin film,
                                                  there will be some resonance absorp-
195. Human Hearing                                tion of the gamma ray, so the detector
The Mössbauer Effect has been used                count will decrease. When the
to determine these actual displace-               eardrum moves opposite, there is no
ments of an eardrum. The Mössbauer                absorption. Because the eardrum
Effect utilizes the recoil-less emission          vibrates in a nonlinear fashion, the
of a 14.4 KeV gamma ray (photon)                  details are somewhat more compli-
from an Fe-57 nucleus, say, and this              cated. From the geometries and the
gamma ray is normally absorbed by                 physical properties of the emission and
an Fe-57 nucleus in another object in             the Mössbauer absorption, the dis-
its path. When the emitters (Fe-57                placement values of the eardrum can
atoms placed on the eardrum) are                  be calculated. The sensitivity of this
moving with the eardrum, the emitted              technique allows eardrum displace-
gamma rays pass through the second                ments that are only fractions of a
object, a cooled thin film of Fe con-              nuclear diameter to be detected.
taining some Fe-57 atoms, to be cap-
tured in a gamma-ray photon detector.
    The important physical property               196. 1908 Siberia
here is that the natural linewidth of the         Meteorite
emitted gamma ray is very narrow,
                                                  Willard Libby and Edward Teller
                                                  explored this event with a reasonable
                                                  hypothesis, since no rocky debris was
                                                  ever found, and the amount of damage
                                                  was enormous. If the meteorite were
                                                  made of antimatter, then the ensuing
                                                  matter-antimatter annihilation in the
                                                  atmosphere and during the ground

                                                                         Answers    251
collision would create plenty of ener-     the 1911 C-14 to C-12 ratio detected
getic photons at 0.511 MeV, 935            in the old oak tree from Wisconsin.
MeV, etc., due to electron-positron        Later, a research group of C. Cowan,
annihilations, proton-antiproton anni-     C. R. Atluri, and W. Libby (1965) did
hilations, etc. Many of these photons      publish a similar result for the analysis
would interact with nitrogen N-14 in       of C-14 content in a 300-year-old
the atmosphere directly to make an         Douglas fir from Arizona showing an
excited N nuclear state or indirectly      increase in C-14 in 1911 with the same
via secondary neutron production in        interpretation, supported by R. V.
the atmosphere. The excited state of       Gentry (1966). However, C-14 meas-
N-14 decays to C-14, which increases       urements of a tree by J. C. Lerman,
the atmospheric concentration of C-14      W. G. Mook, and J. C. Vogel (1967)
in the carbon dioxide that is taken in     nearer the blast failed to show an
by plants immediately after the event.     increase in the 1909 ratio.
    An increase in the C-14 to C-12            Several other interpretations of the
ratio should appear in the radiocarbon     1908 meteorite event are possible.
dating of living organisms such as         One of the main proposals is that an
plants, beginning in the year 1908 for     ice-rock comet struck Earth, much like
local trees, and this increase in the      comet Schumaker-Levy struck Jupiter
ratio should appear a few years later      in 1994. Also, one cannot rule out the
for trees in North America, caused by      possibility that a massive rocky mete-
atmospheric mixing of the C-14. One        orite just burned up completely—that
of us (F. P.) was working for the sum-     is, broke into small fragments that
mer in Willard Libby’s laboratory and      burned up before striking the ground.
was assigned to carefully separate the      Chyba, C., P. Thomas, and K. Zahnle. “The
tree rings from an old oak tree, putting     1908 Tunguska Explosion: Atmospheric
                                             Disruption of a Stony Asteroid.” Nature
pieces into vials, and then coding the       361 (1993): 40–44.
vials so that only I knew which vials       Cowan, C., C. R. Alturi, and W. F. Libby.
contained which tree rings. The sam-         “Possible Anti-matter Content of the Tun-
                                             guska Meteor of 1908.” Nature 206
ples were radiocarbon-dated, and then
                                             (1965): 861–865.
the results were plotted by the C-14 to
                                            Gentry, R. V. “Anti-matter Content of the
C-12 ratio versus the calendar year.         Tunguska Meteor.” Nature 211 (1966):
    W. Libby, E. Teller, R. Berger, L.       1071–1072.
Wood, and F. Potter did not publish         Lerman, J. C., W. G. Mook, and J. C. Vogel.
                                             “Effect of the Tunguska Meteor and
their radiocarbon-dating results, which      Sunspots on Radiocarbon in Tree Rings.”
demonstrated a significant increase in        Nature 216 (1967): 990–991.



252      Answers
197. The Standard                                                   Up quark
                                                                    Down quark
Model
                                           Electron neutrino        Charm quark
As far as we know, no such definitive       Electron                 Strange quark
argument for matching specific fami-
lies exists in the Standard Model of
                                           Muon neutrino            Top quark
Leptons and Quarks and their interac-      Muon                     Bottom quark
tions. As long as six leptons cancel out
the anomalies of the six quarks, for
                                           Tau neutrino             Top' quark
example, all is well! Indeed, one can      Tau                      Bottom' quark
use the second family of quarks to can-
cel the anomaly contributions from
the first family of leptons, the third     mathematics. Although the proposed
family of quarks to cancel the second      scheme successfully predicted the mass
family of leptons, and the first family     of the top quark, this geometrical basis
of quarks to cancel the third family of    for the Standard Model awaits confir-
leptons. In fact, any permutation of       mation of other specific predictions for
the traditional lineup of cancellations    collisions, which are under way at Fer-
would succeed.                             milab and soon to be done using the
    This ambiguity in the cancellations    Large Hadron Collider.
probably indicates that the Standard        Glashow, S. L. “Quarks with Color and
Model as understood is incomplete.           Flavor.” Scientific American 233, no. 4
                                             (1975): 38–50.
One would expect the traditional
                                            Kane, G. “The Dawn of Physics beyond the
scheme, but the conceptual under-            Standard Model.” Scientific American 288,
standing provided by the Standard            no. 6 (2003): 68–75.
Model does not dictate uniqueness.          Liss, T. M., and P. L. Tipton. “The Discovery
                                             of the Top Quark.” Scientific American
    One of us (F. P.) has proposed an        277, no. 3 (1997): 54–59.
interesting mathematical argument for       Potter, F. “Geometrical Basis for the Stan-
matching lepton families to quark            dard Model.” International Journal of
families based on correlations among         Theoretical Physics 33 (1994): 279–306.
finite rotational subgroups of the Stan-
dard Model gauge group for the lep-        198. Spontaneous
tons and quarks. In this scheme, each
lepton family and each quark family is     Symmetry Breaking
in a unique subgroup, and the one-to-      Yes. At least two other methods can
one correlations are dictated by the       achieve the same symmetry-breaking


                                                                    Answers        253
result without requiring the Higgs par-   particle if this mechanism is truly the
ticle. The Standard Model is described    source of symmetry breaking and the
by its continuous gauge group SU(3)C      particle masses. If the Higgs particle
× SU(2)W × U(1)Y. The simplest way of     does not show up, then spontaneous
all is to spontaneously break this con-   symmetry breaking to a discrete group
tinuous group to a discrete symmetry      remains an alternative possibility.
subgroup of the continuous group           Coleman, S. Aspects of Symmetry. Cam-
SU(2). That is, the lepton and quark        bridge, Eng.: Cambridge University Press,
                                            1985, pp. 113–130.
flavor eigenstates would be associated
                                           Icke, V. The Force of Symmetry. New York:
with finite rotational subgroups of          Cambridge University Press, 1999, pp.
SU(2) instead of the continuous group.       232–248.
An analogy from geometry would be          Potter, F. “Geometrical Basis for the Stan-
                                            dard Model.” International Journal of The-
to begin with a sphere and then sym-
                                            oretical Physics 33 (1994): 279–306.
metry-break to a regular tetrahedron,
                                           ’t Hooft, G. “Gauge Theories of the Forces
or a regular octahedron, or a regular        between Elementary Particles.” Scientific
icosahedron. Reconciling discreteness        American 242, no. 6 (1980): 104–140.
with the continuous symmetry group
U(1) of quantum electrodynamics may
be a problem, however, where phases
                                          199. Proton Mass
are assumed to vary continuously.         Quantum chromodynamics describes
Another symmetry-breaking approach        the interactions of the quarks. The
is the quark condensate method, which     up and down quark masses are listed
also does not require a Higgs particle.   as ~ 5 MeV/c2 each. However, these
     At present, no Higgs particle has    “current” quarks are not what is
been detected at the accelerators, even   meant by having them confined inside
though its mass is expected to be below   a proton by the color fields. Instead
200 GeV/c2, within the energy range of    one must use the effective mass—the
the large accelerators. Of course, the    “constituent” mass—which accounts
decay of such a Higgs particle is a fla-   for this confinement and which can
vor-changing neutral current weak         be estimated from the Heisenberg
decay, which means that its decay rate    uncertainty principle. Since δxδpx ≥
is severely suppressed, so only a few     h/4π, and each quark is confined
Higgs decays would have been detected     within the proton radius of about one
among the particle debris so far. When    Fermi, we estimate δpx ~ 100 MeV.
the Large Hadron Collider comes           In three dimensions, the total
online in 2005 or later there should      dp ~ (δpx ) 2 + (δpy ) 2 + (δpz ) 2 ~ 170
be copious production of the Higgs         MeV/c2. So at least 510 MeV/c2 of the

254      Answers
proton mass is to be associated with           Aniol, K. A., et al. “Measurement of the Neu-
                                                tral Weak Form Factors of the Proton.”
the “constituent” mass of the three             Physical Review Letters 82 (1999): 1096.
quarks within the proton. The remain-
der is the energy contributions of the
gluons holding the proton together.           200. Right- and Left-
    Most of the properties of protons,
except the spin, seem to be determined
                                              handed Neutrinos?
by these three “valence quarks,” much         No. The weak interaction is associated
like the valence electrons determine          with the SU(2)-weak part of the Stan-
the important chemical properties of          dard Model gauge group that operates
atoms. However, when the proton’s             in the unitary plane—a plane with two
innards are probed more energetically,        complex axes. That is, particle funda-
more structure is found, up to four or        mental lepton and quark states are
five more particles, called “virtual          defined in this unitary plane. All rota-
quarks.” In addition, up to 30 gluons         tions in the normal unitary plane
can be detected. The proton is reveal-        involve only left-handed doublets and
ing its inner sanctum to investigators,       right-handed singlets, dictated solely by
and the view is becoming quite inter-         the mathematics of the geometrical
esting. Quarks, antiquarks, and gluons        transformation. Mathematicians call
can be said to form a thick “soup”            these transformations right and left
inside the proton, and theoretical and        screw operations. So the physical prop-
experimental physicists are working           erty of left-handed doublet states for
together to figure out the recipe.             the weak interaction is dictated by the
    Today we know that the three              mathematical property of rotations in
valence quarks cannot alone account           the unitary plane. Nature simply
for the proton’s spin. The whole “sea”        “knows” the mathematics!
of quarks, antiquarks, and gluons each             The antiparticle eigenstates are in
possess spin, so one must first deter-         the conjugate unitary plane, which is
mine the contribution made by each            gauge-equivalent (not equivalent) to
individual member of this seething            the normal unitary plane, so the
mass. The results so far suggest that the     energy values of particles and antipar-
sea of quarks makes a minimal contri-         ticles are the same, but all other prop-
bution to the overall spin of a nucleon!      erties are opposites. In this conjugate
                                              unitary plane the mathematics dictates
 Abbott, D., et al. “Measurement of Tensor
  Polarization in Elastic Electron-Deuteron
                                              right-handed doublets and left-handed
  Scattering at Large Momentum Transfer.”     singlets. The existence of two gauge-
  Physical Review Letters 84 (2000): 5053.    equivalent but different 2-D complex

                                                                       Answers       255
spaces conjugate to one another dic-          by summing over all the phase
tates that the universe has both parti-       information from its local environ-
cles and antiparticles. Why there exist       ment. Of course, each particle also
so many more particles than antiparti-        provides phase information to its envi-
cles in our present universe remains to       ronment both near and far. The parti-
be resolved.                                  cle’s new location is the region where
 Altmann, S. L. Rotations, Quaternions, and   the phases match best. The calculation
  Double Groups. Oxford: Clarendon Press,     game requires a dynamic limit to how
  1986, pp. 121–123.
                                              many nearby cells are counted in order
                                              to accumulate a good approximation
201. Physics without                          of the phase information and to main-
                                              tain the local geometrical symmetry.
Equations                                     A proof-of-concept calculation has
The best way to use cellular automata         been done by one of us (F. P.) on a
(CA) on computers is to incorporate           desktop computer using thousands of
the fundamental interactions of the           nodes in a 3-D array, but a good cal-
Standard Model of Leptons and                 culation requires millions of cells or
Quarks plus the gravitational interac-        the equivalent.
tion of the general theory of relativity,          The marriage between physics and
or preferably its quantum gravita-            mathematics has been a happy and
tional version when available. We             fruitful one over many centuries.
know that all these fundamental inter-        Mathematical equations, from simple
actions in nature correspond mathe-           algebraic ones to the more challenging
matically to local phase changes, a           differential equations, have allowed us
process that can be simulated with CA         to summarize an enormous amount of
without using equations by using a            physical phenomena into a simple for-
clever enactment of the path-integral         mat. The underlying fundamental
approach to doing all of physics in real      symmetries of nature have been the
time. Not yet fully achieved except by        true source of many of these equa-
very crude approximation, the physics         tions. However, formulating these
of many-particle interactions will be         symmetries as the Schrödinger equa-
accomplished by large-scale grid com-         tion and Maxwell’s equations, for
putation methods or perhaps by the            example, and solving the equations are
equivalent on a quantum computer.             human processes. We cannot expect
    The fundamental idea is to deter-         Nature to do the same when the sim-
mine the present behavior of a particle       pler process of looking locally for


256       Answers
information is more direct. Therefore        sources, was not needed even though
we think that understanding the uni-         its argument would likewise produce a
verse by combining CA with path inte-        dark night sky.
grals will be the physics of future               The critical quantity is the ratio of
generations.                                 the average lifetime tave of a star to the
 Icke, V. The Force of Symmetry. New York:   time T required for the universe to
   Cambridge University Press, 1999, pp.     reach thermodynamic equilibrium.
   178–206.
                                             Starting with a uniform density of
                                             stars, an observer can appreciate that
                                             after a clock time t = tave there will be
       Chapter 10                            an expanding sphere of burned-out
      Over My Head                           stars beyond which lies a shell of lumi-
                                             nous stars. The radiation from this
                                             shell has a maximum radiation density
202. Olbers’ Paradox                         equal to the surface radiation density
German astronomer Heinrich Olbers            from the average star times the ratio
(1758–1840) was not the first scientist       tave/T as long as the clock time t << T.
to ask “Why is the night sky dark?”          But tave is at most a little more than 10
but his name remains connected to this       billion years, while T can be shown to
paradox. The night sky is dark because       be tens of billions of years, so the night
the time required for the radiation field     sky remains dark. Harrison shows that
to reach thermodynamic equilibrium is        this argument is true for all present
large compared to all other time scales      models of the universe and does not
of interest—that is, the lifetime of stars   require a cosmological redshift. He
is far too short for the sky to be as        argues that Lord Kelvin (1901) was
bright as the paradox suggests. In addi-     the first to give the correct answer,
tion, if all the matter in the universe      which Edgar Allan Poe had antici-
were converted to radiation, the equi-       pated in his qualitative cosmological
librium temperature of the universe          speculations. For the detailed calcula-
would be about 20 K, illustrating that       tions, see the references below.
there is insufficient energy to have a         Harrison, E. R. “Why the Sky Is Dark at
                                               Night.” Physics Today 27, no. 2 (1974):
bright sky. Edward R. Harrison in the
                                               30–36.
early 1970s determined this solution
                                              ———. “The Dark Night-Sky Riddle.”
and also determined that the usual             Science 226 (1984): 941–945.
explanation, based on a cosmological          Pesic, P. “Brightness at Night.” American
redshift of the light from distant             Journal of Physics 66 (1998): 1013–1015.



                                                                     Answers     257
203. Headlight Effect                         also are the different clock rates for
                                              the two frames of reference, so the
In the special theory of relativity (STR)     number of photons emitted per second
a result called the headlight effect          on the star and received at Earth will
occurs. One considers the Lorentz-            differ. Moreover, the spectrum of light
Fitzgerald contraction of distances in        will be different as well.
the direction parallel to the constant         Taylor, E. F., and J. A. Wheeler. Spacetime
velocity and no change in the perpen-           Physics. San Francisco: W. H. Freeman,
dicular direction. If the primed frame          1966, p. 69.
is the vehicle frame, then the angle in
the two frames are related by cos φ =
(cos φ′ + v/c)/(1 + v/c cos φ′ ). Substi-     204. Incommunicado?
tuting the appropriate values tells us        No and yes! No, in the normal sense
that cos φ ~ 1, or φ ~ 0°! Therefore, all     case because the relative velocity can
the light is in a very small solid angle      never exceed the speed of light. The
in the forward direction, and only an         successive pulses may arrive less and
observer directly along the line of           less often, but you will never outrun
motion will see the light. You will not       the light.
see the light from the relativistic vehi-          And yes, you would lose commu-
cle passing nearby unless your eye is         nication contact if we allow the space
within the very narrow light cone.            itself to expand, analogous to the
    In the rest frame of the source, the      expansion of the universe in present
star emits light in all directions, yet the   cosmological models. The addition of
calculation reveals that for an observer      velocities is the old classical physics
of a very fast-approaching star, practi-      one, not the relativistic one. The pho-
cally all its light will be shining along     ton velocity is affected by the local
the direction of motion! A fast-              environment; the local substratum
approaching star or galaxy will pos-          (i.e., coordinate system) “drags” the
sess a very narrow bright headlight           photon along. Imagine two local
beam that could miss Earth. Mean-             regions in rapid recession from each
while, a fast-receding star or galaxy         other. If the person in one region fires
may not be seen at all because its light      a photon toward the other, the sub-
is redshifted out of the visible range        stratum of the first region drags the
and practically all its light shines away     photon along, slowing the photon’s
from us!                                      progress toward its target. If the
    So in observing stars, there is this      expansion rate is high enough, the two
STR headlight effect to consider. There       regions can be receding from each

258       Answers
other at light speed or greater, pre-          four test particles inside their spaceship
venting communication between you              and having the capability to measure
and your friend.                               their relative accelerations, they can
 Higbie, J. “Radial Photon Paths in a Cosmic   succeed in determining all the compo-
  Model: A Student Exercise.” American         nents of the Riemann tensor and
  Journal of Physics 51 (1983): 1102–1107.
                                               decide whether they are in the space-
                                               time curvature of a rotating central
205. Local                                     mass. Note that gyroscopes do not
Accelerations                                  help here because one would need to
                                               check their alignment with stars out-
The presence of the massive body can           side, which is forbidden. The challenge
be determined by the trajectories of the       here is to measure a new effect, called
two test masses upon their release. In         intrinsic gravitomagnetism, introduced
the simple case in which the laboratory        by the GTR, that the space-time geom-
is not moving with respect to the mas-         etry and the corresponding curvature
sive body, when released equidistant           invariants are affected and determined
from the object but separated from             by both mass-energy and mass-energy
each other, the two test masses will           currents relative to other mass—that
move toward each other faster than             is, by mass-energy currents that cannot
their mutual gravitational acceleration        be eliminated by a Lorentz transforma-
as they fall toward the body. In addi-         tion. See the Ciufolini and Wheeler ref-
tion, if they are separated vertically so      erence below for the details.
that one test mass begins closer to the         Ciufolini, I., and J. A. Wheeler, Gravitation
massive body than the other, their ver-          and Inertia. Princeton, N.J.: Princeton Uni-
tical separation distance will change as         versity Press, 1995, pp. 358–360.
                                                Kalotas, T. M., A. R. Lee., and R. B. Miller.
they fall. In a uniform gravitational
                                                 “Einstein on Safari.” The Physics Teacher
field their separation distance would            29 (1991): 122–124.
remain fixed in value in each test.              Martin, J. L. General Relativity: A Guide to
     One can extend this problem to              Its Consequences for Gravity and Cosmol-
                                                 ogy. Chichester, Eng.: John Wiley & Sons,
consider a rotating massive body. Can            Ellis Horwood, 1988, pp. 93–94.
observers inside a spaceship determine
by “local” measurements only—that
is, without looking outside—if they are
                                               206. Twin Paradox
in the field of a rotating central mass,        Both special theory of relativity (STR)
or if they are just moving with velocity       and general theory of relativity
V on a Schwarzschild background                (GTR) explanations for the aging of
metric? Yes they can; by using at least        the space-traveling twin should be

                                                                        Answers       259
considered. If by the STR we consider      spaceship can be approximated by an
inertial reference frames only and         equivalent gravitational acceleration—
ignore any accelerations experienced       that is, using the Equivalence Princi-
by the space traveler, a symmetry          ple—we can expect the traveling clock
would exist between the two frames,        to tick slower during the acceleration.
and the twins must both age at the         And that behavior is why the traveling
same rate. Therefore, the accelerations    twin ages less. As seen by a third
experienced by the space traveler          observer at rest with respect to the
make the difference in the aging.          stars and the stay-at-home twin, the
    One can handle these accelerations     clock on the spaceship is changing its
in the STR or in the GTR. Some peo-        rate of ticking during the accelerations.
ple argue that this twin paradox prob-
lem requires only the STR because
there is no curved space-time in the
                                           207. Twin Watches
problem—that is, both twins can be         A watch ticks at its fastest rate when at
considered to be in a flat space-time       rest and when there is no gravitational
because no gravitational accelerations     field. So there are two effects to con-
near a mass are necessary. Then one        sider: (1) from the special theory of
would handle the accelerations for the     relativity (STR), the motion of the
space-traveler twin in terms of STR        watch with respect to the laboratory
calculations, perhaps via the velocity     frame affects the ticking rate; and (2)
parameter technique. A true GTR            the change in gravitational potential
problem, by contrast, would require        according to the general theory of rel-
the physics of the curved metrics near     ativity (GTR) affects the clock ticking
a massive body.                            rate. For a watch in free fall, the two
    The solution of the twin paradox       effects are exactly opposite and cancel!
using the GTR relies on clocks ticking     The two watches agree again when she
slower in a gravitational potential near   takes the second reading.
a mass. The clock at a far distance            Now for the details. First, are there
from the mass ticks at its fastest rate    any symmetry considerations that
and, if brought closer to the mass,        would simplify the calculation? Yes;
begins to tick slower and slower.          the two parts of the journey for the
Therefore a person closer to the mas-      moving watch—the upward and the
sive body, where the gravitational         downward parts—are time reflections
acceleration is greater, ages slower.      of each other, and these two parts
    In cases where the acceleration of a   require the same elapsed time in the


260      Answers
laboratory frame and in the moving-                 From the GTR, the time interval T
watch frame.                                    between clock ticks at radial distance
     Pick the laboratory frame of refer-        R outside of a body of mass M is given
ence. As the watch goes upward in the           by T = T ′ (1 − 2GM / ( Rc 2 )). In the
lab frame at its maximum velocity ini-          limit of very large R, the clock ticks
tially, the STR makes the watch tick            at its fastest rate. By definition, g =
faster as the velocity decreases, and the       GM/R2 at the surface of Earth. Substi-
GTR makes the watch tick faster as              tute the above heights for the two dis-
greater height is achieved. On the              tances from the massive body and take
downward journey, the watch ticks               the difference. One calculates that T2 ~
slower and slower by both STR and               T1 – T ′ g ∆h/c2, a quantity propor-
GTR effects. So we need only calculate          tional to the change in height and a
the changes in the tick rate when the           quantity from the GTR that changes
watch has gone upward by a small                as fast as the quantity from the STR.
amount—∆h, say.                                     So the total change in the tick rate
     From the STR, the time interval T          going upward is canceled by the total
between ticks at velocity v is given by T       change in the tick rate coming down-
= T ′/ (1 − v 2 / c 2 ) , where T ′ is the      ward, to make no net change when
time interval between ticks of the              they are once again at the same height.
watch in its own reference frame. At            If this argument has any flaws, do not
two different heights—h1 and h2 = h1 +          blame either of my colleagues, Richard
∆h—the time intervals between ticks             P. Feynman (deceased) or B. Winstein,
are T1 = T ′/ (1 − v12 / c 2 ) and T2 =         for they know not what they had
T ′/ (1 − v 22 / c 2 ) ,respectively, because   wrought!
the velocities will be different at the
two heights. Since v << c, and assum-           208. Global Positioning
ing a uniform acceleration approxima-
tion for free fall, by the third golden         Satellites
rule of kinematics, v22 = v12 – 2 g ∆h.         The general theory of relativity (GTR)
Substitute the velocities squared into          plays an important role! Corrections
the watch’s time interval relations and         must be made for clock rates in a grav-
expand the square roots in the denom-           itational field in addition to the special
inators by the Taylor series expansion          theory of relativity (STR) corrections
1/ (1 − ε ) ~ 1 + ε / 2 + . . . . One calcu-    to the clocks for the movement of the
lates T2 ~ T1 – T′ g ∆h/c2, a quantity          satellite. Both relativistic effects foul
proportional to the change in height.           up what should have been a pretty


                                                                        Answers     261
simple geometry calculation relating            correction factor of about 4.8 × 10–10
distance to time and velocity. The              for this GTR effect, a little more than
clocks in the satellites tick at a slightly     four times the STR effect, or about
faster rate than identical clocks on the        14.4 centimeters of error every second.
ground because they are in a slightly           So in 10 minutes even this small effect
weaker gravitational field, being far-           produces an error of about 86 meters
ther from the center of Earth. They             if not accounted for. Who would have
tick slower than the Earth-bound                thought that both STR and GTR
clocks because they are moving faster           effects are big enough to play impor-
with respect to the stars.                      tant roles in such a useful practical sys-
    We can estimate the sizes of these          tem as GPS!
effects. From the STR, the time inter-
val T between the ticks of a clock
moving at velocity v is given by T =
                                                209. Solar Redshift
T ′/ (1 − v 2 / c 2 ) , where T ′ is the time   Even though there may be no relative
interval between ticks of the clock in          radial motion between the Sun and the
its own reference frame. At slow                observer on Earth, there is still a grav-
speeds v << c, one expands the square           itational redshift dictated by the gen-
root by the Taylor series expansion             eral theory of relativity (GTR). Recall
1/ (1 − ε ) ~ 1 + ε / 2 + . . . to obtain T     that the infinitesimal distance ds in a
~ T ′ (1+v2/(2c2)). For satellites orbiting     flat Euclidean space with coordinates
Earth in about 720 minutes, their               (r, θ, φ) is defined by ds2 = c2 dt2 – dr2
speed makes the time correction factor          – r2 dθ2 – r2 sin2 θ dφ2. In the gravita-
about 1.1 × 10–10. Multiplied by the            tional field of mass M, this infinitesi-
speed of light, this time factor corre-         mal distance in the GTR becomes the
sponds to a distance error of about 3.3         Schwarzschild line element ds2 = (1 –
centimeters for each second.                    rg /r) c2 dt2 – (1 – rg /r)–1 dr2 – r2 dθ2 – r2
    From the GTR, the time interval T           sin2 θ dφ2, where rg = 2GM/c2 and G is
between clock ticks at radial distance          the gravitational constant.
R outside of a body of mass M is given               We see that near a massive body
by T = T ′ (1 − 2GM / ( Rc 2 )) . Con-          such as the Sun, the time coordinate
sider two different radii: r1 = 6.37 ×          includes a factor (1 − rg / r ) , where r
106 m and r2 = 2.02 × 107 m. Substi-            is the position of the light measured
tute the two radii for the two distances        from the center of the Sun. By evaluat-
from the massive body and take the              ing this factor at the Sun’s surface and
difference. One calculates a clock              at Earth’s distance, one finds that the


262        Answers
clocks at the two distances are ticking    precession of about 43 seconds of arc
at different rates, faster for bigger r.   per Earth century. There are many
One approach is to assume that the         other precessional effects acting on the
photon does not change its inherent        orbit, including effects from all the
physical properties—for example, it        other planets orbiting the Sun, all
maintains its characteristic frequency     these perturbations amounting to a
established during the emission            whopping 532 seconds of arc per cen-
process at the surface of the Sun. Then    tury, all but the residual 43 seconds
the observer on Earth, who has the         being explained with Newtonian
faster-ticking reference clock with        mechanics.
respect to the stars, will measure a           When the angular change around
lower photon frequency and see the         the orbit is calculated with GTR in the
light as redshifted.                       φ-coordinate and then independently
    A second approach determines           in the r-coordinate, there is disagree-
that the change in r for the photon can    ment, which is the conceptual source
be shown to correspond to a change in      of the effect. Or one can assign an
gravitational potential. The photon        additional equivalent mass distribu-
essentially begins in a gravitational      tion for the energy in the gravitational
potential energy valley and climbs         field surrounding the Sun, creating a
upward to reach Earth. Its total energy    metric that does not correspond to a
must remain constant, so the increase      1/r potential, perhaps 1/r2 or 1/r3, or
in gravitational potential energy is       some other function of r instead of
matched by the decrease in photon          the Newtonian inverse r potential. All
energy—that is, a redshifted photon—       these functions of r will exhibit a pre-
because its energy E = hν.                 cession of the orbit.
 Krane, K. S. Modern Physics. New York:        In addition, a body in orbit such
  John Wiley & Sons, 1983, pp. 438–442.    as a planet orbiting the Sun actually
                                           does not obey Kepler’s third law pre-
                                           cisely. That is, even when we ignore
210. Orbiting Bodies                       the precession of the orbit by assuming
The general theory of relativity (GTR)     its return to the same angle with
in the Schwarzschild metric approxi-       respect to the stars, the period of orbit
mation for the space-time metric           needs correction. This period of orbit
about the Sun predicts a precession of     correction is a so-called fourth inde-
the planetary orbit. Mercury, for          pendent general test of the GTR, in
example, accumulates a total GTR           addition to the gravitational redshift,


                                                                  Answers     263
the deflection of starlight, and the pre-         metric surrounding a star, for exam-
cession of an orbit.                             ple, the path of the light from a distant
      The correction to the period of            star is diverted toward the Sun from a
orbit, as calculated by Preston and              straight-line path when passing near
Weber in the reference below, begins             a massive body such as our Sun. Half
by putting the reference clock at the            of the deflection angle is caused by
center of the orbit. The classical New-          the Newtonian attraction of the Sun;
tonian period of orbit is given by               the second half is caused by the
                                   (GM)
Kepler’s third law: T = 2π a3/2/√ ,          geometrical modification, called cur-
where a is the semimajor axis of the             vature, of space by the Sun. Gravita-
ellipse and M is the mass value of the           tional lensing ideas have been around
central body. For the elliptical orbit of        for about 200 years, but only in the
eccentricity ε, in the GTR one can cal-          past decade or so has gravitational
culate the period of orbit in the radial         lensing played an important part in
coordinate Tr = T (1/α + 3/2 rg/r) and           astronomical measurements.
in the φ-coordinate Tφ = T (1/α – 3/2                If the intermediate massive body is
(rg /r) (ε2/α)), where α = (1 – ε2)3/2. For      a galaxy, then light from distance
orbiting bodies near potential black             sources behind this galaxy will be
holes, this correction can get large as          focused somewhat by the two effects,
the radial distance r approaches the
Schwarzschild radius rg = 2GM/c2.
 Krane, K. S. Modern Physics. New York:                Hubble space telescope
  John Wiley & Sons, 1983, pp. 438–442.
                                                       lensing image
 Landau, L. D., and E. M. Lifschitz. The Clas-
  sical Theory of Fields, 4th ed. Sydney: But-
  terworth-Heinemann, 1987, pp. 328–330.
 Preston, H. G., and J. Weber, “The Period of
  Orbit as a Test of General Relativity.”
  Physics Essays 6 (1993): 465.


211. Gravitational
Lensing
The general theory of relativity (GTR)
tells us that all forms of energy are
affected by a gravitational field,
including the energy carried by pho-
tons of light. In the Schwarzschild

264       Answers
just like the light going through a glass     Wambsganss, J. “Gravity’s Kaleidoscope.”
                                               Scientific American 285, no. 5 (2001):
lens is refracted to a focus. However,         64–71.
the geometry of the light focusing is
much more complicated for gravita-
tional lensing than for a simple sym-        212. Cosmological
metrical convex lens, for several
                                             Redshifts
reasons. The light may be focused
onto a line instead of a point, for          There are three distinct causes for the
example, in an ideal case. Therefore,        spectral shift of light emitted (or
astronomers can use galaxies as lenses       absorbed) by a galaxy: the kinematical
to gather more light from far objects        Doppler shift of the special theory of
better. The focus may be poor, but the       relativity (STR), the gravitational red-
greater intensity allows many spectro-       shift of the general theory of relativity
scopic techniques to work better.            (GTR), and the cosmological redshift
    Usually the image resolution of          caused by the expansion of the uni-
the distant object is quite limited by       verse. These three effects cannot be
inhomogeneities in the intermediate          distinguished from one another by
galaxy. However, these properties of         observing the spectrum of a single
the intermediate galaxy can be exam-         galaxy or other single light source.
ined quite well! In fact, if the classical   One can separate out the kinematical
application of the GTR to calculate the      Doppler shift for a cluster of galaxies,
focusing effects in gravitational lensing    however, via statistical methods.
is correct, then the total mass of the           The standard explanation of the
intermediate galaxy and its mass distri-     cosmological redshift says that the
bution can be determined. Galaxy             coordinate system of the universe is
masses determined with gravitational         expanding while the galaxies remain
lensing have disagreed with the very         at their local coordinate values. One
successful proposal for a modified           can use an expanding balloon to
Newtonian dynamics (MOND) within             “mimic” this type of behavior. Inflate
a galaxy region and have supported the       a balloon enough to enable you to
“dark matter” models. However, the           draw a coordinate system on its sur-
game is not over for MOND because            face. Place some galaxies on the bal-
large-scale gravitational quantization,      loon surface. Now inflate the balloon
either from a version of M-theory and        further. The galaxies are farther apart,
superstrings or from some other quan-        but they maintain the same coordinate
tization scheme, may come to the res-        values. Or one can use Earth. If Earth
cue eventually.                              began to expand, Philadelphia and Los

                                                                    Answers     265
Angeles would move apart, yet each            213. Tired-Light
would retain its present longitude and
latitude. Distances are stretched, so
                                              Hypothesis
wavelengths will become longer.               The only two specific pieces of evi-
    Perhaps another viewpoint will be         dence are the time dilation arising
helpful. One does not need to expand          from the expansion of the universe,
space in this view. According to S.           and the spectral shape of the cosmic
Weinberg in the Chown reference               microwave background. Astronomers
below, simply accept the fact that            see that exploding stars in distant
“every bit of the universe is rushing         galaxies brighten and fade more
away from every other bit”—that is,           slowly than those nearby. If the star
“the galaxies are exploding away from         emits a light pulse on January 1 and a
each other, as any cloud of particles         second pulse on February 1, these two
would do if they are set in motion            pulses are separated by one light-
away from each other.” The matter             month. As they travel toward Earth,
inside the individual galaxies does not       their separation distance increases,
take part in the general expansion            perhaps doubling, so that they are
because the local gravity holds the           received two months apart. The tired-
local matter together. The universe’s         light hypothesis cannot explain this
expansion appears just beyond the             extended time interval. In fact, distant
frontiers of the Local Group of galax-        supernovas are observed to wax and
ies, about 4 million light-years from         wane more slowly than nearby ones.
the Local Group’s center of mass.                 The observed spectrum of the
    M. L. Bedran in the reference listed      microwave background radiation is a
below compares the Doppler redshift           perfect blackbody shape, easily
to the cosmological redshift for a            explained by the expansion of the uni-
galaxy with z = 1, where 1 + z = exp          verse from a thermodynamic equilib-
[v/c], determining that a 2.4 percent         rium condition. For the tired-light
difference between the two redshifts          hypothesis, an initial blackbody spec-
exists for this galaxy with an STR            trum does not remain a blackbody
value of v/c = 0.6.                           spectrum as the light becomes red-
 Bedran, M. L. “A Comparison between the      shifted.
  Doppler and Cosmological Redshifts.”         Croswell, K. The Universe at Midnight:
  American Journal of Physics 70 (2002):        Observations Illuminating the Cosmos.
  406–408.                                      New York: Free Press, 2001, p. 76.
 Chown, M. “All You Ever Wanted to Know
  about the Big Bang.” New Scientist (April
  17, 1993): 32–33.


266       Answers
                                            Hawking, S. W. A Brief History of Time:
214. Black Hole                              From the Big Bang to Black Holes. New
Entropy                                      York: Bantam Books, 1988, pp. 104–110.
                                            Penrose, R., The Emperor’s New Mind.
There should be radiation from the           Oxford: Oxford University Press, 1989,
black hole—that is, from the sur-            pp. 361–363.
rounding space, not from inside the
black hole, because nothing can get        215. Black Hole
out. This Hawking radiation was first
calculated in the 1970s and awaits
                                           Collision
experimental verification.                  Yes. The two black holes should
    By taking quantum mechanics into       coalesce like two liquid drops. We
account, particles and antiparticles are   need to ensure that the entropy after-
being created continually via virtual      ward in the coalesced final state is
pair creation in the vacuum. When this     greater than the entropy in the initial
process occurs near a black hole, one      state. We do that by adding up the
particle of the pair may be “eaten” by     entropy in the two states, separate
the black hole and the other may           black holes versus one larger black
escape. In the thermal equilibrium         hole with gravitational waves carrying
state, the amount of energy that the       away some energy and entropy in the
black hole loses to Hawking radiation      final state.
is exactly balanced by the energy              The black hole entropy is propor-
gained by swallowing other “thermal        tional to the event horizon area, which
particles” that happen to be running       grows as the mass to the fourth power.
around in the “thermal bath” in which      Suppose we take two black holes, one
the black hole finds itself.                of mass M1 and the other of mass M2.
    The temperature of a nonrotating       Their initial total entropy is propor-
black hole is given by T = hc3/(8πkGM),    tional to M 14 + M 24. If they merge and
where h is Planck’s constant and k is      their final total mass is approximately
Boltzmann’s constant. Note that this       M1 + M2, then their final total entropy
expression connects gravitation, ther-     is proportional to (M1 + M2)4, which
modynamics, and quantum mechanics.         you can verify is greater than the orig-
For black holes of a few solar masses,     inal total entropy, so the reaction will
the temperature is only about 10–6 K!      go. In cases where the final entropy is
The smaller black holes with little mass   only slightly greater than the initial
will be at a much higher temperature,      entropy for the black hole parts only,
contrary to intuition.                     one may need to add in the entropy in


                                                                  Answers     267
the gravitational waves to ensure a            so that the axis of the tube and the
bigger inequality                              path of the ray coincide. Measure-
    Thus a bigger mass means a bigger          ments with a straight ruler verify that
surface area, which means a bigger             the axis of the tube is circular, yet
entropy than if the two smaller black          because of the bending of the light
holes remained apart. Simulations in           rays, the tube is seen as absolutely
3-D of colliding black holes and their         straight by an observer on the axis. A
emitted gravitational waves can be             lamp held at the axis by a colleague
seen on the Internet.                          will appear dimmer to you as he or she
 Bekenstein, J. D. “Black-Hole Thermo-         walks away along the axis, but the
  dynamics.” Physics Today 33, no. 1 (1980):   lamp is never obscured, so you must
  24–31.
                                               conclude that the tube is straight.
                                               Therefore, along this circular path one
                                               would expect no centrifugal force
216. Centrifugal Force
                                               effects.
Paradox                                            Instead of the tube being around
The conceptual resolution of this par-         the circular light path, consider the
adox starts with the consideration of          tube to be around a smaller circular
light paths near the black hole. The           path centered on the black hole. With
general theory of relativity (GTR) pre-        rulers one can again verify that the
dicts that there should be light paths         tube curves to the left, with the black
around the black hole that are circular        hole on the left as one walks forward.
at a radial distance of 1.5 times the          The outward direction is to the right.
gravitational radius rg = 2GM/c2.              Everyday experience tells us that the
Around one of these circular light             centrifugal force pushes outward.
paths imagine a circular tube centered         Again your colleague walks away with
exactly on the path of the circular ray        the lamp held along the axis of the
                                               tube. If somehow the light rays were
                                               not bent by the gravitational field of
  Black hole           Light ray               the black hole, you would see the lamp
                                               disappear behind the left side of the
                                               tube, and you would conclude that
                                               the tube bends to the left. If the path is
                                               the one discussed above, the lamp is
                                               always in view. But the tube is so close
                                               to the black hole that the light rays
Tube                                           bend even more than circular rays. So

268       Answers
you actually see the lamp disappear to           In conventional geometry, the
the right. Therefore the outward direc-      geodesic is the shortest curve between
tion is to the left, and you must predict    two points measured by counting how
that the centrifugal force would push        many rulers fit along the curve. In a
to the left!                                 flat space—that is, in a space free from
    Several related paradoxes for paths      gravitational fields—the geodesic is a
near a massive body are discussed in         straight line. In the GTR one can
the references, including the fact that      define the distance between two points
for a rocket to maintain a constant          in space as half the time it takes for
speed the boosters would need to fire         light to travel from one point to the
perpendicularly to the path, but their       other and back, multiplied by the
force is not dependent on how fast the       speed of light. In flat space, the two
rocket is moving.                            definitions agree.
 Abramowicz, M. A. “Black Holes and the          In 4-D space-time, light always
  Centrifugal Force Paradox.” Scientific     moves along geodesics and traces the
  American 268, no. 3 (1993): 74–81.
                                             geometry of space-time. In a 3-D space
 Abramowicz, M. A., and J. P. Lasota. “On
  Traveling Round without Feeling It and
                                             warped by a gravitational field, how-
  Uncurving Curves.” American Journal of     ever, the light rays are curved and do
  Physics 54 (1986): 936–939.                not coincide with geodesics in general,
                                             so the geometry of space is not traced
                                             by light rays.
217. Geodesics and                            Abramowicz, M. A. “Black Holes and the
                                               Centrifugal Force Paradox.” Scientific
Light Rays                                     American 268, no. 3 (1993):74–81.
                                              Abramowicz, M. A., and J. P. Lasota. “On
The two statements are not in conflict.
                                               Traveling Round without Feeling It and
One must always distinguish geodesics          Uncurving Curves.” American Journal of
in four-dimensional space-time from            Physics 54 (1986): 936–939.
geodesics in three-dimensional space.         Misner, C. W., K. S. Thorne, and J. A.
                                               Wheeler. Gravitation. San Francisco: W. H.
Light rays always follow geodesics in
                                               Freeman, 1973, pp. 31–34.
4-D space-time, but these paths are
not necessarily geodesics in 3-D space.
An analogy is helpful. Each great circle
on a globe is a geodesic line on the
                                             218. Galaxy Rotation
two-dimensional surface but, being a         To retain a Newtonian gravitation
circle, the great circle is not a geodesic   explanation for the rotation properties
line in the 3-D Euclidean space in           of a galaxy, additional matter, called
which the globe sits.                        “dark matter,” in a halo surrounding

                                                                     Answers      269
the galaxy has been proposed, its mass      Briefly, the large-scale structure and
being about 10 times the mass of the        behavior of the galaxy may result from
visible mass! Known particles such as       the galaxy being in some quantization
electrons, protons, neutrinos and so        state. In modeling this type of theory,
on cannot be its major constituents;        all the disk stars would be in the same
otherwise the particles in the halo         quantization state independent of posi-
would have been detected already.           tion radially and, by the virial theorem,
Some exotic form of matter/energy is        must have the same tangential velocity
required in this galactic halo. How-        V = GM2/(nJ), where M is the amount
ever, any type or types of possible         of visible mass, n is a small integer, and
“dark matter” in sufficient quantities       J is the total angular momentum of this
has not yet been found.                     visible mass of the galaxy. Substituting
    An interesting proposal that does       reasonable values for our Galaxy (the
not require “dark matter” is called         Milky Way), for example, one obtains
MOND, an acronym for modified               a value near to the measured value of
Newtonian dynamics, which applies           V = 220 km s–1. This theory predicts
whenever the inward radial accelera-        that the next quantization state would
tion in the galaxy is below the value a0    have exactly half the disk tangential
= –1.1 × 10–10 m s–2, an incredibly         velocity and, indeed, in 2003 a mass
small amount by Earth standards but a       current of stars circulating the Galaxy
value that occurs in most galaxies.         just beyond the “edge of the easily vis-
Essentially, MOND replaces the New-         ible disk” with a tangential velocity of
                                √ (gN a0
tonian acceleration gN by g =  ) .    110 km s–1 was determined serendipi-
All galaxies examined so far seem to        tously from data collected by the Sloan
obey the consequences of this ad hoc        Digital Sky Survey (SDSS)! Whether
rule by using the galactic visible matter   this proposed large-scale quantization
only, but its possible origin in terms of   faithfully represents gravitational
fundamental physics principles is           behavior in galaxies and in the uni-
being investigated still. The major         verse remains to be fully examined.
problem for MOND has been its                Cline, D. B. “The Search for Dark Matter.”
inability to accommodate the empiri-          Scientific American 288, no. 3 (2003):
                                              50–59.
cal results on the focusing of distant
starlight by gravitational lensing.          Milgrom, M. “Does Dark Matter Really
                                              Exist?” Scientific American 287, no. 2
    An even more exotic solution has          (2002): 42–52.
been proposed to explain galaxy rota-        Preston, H. G., and F. Potter. “Exploring
tion without requiring “dark matter.”         Large-Scale Gravitational Quantization



270      Answers
  without h-bar in Planetary Systems, Galax-    high temperature now exhibits a low-
  ies, and the Universe.” E-print archive for
  physics papers. http://lanl.arxiv.org/abs/
                                                temperature blackbody spectrum
  gr-qc/0303112 (2003).                         because the expansion of the universe
                                                has “stretched the wavelengths.”
                                                     The stars we see are not in thermal
219. Cosmic                                     equilibrium as a collective whole. One
                                                cannot produce a perfect blackbody
Background Radiation                            spectrum at any temperature by sim-
Amazingly, cosmic background radia-             ply using billions of stars that are not
tion (CBR) has a perfect blackbody              in thermal equilibrium as a whole and
distribution! This CBR is uniform and           adding up their radiation intensities in
isotropic in the universe to one part in        the universe. One cannot obtain a
100,000 and amazingly flat over large            blackbody spectrum from many other
spatial regions—that is, large solid            hypotheses about the cosmological
angles. One suspects that these large           redshift of light from distant objects,
spatial regions, even those in opposite         such as the tired-light effect wherein
directions in the sky, have always              the light loses energy during tranversal
been in communication with each                 of the universe.
other, to make them so uniform. Of                   One could, however, speculate that
course, smaller regions have their indi-        the galaxies have never changed their
vidual characteristic galaxies, clusters        average separations, that there has
of galaxies, and so on.                         been no coordinate expansion as
    The most popular interpretation,            stated in the standard inflationary
the standard inflationary model of the           model. The cosmological redshifts
universe, requires the universe to orig-        would correspond to the redshift pro-
inate with the far regions much closer          duced by an “effective cosmological
to each other in thermal equilibrium            gravitational potential well”—for
for the observed uniformity to develop,         example, in which the source sits
then for a very fast inflation to occur          lower in the well than the observer,
that separated them out of communi-             true for all sources and all observers in
cation reach. Now we see these galax-           the universe. The galaxies would be
ies, once close together, in opposite           closer together at all epochs, in com-
directions in the universe and large            munication with each other, and in
regions with the same large-scale char-         thermal equilibrium, so the measured
acteristics in all directions. Their origi-     uniformity would be expected—that
nal collective blackbody spectrum at a          is, all directions should look the same.


                                                                        Answers      271
One consequence would be that one             proposed by Nottale, including many
could never see galaxies beyond about         sets with larger integers.
12 billion light-years or so, the actual          Extrasolar planetary systems with
distance value depending on the aver-         three planets have been found, but
age matter/energy density of the              their statistical fits allow several sets of
vacuum. The redshifts would be inter-         integers also, so they are not yet the
preted as “effective recession veloci-        definitive test. Unfortunately, we must
ties” that reach light speed at this far      wait for a definite extrasolar system or
distance.                                     a precise laboratory test to resolve
 Hasinger, G., and R. Gilli. “The Cosmic      the issue of whether the patterns are
  Reality Check.” Scientific American 286,    simply numerology or an exhibit of
  no. 3 (2002): 60–67.
                                              part of a new fundamental gravita-
 Peebles, P. J. E. “Making Sense of Modern
  Cosmology.” Scientific American 284, no. 1   tional theory.
  (2001): 54–55.                               Lynch, P. “On the Significance of the Titius-
                                                Bode Law for the Distribution of the
                                                Planets.” Monthly Notice of the Royal
220. Planetary                                  Astronomical     Society   341     (2003):
                                                1174–1178.
Spacings                                       Nottale, L. “Scale-Relativity and Quantiza-
                                                tion of Extra-Solar Planetary Systems.”
The orbital radii of the planets only           Astronomy & Astrophysics 315 (1996):
roughly follow the Titius-Bode law, so          L9–L12.
this specific pattern is probably bogus.        Nottale, L., G. Schumacher, and J. Gay.
However, L. Nottale and his research            “Scale Relativity and Quantization of the
                                                Solar System.” Astronomy & Astrophysics
group have shown that the planets               322 (1997): 1018–1022.
obey a generalized Schrödinger-like
wave equation (with one unknown
parameter) that has solutions dictating       221. Entropy in the
a regular pattern for where orbiting          Big Bang
bodies reach an equilibrium radius.
                                              We quote from the reference listed.
The planets of the Solar System occupy
                                              The “standard” answer attempting to
only these radial positions and leave
                                              explain the paradox is:
some equilibrium radii unoccupied,
perhaps a consequence of their history          True, the fireball was effectively
of formation. However, even though              in thermal equilibrium at the
Nottale’s fits are extremely good, there         beginning, but the universe at that
are several other sets of small integers        time was very tiny. The fireball
that statistically fit as well as the set        represented the state of maximum



272      Answers
  entropy that could be permitted for      several kilometers or more. One could
  a universe of that tiny size, but the    place the rotating laboratory gravita-
  entropy so permitted would have          tional wave source several kilometers
  been minute by comparison with           away or more from the gravitational
  that which is allowed for a universe     wave detector, but the decrease in the
  of the size that we find it to be        radiation field intensity with distance
  today. As the universe expanded,         squared combined with the low sensi-
  the permitted maximum entropy            tivity of the detectors make this
  increased with the universe’s size,      arrangement unlikely to work with
  but the actual entropy in the uni-       present detectors. Therefore, as far as
  verse lagged well behind this per-       we know, there has never been a true
  mitted maximum. The second law           test of the gravitational wave response
  arises because the actual entropy is     of any detector to gravitational radia-
  always striving to catch up with         tion using laboratory sources of grav-
  this permitted maximum.                  ity waves.
                                               There have been two fundamental
This answer cannot be correct if the
                                           types of gravitational wave detectors:
universe will eventually suffer a “big
                                           the Weber bar antenna, named after
crunch,” for then the argument would
                                           pioneering physicist Joseph Weber,
apply again in the reverse direction!
                                           who began this research field in the
We are at an impasse.
                                           1950s with his one meter diameter
 Penrose, R. The Emperor’s New Mind.
  Oxford: Oxford University Press, 1989,   suspended aluminum bar; and the
  pp. 317–330.                             interferometer type such as LIGO, first
                                           analyzed by the same Joseph Weber
222. Gravitational                         and his students. The classical calcula-
                                           tion of the resonant response of the
Wave Detectors                             Weber bar reveals just how limited is
For all kinds of waves, for locations      its sensitivity to gravitational waves
beyond several wavelengths, the solu-      originating in our Solar System and
tions of the wave equation correspond      Galaxy. However, if the Weber bar
to the radiation field transporting        antenna actually behaves differently
energy and momentum from the               than originally expected, as a collec-
source into the surrounding space.         tive quantum oscillator responding
When considering possible sources of       coherently, say, then it will respond
gravitational waves in the Galaxy and      well to all frequencies of the incident
beyond, the wavelengths are typically      gravitational waves. Hundreds to


                                                                  Answers     273
thousands of vibrational modes could       exceed the cubic curve for the three
be excited in a large range of frequen-    types of spaces: positively curved, flat,
cies with an increase in sensitivity of    or negatively curved, respectively.
many powers of ten.                             On a “small” scale, when the total
    At this time there has been no sub-    number of sources is less than a few
stantiated detection of gravitational      hundred, there can be a relatively large
waves by either type of detector.          uncertainty in the general behavior of
Weber reported a twice-daily response      the plotted curve. But as more and
of his two side-by-side almost-identi-     more sources are counted at farther
cal bar antennas for orientations          distances, the asymptotic behavior
pointing toward the center of the          should become apparent. However,
Galaxy during a period of almost two       adjustments must be made for the
decades, but no other researcher has       finite velocity of light and for possible
verified this behavior with an inde-       evolutionary changes in the sources.
pendent detector. So we must wait for      Distant sources are sampled at an ear-
the first detection of gravitational       lier time, possibly at a closer distance.
waves by LIGO or other detectors.               If the universe is actually represen-
Unfortunately, interferometer types of     tative of a discrete space, one can show
detectors such as LIGO and VIRGO           that by counting many sources one can
cannot operate as a collective quan-       determine the curvature in the limit as
tum oscillator.                            the number of sources becomes large.
                                           Think of a lattice of points as one sim-
                                           ple example, such as a regular lattice
223. Space Curvature                       of atoms in a solid. By counting the
The proposed method for determining        nearest neighbors only, then the next
the curvature of space will work for       nearest neighbors, and so on, one even-
both continuous and discrete spaces.       tually approaches asymptotically to a
If we assume a uniform density of          plotted line from which the curvature
stars, or galaxies if we choose to count   can be determined.
galaxies, the number N of this particu-         Of course, in a discrete space, one
lar kind of source within a sphere of      must be careful not to count over and
radius R in a Euclidean space (zero        over the images of the same source.
curvature space) is given by N = ρ         For example, imagine space divided
4πR3/3, where ρ is the uniform den-        into identical cubes next to each other
sity. When N is plotted against the dis-   and filling all space. Standing inside
tance, N will fall short, match, or        one cube, we can look to our right to


274      Answers
see ourselves inside the first cube on             Wolfram, S. A New Kind of Science. Cham-
                                                   paign, Ill.: Wolfram Media, 2002, pp.
our right looking away to the next                 433–540.
cube, and so on. Each successive image
will be dimmer and will be earlier in
time because the light does not travel           224. The Total Energy
infinitely fast. If our real space in the         Yes, there can be the creation of mat-
universe is discrete, the cube size              ter out of nothing with no violation of
would be enormous, certainly way                 conservation laws! First proposed in
beyond the size of our Local Group of            1958 by H. Margenau and later recal-
galaxies; otherwise we would have                culated in more detail by N. Rosen
detected this discreteness already by            and others in 1994, the gravitational
having seen multiple images of our               energy cancels the mass energy in a
own Galaxy.                                      closed, homogeneous universe.
    If space in the universe is curved,              The simplest general approach was
then cubes will not fill the space.              done by Margenau. Consider a finite
Mathematicians point out that one of             spherical universe of radius R filled
the dodecahedral spaces would be the             with matter and radiation of equiva-
simplest space-filling for a negatively           lent total mass M. The gravitational
curved discrete space, the most likely           potential energy is the negative quan-
type of space curvature for the uni-             tity –kGM2/R, where G is the gravita-
verse. However, the curvature of the             tional constant and k is a positive
universe is not known unambiguously              numerical factor not greatly different
yet, although a flat space with no cur-           from 1. The total energy E in the uni-
vature will nicely fit the present data in        verse is then E = Mc2 – kGM2/R. Using
the standard model of an inflationary             representative values such as R = 1.3 ×
big bang universe.                               1026 m and a mass density of 8 × 10–27
 Eckroth, C. A. “Counting Distant Radio          kg/m3, we estimate k ~ 2.4 when E =
  Sources to Determine the Overall Curva-        0. Nathan Rosen and others showed
  ture of Space.” The Physics Teacher 30
  (1992): 92–93.                                 that the gravitational energy cancels
 Gruber, R. P., A. D. Gruber, R. Hamilton,       out the mass energy without resorting
  and S. M. Matthews. “Space Curvature and       to numerical estimates.
  the ‘Heavy Banana Paradox.’” The Physics        Cooperstock, F. I., and M. Israelit. “The
  Teacher 29 (1991): 147–149.                      Energy of the Universe.” Foundations of
 Levin, J. How the Universe Got Its Spots:         Physics 25 (1995): 631–635.
  Diary of a Finite Time in a Finite Space.       Jammer, M. Einstein and Religion: Physics
  Princeton, N.J.: Princeton University Press,     and Theology. Princeton, N.J.: Princeton
  2003, pp. 132–155.                               University Press, 1999, pp. 201–203.



                                                                         Answers     275
 Margenau, H. Thomas and the Physics of        will be overwhelmed by many other
  1958: A Confrontation. Aquinas Lecture
  23. Milwaukee: Marquette University Press,
                                               particle decays into the same final
  1958, p. 41.                                 products. Perhaps when the Large
 Rosen, N. “The Energy of the Universe.”       Hadron Collider is turned on in a few
  General Relativity and Gravitation 26        years, with its very high rate of pro-
  (1994): 319–321.
                                               duction of quarks, the statistics will be
                                               so much better that the b′ quark
                                               should be easy to find.
225. Different                                     If the b′ quark is found, then we
Universes?                                     expect that all other fundamental
                                               physical constants should also be
If the lepton and quark masses are dic-        derivable from mathematical invari-
tated by fundamental mathematical              ants. If the proposed scheme is correct,
quantities, we are behooved to con-            then our universe is the only universe
sider that all fundamental quantities in       possible. Even exotic speculations such
nature have an origin in fundamental           as time travel could be eliminated if
mathematics. There can be no alterna-          the direction of time is one of the
tive universes, each supposedly having         innate properties of the particle state
different fundamental constants, for           definition. However, we must remem-
they must have the same mathematics            ber always that Nature is more clever
dictating the same physical constants.         than we can hope to be, so we must
     In 1994 F. Potter, within the con-        continue to test every reasonable pro-
fines of the Standard Model of Lep-             posal for the truth.
tons and Quarks, related the lepton             Linde, A. “The Self-Reproducing Inflation-
and quark mass ratios to a mathemat-             ary Universe.” Scientific American 279, no.
ical invariant called the elliptic modu-         11 (1994): 48–55.
lar invariant J, which is invariant             Potter, F. “Geometrical Basis for the Stan-
                                                 dard Model.” International Journal of The-
under all linear transformations. The            oretical Physics 33 (1994): 279–306.
critical prediction is a fourth quark           Tegmark, M. “Parallel Universes.” Scientific
family with a b′ quark mass of about             American 288, no. 5 (2003): 40–51.
80 GeV/c2 and a t′ quark mass of
about 2,600 GeV/c2. Although
searches for a b′ quark have been
under way at the Fermilab collider for
several years, its existence cannot be
ruled out yet because the decay reac-
tions have very low probability and

276      Answers
         Chapter 11
        Crystal Blue
        Persuasion

226. Iodine Prophylaxis
The iodine tablets, usually potassium      information, but a little mathematics
iodine, work by “topping up” the thy-      reveals the answer.
roid gland with stable iodine to reduce        Note that the rear wheel of a bike
its accumulation of any radioactive        always points to the place where the
iodine that may have been released         front wheel touches the ground.
into the environment by a nuclear          Therefore, the tangent to the rear
accident. Inhalation of the radioactive    wheel track will always cross the front
iodine in the air is the major route of    wheel track, while the front wheel
entry into the body and the thyroid.       track does not exhibit this geometrical
For maximum benefit, the iodine            property.
tablets should be taken before the             Once we identify the rear track, we
radioactivity fallout reaches your area;   can pick two random points on it and
otherwise the tablets themselves will      extend the tangents to where they
have become radioactive.                   cross the front track in both direc-
                                           tions, measure the segments, and
                                           determine which direction yields seg-
227. Bicycle Tracks                        ments of the same length. Since a bicy-
In “The Adventure of the Priory            cle cannot change its length, we learn
School,” Sherlock Holmes not only          the direction of travel.
draws a map of the neighborhood of
the school but also examines several
sets of tyre tracks on the moor. He
                                           228. Earth Warming
needed to determine the bicycle’s          Yes, there can be fluctuations in the
direction of travel solely from his        thermal energy conducted and con-
inspection of the tracks. Holmes could     vected to Earth’s surface from sources
tell from the depth of the wheel           within. There are several types of
impression which track was made by         thermal energy sources, including
the rear tyre. You don’t have that         radioactive nuclei emitting particles


                                                                  Answers    277
that transfer their kinetic energy to      wires in the process of making contact.
thermal energy, as well as friction        Held near a radio, the small spark
between internal rock flows, which         across the gap can be heard through
could create local hot spots and/or        the radio, indicating that many fre-
temporary changes in the flow proper-       quencies are being emitted. As an
ties of liquid rock or the thermal con-    additional demonstration, one can
ductivity of the rocks. Therefore, small   even move a radio near a small electric
changes in the rates of thermal energy     motor of the kind that has brushes to
transport to Earth’s surface are possi-    hear its rotation frequency because the
ble and most likely happen continu-        brushes make and break contact each
ally. Are these internal sources the       revolution.
culprits in the present slow average           Of course, if one desires to have a
temperature increase? These fluctua-       higher-current spark gap, a car battery
tions are expected to be too small, but    or a transformer can be used with
no one knows for sure.                     proper safety precautions to provide a
                                           healthy current that can be operated in
                                           an intermittent mode or in a continu-
229. Frequency                             ous mode. Nearby radios, televisions,
Jamming                                    and so on will be affected by this
                                           larger-current spark gap device. Even
                                           GPS transmissions between 1,000
                                           MHz and 2,000 MHz may be affected,
                                           so some care must be taken not to vio-
                                           late federal transmission limitations.


A spark gap usually is a wonderful         230. Light Energy
noisy source of electromagnetic waves      A light source emitting light of fre-
at all frequencies simultaneously. The     quency f approaching an observer at
greater the current across the gap, the    constant velocity v will appear blue-
more intense will be the total radia-      shifted by an amount corresponding
tion emitted at each frequency. There      to the relativistic Doppler effect for-
will be a distribution of intensity ver-   mula f ′ = f [(1 − v 2 / c 2 )] / (1 − v / c )
sus frequency that can be “tuned” a        because the clock ticking rates will be
little by adjusting the gap spacing.       different for the source reference frame
     A simple spark gap would be a         and the observer reference frame, and
small battery, such as a D cell, and two   their separation distance is decreasing.

278      Answers
When v << c, we can expand the for-           is defined to have a pH of less than
mula in a Taylor series to obtain f ′ ~ f     5.0, a condition that tends to occur
(1 + v/c – v2/2c2 + . . . ), so the leading   more often in industrialized areas of
term in powers of v/c is positive, cor-       the world than in remote regions.
responding to the blueshift. We                   Human activities can increase the
assume that an acceleration itself does       amount of CO2 in the air, but so do
not produce an additional fundamen-           many natural resources, such as volca-
tal frequency shift, although there will      noes, lightning strikes, cows, bacteria,
be acceleration effects because the           and fires. When industrial and auto-
source is changing instantaneous              mobile exhausts release sulfur com-
comoving inertial frames.                     pounds and nitrogen compounds,
     For a photon, its energy is E = hf       these molecules combine with oxygen
and its momentum is p = E/c, so both          to form sulfuric acid and nitric acid,
energy and momentum are different in          which can harm ecosystems, historic
different reference frames because the        monuments and buildings, and the
observed frequencies are different.           health of people around the world.
Notice that the recoil of the source on       The reduction of sulfur and nitrogen
emission of light and of the observer         compounds released into the air has
on detection are not accounted for in         become a worldwide concern.
the discussion and that the energy and         Trefil, J. The Nature of Science. Boston:
momentum input necessary to keep                Houghton Mifflin, 2003, pp. 6–7.
the relative velocity of the source and
observer fixed must be considered
                                              232. Electrical Current
also. Of course, energy and momen-
tum conservation laws are obeyed in           The electrons in house wiring move at
this example.                                 a snail’s pace, with an average drift
                                              velocity of about a millimeter per sec-
                                              ond. These electrons, which are free to
231. Acid Rain                                move in the metal wires, are distrib-
Not so! The falling raindrops will not        uted throughout, so when the switch is
remain neutral at pH 7. Pure rainwa-          closed to make a complete circuit, they
ter falling through unpolluted air is an      move en masse, sort of like water pass-
acid, with a pH of about 5.6, because         ing through a continuous hose that
as the drops form and fall they dis-          closes on itself. The electron velocity is
solve carbon dioxide in the air and           limited because its negative electric
react to produce carbonic acid,               charge interacts with the lattice of pos-
H2CO3. Officially, therefore, acid rain        itive ions during the movement.

                                                                      Answers     279
     In addition to having a drift veloc-     234. Crystal Growth
ity, the electrons experience a random
sequence of pinball-like collisions to        The speed and precision of the crystal
change their speeds and directions,           growth depend on many factors,
essentially behaving as a free electron       including the temperature, concentra-
gas. Consequently, the host metal             tion, and purity of the solution.
gains some thermal energy, and its            Assuming the ideal solution, each
temperature rises. If a lamp is the           additional atom to be added from the
incandescent type, the tungsten alloy         solution must first find a location on
filament will gain enough energy by           the growing surface of the developing
this process to dramatically increase its     crystal. But these atoms in solution are
temperature to a new equilibrium tem-         randomly moving about, making ran-
perature of about 2000 K to glow in           dom collisions with the crystal at ran-
the visible and the infrared.                 dom locations on the surface. How
                                              can they build a perfect single crystal?
                                                  Their little secret is that some
233. Earth’s Orbit                            atoms that have been added at mar-
Although the general theory of relativ-       ginal locations, say, can escape from
ity dictates a precession of the perihe-      these surface locations to allow other
lion for all planets, including Earth,        atoms from the solution to find a bet-
this effect is very small compared to         ter location nearby, “better” here
perturbations provided by gravitational       meaning to be held electrostatically
influences of all the planets. Apparently      tighter to the crystal. But these better
Earth’s elliptical orbit will pass through    locations do not occur in chronologi-
a repetitive cycling about every 93,000       cal order because they are determined
years from its present ellipse and orien-     by the collective influence of numerous
tation with respect to the stars, to a cir-   atoms already in the crystal, and the
cle, then back to an ellipse with an          best position one microsecond ago
orientation perpendicular to the pres-        may not be the best position for an
ent orientation, to a circle again, back      atom now. Therefore, the addition and
to an ellipse perpendicular again, etc.,      subtraction of atoms from the growing
until the present elliptical orientation is   crystal surface proceeds almost by trial
recovered approximately. Of course, all       and error! Consequently, one cannot
the planets are experiencing these            write down an algorithm for placing
perturbation effects simultaneously, so       atoms from the solution onto the
the detailed calculations become quite        growing crystal.
interesting!                                      When the crystal grows slowly,

280       Answers
there is plenty of time for the sampling        Sapphire is aluminum oxide with
process to proceed to fruition, and the     trace impurities of iron and titanium
crystals tend to form with fewer dislo-     atoms, which are responsible for the
cations and inclusions. When the crys-      deep blue color shades most people
tal grows rapidly, errors in the crystal    associate with sapphire. Several other
structure become trapped, and these         colors of corundum, such as yellow,
crystals tend to have many disloca-         reddish-orange, and violet, also are
tions and inclusions.                       classified as sapphire. Synthetic sap-
                                            phires have been produced commer-
                                            cially since 1902 and are used for
235. Ruby, Sapphire,                        scratch-resistant watch crystals, opti-
                                            cal scanners, and in applications where
and Emerald                                 physical strength and transparency to
How are ruby, sapphire, and emerald         ultraviolet irradiation are important.
crystals related? Ruby and sapphire             Emeralds are quite different from
are color variations of the same min-       rubies and sapphires in that emeralds
eral, corundum. Rubies contain a            are the green form of beryl, colored by
small amount of chromium. Pure              the presence of chromium or vana-
corundum is a colorless, trigonal crys-     dium. The crystal structure of beryl
tal that occurs in a wide variety of col-   emeralds is hexagonal (six-sided), with
ors due to infiltrations of other           a hardness slightly higher than quartz
elements. All color variations of           but considerably less than diamond.
corundum, with the exception of ruby,       Emeralds are notorious for containing
are called sapphires.                       flaws, and flawless stones are rare and
     Rubies are red variations of the       greatly valued.
mineral corundum, a crystalline form            The colors in these gemstones are
of aluminum oxide and one of the            produced by characteristic downward
most durable minerals that exists.          atomic transitions involving F-centers
Only diamonds are harder. Rubies’           (from the German word farbe, mean-
rich, red colors arise from the substi-     ing color) at the chromium or other
tution of a small number of aluminum        atoms. In a simple view of an F-center,
atoms by chromium atoms. When               the ambient light excites one electron
exposed to high temperatures, rubies        in the chromium atom, for example,
turn green, but they regain their origi-    so the atom can be treated analogous
nal color after cooling. Some rubies        to the hydrogen atom, with a large
phosphoresce with a vivid red glow          average radius for the location of the
when illuminated by ultraviolet light.      electron away from the chromium

                                                                  Answers     281
nucleus. The electron transition back              Applying Kepler’s third law for a
to a lower energy state causes the             particle of mass µ between Earth’s mass
emission of a photon in the visible.           m and the Sun’s mass M orbiting with
 Nassau, K. “The Causes of Color.” Scientific   Earth’s period T, one obtains after sev-
  American 243, no. 4 (1980): 124–154.         eral steps GMµ/(r–R)3 – Gmµ/(R2(r–R))
 Perkowitz, S. “True Colors: Why Things        = GMµ/r3 where r is the Earth–Sun dis-
  Look the Way They Do.” The Sciences
  (May–June 1991): 22–28.                      tance and R is the Earth–particle dis-
                                               tance. The particle at L1 will be about
                                               0.01 times the distance to the Sun. The
236. Kordylewski                               L3 point on the night side of Earth can
Clouds                                         be calculated in the same way, replac-
                                               ing r–R with r+R. However, the other
Joseph L. Lagrange in the late 1700s
                                               three points must be calculated with
calculated via Newton’s laws that
                                               the gravitational attractions of the
there are five special positions for
                                               other planets included.
objects bound by any two-body
                                                   Similar calculations have been
system. Now called Lagrange points,
                                               done for the five Lagrange points for
positions L1, L2, and L3 are unstable,
                                               the Earth-Moon system. Polish
while L4 and L5 are stable. Several
                                               astronomer K. Kordylewski in 1961
spacecraft have been placed at or
                                               reported the observation of dust clouds
near these Lagrange points, and
                                               at the L5 point, but some observers
there have been proposals for
                                               have not seen them. Particles here may
building space colonies at the L4 or L5
                                               not remain long before being ejected,
positions.
                                               according to calculations.
                         L4                     Kordylewski, K. “Photographische Unter-
                                                 suchungen des Librationspunktes Lim5
                                                 System Erde-Mond.” Acta Astronomica 11
                                                 (1961): 165–169.
                                                O’Neill, G. K. “The Colonization of Space.”
                                                 Physics Today 27, no. 9 (1974): 32.
 L1                           L2       L3

                                      Moon     237. Twist Scooter
              Earth
                                               If the plane of the V arms of the twist
                                               bike remained horizontal at all times,
                                               there would be no forward motion
                                               except by pushing with a foot on the
                         L5
                                               ground. By tilting the vertical shaft of

282       Answers
the handlebar sideways about 10             phase with the photons from the right
degrees or so, the front of the scooter     side. In other words, the photons are
is lowered a bit, and a slight unbal-       not expected to be in phase. Their
anced outward push of the arms of the       phase difference depends on three
V by the rider’s legs provides a for-       parameters: their initial phase differ-
ward net force similar to the resulting     ence, the distance to the star, and the
net force exercised by an ice skater.       diameter of the star. By slowly chang-
The initial forward movement from           ing the separation of the two photode-
rest may be difficult on an upslope         tectors on the arms of an intensity
exceeding a particular angle deter-         interferometer, one can sweep across a
mined by the size of the wheels and by      range of phase differences to determine
the possible tip angle of the handlebar.    the diameter of the source. One labo-
                                            ratory analogy might be considering
                                            how one would determine the spacing
238. Unruh Radiation                        between the slits of a two-slit interfer-
The Equivalence Principle tells us that     ence experiment with a similar appara-
a particle accelerating in the vacuum is    tus. Ultimately, the amplitudes and not
equivalent to a particle at rest in a       really the intensities interfere. How-
uniform gravitational field. If there is     ever, the phase correlations depend on
radiation in one case, there must be        the product of intensities, in contrast
radiation in the other equivalent case.     to the two-slit interference example.
The former is called Bekenstein radia-          The original experiment is known
tion, the latter Unruh radiation,           as the Brown-Twiss experiment,
named after physicists who studied the      named after the two researchers who
properties of the radiation theoreti-       first succeeded in using the technique
cally. No one has ever measured this        to determine a star diameter back in
radiation because its intensity is many     1957. Interference associated with the
powers of 10 too faint to be detected.      superposition of separate light intensi-
                                            ties was viewed with considerable
                                            skepticism. Apparently, as the story
239. Star Diameters                         goes, one of the original researchers
One can determine the interference          was giving a physics talk at Caltech
diameter of a distant star even when        soon after their first measurements. In
optical parallax resolution of its diam-    those days, several Nobel physicists
eter is impossible by utilizing quantum     would sit in the front row along with
interference between the photons from       Richard Feynman and other prominent
the left side of the star arriving out of   physicists. About 10 minutes into the

                                                                  Answers      283
talk, Feynman walked out, much to the          stimulate the emission of a second
dismay of the speaker. About 40 min-           photon from a nearby atom, and the
utes later, just near the end of the talk,     two can stimulate the emission of a
Feynman walked back in and sat down            third photon, and so on. In principle,
in his seat again. The speaker then            the photons arriving at the receptor
asked why he walked out and then               can be single, double, triple, and so on,
returned. Feynman responded that he            the actual photon state depending on
had walked out because he did not              how many stimulated photons were
believe that the physics was correct. He       picked up before escaping the light
explained that he had gone back to his         source. The receptor receives a differ-
office and worked out the problem,             ent energy burst with each absorption.
only to discover that the physics had          Since the probability for stimulated
been done correctly. He then returned          emission into the same final state is
to acknowledge the cleverness of the           proportional to the number of pho-
speaker and his colleague. Now the             tons in that state already, these
speaker was in dismay again, amazed            multiple photon processes occur quite
that someone could have worked out             readily.
the many details in so short a time!               Real light sources such as incan-
 Brown, R. H., and Twiss, R. Q. “A New         descent bulbs have huge physical
  Type of Interferometer for Use in Radioas-   dimensions compared to the wave-
  tronomy.” Philosophical Magazine 45
  (1954): 663.                                 length of light. There will be numerous
 Silverman, M. P. A Universe of Atoms: An      ideal chaotic sources along the fila-
  Atom in the Universe. New York: Springer-    ment wire simultaneously and ran-
  Verlag, 2002, pp. 102–126.
                                               domly emitting photons toward the
                                               detector. These photons tend to arrive
                                               in bunches, with the photons within
240. Glauber Effect                            any one bunch coming from several
Yes, a standard incandescent lightbulb         places in the source. Very seldom does
does emit single photons, and some-            one find a steady stream of photons
times there are photon pairs, and              with nearly equal time spacing arriv-
triplets, and so on. In the ideal chaotic      ing from the lightbulb when one looks
photon source—a hot, incandescent              on the nanosecond time scale.
wire that has physical dimensions               Loudon, R. The Quantum Theory of Light,
smaller than a wavelength of the                 3rd ed. Oxford: Oxford University Press,
                                                 2000, chap. 6.
emitted light, for example—the first
spontaneously emitted photon can


284       Answers
241. Bird Sounds                                 243. Hair-Raiser
Some birds can emit just a fundamen-             Function
tal frequency with no harmonics. Just            For the HRF of a non-integer, one
how the bird eliminates the harmonics            needs to write down a few more exam-
originally generated within is being             ples of the given integer description.
investigated. The present conjecture is          Then take the logarithm of each exam-
that a cavity resonance amplifies just            ple to discover that they all can be
the fundamental before the sound is              expressed as log N = nn–1 log n. By tak-
emitted. If the fundamental frequency            ing the exponential of both sides with
changes, then the cavity must change             the proper grouping, the final expres-
to accommodate the new fundamental               sion becomes N = (n)^(nn–1)—that is, n
in “live time.”                                  to the power (nn–1). With HRF(x) =
                                                 (x)^(xx–1), the HRF of non-integer val-
                                                 ues for x becomes an easy calculation
242. Spouting                                    with the appropriate calculator, one
Alligator                                        capable of many decimal places. What
                                                 is the limit as n approaches zero?
To eject water droplets upward, the
                                                 Complex numbers can be used, as well
alligator head vibrations must provide
                                                 as irrationals such as π.
the initial energy to create nearly
                                                     A plot of the HRF using integers
standing waves in the shallow water
                                                 shows a remarkably steep rise for even
on the back of its head. As wave crests
                                                 small integers; hence its name! You
become larger, droplets of water break
                                                 might want to compare its rise to an
off and are projected high above the
                                                 exponential function. And if all you
surface. One can simulate this effect
                                                 desire is an approximate value for the
by sliding a styrofoam cup filled with
                                                 inverse or for the HRF of a non-integer,
water across a finished wooden sur-
                                                 the plot provides a visual image and a
face at about 10 centimeters per sec-
                                                 means to satisfy your curiosity.
ond. Water droplets will shoot upward
                                                     However, as far as we know, the
to about 20 centimeters.
                                                 inverse HRF is awkward, and no easy
 Jargodzki, C., and F. Potter. “Spouting Water
  Droplets.” In Mad about Physics: Brain-        calculation algorithm is known. We
  twisters, Paradoxes, and Curiosities. New      don’t even know whether the inverse
  York: John Wiley & Sons, 2001, p. 39.          can be expressed as the limit of a
                                                 series! One can determine the inverse
                                                 by successive approximation to any


                                                                       Answers     285
number of decimal places with the             required. However, during operation
appropriate calculator.                       the system does not require fuel or
     Of what use is the hair-raiser func-     other mass be expelled into the envi-
tion? The question reminds us of two          ronment to move in space. The sys-
classic quotes from Michael Faraday           tem is designed to operate in two
when he was attempting to explain a           operational modes: in Mode I the
discovery to the visiting prime minis-        system incrementally moves the pay-
ter. He was asked: “But, after all, what      load platform forward with each
use is it?” To which Faraday replied,         operational cycle. In this first mode,
“Why, sir, there is the probability that      the velocity imparted to the payload
you will soon be able to tax it.” And         platform is not additive. In Mode II
when the prime minister asked of a            the payload platform accelerates for-
new discovery, “What good is it?,”            ward a discrete increment of velocity
Faraday replied, “What good is a new-         during each operational cycle. In this
born baby?”                                   second mode the increments of
                                              velocity are additive.

                                                There is no problem with energy
244. Space Crawler                          conservation because the onboard bat-
                                            tery supplies the energy. The inventor
The U.S. Patent Office awarded patent
                                            Virgil Laul claims that this propulsion
5966986 in 1999 to this propulsion
                                            device when attached to spacecraft
device. We quote the patent abstract:
                                            will be able to propel spacecraft out in
  A propulsion system which is              space. We leave this problem as a final
  designed to be used on a payload          challenge. What is the physics here?
  platform such as a spacecraft, satel-     Are any conservation laws violated?
  lite, aircraft, or an ocean vessel. To    Will the device work in space as well
  operate the system electrical power is    as it does on the air table?




286       Answers
Index
absorption                    conservation of, 55         Rutherford, E., 73
  atmospheric, 216            electron in atom, 67      axis of rotation, 24, 31
  cancellation, 212           of flywheels, 54
academic                      quantization, 67          b’ quark, 276
  disputes, 44                quantum, total, 67        Babylonians, 13, 20
  types, 118                annihilation, 87, 97,171    ball, bouncing, 87
acceleration                Anthropic Principle, 96     baseball bat, 53
  angular, 144              antiparticles, 87, 255      BBs, 37
  car driver, 146           Apollonius, 21              Be-8 synthesis, 95
  collision time, 165       archaeology, 53             beans, parboiling, 6
  free fall, 30             Archimedes                  beets, peeling, 7
  local, 107                  gravestone, 22            Bekenstein radiation, 283
  movie of, 17                principle, 3              Bell inequalities, 87
  pseudo-force, 168         Aristotle, 50               benzene
  rocket ship, 107          arm contortions, 25           carbon atoms, 69
  in sandglass, 144         asteroid, 36                  energy levels, 69
  skiers, record, 167       astronauts, 123             beta decay, 244, 245
  time reversal of, 16      astronomers, 30             bicycle tracks, 118
  twin paradox, 108         atmosphere                  big bang
  uniform, 44                 absorption by, 216          entropy, 114
acid rain, 119                carbon-14 in, 93            Hoyle, F., 114
action-at-a-distance          cosmic rays, 124            inflationary, 41, 112
  Boscovich, R. J., 66        expansion of, 35            microwaves, 99
  Riemann, G. B., 111         greenhouse, 70, 71        billiards table, 20
air                           ozone layer, 70           bird sounds, 122
  boundary layer, 197         refraction by, 140, 152   birth dates, 15
  can, air pressure, 7        UV, 103                   black hole
  density, 169                wings in, 31                collisions, 110
  engine, 50                atom                          entropy, 110
  pressure, 125, 126, 132     absorption by, 135          forces, radial, 111
  soup can, 129               Bohr, N., 68                and information, 89
  in straw, 128               Bose-Einstein, 74           time symmetry, 150
  viscosity, 144              carbon, 69, 70            blackbody spectrum,
algebra, 103                  clusters, 55, 59            cosmos, 112, 266, 271
alien beings, 31              emission by, 66, 67         plasma, 218
alligator, spouting, 122      quantum dots, 74          blueberry muffins, 4
alpha particle, 96            X-ray laser, 73, 222      body cushion, 30
amino acid triplet, 149     atomic model                Bohr, N.
ammonium maser, 68            Bohr, N., 67                atom, 68
angular momentum              Nagaoka, H., 73             completeness, 83


                                                                                   287
Bohr, N. (continued)       carbon                      collision
  double-slit, 232           cycle, in stars, 245        asteroid, 36
  habits, 70                 nuclear levels, 95          body cushion, 30
  quantization, 67           synthesis in stars, 95      bullet, 30
  religion, 65             carbon-14 dating, 93          molecules, 167
boiling point, 134         carbon dioxide                spaceships, 46
boiling water                air amount, 279             wall, in cartoons, 30
  altitude effects, 5        in bread, 126             color
  beets in, 7                greenhouse gas, 70, 71      cadmium, 59
  in ice, 8                  moderator, 98               F-centers, 281
  salt added to, 7           plants, 93                  nanophase, 59
  watched pot, 8             in water, 9                 ruby, 120
Boltzmann, L., 44          carbonic acid, 279          communication
book rotation, 25          Carnot cycle                  black hole, 106
Bose-Einstein                ferrofluid, 52               delays, 175
  condensate, 74, 208        photon, 61                  jamming, 118
  Cooper pairs, 84           quantum, 61                 spaceships, 43, 106
  He-4, 85                 cartoons, 30, 35            computer
  statistics, 177          Casimir effect, 84            atomic, 73
bosons, 85, 177            cellular automata, 104        DNA, 221
Bragg scattering, 74, 79   centrifugal force, 111        java quantum, 221
brain                      CFC, 216                      nuclear spins, 221
  connections, 23          chaotic systems               quantum, 79
  magnetic field, 85          competition, 57           concrete, 239
  power needs, 54            hot wire, 284             conductivity
bread kneading, 3                                        electrical, 200, 220
                             identical, 57
Brownian motor, 50                                       thermal, 130, 131, 132
                           Chernobyl, 179
bubble collapse, 218                                   conics, 21
                           Chinese cooking, 6
bullet                                                 consciousness, 86
                           chlorophyll, 135
  fireworks, 34                                         convection, 132
                           circadian rhythm, 149
  impact, 30                                           Cooper pairs, 84
                           classical mechanics, 51
Bushmen, 172                                           copper
                           clocks,
butter, measuring, 3                                     cladding, 175
                             atomic, 15                  nanophase, 55
cadmium selenide, 59         eternal, 15                 oxide tunneling, 238
caffeine, 7, 79              identical, 43               X-ray laser, 73
calendar                     light, 16                 Coriolis, G., 55
  Gregorian, 142             molecular, 17             cosmic rays, 93, 124, 221
  Julian, 142              coal burning, 98            Coulomb
  lunar, 14, 143           coffee, 7, 79                 barrier, 244, 245, 248
  Mayan, 52                coherent                      blockade, 56
  rice planting, 14          light scattering, 227     Crab Nebula, 182
calories                     scattering, 80, 81, 176   creative thinkers, 85
  from fat, 5                X-rays, 227               crystal structure
  human needs, 4           coin tosses                   Bragg scattering, 79
campfire, igniting, 134       random walk, 50             diamond, 70
can, pressure in, 7          randomness, 50              graphite, 70
car driver, 15, 42         cold fusion, 98               growth, 120


288        Index
  lattice, 120, 248             fractional, 24              height, 69
  quantum dots, 74              space, 21, 22, 24           letter to Mileva, 43
crystalline, 55, 198            space-time, 27              light bending, 106
cubes                         dimples in bat, 54            mass-energy, 41
  array, 21                   dipole-dipole, 135            paper in 1905, 46
  space-filling, 22            dispersion, 59                photon emission, 68
  symmetry, 24, 160           distance, 20                  reality, 114
cup, rotating, 27             DNA                           relativity 1905, 27
cuprates, 55                    clock, 17                   space-time, 89
Curie, M., 93, 95, 102          computer, 221             electrical
Curie, P., 93, 95, 102        dodecahedron, 22, 24, 153     circuit, 58
Curie temperature, 102, 199   Doppler shift, 65             conductor, 200
cylinder, 22                  double-slit, 83, 213, 232     current, house, 119
                              driving, switching, 15        diode, pickle, 137
dark matter, 269                                          electrical charge
dates, 9                      Earth                         fractional, 200
day length                      atmosphere, 35              on pinhead, 56
  Egyptians, 12                 climate, 140                transferred, 200
  hours in, 12                  magnetic field, 241          tunnel junction, 56
  sidereal, 37                  Moon, seen from, 77       electron
  solar, 37                     orbit change, 119           in a box, finding, 78
  tidal effects, 57             rotations in year, 13       Cooper pairs, in, 84
  winter, 61                    speed in orbit, 12          density, metals, 130
de Broglie λ, 67, 222, 234      temperature, 178, 216       house current, 119
decay                           warming, 118                single, tunneling, 200
  competitive, 57             ecliptic, 14                  spin, vacuum effect, 84
  neutron, 95                 eddy                          Standard Model, 101
  nuclear, 86                   current, 199              elements
decoherence, 224                turbulent, 197              pun, 122
defrosting tray, 5            efficiency                     synthesis of, 94, 95, 123
demonstrations, 34              light bulb, 209           ellipse, 21, 119
density, cream, 127             photon engine, 61         elliptic functions, 116, 276
Descartes, R., 57               thermal, 52, 61           emerald, 120
detector                      egg and bottle, 2           energy
  calibration, 114            Egypt                         Earth warming, 118
  gravitational, 80, 114        day length, 12              field, charge, 47
  neutrino, 81                  pyramids, 54                kinetic, 51, 55
determinism, 88               Einstein, A.                  mass-energy, 40, 42
deterministic                   Beiblätter, 44              rotational, 99
  behavior, 57                  Bible study, 107            in universe, 115
  chaos, 57                     completeness, 83          energy levels
detonation, 37                  Curie letter, 104           benzene, 69
deuteron, 96                    deductions by, 45           C-12 nucleus, 95
diamond, 70, 220                determinism, 88             hydrogen, 67
digital timer, 15               dice quote, 100             nuclear, 93
dimensions                      emission, 208               quantum dot, 223
  defined, 159                   EPR paradox, 87           engine
  fractal, 159                  God, 100, 110               air-driven, 50



                                                                         Index    289
engine (continued)           nuclear shell, 93            geodesics, 111
  Brownian, 50             fission                         glass
  magnetocaloric, 52         Pu-239, 246                    absorption in, 135
  photon, 61                 U-235, 99, 245                 old window, 54
entropy                      U-238, 246                   Glauber effect, 122
  asymmetry in, 18         Fitzgerald, G. F., 96          gluons, 255
  big bang, 114            fluid                           glycemic index, 9
  black hole, 110            binary, 52                   gold density, 128
  disorder, 58               ferrofluid, 52                GPS signals, 108
  final state, 239            inmiscible, 52               grain
  heat flow, 58               magnetorheological, 52         boundaries, 200
  liquids, 196             fluorescence                      sand, atoms in, 65
eponymy, 24                  lights, 66                     size, 55, 65
EPR paradox, 87              quantum dots, 74             graphite, 70
equations, time in, 18     flywheels, coupled, 54          grasshoppers, 57
equinoxes, 12, 14          FM radio frequency, 69         gravitational
eternal clock, 15          force                            bound system, 113
evaporation, 9               aperiodic, 150                 clock rate, 60, 108, 260
explosions in space 33       applied, 44, 51, 55            detectors, 80, 114
exponential growth, 57       buoyant, 170                   lensing, 109
                             centrifugal, 111               tidal effects, 205
faces, resolution of, 80     field, in movies, 36            twins, 60
fame, 59                     fluctuating, 51                 waves, 80
Faraday, M.                  geometry, 111                gravitomagnetism, 259
  flux law, 202               torque, 144                  gravity
  quotes, 286                van der Waals, 135             artificial, 31
farmer                     forensics, paint, 65             in galaxies, 112
  goose chase, 23          4-D, 24, 27                      geodesics, 111
  rice, planting, 14       four-momentum, 40                geometry, 111
faster than c              free energy, 196, 202          greenhouse effect
  spotlight, 41            free fall                        gases, 70, 71, 216
  quasar, 41                 acceleration, 30               water vapor, 216
  wave function, 225         body cushion, 30             Groundhog Day, 14
Fe-56 stability, 94          cartoons, in, 30             group velocity, 213
Fermi, E.                    Galileo, 59                  growth, 57, 120
  energy, 220                Philoponus, J., 50           gunfight, 30
  exclusion, 242             twin watches, 108
  Golden Rule, 87          friction, static, 128          hair-raiser function, 123
Fermi-Dirac, statistics,   Friday 13th, 79                Hall effect, quantized, 72
    237                    fullerene, 215                 halogen lamp, 138
fermions, 177, 237                                        hamburgers, 131
ferromagnetism, 54, 102    g-2, muon, 236                 hardness, metals, 55
Feynman, R. P.             Galaxy center, 274             Hawking radiation, 267
  Brown-Twiss, 283         galaxies, 110, 111, 258, 265   He-4
  ratchet, 50              Galileo, G., 59, 142             burning in stars, 95
  twin watches, 108        galvanometer, 58                 superfluid, 85
field theory                gamma rays, 250, 251             synthesis in stars, 96
  Boscovich, J., 66        Gamow, G., 109                 headlight effect, 106



290       Index
heat bath, 61                 icosahedron, 22, 25, 160   leap years, 13
heat engine                   ignition                   LED, 71
  air, 50                       oxygen, 133              lens, 66
  Brownian, 50                  spontaneous, 133         Lenz’s law, 199
  photon, 61                  inertia, 56                Lewis, G., 71, 83
Heisenberg, W.                information                Libby, W., 93, 100
  matrix model, 84              black hole, and, 89      life, 120, 122, 133
  on reality, 78                phase, 214, 256          light
  uncertainty principle, 83     quantum, 89                 ambient, 97
helium burning, 95              speed of, 182               approaching, 106
helium liquid, 72, 85           storage, 73                 bending of, 106, 109
Hubble telescope, 264         infrared                      bulb, 122, 209
human                           images, 35, 80              and car driver, 42
  aging, 45, 47, 60             resolution, 176             ceramic response, 59
  brains, 23, 54, 85          insulation, fat, 10           clock, 16
  collision damage, 166       integrated circuits, 73       cosmic background, 112
  DNA, 17                     interference                  efficiency, sources, 209
  energy needs, 4               double-slit, 83             faster than, 32, 41, 182
  faces, 35, 80                 intensity, 121              flash, 15, 33, 34, 71, 173
  health in UV, 67              laser, 213                  forensics, paint, 65
  hearing, 100                  quantum, 75                 geodesics, 111
  height, 69, 123               star light, 121             infrared, 80
  memory, 156                 Internet, game play, 34       moving source, 119
  radioactivity, 97           invariants, 40, 41, 51        oven, 10
  SAD, 17                     iodine, 118, 179              sabers, 36
  shrunken size, 32           iron filings, 52               scattering, 59, 80, 174, 177
  space debris, 32                                          signals, 43
  space journey, 45, 46       Josephson effect, 85          squeezing, 84
  UV need, 210                                              therapy, 17
  walking, 59                 Kepler’s 3rd law, 264         tired-light, 110
Huxley, A., 119               Kerr effect, 59               tweezer, 66
Huygens, C., 51               kinetic energy                UV, 80
hydrogen                        discovery, 51, 55        light tweezer, 66
  atom, 67, 164                 vis viva, 55             lightning, 33
  bonds, 126, 196             Kordylewski cloud, 120
                                                         LIGO, 229
  burning, star, 97
                                                         Lincoln, A., 32
  MRI, 82                     Lagrange points, 282
                                                         lipstick color, 59
  nuclear device, 37          Lamb shift, 84
                                                         liquid helium siphon, 72
  in space, 174               Landau levels, 220
                                                         locality violation, 240
  stability of atom, 27       Larmor, J., 41
                                                         Lorentz, H.,
                              laser
                                                            contraction, 42
ice                             beams in space, 34
  bond angles, 8                kinetic, 68                 relativity, 41
  floating, 6, 127               noninversion, 68
  fluorocarbons, 216             security system, 34      magnetic
  in microwave, 8               strobe, 146               dipole, 198
  polar melt, 33                Weber, J., 68            magnetic
  in ponds, 133                 X-ray, 73, 212            field, 52, 72, 85
ice cream, 5                  LCD, 71                     induction stove, 10



                                                                          Index     291
magnetic (continued)         meteor, 120                MSG, 2
 resonance, 82               meteorite in 1908, 100     muon, 101, 236, 248
Mars                         methane, 70, 71
 debris on, 116              metronomes, two, 18        Nagaoka, H., 73
 moons, 30                   Michelson, A., 42          nanophase cluster
Martians, 33                 microtubules, 156            cadmium, 59
maser, 68                    microwaves                   copper, 55
mass quadrupole, 80           absorption, 135             energy levels, 203
mass-energy                   background, 99, 266       neurons, 23, 95
 confusion, 42                heating water, 3          neutrino
 creation of, 115             ice absorption, 8           detectors, 81, 231
 definition, 40, 41            metals in, 6                handedness, 103
 electron, 47                 oven, 6, 10                 mass, 45, 103
 particles, system of, 42    milk and cream, 3            proton cycle, 97
 Poincaré, H., 45, 47        Milky Way, 112, 113, 274     scattering, coherent, 81
 proton, 103                 Miller, J. S., 2             solar, 81
material                     Minkowski, H., 27            supernova 1987A, 173
 ceramic, 59                 minute, origin, 13           Standard Model, 101
 dispersive, 59              mirror types, 67, 211      neutron
 empty space in, 68          miscible liquids, 52         capture, 246, 249
 glass flow, 197              modular function J, 276      decay of, 95
 hardness, 55                molecular                  Newton, I.
 magnetic, 54                 clock, 17                   birth date of, 142
 nanophase Cu, 55             displacement, 176, 177      comets, 110
 smart, 203                   scattering IR, 177          firstborns, 122
 stretching of, 35           momentum, 57, 164, 170       height, 69
mathematicians, 118, 121     MOND, 265, 270               mass-energy, 42
Matthew effect, 59           Moon                         Principia, 23
Maxwell’s equations           calendar, 14                quip about, 54
 failure of, 208              in daylight, 142          Newton’s Laws
 invariance of, 41            distance, 20, 206           2nd law, 125, 128, 132
 symmetry of, 102             Easter, 143                 3rd law, 170
measurement                   full, 13                    torques, 144
 Bohr, 232                    size of, 25               Ni isotopes, 94
 Moon distance, 20            tidal effects, 57         night sky, 106
 quantum, 78, 83             Mössbauer Effect           nitrogen, liquid, 5
 simultaneous, 191            eardrum, 251              NMR, 226
 uncertainty principle, 83    gamma rays, 208           noise
meat                          Weber bar, 230              jamming with, 118
 Chinese style, 6            movies                       robotic, helpful, 60
 cooking roast, 5             battle sounds, 33           shot, electrical, 84
 preserving, 5                body cushion, 30          north, finding, 18
Meissner effect, 55           gunfight, 30               nuclear
memory, 17                    sounds, 33                  beta decay, 244, 245
men, 4, 10                    spaceships, 32              binding energy, 242
metals                        warp speed, 32              decay, 86
 density of, 128              western, 165                device, 36, 37, 94, 180
 thermal conductivity, 130   MRI, 82                      emergency, 118



292       Index
 energy levels, 93, 95         random, 82, 227             positronium, 87
 fission, 99                    space, 239                  pot, watched, 8
 gamma rays, 250               uncertainty, 235            potassium-40, 98
 reactor, 97, 98, 179          velocity, 213               potato, 4, 9
 spin flip. 226               philosophical, 74             Potter, F., 160, 253, 256, 276
 spin-orbit, 242             phosphorous, 175              power plant, coal, 98
 submarine, 36               photon                        precession, 109, 119, 139
 synthesis, 94, 95             absorption, 65              preserving food, 5
 waste, 98                     bunching, 122               prism, 87
nucleus                        emission, 65, 67            probability
 large, 99                     engine, 61                    amplitude, 75
 models, history of, 73        scattering, 78, 177           quantum, 75, 76
 neutron in, 95                UV, 66, 210                   reaction rate, 87
 rotation of, 250            photosensitivity, 143         procrastinating, 13
 shell model, 93             photostrictive effect, 203    projectile impact, 30
 stable, 94, 99              physicist, 18, 20, 121, 165   protein, 148
                             physics                       proton, 103, 161, 255
octahedron, 22, 24, 160        equationless, 103           psychology, 165
Oklo nuclear reactor, 97       mathematics, 114, 116       Pu-239, 99, 180
Olbers’ paradox, 106         pi, 22, 23, 26                pyramids, 54
optical                      pickle, electric, 9           Pythagoras, 20
  cavity, 61                 pinhead, charge on, 56        Pythagorean
  color, lipstick, 59        Planck, M., 115                 theorem, 24
  solitons, 59               planet orbits                   triplets, 20
orbits,