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Special_Relativity Powered By Docstoc

Between 1866 and 1870 James Clerk Maxwell developed a theory of
electromagnetic disturbances and determined that they travel at a speed very
near that of light. Although the speed of light had been experimentally
determined prior to Maxwell's work by several scientists, it was in the
publication of his book Electricity and Magnetism in 1873 that the four
differential equations now known simply as Maxwell's equations first
appeared together and presented a unified view of both electric and
magnetic fields in space as well as a theoretical value of th speed of light.
According to the groundbreaking work of Maxwell, light is nothing more
than a particular type of electromagnetic radiation! What his calculations
also told him, though, was that electromagnetic waves—visible light
included—can never stop or slow down. Light in a vacuum always travels at the speed of light.

And God said

...and there was light.

Maxwell, however, lived in a time that saw the beginning of the industrial revolution and his view of a
mechanical world in which all motion was determined by prior causes and played out upon an absolute
frame of reference was true to his time. He could not understand the propagation of light without a
medium to carry the electromagnetic wave nor could he understand how light could always maintain the
same speed regardless of an observer's motion relative to the light beam. Additionally, light traveling
through a vacuum was as inconceivable as the wake of a boat propagating without water! Hence, the
lumeniferous ether was postulated as the medium needed to transport electromagnetic radiation. It was
believed to be invisible and massless and pervaded all of space. Its sole function was to carry
electromagnetic waves and it very nicely fit with the mechanistic Newtonian view of an absolute
coordinate system from which all motion through space could, in theory, be measured.

This Newtonian view of motion had been extremely successful at describing a large number of physical
phenomena and could be traced all the way back to Galileo's investigations of inertia. This concept of
inertia—incorporated by Newton into his first law of motion—states that every body perseveres in its
state of rest, or of uniform motion in a right line, unless it is compelled to change that state by forces
impressed upon it. According to this world view, time and space are absolute and there exists some frame
of reference (such as the coordinate system needed to identify the position of a desk in a multistory
building) that is also absolute. Furthermore, experiments performed are meaningful only to the extent they
can be related to the same experiment performed in that absolute coordinate system. According the
experiments performed by Galileo, if an observation is made in an inertial reference frame—a reference
frame that is stationary or moving at a constant velocity—that observation can be described in another
inertial frame of reference by simply adding velocities. A procedure knows as a Galilean transformation.

In an attempt to determine the motion of Earth through the ether, and thus determine its position relative
to the absolute reference frame of space and time, the American physicists Albert Michelson and E. W.
Morley conducted a series of interferometry experiments beginning in 1887 (see below.)


                                                               The experiments were based on the simple
                                                               concept that light should be affected by
                                                               Earth's motion through the ether. To
                                                               understand this, consider what happens to a
                                                               swimmer who wants to cross a stream that
                                                               has a swiftly-moving current. As the
                                                               swimmer starts across the stream, the current
                                                               will push against her and cause her path to
                                                               veer downstream instead of straight across
                                                               the river. This will cause the swimmer to
                                                               swim more slowly as she battles against the
                                                               current and swim faster as she swims with the
                                                               current. Using this reasoning, Michelson and
Morley believed that a light beam split by a mirror into perpendicular paths will travel at different speeds
since one beam will be impeded by the "current" produced by Earth's drift through the ether. When the
two light beams recombined, they hypothesized, the beams would be out of phase due (see figure on right
below) to the effect of Earth's motion relative to the ether and an interference pattern would be observed.
Even using their very elaborate interferometer, however, the two scientists were unable to detect any
movement of the earth through the ether and gradually the scientific community was forced to accept the
notion that space and time were not absolutes in any conventional sense and that there is no external
standard of rest from which all motion through space can be measured absolutely.

In order to explain the "null" result of the Michelson-Morley
experiment, young Irish physicist George FitzGerald hypothesized
that the physical dimension of an object is deformed in the direction
of motion. Thus, the distance between the mirrors of the
experimental apparatus is contracted in the direction of motion by
exactly the amount needed to account for the lack of interference of
the light beams. The world-renowned physicist Hendrick Lorentz reached the same conclusion, and in an
act of almost unheard of generosity, shared credit with the less-well known FitzGerald.

By the end of the nineteenth century, physics was faced with a crisis of immense proportions. Maxwell's
equations, which so beautifully combined the equations describing electric fields with those of magnetic
fields while theoretically providing a value for the speed of light, produced contradictory results
depending on one's frame of reference. If we consider the case of electromagnetic induction—which
causes an electric field in a wire loop when a magnet passes through the loop, the accepted view of
Maxwell's equations result in a paradox. A moving magnet passing through a stationary wire loop yields a
different result than that of a moving wire loop passing over a stationary magnet. The young patent clerk
Albert Einstein believed that a single phenomenon—wire loop and magnet approach each other—
demanded a single explanation. As far as he was concerned, the vast body of work that was being created
on the subject (and was also leading some of the greatest scientific minds down dead-end paths) was
simply a misinterpretation of Maxwell's equations based on an erroneous belief in the reality of the
luminiferous ether. The solution, according to Einstein, was to give up the concept of the ether and accept
the speed of light is constant for all observers.

According to Galileo and Newton, if you speed up, an object's speed relative to you decreases and
eventually appears to stand still once you reach the same speed as the object. But, according to Maxwell
and countless experiments, light can never stand still.

Einstein realized there was a conflict between the type of relative motion between two inertial frames of
reference—called a Galilean transform—and the relative motion between an observer and a beam of light


that could only be resolved with a reinterpretation of the type of transform suggested by Lorentz-
FitzGerald contraction. In his 1905 paper On The Electrodynamics of Moving Bodies, Einstein resolved
the conflict between intuition (based on measurements made in our slow-moving, everyday world) and
the measurable properties of light, but the resolution came at a price: Individuals who are moving with
respect to each other will not agree on their observations of space and time!

To understand the implications more fully, consider the following thought experiment courtesy of Brian
Greene's book The Elegant Universe.

Imagine two countries that have been at war are sitting down to sign a treaty ending hostilities while
traveling aboard a train that is moving at a constant velocity. The catch is that neither country's delegate
wants to sign the treaty before the other delegate and thus, a simple system is devised to ensure that both
delegates sign the peace treaty simultaneously. The solution involves setting a light bulb at the center of a
table in such a way that the light bulb is exactly between the delegate from Forwardland (who is facing
the direction the train is traveling) and the delegate from Backwardland (who has her back to the direction
the train is traveling). When the light bulbs lights up, that is the signal for both delegates to sign the

This setup is agreeable to all parties on the train and to both security councils in the countries' respective
capitals. Once the bulb lights up and the delegates have simultaneously signed the peace treaty, everyone
on the train celebrates the cessation of hostilities, but they are perplexed to discover that fighting has
broken out anew between the two countries. The reason given is that the delegate from Forwardland was
tricked into signing the treaty before the delegate from Backwardland! How can this be?

The solution lies in the fact that the speed of light is constant for all observers in inertial frames of
reference. In this case, one reference frame is the train moving at a constant velocity and the other
reference frames are the security councils sitting at their tables in the two capital cities. On board the
train, the distance from the light bulb to each delegate is fixed and the light reaches each delegate at the
same time. But from the point of view of the two security councils, once the light leaves the bulb, the
delegate from Forwardland is moving towards the approaching light at the velocity of the train and it
reaches him earlier than the light traveling toward the delegate from Backwardland. This is because the
delegate from Backwardland is moving away from the approaching light at the velocity of the train.
Because the speed of light is constant, it reaches the delegate from Forwardland first according to the
observers on the two security councils! Events that are simultaneous in one inertial frame of reference are
not simultaneous in other inertial reference frames!

When Einstein formulated his special theory of relativity—it is
"special" in the sense that it deals with Euclidean, or flat-space
geometry—he developed two simple, yet profound principles: one
dealt with the nature of light, and will be described a bit later, while
the second dealt with the nature of physical laws in a more general

The principle of relativity is not concerned with a specific physical
law, but rather with all physical laws and is based on one very simple
premise: When describing an object's speed or velocity (the object's
speed and direction of motion), it is essential to specify who or what


is doing the measuring. In other words, we can speak about the motion of an object, but only relative to or
by comparison to another object. Saying "Marianne is traveling at 65 miles per hour" is meaningless, but
saying "Marianne is traveling at 65 miles per hour past Ginger" does have meaning. In other words, there
is no absolute notion of motion. All motion is relative to some external reference.

Imagine Marianne has boarded an airplane and settled comfortably into her seat. After picking up a
magazine she begins reading while waiting for the airplane to depart for its destination. Momentarily she
looks out the window and notices the aircraft next to her plane moving forward slowly. Unsure of what
has happened, she cannot be sure if the airplane seen out the window is slowly taxiing forward or if her
own airplane has been pushed back from the gate and is slowly backing up. Likewise, as Ginger looks out
the window of the other plane, she may experience the same momentary confusion. Is her plane moving
forward or is Marianne's plane pushing back from the gate?

This example describes one and the same situation but from two different and equally valid points of
view. If the push-back from the gate is very gentle, both passengers will still feel she is stationary and
perceives that the other is moving. As stated, there is no one view point that is correct and the other

Although the example described above is a bit contrived, it does serve to highlight the essential aspects of
the principle of relativity because airplanes taxi slowly and it is quite possible that passengers may not
feel the initial "push back" from the boarding gate. The example can easily be reshaped to describe two
astronauts floating freely in the vacuum of space. Marianne would have no way of knowing whether she
is motionless in space or if she is moving. Even if she suddenly notices Ginger in the distance and sees
that the distance between the two of them is decreasing, Marianne would have no way of knowing if she
is stationary and Ginger is approaching her, if Ginger is stationary and Marianne is moving towards her,
or if both are moving.

You may have figured out the essential element in the scenarios described so far and that key element is
that neither Marianne or Ginger is being acted upon by an external force. Both exist in their own inertial
frame of reference in which they are either stationary or moving at a constant velocity. In this respect,
inertial reference frames are force-free frames of reference. This is a crucial consideration because if
either Marianne or Ginger had a rocket pack, she could fire the rocket and feel the force that would give
away the fact that she is moving. Constant velocity motion is relative, but accelerated motion is not.

Einstein realized that there was no experiment that a person in an inertial reference frame could perform
that would give away whether the experimenter was stationary or moving at a constant velocity. He then
went on to make an even grander claim about nature: the laws of physics must be absolutely identical for
all observers undergoing constant velocity motion.

Combining concepts Einstein arrived at through his own "thought experiments" and those contributed by
some of the greatest scientific minds of the 19th century, Einstein formulated two postulates:

         The speed of light has the same value in all inertial reference frames
         The laws of physics are identical in all inertial reference frames

While the second postulate is simply a restatement of the principle of relativity, the first postulate is
dramatically different than what common sense dictates.

Experience tells us that a fast car can catch up to a slower car or that gravel that falls out of the back of a
dump truck actually has a speed that equals the speed of the dump truck plus the speed of an approaching
car as the gravel bounces on the pavement. The same is not true for light, however. The light emitted from


the dump truck's tail light would be measured to travel at 300,000 kilometers per second regardless of
whether the light was measured by someone riding in the truck, someone following the dump truck in a
car traveling at 60 miles per hour, or by someone standing by the side of the road. Taken to an extreme, a
spaceship traveling at half the speed of light (0.5c) could fire a laser weapon at an enemy spaceship, but
the speed of the laser beam would be simply the speed of light and not the speed of light plus the speed of
the pursuing spaceship. This would be true for all observers regardless of whether the observer was on the
spaceship, being fired upon, or simply watching from a nearby space station. As stated before, the
constancy of the speed of light was implied by Maxwell's equations and a number of experiments hinted
that this was true, but the thought experiments described on this page and the previous pages point out the
bizarre results of this simple fact.

Let's now investigate some of the weirdness associated with special relativity by using the example of a
Formula 1 car driven by Marianne at the fabled site of the Belgian Grand Prix—Spa-Francorchamps. The
vital statistics of the car are given in the table below:

                                        Length: 14.8 ft
                                        Mass: 1,323 lbs*
                                        Kemmel Straight: 214 mph
                                        Time to 125 mph: 3.8 sec
                                        *includes Mariannne

As the car is zooming around the course, let's compare the results of measurements made by Marianne as
she is driving the car and by Ginger as she watches the car go around the track:

Marianne's Frame of Reference

                                        L: 14.8 ft
                                        m: 1,323 lbs
                                        Kemmel Straight: 214 mph
                                        t to 125 mph: 3.77777777777751 sec

Ginger's Frame of Reference

                                        L: 14.7777777777773 ft
                                        m: 1,323.00000000000059 lbs
                                        Kemmel Straight: 214 mph
                                        t to 125 mph: 3.8 sec

Okay, it's not a big deal, but something is amiss. Marianne and Ginger cannot agree on the length of the
car, its mass, or how long it takes the car to accelerate to 125 mph! The discrepancy is infinitesimally
small, however, and hardly anything to worry about in everyday racing. Things get much stranger,
though, if we consider the extreme case of a race car that can travel at a significant percentage of the
speed of light.


Imagine the scenario of a car that can travel through space in the Solar Grand Prix. Before the start of this
extraordinary race, Marianne and Ginger carefully measure their car and calculate the time needed to
reach the Sun. The results of their measurements are shown below:

                                         L: 14.8 ft
                                         m: 1,323 lbs*
                                         Speed: 580,000,000 mph (0.87c)
                                         t to Sun: 9 min 58 sec
                                         *including Marianne

Now things are getting interesting. Aerodynamics and wind tunnel testing are essentiall irrelevant when
space travel is concerned, but the g-forces would be substantial. Anyway, let's check in and see what
happens after the race is underway and Ginger and Marianne double-check their measurements and

Marianne's Frame of Reference

                                         L: 14.8 ft
                                         m: 1,323 lbs
                                         Speed: 0.87c
                                         t to Sun: 4 min 38.5 sec

Ginger's Frame of Reference

                        L: 7.4 ft
                        m: 2,638 lbs
                        Speed: 0.87c
                        t to Sun: 9 min 58 sec

Either Marianne's traditional pre-flight breakfast had a dramatic effect on her or something was seriously
amiss. As the car sped towards the Sun, Ginger thought the car looked only half as long as it did before
the race and it had doubled in mass! When Ginger told Marianne of her findings, Marianne was certain
Ginger screwed up her measurements and couldn't use her calculator correctly. As far as Marianne was
concerned, the car was still 14.8 ft long and there was no change in its mass, either. What was strange,
though, was Marianne's watch seemed to have lost 5 minutes and 20 seconds when she compared it to
Ginger's after she arrived at the Sun! What the two ladies have experienced is the weird reality of life near
the cosmic speed limit.

After discussing the problem with their crew chief and consulting the team astronomer, the women were
reassured that there was nothing to be concerned about. The car was performing perfectly. To understand
the discrepancies in length, mass, and length of trip, the women should have paid attention to the special
relativity lecture in driver's school instead of daydreaming while they looked out the window.
Specifically, they should have paid attention to the discussion of Lorentz transformations.


Lorentz Transformations
The Lorentz transformations are mathematical equations that allow us to transform from one coordinate
system to another. Why would we want to do this? Because special relativity deals with inertial frames of
reference. When comparing physical quantities in one frame to the same
quantities in another, it is necessary to first transform from one coordinate
system to another. Thus, we can utilize the Lorentz transforms to convert
length and time from one frame of reference to another.

For example, if you are flying in an airplane and a friend is standing still on the
ground, you could apply the transformations to transform your friend's frame of
reference into your frame of reference and she could do the same for you in
your frame of reference. The previous statements imply that lengths and times
are not the same for objects that are in motion with respect to each other.
Using simple Galilean transforms is fine for everyday situations, but they break
down when observations involve electromagnetic radiation. This forced
Einstein to utilize Lorentz transformations because they provide a method of
translating physical quantities from one frame of reference to another when the
speed of light is held constant in both.

The result of applying Lorentz transforms of special relativity (see equations above), however, is that
dimensions along the direction of motion contract (this is called Lorentz-FitzGerald contraction) and this
observation became one of the consequences of special relativity. For an object traveling at velocity v, the
mass, length, and rate of the passage of time measured by someone watching the object pass by are m, L,
and t. The person onboard the moving object would measure mass, length, and rate of the passage of time
as mo, Lo, and to, however.

In summary, remember the results of the extraordinary race to the Sun:

         As seen by an observer in a different inertial reference frame, the length of a moving
          object will be shortened along the direction of motion and the mass will be larger.
         The clocks onboard the moving object will run slower (as will all processes!) when
          compared to the clocks of a different inertial reference frame.

These results may seem very difficult to explain or understand given our common sense about how the
world works, but the concepts really are not that difficult to understand so long as we are willing to let go
of our "common sense" and follow the logical arguments and mathematics carefully. To help us
understand this phenomenon of time dilation more fully, let's construct the simplest clock of all: a clock
consisting of two mirrors and a single photon of light reflecting between them.

The Light Clock
This odd device can be thought of as a clock if we use the length of time required for the photon to reflect
off the second mirror and return to its starting point as one "tick" of the clock. For example, if the mirrors
are six inches apart, the round trip time for one tick of the clock is 1 billionth of a second. Therefore, one
second is equivalent to 1 billion cycles (ticks) of the light clock.

In the diagram below, we see two light clocks—one stationary and one moving to the right at a significant
percentage of the speed of light. In each clock, photons of red light having the same wavelength (or
energy, if you prefer) are used.


                                                                It makes perfect sense that the photon
                                                                reflects back and forth between the two
                                                                mirrors of the stationary clock, but the
                                                                photon of the moving clock must behave
                                                                the same way. Remember that the principle
                                                                of relativity states that it is not possible to
                                                                determine one's state of motion or rest in an
                                                                inertial reference frame and that the laws of
                                                                physics must be the same for all non-
                                                                accelerated observers. Therefore, the law of
                                                                reflection must hold regardless of whether
                                                                the clock is stationary or moving at
                                                                constant velocity.

                                                                  Since the moving clock is traveling at
                                                                  velocity v, the actual path the photon
travels as it moves between the two mirrors is considerably longer than the path length of the stationary
clock since the moving clock actually travels a distance d during one tick. At everyday speeds, such as 10
mph, the increased path length of the moving clock is impossibly small since the moving clock gets only
15 billionths of an foot to the right before the photon is reflected back to the bottom mirror. Nevertheless,
the increased path length means it takes the photon of the moving clock longer to return to the bottom
mirror and the clock ticks slower. If, on the other hand, the moving clock were traveling at 75% of the
speed of light (0.75c), the moving clock would tick at only 2/3 the rate of the stationary clock That's why,
in our example of the Solar Grand Prix, Marianne's clock was slow compared to Ginger's—a clock
moving at 0.87c would tick at only half the rate of a stationary clock. Marianne's watch, her car engine's
rpms, and her biological processes were all occurring at half the rate of Ginger's!

From Marianne's perspective, however, she is completely oblivious to the time discrepancy. Only after
Marianne makes the return trip to Earth (also traveling at 0.87c) and the two women compare notes do
they realize that they experienced the passage of time differently. In fact, when Marianne gets back to
Earth, her clock will be a total of 10 minutes and 40 seconds slow and she will have aged that much less
than Ginger. Separately, each had a legitimate claim that they were experiencing time normally since
neither could tell if she was stationary or moving at constant velocity. Both were in force-free (inertial
reference frames). This discrepancy in the passage of time is often referred to as the Twin Paradox
because if Marianne and Ginger were identical twins, we have discovered a way for the twins to age at
separate rates and no longer be the same age!

If you've been following closely, you may have wondered why Marianne's clock has lost time when
Einstein said motion is relative for observers in inertial reference frames. Don't both women have an
equal right to claim her clock is slow since neither one could perform any experiment that proves she is at
rest or in force-free motion? Is the principle of relativity only an illusion? Let's investigate the situation
more fully to see if there is a logical solution or if special relativity breaks down into absurdity.

Imagine that Ginger and Marianne agree to perform a time trial. In this case, however, Marianne begins
her "lap" from the orbit of Mars (distance from the Sun = 1.5 AU) and at the precise instant that she and
Ginger meet at Earth, they synchronize their clocks to 0:00. As Marianne and Ginger get farther apart,
each woman can make a valid claim that her clock is the slower clock. To resolve the dilemma head-on,
Marianne must return to Earth so that she and Ginger can compare times. Therein lies the resolution to the
apparent problem: Marianne must brake and turn around or simply make a wide sweeping turn at constant


speed. In either case, Marianne is no longer moving in an inertial reference frame but accelerates. Her
motion is no longer force-free and she feels her motion through the force exerted on her body. By turning
around and returning to Earth, Marianne gives up any claim to being at rest relative to Ginger.

Just how much slower can Marianne's clock and biological processes be? Let's consider a case where
Marianne is able to get her spacecar up to 99.5% of the speed of light (0.995c) and that she and Ginger
again synchronize their clocks to 0:00 as Marianne and Earth pass. To fix our scenario in time, imagine
that Ginger and Marianne are both 22 year-old recent college graduates. Let's also imagine that Marianne
has sufficient fuel reserves to pilot her vehicle for 3 years before turning around and heading back to
Earth. If she returns to Earth at the same speed, the total time she is away from Earth will be 6 years and
she will be a still youthful 28 years old when she meets Ginger. Ginger will have to search the dim
recesses of her memory for any recollection of the young woman greeting her, however, because Ginger
will be enjoying her retirement years at the stately age of 82! Amazing as it may seem, Marianne will
have been away from Earth a total of 60 years and traveled a distance of 30 light-years from Earth before
heading for home.

These Are The Voyages...
One truism of science fiction plots centering around space travel is that spaceships must travel faster than
the speed of light. It certainly makes voyages occur at time scales familiar to the viewing audience, but
special relativity makes faster than light travel unnecessary (and as far as the majority of scientists have
concluded, downright impossible!), as the following example will demonstrate.

Consider if you will, a Constitution class starship with a crew of 430 men and women. The specifications
of the ship are

Length: 947 ft
Mass: 150,000 tons

The intrepid crew is given the assignment of exploring the spectral type M5 red dwarf known as Luyten
725-32. This dim neighbor of our Sun is 12.5 light-years from Earth and lies in the constellation Cetus—
the sea monster sent to devour the sacrificial maiden Andromeda—at right ascension 1hr 10min and
declination -17° 47'. Hoping to minimize the quantity of food, water, and other supplies necessary to
sustain the crew, the Lorentz transformations are used to calculate various voyage times.

The results of the calculations for a 25 light-year round-trip mission are summarized in the table below:

 Velocity            Mass                 Length         Length of Mission       Length of Mission
                     (tons)               (ft)           (years from Earth)      (onboard)
 100,000 mph         150,000              947            167,000                 167,000 yrs
 0.25c               154,900              907            100                     97 yrs
 0.50c               173,000              811            50                      43 yrs
 0.90c               344,000              408            27.8                    12 yrs, 1.5 months
 0.98c               754,000              186            25.5                    5 yrs, 2 weeks
 0.999c              3,350,000            41.9           25.0                    1 yr, 1.5 months
 0.9999c             10,700,000           13.2           25.0                    4 months, 1 week

Certainly, achieving the highest velocity is desirable if the storage space for supplies is limited and being


able to accelerate to 99.99% of the speed of light would require less than half a year of supplies! Of
course when the crew returns to Earth, loved ones left behind may have changed considerably. A mother
leaving a toddler with her father on Earth would return to find that her daughter may have already
graduated from college! She would have aged only a few months and in a very real sense would have
traveled into her daughter's future.

Also notice the profound effect traveling at velocities near that of light has on the mass of the spacecraft.
As velocity increases, so does the starship's mass and thus the amount of thrust needed to accelerate the
vessel to ever higher velocities. The mass increases exponentially past 99.99% of the speed of light and at
exactly the speed of light, the mass would theoretically become infinite. Even a simple application of
Newton's second law of motion (usually expressed as F = ma) dictates that an infinite force is needed to
accelerate a starship to the speed of light because some of the ship's energy is converted into an ever-
increasing mass. It is not a technological barrier that stops us from reaching the speed of light and beyond,
it is a fundamental property of the behavior of nature. The consolation is that we do not need to break the
speed of light to voyage to the stars, we only need to approach it closely.

While the prospect of voyaging to the stars sparks the imagination, it is also true that you can't go home
again. Our little example points out the huge dilemma space voyagers will face should they embark on a
journey to even a neighboring star. An expedition to a distant star such as Betelgeuse—at a distance of
500 light-years—would take a mere 14 years to return home at 99.99% of the speed of light (0.9999c),
but the crew would return to Earth after everyone they knew had been dead for nearly 1,000 years. Being
able to achieve velocities that are very near the speed of light is essential if such voyages are to be
feasible from the standpoint of the crew.

In reality, however, future human exploration of space will require a commitment far beyond anything
earlier generations of explorers had to endure. Even if we could reach the technological milestone of
constructing a starship capable of accelerating to 0.10c, a multi-generational crew would be required to
complete a mission. Just traveling to our sun's nearest stellar neighbor Alpha Centauri would require 85
years! It's quite possible that entire generations would be born and die never having set foot outside of
their spacecraft during such a voyage! Interestingly, there was a symposium during a recent meeting of
the American Association for the Advancement of Science (AAAS) at which a group of physicists,
biologists, anthropologists, and engineers contemplated the challenges of undertaking just such a space

Before leaving the topic of special relativity, let's consider a concept proposed by the mathematician
Hermann Minkowski and later by Einstein. From the time that we first roll over or attempt to stand, we
intuitively have a sense that we move through space. What Minkowski and Einstein advocated was
thinking of time as a fourth dimension (in addition to the three dimensions of space).

If we think about this we can understand the logic of their reasoning. If we wish to meet someone, it is not
enough to say where—the Java House at 713 Mormon Trek Blvd, for example, we must also specify
when we will meet (7:30 p.m. on March 29, 2007). In a very real sense, we must specify where in time-
space we will meet. We've already discussed the bizarre result that perceptions of time and space are
dependent on the relationship between the observer and the observed, but motion through time and its
relation to motion through space can be elegantly explained using another example from Brian Greene's
The Elegant Universe (see Suggested Reading page.)

Let's imagine a very impractical car that is capable of attaining only one speed (say, 100 mph) and it
maintains that speed until its engine is shut down and the car coasts to a stop. Now let's imagine that
Marianne is testing this car on a 1/4-mile stretch of a wide-open salt flat. After she crosses the start line


going 100 mph, Ginger times Marianne and finds that she covers the 1/4-mile in exactly 9.0 seconds.
Time after time Marianne repeats the test and the time is always 9.0 seconds. Until late in the afternoon,
that is. As the afternoon progresses the times get longer and longer; first 9.2 seconds, then 9.5 seconds.
Eventually the times become longer than 10 seconds. Frustrated at the increasing test times, Marianne and
Ginger check the car out but find no mechanical problems to explain the longer 1/4-mile times. Looking
at the fresh tire tracks in the salt, Marianne suddenly realizes the problem: as the afternoon progressed the
Sun was affecting her ability to see the distant mountains and she couldn't tell which direction was due
west anymore. Instead of driving in the same direction trial after trial, she began to veer slightly further to
the southwest with each 1/4-mile run. The car was not getting slower—the distance to the finish line was
getting slightly longer because a small southward component was being added to the direction of travel.
Some of the 100 mph went into traveling west,but some speed was also expended in the southerly
direction. The car was actually going slightly farther than 1/4-mile with each successive time trial!

Einstein and Minkowski envisioned our journey through space and time much like Marianne's imaginary
quarter-mile test drives. Like the impractical car with only one speed, Einstein proposed that every object
in the Universe always travels at the speed of light through spacetime but that speed is a combination of
motion in two dimensions. In other words, our velocity through spacetime is a combination of our speed
through space and our speed through time. In everyday life, we move at a very slow speed through space
and thus travel mostly through time. For objects that travel through spacetime at very high speeds, the
passage of time slows because most of the object's motion is through space. Time passes fastest for
objects at rest while time stands still for photons of electromagnetic radiation that travel at the speed of