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Relativity - DOC

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Introduction


T
        he first person to understand the                   Quantum Physics pertains to the incredibly
        relationship between space and time was             small-scale structure of atoms and nuclei.
        Albert Einstein. Three important papers             Relativity applies to the incredibly large-scale
were published in the early part of the 20th                structure of the entire universe. There are two
Century that established Einstein as one of the             parts to relativity: Special relativity describes
greatest scientists of all time. There were also            how space and time are related at constant
many other very talented scientists who laid the            velocity, and General Relativity describes how
groundwork for Einstein‟s research, and many                space and time are related in accelerating
other scientists working after to prove his                 reference frames, a more „general‟ case. Today,
theories in the laboratory. The physics of Isaac            scientists are attempting to develop a theory that
Newton do not apply in the relativistic world of            encompasses everything, both the very big and
Einstein, nor do they apply in the sub-atomic               very small.
world -–the other great 20th Century scientific
revolution. Newton‟s world of Classical Physics
is the everyday ordinary medium sized universe.

Reference Frames

       et‟s say you are on a train with absolutely

L
                                                            have a lap full of tea. Uncover the windows and
       smooth tracks. On board you can play                 only now can you detect your motion.
       catch with a football, pour tea into a cup,
and jump up and down like you could at rest.                Imagine you and your friend are playing catch on
The ball, while in the air, does not suddenly hit           the moving train, and a third friend is outside not
the rear of the train, nor does the tea or you. You         moving on the station platform. How would this
all continue to move with the train. In fact, if the        stationary person detect and measure the motion
windows were all covered you couldn‟t even                  of the ball?
determine if you were moving or not. Until, of
course, you turn the corner or hit that bump and

Question
1. Suppose you can throw a ball at 80 km/h.
                                                        80 km/h
     a)        You throw the ball from the back of a
               stationary truck. How fast is the ball
               moving relative to your catcher friend
               when he catches the ball? 80 km/h


          b) Now suppose you throw the ball toward       140 km/h
             your friend from the back of a moving
             truck. The truck is moving toward your
             friend at 60 km/h. How fast is the ball
             moving as your friend catches the ball?
             140 km/h

          c)     You throw the ball from the             20 km/h
                 back of a truck moving away
                 at 60 km /h from your
                 friend. How fast is the ball
                 moving as your friend
                 catches the ball?
Postulates of Special Relativity


A
         ccording to Einstein, these examples of           motion of the truck. If a rocket travelling at 0.9c
         adding and subtracting velocities                 (90% speed of light) were to fire missiles at 0.9c
         developed by Galileo are wrong for                relative to the motion of the rocket, a stationary
objects that travel near the speed of light. A             observer would see those missiles travelling at
beam of light shines at the speed of light                 0.99c.
regardless of the motion of the source of light. If
you and your friend were to perform the same               Einstein was an inquisitive thinker. He asked the
experiment as before but this time shining light           question: “What would it be like to travel
rather than throwing balls the results would not           alongside a beam of light?” From thinking about
be the same. If you were able to measure the               and answering that question he formulated the
speed of light coming from a flashlight, it would          Two Postulates of Special Relativity
have the same measured value regardless of the

         First Postulate                All the laws of nature are the same in all uniformly moving
                                        reference frames.

         The consequence is that you cannot detect your own motion. Coins flip as if they would when
         travelling near the speed of light or at rest. This is where the term „relativity‟ originates: All
         motion is relative. If you are travelling in one spaceship past another spaceship, it would be the
         same as, and indistinguishable from, another spaceship travelling backwards relative to your
         stationary spaceship.

         Second Postulate               The speed of light in empty space will always have the same value
                                        regardless of the motion of the source or the motion of the observer.

         The consequence is that there is an upper speed limit in the universe, the speed of light, and
         nothing can travel faster.

Time Dilation


T
       ime runs more slowly on moving objects               Proof: High-speed radioactive cosmic particles
       as measured by stationary observers.                 decay more slowly, as measured by us in the
       However, on the spaceship, everything                stationary laboratory.
appears normal. And according to the spaceship,
everything is moving around the stationary
spaceship, and all of their clocks run slowly.

Twin Trip

One twin rides a high-speed spaceship, the other            on the whether the spaceship is receding or
stays at home. If the travelling twin maintains a           approaching, according to the Doppler Effect.
speed of 0.87c for one year, 2 years elapses on               Receding          10 pulses @12 min         = 120 min
the Earth. At 0.995c for one year, 10 years will
                                                               Approaching       10 pulses   @ 3 min          = 30 min
elapse on then Earth. The travelling twin will
come back younger than the twin who stayed on                  Total                                          2 ½ hours
the Earth. This is a result of the constancy of the
                                                           More time has elapsed on the Earth. This makes
speed of light.
                                                           time travel possible! (Sort of) If you travel near
Proof: Let the spaceship send a pulse of light
                                                           the speed of light for a long time, say 5 years,
every 6 minutes. 20 pulses of light at 6 minutes
                                                           you could travel 1000 light years distance, as
are 120 minutes or 2 hours. On the Earth the
                                                           measured relative to Earth distance. How is this
pulses are received at different rates depending
                                                           possible? This is a result of length contraction.
Length Contraction


M
           oving objects undergo changes in           length contraction. The muon would measure
           length as well as time. Moving objects     the distance travelled as shorter.
           appear to contract along the length of
motion. To explain how this works take the             Earth measures        9000 m     @ 30 s        300 m/s
example of the radioactive cosmic particle the         the muon travel
muon. The muon half-life at rest is 2 s and the
half-life while moving as measured by stationary       Muon measures         600 m      @ 2 s         300 m/s
                                                       itself travel
observers on the Earth is 30 s. Therefore, as
the muon decays 30 s passes on the Earth, and        Both measure the same relative motion,
2 s passes for the muon. But how does the            therefore, this example still obeys Einstein‟s
muon see this change in time. How is this             First Postulate.
explained? Einstein‟s First Postulate must be
obeyed, which states that everyone in all
reference frames moving or not must get the
same results. This problem is explained using
Relativistic Momentum


F
       or a stationary observer they will measure     electron is deflected less than what would be
       objects moving with an increased               predicted using classical Newtonian Physics.
       momentum. Indeed, the momentum, as
                                                                            Magnet
measured by a stationary observer, approaches
infinity as the object travels closer and closer to
the speed of light. That object, of course, would
not measure any change in its own momentum.                                                       Relativistic (observed)
Consequently, no object of any mass can travel
at the speed of light.                                                      Magnet             Classical prediction
Proof: In a particle accelerator, very small                                                   (not observed)
nuclear particles routinely travel near the speed     To the particles, they see no change in their own
of light. The physicists who study these              momentum. So how is it explained to be
particles must use the concepts of relativity to      consistent with the First Postulate? To the
explain their results. In a particle accelerator      particles, the distance travelled would decrease;
charged particles are deflected when they pass        therefore there is less deflection.
through a magnetic field. A faster moving

Equivalence of Mass and Energy


M
         ass is simply a form of energy. Even         Rest energy, like other forms of energy, can be
         if an object is not moving it has rest       converted into other forms. When striking a
         energy. The relationship between             match the mass of the products is slightly less,
mass and energy is in the form of the 20th            about 1 part in a billion, because of the kinetic
Century‟s most famous equation:                       energy of the products (the fiery hotness). For
                                                      nuclear reactions the mass difference (mass of
         E0 = mc2.                                    products < mass of reactants) is more substantial
                                                      which results in a huge release of energy
E0 is the rest energy of the object, m is the mass    (enough to obliterate entire cities). Hot tea has
and c is the speed of light. Because c is such a      more mass than cold tea, because there has been
large number already, its square is even larger,      an increase in the energy of the tea.
the rest energy of even the smallest amount of
matter is enormous.
Review Questions

1.   The speed of a ball you catch that is thrown from a moving truck depends on the speed and direction
     of the truck. Does the speed of light measured from a moving source depend on the speed and
     direction of the source? Explain. Yes, the speed of the truck and ball matters at these low
     velocities; the velocity is either added or subtracted depending on the direction.
     But the speed of light does not matter as the speed of light is the same in all
     reference frames, moving or not.
2.   What are Einstein‟s Two Postulates of Special Relativity?
         First Postulate                 All the laws of nature are the same in all uniformly
                                         moving reference frames.

         Second Postulate                The speed of light in empty space will always have the
                                         same value regardless of the motion of the source or
                                         the motion of the observer.
3.   If a spaceship moves away from you at half the speed of light and fires a missile at half the speed of
     light relative to the spaceship, common sense may tell you that the missile moves at the speed of light
     relative to you. But it doesn‟t. The relativistic addition of velocities is given by:
             v1  v 2
         V 
                vv
            1  1 22
                 c
                0.5c  0.5c     1.0c
          V                           0.8c
                 (0.5c)(0.5c) 1  0.25
             1
                      c2
         Substitute 0.5c for both velocities to determine the velocity of the missile relative to the stationary
         observer.

4.   Use the equation in question #3 to show that for small everyday velocities this equation is practically
     the same as v1+v2.
                                 m
         V  0.01c  3 106         3000km / s
                                 s

                 0.01c  0.01c      0.02c
         V                                 0.02c
                  (0.01c)(0.01c) 1  0.0001
               1
                        c2
5.   Use the equation of question #3 to show that even if the spaceship could move at the speed of light
     and fired a missile at the speed of light, the velocity relative to the stationary observer is still the speed
     of light. There is indeed an upper speed limit in the universe.
                cc     2c
         V               c
                 c  c 11
               1 2
                  c
6. If we view a passing spaceship and see their clocks running slow, how do they see our time running?
   They also see our clocks as running slow. They have to, because every reference
   frame must have the same results.
7.   Is it possible for a person with a 70 year life span to travel farther than light travels in 70 years?
                                                                                     but
     Explain. Yes, because the clocks on the spacecraft will not have elapsed 70 years,
     something less. To accommodate this, the spacecraft does not travel 70 light years,
     but due to length contraction, it has traveled some distance less.
8.   Suppose you were travelling in a smooth riding train with no windows could you sense between
     uniform motion and rest? Between accelerated motion and rest? Explain how you could do this with a
     bowl filled with water. You cannot tell the difference between uniform motion
     (constant velocity) and rest. All the laws will be the same. You can juggle, pour tea
     or walk with the same results at rest as you would on a train. But you can tell
     when you are accelerating, this is not uniform motion. A bowl filled with water
     will slosh back when accelerating and slosh forward when stopping. Einstein’s
     Theory of General Relativity says that an accelerating frame of reference is
     indistinguishable from gravity.
9.   If you were travelling in a high-speed spaceship, how would the meter sticks on board appear? How
     would these meter sticks on board appear to a stationary observer outside? The meter sticks on
     board traveling with you would be the same length as at rest – one meter. But if a
     stationary observer were to see those meter sticks moving past, they would be
     shorter than one meter.
10. How does the distance to Pluto as measured by a fast moving spaceship compare to our measurement
    on a stationary Earth? The distance to Pluto as measured by a fast moving spaceship
     would be less than when measured by a stationary observer on the Earth.

                                        nuclear fusion within the Sun. When hydrogen
11. Where does solar energy originate? From
     nuclei combine to form helium nuclei, the mass of the products is less than the
     mass of the reactants. The missing mass is energy.
12. What would be the momentum of an object as it is pushed toward the speed of light? How much
    energy would it take to get objects of any mass to travel at the speed of light? The momentum         of
     an object as measured by a stationary observer would approach infinity. So it
     would take a corresponding infinite amount of energy to get a massive object to
     travel at the speed of light. Only mass less particles – photons – travel at the speed
     of light.
13. The two-mile long particle accelerator at Stanford University in California appears to be less than a
    meter long to the electrons that travel it. Explain. This is due to length contraction. The
     electrons measure the length they travel as less than the stationary observers.

14. Does the concept of relativity mean that all of Newton‟s equations are wrong? Explain.
                                                                                 No,
     Newton’s equations work perfectly well for objects traveling at slow speeds (non-
     relativistic speeds, that is, no where near the speed of light.) Our everyday world of
     science on the earth and planets fits this model very well.

                                                                             Any research
15. Describe the types of physics research in which knowledge of relativity is required.
     that explores very fast particles must use the concept of relativity. As well as any
     cosmology, the large scale structure of the universe.

				
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