Special Relativity

[1.1] THE ETHER WIND / DOPPLER SHIFT

The second principle arose from considerations of the observed behavior of light. This principle was articulated by Albert Einstein as a postulate in response to a very difficult problem in physics. What Einstein said was that the speed of light is an absolute constant. Why he said this, and why it was such an astonishing thing to say requires a little explanation.

In classical physics, a wave is a disturbance through a propagating medium. For example, a sound wave consists of variations in pressure propagating through the air, while a water wave consists of variations in height of water propagating across a body of water.

If electromagnetic radiation was in the form of waves, then by classical thinking it had to be a disturbance of some sort of propagating medium. The only problem was that light propagated through free space where nothing of substance could be detected. Physicists therefore suggested the existence of an "luminiferous (light-bearing) ether" as the propagating medium.

Although the ether was invisible and undetectable, it was thought to fill the entire Universe, with the planets and stars moving through it unimpeded. This implied that the ether established an "absolute frame of reference" for the motion of objects through it. The ether was universal and fixed, and so the motions of heavenly bodies could in principle be measured relative to it.

By this time, the speed of light had been determined to be about 300,000,000 meters per second. The question quickly arose: just how fast is the Earth moving relative to the ether? Experiments were conducted to determine the velocity of the "ether wind". The most famous of these experiments was conducted by the brilliant American experimental physicist Albert A. Michelson and his colleague E.W. Morley.

An optical interferometer operates by splitting a beam of light, running it through two paths at a right angle to each other, summing the two beams back together, and then observing the interference effects between them. Shifts in light propagation through the two paths are easily detected. Optical interferometry is extremely sensitive, and is now often used an as element in highly accurate sensor and instrument systems.

The Michelson-Morley interferometer was built on top of a heavy stone slab to ensure stability and reduce nose. The slab could be rotated around the vertical axis. A light beam was split, reflected along two paths at right angles to each other, and then summed again to generate interference effects.

If the reflected light beam was in the direction of the ether wind, the velocity of the ether wind would be subtracted from the velocity of light in one direction, then added in the other. There would be no net effect on the round-trip time.

However, if the reflected light beam was crossways to the ether wind, then the effect of the ether wind would be to slow the beam down whether it was coming or going. The relationship would be that of a right triangle, where the speed of light (C) in the absolute frame of reference was the hypotneuse of the triangle, the speed of the ether wind (V) was the far side of the triangle, and the measured speed of light (Cm) was the adjacent side of the triangle. By the Pythagorean theorem:

    C^2 = V^2 + Cm^2
 

    Cm =  SQRT( C^2 - V^2 )
 

    Cm/C = SQRT( 1 - (V/C)^2 ) = SQRT( 1 - Cf^2 )
 

The idea behind the Michelson-Morley apparatus was to observe the interference effects when the apparatus was in one orientation, and then observe their change when the apparatus was rotated 90 degrees. This would in principle reverse the pattern of delay between the two arms of interferometer, and so reverse the interference pattern.

However, when the experiment was actually performed, the Earth didn't seem to have any appreciable speed relative to the ether wind. The speed of light seemed to be the same along both arms of the interferometer, no matter what the orientation of the device was.

There was the suspicion that this result might be an unbelievable coincidence, that the Earth just happened to be at a place in its orbit where its orbital velocity put it in a frame of reference that matched that of the ether. Although this was implausible, it could be tested. If the Earth completed another half orbit around the Sun from the point where the experiment was performed, it would be going in the opposite direction relative to the ether, and its velocity in comparison to the fixed ether would be twice its orbital velocity.

The experiment was performed six months later, and once again the results were negative. Physicists were baffled and went back to the drawing board.

The Doppler effect is the change in frequency caused by the motion of an object. If a train approaching at high speed blows a whistle, the frequency of the whistle is higher than it would be if the train were at rest. Similarly, if the train is moving away, the frequency of the whistle is lower.

If the time of the period of the whistle's wavelength when the train is at rest is "T" and the period when the train is moving at velocity "V" is "Tm", then if the speed of sound is given by "S", the change in period due to the Doppler shift is given by:

    Tm  =  T * ( 1 - V/S )
 

    Tm  =  T * ( 1 + V/S )
 

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[1.2] EINSTEIN'S POSTULATE

That year he published three scientific papers, one on the photoelectric effect, one on Brownian motion, and one on Special Relativity. The first two are of no great importance in this document, though the paper on the photoelectric effect was one of the studies that led to modern quantum mechanics, and also won Einstein the Nobel prize in 1921.

The paper on Special Relativity considered a very simple but disturbing postulate and examined its implications. Einstein's postulate, mentioned in the previous section, was that the speed of light was an absolute constant. No matter how fast you were moving, any beam of light from any source would be measured to have exactly the same velocity of about 300,000,000 meters per second by any observer. This is why the Michelson-Morley experiment gave negative results.

This seems like an innocent statement until you think about the implications. Supposed someone is coming towards you in a rocket ship at half the speed of light, and then fires off a flash bulb. By classical physics, the flash would reach you at 1.5 times the speed of light. No, said Einstein, it still comes at you at the speed of light, no more or less.

If the rocket ship is moving away at half the speed of light, classical physics says that the flash will move toward you at half the speed of light, but according to Einstein, it still comes at you at the speed of light.

Of course, the speed of the flash as seen from the frame of reference of the rocket ship is the speed of light as well, and the flash propagates away from the ship in all directions at that speed as seen by the crew.

By classical physics, this is completely absurd. If you are driving in a the back of a truck at 80 KPH and shoot a pebble with a slingshot in the forward direction at a velocity of 50 KPH, then ignoring wind resistance the pebble's velocity as seen by someone at the side of the road is 130 KPH. If you shoot the pebble in the backward direction, it will be seen as flying at 30 KPH.

However, what Einstein pointed out was that it was classical physics that was illogical as far as the propagation of light was concerned. If the speed of light was relative to the motion of a specific inertial frame of reference, then there could be an inertial frame of reference where light would stand still or go backwards. You couldn't see yourself in a mirror.

What Einstein did was extend the concept of Galilean Relativity to the measured behavior of light. Galileo observed that there was no way to tell the difference between an object at rest and an object in uniform motion. Einstein said this also applied to light. You could not determine if you were in motion or were at rest by measuring the speed of light relative to you. The equivalence principle said it would always be the same.

Suppose you are on a spaceship and emit a flash of light. By the equivalence principle, it propagates away from you at the speed of light in all directions, and obviously precedes you in the direction of your motion. Suppose somebody on Earth sees that flash. The light of the flash has necessarily arrived before you have, and since the flash is moving at the speed of light, you must be moving more slowly than that.

The second implication was that it undermined traditional concepts of simultaneous action. Suppose you are standing on Earth and watching a friend fly by across your line of sight at great velocity in a glass box. Suppose as your friend flies by, she fires off a flashbulb positioned in the center of the box.

To your friend in the box, the light has to move an equal distance to hit the front and back walls, and so the light hits both walls simultaneously. To you, however, as the flash moves backward, the back wall of the box moves toward it, and as the flash moves forward, the front wall of the box moves away from it. The light from the flash strikes the back wall before it strikes the front.

This says that two events that are separated in space that are seen as simultaneous in one frame of reference are not necessarily simultaneous when seen in another, and any calculations in physics cannot rely on the simultaneity of distant events as a basic assumption.

More interestingly, suppose that one frame of reference is moving relative to another, and that both of these frames contain a clock. Suppose also these clocks are designed to start ticking when one moves right next to the other. The time at which the two clocks start is synchronized, since they are effectively in the same place at the same instant, but once they move apart, they are no longer necessarily synchronized.

This meant that the time they kept wasn't necessarily the same, either. In fact, time might vary between frames of reference.

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[1.3] TIME DILATION

To see how this works, consider watching a spaceship flying across your line of sight. Suppose in this spaceship they have a "clock" consisting of a pair of mirrors oriented at a vertical right angle to your line of sight, with the "ticks" of the clock defined by the time it takes a light pulse to bounce back and forth between the mirrors:

By the equivalence principle, this clock keeps perfect time on the spaceship. The light pulse moves back and forth between the mirrors at a constant speed, and the distance between the mirrors remains constant as well. If we designate the speed of light as usual by "C" and the distance between the mirrors as "D", then the tick time "T" is given by:

    T = 2 * D / C 
 

Let's designate this longer measured time by "Tm" and the velocity of the spaceship by V. Then the relationship between Tm and T is given by:

The distance traveled by clock in the time it takes for the pulse to make a one-way trip from one mirror to the other is:

    Tm * V / 2
 

    Tm * V  
 

    D = T * C / 2
 

    T * C 
 

    Tm = SQRT(( T * C )^2  +  ( Tm * V )^2) / C
 

    Tm^2 * C^2  =  ( T * C )^2  +  ( Tm * V )^2
 
    Tm^2 * C^2  -  Tm^2 * V^2  =  T^2 * C^2
 
    Tm^2 * ( C^2 - V^2 )  =  T^2 * C^2
 
    Tm^2  =  T^2 * C^2 / ( C^2 - V^2 )
 
    Tm^2  =  T^2 / (( C^2 - V^2) / C^2 )
 
    Tm  =  T / SQRT( 1 - V^2 / C^2 )  =  T / SQRT( 1 - Cf^2 )
 
        =  ( 2 * D / C ) / SQRT( 1 - Cf^2 ))
 

    1 / SQRT( 1 - Cf^2 )
 

    Cf       factor
    _______  ______
 
    0.1        1.01
    0.5        1.15
    0.8        1.67
    0.9        2.29
    0.95       3.20
    0.99       7.09
    0.999     22.37
    0.9999    70.71
    0.99999  223.61
    _______  ______
 
 

To the crew of the spaceship, the distance between the mirrors remains the same and the speed of light remains constant. To the observer on Earth, the clocks on the spaceship all run more slowly by the same factor. If they didn't, one clock could be turned on its side, left to run for a while, and then righted again, and would have a different time than a second clock that remained upright. That doesn't and can't happen in the frame of reference of the spaceship.

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[1.4] LENGTH CONTRACTION

Suppose the spaceship is going to a star system a distance D away from Earth at a velocity V, and this star system is not moving at any appreciable speed relative to the Earth. The time T it takes the spaceship to reach the star system is simply:

    T = D / V
 

However, the Earth observer also notices that the spaceship's clock is running more slowly, and in principle the observer could count all the ticks from departure from Earth to arrival at the remote star system to prove that indeed time has run more slowly for the crew of the spaceship. Time dilation is not an optical illusion.

This means that the crew of the spaceship has traversed the distance D in a time Tm, which is shorter than T as given by the formula we have already derived:

    Tm  =  T / SQRT( 1 - Cf^2 )
 

    Dm  =  D * SQRT( 1 - Cf^2 )
 

At that speed, we observe the pole as being length-contracted by 60%, reducing its length to 9 meters. The Flash enters the barn through an open door, while the door on the opposite side through which he intends to exit is closed. The entry door is closed automatically when the tail end of the pole clears it, and then the exit door is opened in sequence quickly enough to allow him to pass through without obstruction.

This scenario makes perfect sense. We observe a 9 meter pole inside a barn 15 meters long, and it obviously fits inside with both the doors closed.

However, let's look at it from the point of view of the Flash. He can consider himself standing still in his frame of reference, with the barn approaching him at 80% of the speed of light. That means that the barn is length-contracted to 9 meters, while his pole still remains 15 meters long. How can it possibly fit inside the barn with both doors closed?

The trick in this case is the non-intuitive fact, which we've already established, that what is simultaneous in one frame of reference may not be simultaneous in the other. As the Flash passes through the entry door, he sees the exit door open in front of him even before he clears the entry door, and the pole is already partway out the exit door before the entry door closes.

• The tip of the pole enters the open door 1, which emits a pulse.

• The end of the pole leaves door 1 and the door closes.

• When door 2 receives the light pulse, it opens to admit the pole.

This analysis will ignore delays in sensing the pole position; triggering a light pulse; sensing a pulse; and opening or closing doors. By assuming increasingly microscopic dimensions for the diameter of the pole and doors and so on, these delays could be in principle made as short as needed.

First, let's see what happens in the rest frame of the barn. Since the speed of light is 300,000,000 meters per second, then light travels a meter in 3.33 nanoseconds, and so it takes exactly 50 nanoseconds for a light pulse to travel across the barn. As the pole is moving at 80% the speed of light, it takes 62.5 nanoseconds for the tip of the pole to travel across the barn, and 37.5 nanoseconds for the pole to pass through door 1.

This gives the time sequence, with intervals in nanoseconds:

     00.0    pole enters door 1, pulse emitted
     37.5    pole leaves door 1, door 1 closed
     50.0    door 2 senses light pulse from door 1, door 2 opened
     62.5    pole enters door 2
 

In this scenario, the pole is 15 meters long and the barn is meters long, and so it takes 37.5 nanoseconds for the tip of the pole to go from door 1 to door 2, and 62.5 nanoseconds for the entire pole to pass through door 1.

Where the trick comes in is in the propagation of the light pulse from door 1 to door 2. Since door 2 is moving towards the light pulse at 80% of the speed of light, then the time T for the pulse to arrive at door 2's sensor is given by the algebraic expression:

    ( C * T )  +  ( 0.8 * C * T )  =  9
 
    1.8 * C * T  =  9
 
    T  =  9 / ( 1.8 * C )  = 16.7 nanoseconds
 

     00.0    pole enters door 1, pulse emitted
     16.7    door 2 senses light pulse from door 1, door 2 opened
     37.5    pole enters door 2
     62.5    pole leaves door 1, door 1 closed
 

The first, entirely sensible, instinct of a reader when given a scenario like this is that it's double-talk and trickery, and to find holes in it. For example, the Flash would say that from his frame of reference, he would think that in the barn's frame of reference light passes across it at 1.8 times the speed of light.

However, we know that in the barn's frame of reference, the speed of light remains exactly the same. This is illogical, right? But we've already addressed this issue, since it's one of the two fundamental postulates of Special Relativity: the speed of light is the same in all frames of reference, and this postulate is provable by experiment. What Special Relativity does is give the tools to work out the implications, and no matter how non-intuitive those implications might seem, they will always work out.

It also implies that if it were possible to go faster than the speed of light, there would be frames of reference where an effect would happen before its cause. Causality is a notion accepted on faith plus the experience that it is never violated in practice, and the idea that a violation of causality is possible is about as scientifically credible as perpetual motion machines. In fact, it's just about the same thing.

First, we'll make the assumption that the distance D between the mirrors is the same as it is when it's upright, even though this assumption has just been shown to be false.

Obviously, when the light beam is moving in the direction of the motion of the clock, the amount of time it takes to move from one mirror to the other will be much longer than it will be when moving against the direction of motion.

We'll designate the time of the forward transit as Tf and the time of the reverse transit as Tr. This means the total tick time is:

    T  =  Tf + Tr
 

    Tf  =  ( D + V * Tf ) / C
 

    Tf * C  =  D + V * Tf
 
    Tf * C - V * Tf  =  D
 
    Tf * ( C - V ) = D
 
    Tf = D / ( C - V )
 

    Tr  =  ( D - V * Tr ) / C
 

    Tr = D / ( C + V )
 

    T  =  Tf + Tr  
 
       =  D / ( C - V ) + D / ( C + V )
 
       =  ( D * ( C + V )  +  D * ( C - V ) ) / ( C^2 - V^2 )
 
       = 2 * D * C / ( C^2 - V^2 )
 
       = 2 * D / ( C * ( 1 - Cf^2 )
 

    Tm  = 2 * D * SQRT( 1 - Cf^2 ) / ( C * ( 1 - Cf^2 ))
 
        = ( 2 * D / C ) / SQRT( 1 - Cf^2 ))
 

This seems a little counterintuitive. If the clock's length is shorter when it is lying on its side, then shouldn't the clock run faster when it's in motion since the light doesn't have as far to travel? However, the trick is that the second mirror of the clock moves away from the first as the light pulse moves forward to catch it, and at high fractions of lightspeed this results in a substantial increase in the distance the light pulse has to travel. To be sure, when the pulse bounces back to the first mirror, the length and trip time is shortened, but not enough to cancel out the increased trip time in the forward direction.

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[1.5] THE TWIN PARADOX

This idea seems contradictory. If each twin sees the other's clock is running more slowly than their own and by the same factor, how can one age less than the other? Nonetheless, that is in fact the case. Seeing why is tricky, partly because although time dilation is not an optical illusion, optical illusions caused by the Doppler effect are also involved.

The phenomenon of relativistic time dilation was derived above in the case of a spaceship moving across our line of sight, in which the clock appears to run more slowly. However, that does not mean that the clock will be seen to run slower in all circumstances, which is a common misconception about Special Relativity. If Alice's spaceship is moving toward Bob, then the Doppler shift will make her clock seem to run faster than his. If Alice's spaceship is moving away from Bob, her clock will seem to run even more slowly than it would if her spaceship were running across his line of sight.

The classical Doppler shift, as derived previously and applied to light, is given by:

    Tm  =  T * ( 1 - Cf )      ! if approaching
    Tm  =  T * ( 1 + Cf )      ! if receding
 

    Tm  =  T * ( 1 - Cf ) ) / SQRT( 1 - Cf^2 )
 
        =  T * SQRT( ( 1 - Cf )^2 / ( 1 - Cf^2 ) )
 
        =  T * SQRT( ( 1 - Cf ) * ( 1 - Cf ) / ( 1 - Cf ) * ( 1 + Cf) )
 
        =  T * SQRT( ( 1 - Cf ) / ( 1 + Cf ) )
 

    Tm  =  T * SQRT(( 1 - Cf ) / ( 1 + Cf ))
 

    Tm  =  T * SQRT(( 1 + Cf ) / ( 1 - Cf ))
 

This means that in Earth's frame of reference, her spaceship spends 20 years in flight to the distant star system. Since light takes 10 years to travel back from the distant star system, Bob will be able to see her spaceship arrive at its destination 30 years after it left Earth.

Suppose Alice's spaceship automatically emits a pulse of light every month of a year by ship time. Bob will have an automatic system to do the same thing from Earth, though for the moment we won't worry about that.

By the relativistic Doppler effect, the pulses from Alice's spaceship are slowed down by the factor:

    SQRT( ( 1 + 0.5 ) / ( 1 - 0.5 ) )  =  1.73
 

  ( 30 * 12 ) / 1.73  =  208 pulses
 

    SQRT( ( 1 - 0.5 ) / ( 1 + 0.5 ) )  =  0.577
 

    ( 10 * 12 ) / 0.577  =  208 pulses
 

    ( 208 + 208 ) / 12 =  34.6 years 
 

If the time axis is measured in years, the space axis is measured in light-years, and the two axes have equal increments, then light always travels at a 45 degree path on the diagram. An object at rest in the Earth's frame of reference follows a vertical path, since it passses through time but not space, while an object moving closer and closer to the speed of light approaches the 45 degree angle.

The light pulses sent by Alice on her outbound leg are marked in green, while those on her inbound leg are marked in violet. Of course, to reduce clutter only a sample of the pulses are actually shown on the diagram.

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[1.6] THE TWIN PARADOX REVISITED (1)

    SQRT( 1 - 0.5^2 ) = 0.866
 

The tricky issue arises when you consider the equivalence principle. Alice's spaceship can be just as validly regarded as being at rest and the Earth as being in motion. In fact, Alice sees Bob's clocks running at the same slow rate as she flies away from the Earth and at the same fast rate as she approaches the Earth.

However, the two scenarios, though equivalent in the strict sense, are not evenly balanced. To Bob back on Earth, Alice's spaceship is moving at half the speed of light, and her spaceship is length-contracted. To Alice, the rest of the Universe is moving at half the speed of light, and the rest of the Universe is length-contracted.

This is the critical point of the Twin Paradox. Alice is taking a trip through a Universe that appears length-contracted to her, and so a trip at half the speed of light takes a shorter time.

In more detail, to Alice the distance to the star system is only 8.66 light years, and it only takes her 17.3 years to get there. During this time, she is observing pulses emitted every month from Earth, and by the equivalence principle, these pulses are delayed by exactly the same factor of 1.73 as the spaceship's pulses are seen as delayed by Earth.

In 17.3 years, the the number of pulses observed by Alice on board the spaceship is:

    17.3 x 12 / 1.73  =  120
 

Now Alice turns the spaceship around and comes back home. This is another difference between the frame of reference of the Earth and that of the spaceship. The Earth will not begin receiving the fast pulses emitted by the approaching spaceship until 10 years have passed in Earth time after the spaceship turns around. In contrast, the spaceship is doubling back on its own path and will immediately begin to receive the fast pulses sent by Earth. These pulses are sped up by exactly the same factor of 0.577 as the Earth observes emitted by the spaceship.

The spaceship's flight time back is 17.3 years, the same as before, and so the number of pulses observed by Alice is:

    17.3 x 12 / 0.577  =  360
 

There is also an imbalance in the fraction of time that slow and fast pulses are observed between the two frames of reference. Bob sees slow pulses for 30 years, or three-quarters of the trip time as seen by Earth, because of the ten-year time lag before the Earth observer sees the spaceship turn around, and sees fast pulses for ten years. Alice, in contrast, sees slow pulses for half the journey and fast pulses for the other half.

Both see clocks running at slow speeds on the outbound leg of the journey and running fast on the return, but the Earth sees the slow clock for a longer proportion of time. Bob sees 34.6 years pass on the spaceship, and Alice sees 40 years pass on Earth.

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[1.7] THE TWIN PARADOX REVISITED (2)

In this scenario, Alice's spaceship will fly at 50% of the speed of light for 40 years by its own clock, without stopping and turning around. From Bob's point of view, the spaceship's clocks are running slow by a factor of 1.15. This means the trip actually takes 46.2 years from the Earth's frame of reference, and the total distance covered by Alice is 23.1 light years in the Earth's frame of reference as well.

Now suppose that after 17.3 years, Bob decides to use an improved, faster spaceship to catch up with Alice and rendezvous after 40 years of flight by Alice's clock. That's a total of 46.2 years by Bob's clock, so he calculates that it will take:

    46.2 - 17.3  =  28.9 years 
 

    23.1 / 28.9  = 0.8 C
 

Bob's calculation that it will take 28.9 years to catch up was performed assuming the Earth's frame of reference. However, once he takes off in his spaceship at 80% of the speed of light, time slows down by a factor of 0.6, and so Bob actually only takes:

    28.9 * 0.6  =  17.3  years
 

When Bob sets off in his spaceship, he immediately sees the clock on the Alice's spaceship change from ticking slower to ticking faster due to the Doppler shift. You can do the math if you like and see that, due to the time lag of the speed of light, Alice won't see Bob depart Earth until 30 years after her own departure by the spaceship's clock, and so will only see Bob's clock ticking faster due to the Doppler shift for ten years.

Similarly, Alice will measure the velocity of Bob's spaceship as 50% of the speed of light and the distance of its travel as 10 light years. Once again, the equivalence principle is preserved.

Of course, since both spaceships have been moving relative to the Earth, they are both out of timestep with it. If they were both to stop when they rendezvous, 40 years would have passed for Alice, and 34.6 years would have passed for Bob -- while 46.2 years would have passed back on Earth. There are now three frames of reference involved, not just two, and that leads to the next paradox, to be discussed in the following chapter.

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