Variable speed of light

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  • #101
Hurkyl
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The discontinuity is the instantaneous acceleration. The net effect of the acceleration is that the spacebound twin instantaneously changes his reference frame.

If you compute the space and time coordinates of earth before and after this change, you'll find that there's a large difference in the time coordinate. (the difference is proportional to the distance to earth)

If, instead of instantaneously turning around, the spacebound twin accelerated gradually, he'd observe the clock on Earth is running really, really fast.
 
  • #102
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In the example given by Ambitwistor, on page 2, the twin in motion measures the earth based clock running slowly over the entire trip. We can make his trip as long as we want. The correction to the clock on earth as observed by the twin in motion never seems to happen except during the turn around and the final deceleration.

If you travel towards someone or away from someone, you will observe that person's clock as running more slowly than your own; that's ordinary SR. What happens at turnaround, though, is that you switch inertial frames, and so you switch surfaces of simultaneity: you can't forget the relativity of simultaneity. All of a sudden, what the distant clock reads "now" jumps, because you have suddenly redefined what "now" means by switching frames. The Earth-based clock doesn't do this. This is what the "time gap objection" in the FAQ is about.
 
  • #103
Ivan Seeking
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There is still one element of the time gap explanation that I have never been able to reconcile. Now, if I read the FAQ explanation correctly, the traveling twin sees the earth based twin age nearly the entire 14 years [in this case the real elapsed earth time] during the return leg of the trip. This throws me a bit.

Let’s make this simpler and assume that our traveling twin travels out from earth for a designated period of time, and he then stops and waits until he sees light pulses from earth. We plan all of this so that he should see the first pulse just after he reaches a complete stop. Then he accelerates to near C and heads home. Now, after compensating for the Doppler shift, according to the traveling twin the earth based clocks are running slowly. Right??? The linked page seems to say otherwise. Now, let’s further assume that we can decelerate the twin’s ship very quickly; say in just a few hundred meters. Doesn’t he see the earth clocks catch up to real earth time [ie. so that both twins count the same number of flashes over the entire return trip] entirely during the deceleration?

If true, this is the part that loses me:

Until the twin decelerates, where are the missing light pulses?

In the earth frame, the pulses left long ago and are clearly not contained in the space between us and the twin.
 
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  • #104
Hurkyl
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From the spacebound twin's point of view (measurements of Earth are assumed to be corrected for the propagation of light, including the Doppler effect):

On the outward journey, he will observe earth's clocks running slowly.

As he decelerates to come to a "stop" (I presume you mean that he comes to rest in Earth's reference frame), he observes Earth's clocks running very fast.

When he comes to a stop, he will observe that Earth's clocks are now ahead of his own.

He receives the first laser pulse, so he begins his acceleration. During this acceleration, he will see Earth's clocks run very fast.

During the return journey, Earth's clocks are very nearly at a standstill, however, due to the doppler effect, he receives laser pulses at a rapid rate.

Finally, he decelerates when he gets back to Earth, which has virtually no effect on any observations about Earth or laser pulses.
 
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  • #105
Ivan Seeking
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So I did read this correctly. It seems that the form of the relativistic Doppler shift equation never sank in back in college; and I never looked it up. This along with a fundamental misunderstanding of the space-time interval seems to have led me astray.

So, we can sum this up as follows:

1). First and foremost: measurements on clocks and rods in the inertial frame that are made by observers in the inertial frame are real. There are no illusions.

2). With the exception of gedunken experiments that require infinite path lengths or that are in some other way impractical, we observe no discontinuities in the behavior of clock and rods that require any tricks of math to resolve.

3). Finally, the apparent strangeness of the statement that each observer sees the other’s clock running slowly is really resolved by the Relativistic Doppler shift equation, and of course the relativity of simultaneity.

One interesting aside here is that objects in motion are viewed as rotated. A cube in motion appears rotated to reveal one side otherwise parallel to the observer’s line of sight. Spheres in motion do not appear contracted as ellipsoids; they still look like spheres!

For me there is only one question remaining. From what I understand of SR and GR, when we dig deep enough we find that the constancy of C for all observers lies at the base of both theories. Can we say why C is constant for all observers? Is there a more fundamental statement in this regard?

Thanks again Hurkyl, Chroot, and Ambitwistor. I don't think I have ever had the chance to so fully explore some of the finer point of SR.
 
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  • #106
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From what I understand of SR and GR, when we dig deep enough we find that the constancy of C for all observers lies at the base of both theories. Can we say why C is constant for all observers? Is there a more fundamental statement in this regard?

Not really. There are other formulations of relativity in which the "speed of light is constant" axiom is derived, not postulated ... but you can always play that sort of game: pick a few theorems of a theory, take them as axioms, and derive the old "axioms" as theorems from the new axioms (previously theorems). It's not really anything deep that goes beyond what we already know. But it depends on your sense of aesthetics. For instance, if you really don't like instantaneous action at a distance, then you might take causality axioms as fundamental, and then that would "explain" the constancy of the speed of light. (In non-relativistic theories, effects can propagate at arbitrarily high speeds; if you set a speed limit, then you can assume some additional symmetry principles that force the speed limit to be universal for all observers.)
 

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