I am not a physicist by training, but I have studied special relativity enough to understand why simultaneity is a relative concept, and so why if you synchronize clocks in one inertial frame, an observer in another inertial frame wouldn't agree that they are synchronized. Usual textbook example is this: Consider two clocks located at opposite ends of a train which is moving with constant velocity. There is a light source in the middle of the train. It sends two light pulses towards the two clocks simultaneously and clocks begin to run when they receive these light pulses. Observer on the train says that the two clocks are synchronized because for him, light pulses have traveled equal distances. But an observer on the platform disagrees because for him light pulses have traveled unequal distances. This happens because light speed is the same for all inertial observers.

This example makes me wonder if this peculiarity arises because we have used light wave, which does not require a medium to travel, and hence must have constant speed for all inertial observers. What if we used some material wave instead, say sound?

Say there is a long horizontal tube closed off at both ends, and filled with water. The two clocks are placed at opposite ends of the pipe. A sound wave is generated at the center of the tube which travels through water towards both ends, and clocks begin to run once they receive the sound wave. The entire setup is inside the moving train. Observer on the train will as before say the clocks are synchronized.

But how can the observer on the platform now disagree that the two clocks are synchronized?

  • $\begingroup$ The observer on the train still says that the two clocks are synchronized, i.e. they are in exactly the same state as the ones synchronized by light. What's the difference between "there is no difference" for the observer on the platform? $\endgroup$ – CuriousOne Jul 14 '16 at 7:03
  • $\begingroup$ @CuriousOne I don't get your question (then may be you didn't get mine :D). What I am saying is this: relativity theory says that observer on the platform must find that the clocks are not synchronized, no matter the procedure followed. Is that true if some material wave were to be used instead of light wave in synchronizing clocks on the train? $\endgroup$ – Deep Jul 14 '16 at 8:02
  • $\begingroup$ The observer on the train does not know that the train is moving relative to the platform, so for them synchronization always means the same thing. If you put two sets of clocks next to each other, one synchronized with light, the other pair synchronized with some other method (whatever method!), the two pairs of clocks will show exactly the same times. Now, if you look at two clocks in the same position, both of which show exactly the same time, they will show exactly the same time for every observer. $\endgroup$ – CuriousOne Jul 14 '16 at 8:06
  • $\begingroup$ @CuriousOne I am not comparing two clocks placed next to each other. I am not really interested in inference of the observer on the train. What I am doing is this: First, I am comparing two different methods of synchronization, one is the standard textbook case of using light waves and the other is by using material waves.Second, I am interested only in the inference of observer on the platform. He seems to infer that clocks are not synchronized when light wave is used. My question is, would he infer the same when sound wave is used? $\endgroup$ – Deep Jul 14 '16 at 8:17
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    $\begingroup$ You can place two different synchronization setups next to each other. They both lead to clocks that are showing exactly the same time. An outside observer can't tell the difference between one clock that shows 1pm and another clock that shows 1pm, irrespective of how they were made to show 1pm. $\endgroup$ – CuriousOne Jul 14 '16 at 18:52

The conclusion would be the same, but quantitative results would be harder to derive.

Why the conclusion would be the same:

The following relies on the assumption that the speed of sound in the train frame is known to be lower than the speed of light. Now imagine the train observer, in the middle of the train, sending out both light and sound waves simultaneously. He sees the forward and backward waves, of both kinds, reach the respective clocks simultaneously. But the sound waves will obviously arrive after the light waves in each location. The platform observer must then also see the sound waves reach the ends of the train after the corresponding light waves, whatever else he might observe. But we already know that the backward light wave reaches the back end before the forward one reaches the forward end. Hence the same must hold for sound waves, and the platform observer must see the train clocks at different times.

Why quantitative results would be harder to derive:

While light waves are observed from the platform to propagate at the same speed in both directions, and propagation times are then easy to calculate, forward and backward sound waves would be observed to have different speeds, which we haven't a clue how to calculate correctly. The latter statement follows from the fact that adding another velocity to that of a light signal still results in the original velocity of the light signal. The simple Galilean addition of velocities is incompatible with this, and deriving the (nonlinear) relativistic velocity addition law from scratch, without the Lorentz transformations, is challenging - to say the least.

  • $\begingroup$ Speed of sound in water is a constant (depending only on its thermodynamic state). Why then do you find it hard to calculate sound speed w.r.t. observer on the platform? $\endgroup$ – Deep Jul 15 '16 at 4:18
  • $\begingroup$ Upvoted your answer but I have doubts expressed above. Also if, as far as synchronization is concerned, behavior of sound is identical to that of light, doesn't that imply that speed of sound also must be the same for all inertial observers? $\endgroup$ – Deep Jul 15 '16 at 4:41
  • $\begingroup$ So the image that troubles you is observers moving in a medium that propagates both light and sound waves. In this case any waves would have the same speed wrt the medium at rest. If the platform is at rest in the medium, the train must see the medium in motion and must be able to detect this flow. $\endgroup$ – udrv Jul 15 '16 at 14:18
  • $\begingroup$ If so, the speed of sound wrt the train would not be the same as for the platform, because it would compound sound speed in the medium at rest plus the motion of the train relative to the medium. This is what the Michelson-Morley experiment tested, and the conclusion was that no moving medium can be detected as far as light is concerned. $\endgroup$ – udrv Jul 15 '16 at 14:18
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    $\begingroup$ The alternative is that the train carries with it the medium for sound propagation, be it air in the train car or water in a pipe, etc. But in this case the medium is at rest wrt the train, and in motion wrt anything in relative motion relative to the train, including the platform. Then a sound wave is in motion wrt both train and platform, and the same rules must apply to it as for any other object in relative motion wrt both train and platform. And we would have to figure out first how sublight velocities add up under the assumption that light speed remains the same for all observers. $\endgroup$ – udrv Jul 15 '16 at 14:20

Consider two clocks $A$ and $B$ at the two ends of your pipe, and two identical clocks $A'$ and $B'$ right next to $A$ and $B$. An observer stationary with respect to the pipe then synchronizes $A$ with $A'$, then (all at once) synchronizes $A$ with $B$ using light and $A'$ with $B'$ using sound.

Now, according to that observer, all four clocks are synchronized. $A$ and $A'$ are identical, so they stay synchronized. And everyone must agree on this because $A$ and $A'$ are right next to each other. Ditto, everyone agrees that $B$ and $B'$ are synchronized.

Now the observer on the platform, as you know, cannot agree that $A$ and $B$ are synchronized. But he agrees that $A'$ is synchronized with $A$ and $B'$ is synchronized with $B$. Therefore, he cannnot agree that $A'$ and $B'$ are synchronized.

In other words, the very possibility of synchronizing with light is enough to get the relativity of simultaneity, even if, in practice, you synchronize some other way.

  • $\begingroup$ I upvoted your answer but it isn't the complete answer. You begin with the premise that whatever is true for light (as far as synchronization is concerned) must be true of sound. Someone who was suspect of special relativity would turn the argument around, and say, since sound reaches both clocks simultaneously according to every observer because it is traveling in a material medium, the clocks are synchronized, and this must be true no matter which other method was used. $\endgroup$ – Deep Jul 15 '16 at 4:32
  • $\begingroup$ Can you please explain, why sound wave should reach one clock earlier than the other (as per platform observer), without calling upon a comparison with light waves? $\endgroup$ – Deep Jul 15 '16 at 4:33

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