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Suppose I am in a spaceship traveling inertially at a velocity $v$ that is of the same order as $c$. As I pass by a metal bar that is oriented parallel to $v$, someone hits it with another metal bar, causing it to vibrate. The vibration waves travel down the bar with a velocity of $u$. Observing the bar from my spaceship, I see the waves travelling with velocity $u+v\over 1+uv$.

  1. Suppose that the waves are longitudinal, so that $v$ and $u$ are parallel. The atoms in the bar are individually moving back and forth, with a maximum velocity $w$ which is quite different from $u$. $w$ and $u$ are related in a way that depends on the material properties of the bar; I think on its elasticity its density. By comparing $u$ and $w$ I can calculate these properties of the bar. If I observe the motion of the individual atoms, I will see them moving back and forth with a maximum velocity of $v+w\over 1+uv$ and by comparing this with the observed velocity of the travelling wave I will obtain a measurement of the stiffness and density of the bar. But this value will be different from the value that an observer would calculate at rest. What explains this discrepancy? Is it due to relativistic dilation of the bar's mass and volume, and hence its density? Is there some other effect?

  2. Now consider transverse waves. Again $v$ and $u$ are parallel. But this time the individual atoms are moving back and forth perpendicular to $u$ so the observed dilation of their velocity $w$ is different than it was for longitudinal motion. If I do the same calculations as in (1) I should get different results for the physical properties of the bar: I see the transverse and longitudinal waves moving at the same speeds in the bar, but when I look closely at the motion of atoms, and try to calculate material properties of the bar I get two different sets of results, because the atomic motion causing the longitudinal wave is relativistically dilated to a different degree than is the atomic motion causing the transverse wave. What is going on here? What does it really look like?

Presumably there is no discrepancy and I see a consistent picture regardless of $v$. How do all these apparently conflicting measurements iron out in special relativity?

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2 Answers 2

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I guess Ron is saying that by definition the bulk modulus is the value measured in frame at rest relative to the bar, and therefore it makes no sense to ask how the bulk modulus depends on speed. It's a bit like relativistic mass: few physicists assign any significance to relativistic mass - we just use rest mass, which is defined in the rest frame.

Having said this, I think your question is interesting, though I doubt it exposes any fundamental insights into relativity. You ask:

Is it due to relativistic dilation of the bar's mass and volume, and hence its density? Is there some other effect?

I started doing a few back of the envelope calculations and quickly concluded that the answer is "all of the above". However I also concluded that it would take more time than I have available in my coffee break to give you a detailed answer. The problem with SR is that it's easy to throw around ideas like velocity addition or time dilation, but doing this casually is to tread a dangerous path. To really work out what's going on you need to take your system, i.e. the bar, and apply the Lorentz transforms to calculate exactly how it looks in your frame. If you do this you'll find that the atomic motion, interatomic forces and interatomic spacing all change, so the bulk modulus and the density both change, and unsurprisingly the speed of sound changes as well. I did Google to see if anyone had done this calculation, but without any success.

Re your second point, the speed of shear and compression waves are usually different. In fact they are different in steel even without raising the spectre of relativistic motion. It's unsurprising that they would appear to be different when viewed from a relativistic frame.

I hope this helps; I feel it's a bit of a cop out since I haven't actually given you a straight answer. I think the answer would be a lot of work (if straightforward maths) and wouldn't reveal any fundamental insights. It's sort of the Physics equivalent of a crossword puzzle.

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Thanks for your answer and for the time you took to search Google and to do the envelope calculations. I did not expect it to expose any fundamental insights into relativity. –  MJD May 15 '12 at 16:59
    
Can you recommend a textbook if I want to learn to do this kind of calculation myself? Assume that I have or can acquire the necessary mathematical background, and all I need is the physics-specific part of the instruction. –  MJD May 25 '12 at 15:34

The calculations of the speed of sound (not the speed of a "shock wave", this is just sound you are describing) are done in the rest frame. They are dependent on the change in relative position of neighboring atoms. The analysis does not apply in a moving frame. In the moving frame, the speed of sound is the boost of the speed of sound in the stationary frame.

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I don't understand how this answers any of my questions. –  MJD May 11 '12 at 0:42
    
You asked "why does the relation between the atomic velocity and the velocity of sound not seem to work in a moving frame". The answer is because it is derived in the rest frame, and the material picks out the rest frame. So why should you expect it to work? You didn't show a contradiction, you just said "how do you calculate the speed of sound in the moving frame"--- well, you go to the rest frame and find the speed of sound, then boost the speed of sound as you did. There is no question I can see. –  Ron Maimon May 11 '12 at 0:45

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