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In SR, there are a couple of paradoxes such as the bug-rivet paradox that require us to take into account a finite speed of impulse.

Suppose I have a rod at rest, and I yank one end to some speed. I know that there has to be a period of time where the other end stays stationary until a signal arrives.

It may help to imagine that the rod is one light-year long, for example. If I pull on one end, it would have to take at least a year until the other end is pulled as well. In thought experiments it can simplify things to imagine that this signal travels at the speed of light. However, it seems that because the rod is realistically held together by atomic forces that may take time to react to changes, the signal would actually travel slower than $c$.

I've heard one person posit that the signal would travel at the speed of sound for the object. However, I can't determine whether this is right or not.

Is there any reference to a paper/book/link that investigates this question for realistic materials? Is there any way to calculate the speed of the impulse?

Ashwin Balaji
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Maximal Ideal
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    Related: https://physics.stackexchange.com/questions/303680/does-special-relativity-predict-superluminal-speeds – Mohammad Javanshiry Jul 22 '20 at 06:15
  • There are no solid objects at those scales. Even something the size of the Earth is a fluid. Mechanically speaking, the rod wold have all the properties of a narrow column of gas. – m4r35n357 Jul 22 '20 at 08:47

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The speed of an impulse in the material is by definition the speed of sound. If you push or pull one end of the rod with a faster speed, you would create a shock wave in the material similar to the one created by a supersonic jet in the atmosphere:

enter image description here

This would change the properties and integrity of the solid material and eventually destroy it.

Even if your rod is made of diamond, the speed of sound in it is $12$ km/s or $25$ thousand times slower than light. The signal in your scenario would take $25,000$ years to travel the distance of $1$ light-year.

For details, please see Speed of sound.

Hiroyashu
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