autodidude said:
Whoa, that's pretty crazy. Is there a limit on how contracted you can make it? Hypothetically, if you could push it faster than the propagation of the compression wave, what would happen?
You would probably cause a shock wave at some point. If you push something faster than the speed of the low amplitude limit of the compressional wave, eventually a discontinuity forms that moves faster than the low amplitude limit of the compressional
One type of “nonlinear effect” would be a shock wave. A shock wave could result in a near discontinuity in density that travels faster than the canonical speed of sound. A sonic boom starts out as a shock wave. When an airplane moves faster than the speed of sound, the air right on the airplane wing has to move faster than sound. The compression at contact has to keep up with with the airplane. This creates a build up of pressure, which results in a shock wave, which breaks down into normal sound waves.
When you refer to the propagation of "the compressional wave", you are probably talking about the low amplitude limit of compressional waves. The speed of compressional wave increases with amplitude, but slowly. The speed of sound is actually the low amplitude limit.
The shock wave still wouldn’t travel faster than the speed of light in a vacuum. Thus, you couldn’t beat relativity using shock waves. However, you could send signals faster than the true speed of sound using a shock wave. Hence this is not really a topic in relativity.
If you push that “rigid” rod hard enough, you could get a shock wave. A large amplitude compressional wave could become a shock wave. The shock wave would travel faster than the low amplitude limit. Thus, you may be able to send a signal faster than the speed of sound. Light waves in the vacuum would race ahead of it.
Furthermore, the shock wave would probably cause damage in the rod. There could be some “breakage” that follows the discontinuity. Part of the rod may vaporize. However, that would be an entirely different subject. The signals in a shock wave would not be faster than c.
See this link on shock waves.
http://en.wikipedia.org/wiki/Shock_wave
“A shock wave (also called shock front or simply "shock") is a type of propagating disturbance. Like an ordinary wave, it carries energy and can propagate through a medium (solid, liquid, gas or plasma) or in some cases in the absence of a material medium, through a field such as the electromagnetic field. Shock waves are characterized by an abrupt, nearly discontinuous change in the characteristics of the medium.[1] Across a shock there is always an extremely rapid rise in pressure, temperature and density of the flow. In supersonic flows, expansion is achieved through an expansion fan. A shock wave travels through most media at a higher speed than an ordinary wave.”
There is no strict limit to how much a material can be compressed. However, the material will undergo phase changes as the pressure and temperature of the material increase. In practice, your rigid rod is going to break in a way consistent with "classical physics" way before you have to worry about relativity. The material could break, vaporize, ionize, and collapse into any number of degenerate fluids. The nucleii of the material could fission, fuse, emit radiation, or any number of strange effects. However, most of these effects would occur long before SR became an issue.
The only strict SR rule is that no signal can go faster than light. Therefore, there have to be hypothetical limits to materials that keep the signals from going faster than c. Hence, there may be a relativistic limit to the compressibility of the material at some point. For instance, one couldn't achieve a density greater than that of a black hole without the material collapsing into a black hole or singularity state. However, the rod would have been rendered unrecognizable way before this point.
