If I push a rigid object at one end, the whole rod should move instantaneously. Would this mean that a signal has been transmitted from the end which is pushed to the the far end faster than the speed of light?
There is no such thing as a perfectly rigid object. The rod will compress when you push it and that compression will travel the length of the rod at the speed of sound for the rod's material (at less than the speed of light), until it reaches the end and uncompresses. Net effect, you push your end, and at sometime later the other end of the rod moves. the time delay between the movement of the ends of the rod will always be longer than it would take for light to traverse the length of the rod.
I've seen this "thought experiment" come up on several forums, and it usually generates a lot of argument. I think the problem is that this question doesn't really have an answer, because once you start applying physical laws as restrictions to help you narrow down the possibilities, you have to change the nature of the question. I think it's really designed to get you thinking about what the laws really require of reality, and what the consequences are of the gaps between quantum mechanics and relativity. Not to get an interesting definite answer.
The short answer is that it will take the amount of time required for the eletromagnetic force to transfer your pressure at one end from one atom to the next all the way down the rod. In other words, as someone already stated, the speed of sound is the factor. However, this assumes a certain amount of compressibility at the atomic level. Which means the rod isn't truly rigid. If the rod were perfectly rigid, then the answer to your question would be "yes". However, it would also have no physical significance, as the scenario varies so widely from anything possible in reality. Except maybe a neutron star, which some people think may be regarded as being as close as possible to being incompressible without collapsing to a black hole. It is thought that the speed of pressure waves (and hence sound) through neutron star material is very close to the speed of light.
"Rigid" materials are good for messing around with ideal scenarios of laws which don't deal specifically with phenomena that are pushed to their limits by "rigid" materials.
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