If you were able to construct a rigid beam 25 million miles long wouldn't you be able to transmit data (push a button) on the other end faster than the speed of light?
So, in the best case, this method of communication is equivalent to communicating with electromagnetic waves. And they have a speed limit: the speed of light :)
So this wouldn't be faster than light communication.
Yes it does. And in fact that is the case: The beam has compressed. A fully rigid beam is impossible.
To think about it, if you have a sufficiently long beam you won't be able to give it any angular momentum at all, regardless of how much force you apply. It will be fixed in space. Perfect starting point if your hobby is moving stars with a combination of chains and pulleys.
A beam made of matter is not rigid (as we think of the word) on that scale, as others in the thread have pointed out.
Except you can't. A existence of sound traveling through the beam proves without a doubt that the beam is not rigid.
So yes, relativity dictates that you can't create a rigid beam 25 molecules long, much less 25 million miles long. There can be no such thing as a perfectly rigid beam of any length.