Assume that we have a long Born-rigid rod with a length $L_0$ in which the speed of light is the ultimate speed for sending information. If there is a tiny missile at the right end of the missile that exerts an infinite force on the rod to move it to a significant portion of the speed of light in no time, what would be rod's left end speed?!
Indeed, when the missile is fired, the left end has not been noticed yet that the far right end of the rod has moved till a signal reaches the left one after a long time of $L_0/c$. During this time, the lab observer confirms that the rod has extremely been stretched. On the other hand, special relativity predicts that the rod should be Lorentz contracted from the perspective of the lab observer after the signal of motion reaches the left end becoming very short in length. In this case, the lab observer confirms that the left end must travel FTL even with an infinite velocity to reach close to the far right end with a tiny Lorentz contracted distance between. Where is the problem?!