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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?!

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    $\begingroup$ Certainly at least related to physics.stackexchange.com/q/2175 $\endgroup$
    – David Z
    Jan 8 '17 at 8:22
  • $\begingroup$ Thanks David, but my question slightly differs from the question at the link above. $\endgroup$ Jan 8 '17 at 8:29
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    $\begingroup$ I think the question needs to be made at least physically plausible: 'infinite force' and '[acceleration in] no time' are physically absurd. I am not saying that the question can't be made physically reasonable. $\endgroup$
    – user107153
    Jan 8 '17 at 8:29
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    $\begingroup$ Note that Born-rigidity isn't (can't be) an intrinsic property of an extended object, but is instead arranged by the strategic application of forces is a way that remains synchronized in the body frame. That has to be arranged ahead of time by way of signalling that respects the speed of light, so no violation of causality is possible. $\endgroup$ Jan 8 '17 at 9:07
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    $\begingroup$ Also, the rod's putative Born rigidity is almost certainly incompatible with the kind of single-point external force application you propose. Born rigidity is in general n0t a "material" property or a property of the "rigid" object alone: its applicability also depends on the system of forces on the body in accelerated motion. A different set of forces would impart, in general, non Born rigid motion. $\endgroup$ Jan 8 '17 at 9:08
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Assume that we have a long Born-rigid rod with a length 𝐿0 ... Where is the problem?!

The problem is right in the beginning. There is no such thing as a Born-rigid rod, there is only Born-rigid motion.

Even a rod made of unobtanium cannot move in Born rigid motion by being pushed on one end. However, even a rod made of jell-o can have Born-rigid motion if it is accelerated uniformly at every point along the rod. In fact, I find that often in relativity problems it is best to think of all materials as being made of jell-o.

So for your post, to make a Born rigid motion you cannot have merely one missile on the end, but rather an array of missles all along the rod whose engines are synchronized in advance to all fire at the proper time.

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    $\begingroup$ I think I saw a sale on unobtanium jello-o at Kroger the other day $\endgroup$ Mar 26 '19 at 15:03
  • $\begingroup$ @Dale: Okay. Let's forget about the Born-rigid body or any Born-rigid motion. Repeat the experiment using a real "iron" rod. Replace the speed of light with that of sound (in iron); and now, answer the question, please! $\endgroup$ Mar 26 '19 at 16:36
  • $\begingroup$ I am not sure what the question is in that case. The rod is simply distorted. As the right end moves the left end stays stationary. Its shape is changed and it no longer has a single rest frame. $\endgroup$
    – Dale
    Mar 26 '19 at 16:59
  • $\begingroup$ It is clear that the iron rod's left end, in one way or another, would also get the same constant velocity as the right end, otherwise relativity is not capable of explaining the behavior of a real rod even after reaching a constant velocity! Do you mean by your last sentence that "proper length" is a relative measure? $\endgroup$ May 25 '19 at 8:23
  • $\begingroup$ Proper length is not relative, but it is only defined for an object undergoing Born rigid motion. The rod is not moving rigidly, so it does not have a definite proper length. $\endgroup$
    – Dale
    May 25 '19 at 11:00

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