A paradox in special Relativity regarding elastic rod

Question

A certain piece of elastic breaks when it is stretched to twice its unstretched length. At time $t = 0$, all points of it are accelerated longitudinally with constant proper acceleration $α$, from rest in the unstretched state. Prove that the elastic breaks at $t =√3c/α$.

My question is: We can choose the left end point or the right end point of the rod as an instantaneously moving frame S' to solve the porblem. Say, If we choose the left end as S', due to length contraction, we know the right end must be stretched, and vise versa. But paradox comes when we consider the whole rod as frame S', at every time t', both end are stationary, how do they end up with break? You have a frame at every times the rod is stationary, why would it break? Regarding the whole rod as a frame, what I mean is that in this frame, every point in the rod is stationary because it has the same velocity in frame S, so it must be relative rest to each other at S'. The second question is that how do the observer in S(stationary) interpret the break? They don't observe any stretch! is the rod in the instantanously accelerating frame S' stretching or not? It seems that at every times in frame S', the points in rod is relatively stationary, if so, why would the rod stretch?

Answer 1

If the rod is not to break, each point on the rod must have a different proper acceleration.

There is a coordinate system, the Rindler coordinates, that is appropriate for accelerated observers that maintain constant relative distance. For such observers, there is a "paradoxical property". From the Wikipedia article Rindler coordinates:

Note that Rindler observers with smaller constant x coordinate are accelerating harder to keep up. This may seem surprising because in Newtonian physics, observers who maintain constant relative distance must share the same acceleration. But in relativistic physics, we see that the trailing endpoint of a rod which is accelerated by some external force (parallel to its symmetry axis) must accelerate a bit harder than the leading endpoint, or else it must ultimately break. This is a manifestation of Lorentz contraction. As the rod accelerates its velocity increases and its length decreases. Since it is getting shorter, the back end must accelerate harder than the front. Another way to look at it is: the back end must achieve the same change in velocity in a shorter period of time. This leads to a differential equation showing, that at some distance, the acceleration of the trailing end diverges, resulting in the Rindler horizon.