A paradox concerning rigid body interactions was described by F. W. Sears some time ago. He considered a rigid rod at rest in a Lorentz frame S.
The rod is simultaneously acted on at each end by collinear equal and opposite time dependent forces. Since the net force is zero, the rod will remain at rest in frame S.
In a frame S′, moving with constant velocity −v with respect to S, the rod will have a constant velocity +v.
However, due to the relativity of simultaneity, the two forces at the ends of the rod will now be equal at diﬀerent times.
The paradox is, “How can the velocity of the rod remain constant in frame S′ where the forces are no longer equal and opposite at the same time?”
Although this paradox was posed by Sears many years ago, somewhat surprisingly there do not seem to be any subsequent papers addressing the issue he raised…
… we resolve the question of what really happens in the frame of the moving rod …
Read more http://arxiv.org/pdf/1405.5180v1.pdf