Jerrold Franklin analyzed rigid motion and collisions in special relativity. His treatment distinguishes maintaining proper separations from maintaining the same coordinate separation at one observer's simultaneous times.[ref]
Rigid body motion in special relativity by Jerrold Franklin. Relativistic rigidity and straight-line acceleration
For straight-line rigid acceleration, different points generally require different proper accelerations. A rule that simply gives every point the same acceleration in one inertial frame need not preserve the object's rest-frame length. The idealized kinematic prescription also does not provide a mechanism for instantaneous stress transmission through a real material.
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