Two Permanently Congruent Rods May Have Different Proper Lengths
  We scrutinize congruence as one of the basic definitions of equality in geometry and pit it against physics of Special Relativity. We show that two non-rigid rods permanently kept congruent during their common expansion or compression may have different instantaneous proper lengths (when measured at the same time of their respective reference clocks) if they have different mass distributions over their lengths. Alternatively, their proper lengths can come out equal only when measured at different but strictly correlated moments of time of their respective clocks. The derived expression for the ratio of instantaneous proper lengths of two permanently congruent changing objects explicitly contains information about the objects mass distribution. The same is true for the ratio of readings of the two reference clocks, for which the instantaneous measurements of respective proper lengths produce the same result. In either case the characteristics usually considered as purely kinematic depend on mass distribution, which is a dynamic property. This is a spectacular demonstration of dynamic aspect of geometry already in the framework of Special Relativity. 
Moses Fayngold