Two Permanently Congruent Rods May Have Different Proper Lengths

TL;DR

This paper reveals that two rods kept congruent during expansion or compression can have different proper lengths depending on their mass distribution, challenging the notion that congruence implies equal length in relativity.

Contribution

It demonstrates that proper length equality depends on mass distribution and measurement timing, highlighting a dynamic aspect of geometry in Special Relativity.

Findings

Proper lengths depend on mass distribution.

Congruent rods can have different proper lengths.

Measurement timing affects length equality.

Abstract

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.