Center of mass, spin supplementary conditions, and the momentum of spinning particles

TL;DR

This paper analyzes various spin supplementary conditions in general relativity, examining their physical significance, the momentum-velocity relation, and the concept of hidden momentum, demonstrating their equivalence at dipole order.

Contribution

It provides a comprehensive analysis of different spin conditions, clarifies their physical implications, and shows their equivalence in describing spinning particle motion in curved spacetime.

Findings

Different spin conditions are physically equivalent at dipole order.

The non-parallelism between velocity and momentum is explained by hidden momentum.

Simple examples illustrate the equivalence of various descriptions.

Abstract

We discuss the problem of defining the center of mass in general relativity and the so-called spin supplementary condition. The different spin conditions in the literature, their physical significance, and the momentum-velocity relation for each of them are analyzed in depth. The reason for the non-parallelism between the velocity and the momentum, and the concept of "hidden momentum", are dissected. It is argued that the different solutions allowed by the different spin conditions are equally valid descriptions for the motion of a given test body, and their equivalence is shown to dipole order in curved spacetime. These different descriptions are compared in simple examples.