Time parameterizations and spin supplementary conditions of the Mathisson-Papapetrou-Dixon equations

TL;DR

This paper examines how different time parameterizations and spin supplementary conditions affect the Mathisson-Papapetrou-Dixon equations, revealing that certain constants of motion depend on the chosen affine parameter and clarifying their relationships.

Contribution

It provides a generalized formulation of the MPD equations without fixing the affine parameter and analyzes the impact of different SSCs and parameterizations on conserved quantities.

Findings

Constants of motion vary with affine parameter for MP and TD SSCs.

For OKS SSC, the two affine parameters are equivalent.

Evolved MPD equations clarify the relation between parameters for TD SSC.

Abstract

The implications of two different time constraints on the Mathisson-Papapetrou-Dixon (MPD) equations are discussed under three spin supplementary conditions (SSC). For this reason the MPD equations are revisited without specifying the affine parameter and several relations are reintroduced in their general form. The latter allows to investigate the consequences of combining the Mathisson-Pirani (MP) SSC, the Tulczyjew-Dixon (TD) SSC and the Ohashi-Kyrian-Semer\'{a}k (OKS) SSC with two affine parameter types: the proper time on one hand and the parameterizations introduced in [Gen. Rel. Grav. 8, 197 (1977)] on the other. For the MP SSC and the TD SSC it is shown that quantities that are constant of motion for the one affine parameter are not for the other, while for the OKS SSC it is shown that the two affine parameters are the same. To clarify the relation between the two affine parameters in the case of the TD SSC the MPD equations are evolved and discussed.