Spinning particles in vacuum spacetimes of different curvature types

TL;DR

This paper investigates the motion of spinning test particles in various vacuum spacetimes, analyzing how the curvature type affects their dynamics using the MPD equations and Newman-Penrose formalism.

Contribution

It provides a detailed analysis of spinning particle motion in different Petrov types, introducing a natural choice of tetrads and examining the influence of curvature on the equations of motion.

Findings

Motion depends on spacetime's algebraic type, with special cases simplifying the equations.

A natural choice of tetrad can be made when particle velocity and momentum are not parallel.

The study offers relations valid generally and discusses the case of parallel velocity and momentum.

Abstract

We consider the motion of spinning test particles with nonzero rest mass in the "pole-dipole" approximation, as described by the Mathisson-Papapetrou-Dixon (MPD) equations, and examine its properties in dependence on the spin supplementary condition added to close the system. The MPD equation of motion is decomposed in the orthonormal tetrad whose time vector is given by the four-velocity $V^\mu$ chosen to fix the spin condition (the "reference observer") and the first spatial vector by the corresponding spin; such projections do not contain the Weyl scalars $\Psi_0$ and $\Psi_4$ obtained in the associated Newman-Penrose (NP) null tetrad. One natural choice of the remaining two spatial basis vectors is shown to follow "intrinsically"; it is realizable if the particle's four-velocity and four-momentum are not parallel. To see how the problem depends on the curvature type, one first identifies the first vector of the NP tetrad $k^\mu$ with the highest-multiplicity principal null direction of the Weyl tensor, and then sets $V^\mu$ so that $k^\mu$ belong to the spin-bivector eigenplane. In spacetimes of any algebraic type but III, it is possible to rotate the tetrads so as to become "transverse", namely so that $\Psi_1$ and $\Psi_3$ vanish. If the spin-bivector eigenplane could be made coincide with the real-vector plane of any of such transverse frames, the motion would consequently be fully determined by $\Psi_2$ and the cosmological constant; however, this can be managed in exceptional cases only. Besides focusing on specific Petrov types, we derive several sets of useful relations valid generally and check whether/how the exercise simplifies for some specific types of motion. The particular option of having four-velocity parallel to four-momentum is advocated and a natural resolution of nonuniqueness of the corresponding reference observer $V^\mu$ is suggested.