Spinning particles in vacuum spacetimes of different curvature types: Natural reference tetrads, and massless particles

TL;DR

This paper analyzes the motion of spinning particles, both massive and massless, in vacuum spacetimes with different curvature types, using various tetrad choices to decompose the MPD equations and relate them to Weyl scalars.

Contribution

It introduces new interpretation tetrads based on the particle's world-line and compares their effectiveness in decomposing the MPD equations for massive and massless spinning particles.

Findings

Decomposition using world-line-based tetrads is less flexible for massive particles.

Massless particles' equations involve only $ ext{ extPsi}_1$ and $ ext{ extPsi}_2$ Weyl scalars.

Principal null directions can simplify the equations further.

Abstract

In a previous paper, we considered the motion of massive spinning test particles in the "pole-dipole" approximation, as described by the Mathisson--Papapetrou--Dixon (MPD) equations, and examined its properties in dependence on the spin supplementary condition. We decomposed the equations in the orthonormal tetrad based on the time-like vector fixing the spin condition and on the corresponding spin, while representing the curvature in terms of the Weyl scalars obtained in the Newman--Penrose (NP) null tetrad naturally associated with the orthonormal one; the projections thus obtained did not contain the Weyl scalars $\Psi_0$ and $\Psi_4$. In the present paper, we choose the interpretation tetrad in a different way, attaching it to the tangent $u^\mu$ of the world-line representing the history of the spinning body. Actually {\em two} tetrads are suggested, both given "intrinsically" by the problem and each of them incompatible with one specific spin condition. The decomposition of the MPD equation, again supplemented by writing its right-hand side in terms of the Weyl scalars, is slightly less efficient than in the massive case, because $u^\mu$ cannot be freely chosen (in contrast to $V^\mu$) and so the $u^\mu$-based tetrad is less flexible. In the second part of this paper, a similar analysis is performed for massless spinning particles; in particular, a certain "intrinsic" interpretation tetrad is again found. The respective decomposition of the MPD equation of motion is considerably simpler than in the massive case, containing only $\Psi_1$ and $\Psi_2$ scalars and not the cosmological constant. An option to span the spin-bivector eigen-plane, besides the world-line null tangent, by a main principal null direction of the Weyl tensor can lead to an even simpler result.