Conserved quantities and integrability for massless spinning particles in general relativity

TL;DR

This paper explores the conservation laws and integrability of massless spinning particles in general relativity, focusing on hidden symmetries and their implications in specific spacetime classes.

Contribution

It derives generalized conservation laws from conformal Killing-Yano tensors and demonstrates integrability of the spin Hall equations in certain spacetimes.

Findings

Massless spinning particles exhibit conserved quantities linked to hidden symmetries.

The spin Hall equations are fully integrable in a large class of type D spacetimes.

For massive particles, the generalized Carter constant remains conserved regardless of spin conditions.

Abstract

In general relativity, the dynamics of spinning particles is governed by the Mathisson-Papapetrou-Dixon equations, which are most commonly applied to massive bodies, but the framework also works in the massless case. Such massless versions naturally arise, for example, in the description of energy centroids of high-frequency wave packets. In this work, we consider massless spinning particles in spacetimes with hidden symmetries and we derive the generalized conservation laws associated with conformal Killing-Yano tensors. We then show that the spin Hall equations, a particular case of the Mathisson-Papapetrou-Dixon equations restricted to massless particles with longitudinal angular momentum, are completely integrable in a large class of type D spacetimes. Additionally, we also show that for massive spinning particles, the generalized Carter constant associated with Killing-Yano tensors is conserved independently of the choice of spin supplementary condition.