Conserved quantities and integrability for massless spinning particles in general relativity

TL;DR

The paper extends the Mathisson–Papapetrou–Dixon framework to massless spinning particles in curved spacetimes by deriving generalized conservation laws from conformal Killing–Yano tensors and establishing a weak form of complete integrability for the massless spin Hall equations in a wide class of type D backgrounds. It shows that, in these spacetimes, quantities built from CKY data—such as a generalized Carter constant—are conserved up to perturbative orders in spin, enabling integrability on a constraint surface where the momentum is approximately null. For massive particles, the authors demonstrate a generalized Carter constant independent of the spin supplementary condition, strengthening the role of hidden symmetries in particle dynamics. The results have potential implications for high-frequency wave packet propagation, spin optics, and observational signatures around Kerr-like black holes, and they chart avenues for further relaxing assumptions and extending to broader type D geometries.

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.