Complete set of quasi-conserved quantities for spinning particles around Kerr

TL;DR

This paper investigates the conserved quantities of spinning particles in Kerr spacetime, revealing the absence of certain symmetries and implications for particle motion, extending understanding of quasi-conserved quantities.

Contribution

It establishes the non-existence of specific mixed-symmetry Killing tensors on Kerr, clarifying the structure of quasi-conserved quantities for spinning particles.

Findings

Two non-trivial quasi-conserved quantities are identified.

No stationary and axisymmetric mixed-symmetry Killing tensor exists on Kerr.

Quasi-constants of motion are not in complete involution.

Abstract

We revisit the conserved quantities of the Mathisson-Papapetrou-Tulczyjew equations describing the motion of spinning particles on a fixed background. Assuming Ricci-flatness and the existence of a Killing-Yano tensor, we demonstrate that besides the two non-trivial quasi-conserved quantities, i.e. conserved at linear order in the spin, found by R\"udiger, non-trivial quasi-conserved quantities are in one-to-one correspondence with non-trivial mixed-symmetry Killing tensors. We prove that no such stationary and axisymmetric mixed-symmetry Killing tensor exists on the Kerr geometry. We discuss the implications for the motion of spinning particles on Kerr spacetime where the quasi-constants of motion are shown not to be in complete involution.