Generalized Carter & R\"udiger Constants of $\sqrt{\text{Kerr}}$

TL;DR

This paper explores hidden constants of motion for charged spinning particles in a generalized Kerr background, revealing they exist only under specific multipole parameter conditions.

Contribution

It identifies conditions under which Carter and Rüdiger-like constants of motion exist for spinning probes in a generalized Kerr field.

Findings

Two hidden constants of motion exist only for specific multipole parameters.

These constants are analogous to Carter and Rüdiger constants in Kerr spacetime.

Existence depends on the spin-exponentiation of effective Compton amplitudes.

Abstract

We consider the motion of a charged spinning test/probe particle -- governed by the Mathisson-Papapetrou-Dixon equations with generic, adiabatic, and conservative spin- and field-induced multipole moments -- in a background $\sqrt{\text{Kerr}}$ field on flat spacetime: the electromagnetic field of a charged spinning ring-disk singularity obtained from the $G\to 0$ limit of the Kerr-Newman solution for a charged spinning black hole. We investigate the existence of two extra hidden constants of motion, analogous to the Carter constant (for geodesic motion in a Kerr spacetime, or for its spinning-probe generalization) and R\"udiger's linear-in-spin constant for a spinning probe in a Kerr background. We find that these two constants exist only when the Wilson coefficients parameterizing the probe's multipole structure take the particular values corresponding to spin-exponentiation of the effective Compton amplitudes through second order in spin.