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Generalized Carter & Rüdiger Constants of $\sqrt{\text{Kerr}}$

Christopher de Firmian, Justin Vines

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 Rüdiger'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.

Generalized Carter & Rüdiger Constants of $\sqrt{\text{Kerr}}$

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 field on flat spacetime: the electromagnetic field of a charged spinning ring-disk singularity obtained from the 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 Rüdiger'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.
Paper Structure (8 sections, 36 equations, 1 figure)

This paper contains 8 sections, 36 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The Worldline Action & the Electro-Magnetic MPD Equations
  3. The Root Kerr Field
  4. Carter & Rüdiger Ansätze
  5. OS2 Ansatz
  6. Reduction to Basis
  7. Carter, Rüdiger Results & Implications for Compton Scattering
  8. Discussion, Conjectures & Future Directions

Figures (1)

  • Figure 1: Double Copy & Newman Janis Shift