Note on the Kerr Spinning-Particle Equations of Motion

TL;DR

The paper develops a probe version of the Newman-Janis algorithm to derive a unique spinning-particle EoM for Kerr in self-dual spacetimes by Wick-rotating the exact geodesic deviation equation. This heavenly (self-dual) framework exhibits exact hidden symmetry and, for positive helicity, yields all-multiplicity spin exponentiation of gravitational Compton amplitudes, with an explicit Lagrangian and a linearized coupling that reproduce Levi-Steinhoff-type dynamics and Kerr multipole structure. uplift to real (Earth) spacetime introduces star-shifting ambiguities and contact terms, rendering the earthly Compton amplitudes sensitive to mixed-chirality deformations, though the self-dual sector provides a robust, highly constrained backbone. Overall, the work identifies a principled, self-dual mechanism behind Kerr’s simplicity in the effective theory and highlights the subtle but important role of duality and holomorphic worldlines in gravitational scattering and spinning-body dynamics.

Abstract

We implement a probe counterpart of Newman-Janis algorithm, which Wick rotates the all-orders geodesic deviation equation into a part of exact spinning-particle equations of motion. Consequently, the gravitational dynamics of the Kerr black hole in its point-particle effective theory is completely constrained in the self-dual sector for a hidden symmetry, implying the spin exponentiation of same-helicity gravitational Compton amplitudes to all multiplicities.