Dynamical Implications of the Kerr Multipole Moments for Spinning Black Holes

TL;DR

This paper explores how the Kerr black hole's multipole moments determine gravitational couplings and affect scattering amplitudes, providing new insights into the structure of spinning black holes and their interactions.

Contribution

It establishes that Kerr solutions uniquely determine all multipole moments and related couplings, and derives all-order expressions for Compton amplitudes of spinning bodies.

Findings

All multipole moments of Kerr black holes are fixed by the solution.

Additional couplings do not affect three-point amplitudes but influence Compton scattering.

Explicit expressions for electromagnetic and gravitational Compton amplitudes are provided.

Abstract

Previously the linearized stress tensor of a stationary Kerr black hole has been used to determine some of the values of gravitational couplings for a spinning black hole to linear order in the Riemann tensor in the action (worldline or quantum field theory). In particular, the couplings on operators containing derivative structures of the form $(S\cdot\nabla)^n$ acting on the Riemann tensor were fixed, with $S^\mu$ the spin vector of the black hole. In this paper we find that the Kerr solution determines all of the multipole moments in the sense of Dixon of a stationary spinning black hole and that these multipole moments determine all linear in $R$ couplings. For example, additional couplings beyond the previously mentioned are fixed on operators containing derivative structures of the form $S^{2n}(p\cdot\nabla)^{2n}$ acting on the Riemann tensor with $p^\mu$ the momentum vector of the black hole. These additional operators do not contribute to the three-point amplitude, and so do no contribute to the linearized stress tensor for a stationary black hole. However, we find that they do contribute to the Compton amplitude. Additionally, we derive formal expressions for the electromagnetic and gravitational Compton amplitudes of generic spinning bodies to all orders in spin in the worldline formalism and evaluated expressions for these amplitudes to order $S^3$ in electromagnetism and order $S^5$ in gravity.