Horizon Multipole Moments of a Kerr Black Hole

TL;DR

This paper compares two definitions of horizon multipole moments for Kerr black holes, deriving explicit formulas and analyzing their properties and differences, especially in the small spin limit.

Contribution

It provides a detailed comparison and closed-form expressions for horizon multipole moments under two different definitions for Kerr black holes.

Findings

The multipoles share properties with Hansen field multipoles.

Closed-form expressions are derived for both definitions.

The two definitions differ for l >= 1 when a ≠ 0.

Abstract

The horizon multipole moments of a Kerr black hole are computed from two distinct definitions that have been proposed in the literature. The first one [Ashtekar et al., Class. Quantum Grav. 21, 2549 (2004)] regards axisymmetric isolated horizons, while the second one [Ashtekar et al., J. High Energ. Phys. 2022, 28 (2022)] applies to generic (i.e., not necessarily axisymmetric) non-expanding horizons. We review these definitions in a common frame and perform a detailed study of the resulting multipole moments for the Kerr event horizon. The horizon multipoles are found to share several properties with the (Hansen) field multipoles, including parity constraints and the leading scaling behavior with respect to the Kerr spin parameter a in the regime of small a. For the axisymmetry-based definition, we have obtained a closed-form expression of the multipole moments in terms of a and the spherical harmonic degree l. For the generic definition, we have established closed-form expressions for the conformal unit round metric, the `electric' and `magnetic' potentials related to the multipoles, and the values of the multipoles in the small a limit. We show that the two definitions lead to different values of the Kerr horizon multipoles as soon as l >= 1 (generic nonzero value of a) or l >= 2 (small a limit).