Multipole Moments of Isolated Horizons

TL;DR

This paper defines diffeomorphism-invariant horizon multipoles for axi-symmetric isolated horizons by introducing geometric moments I_n and L_n built from the horizon curvature and Im Ψ_2, and then elevates them to physical mass and angular momentum multipoles M_n and J_n through natural rescalings with the horizon mass M_Δ and radius R_Δ. It provides an explicit reconstruction procedure showing that area and the geometric multipoles determine the horizon geometry up to diffeomorphism, and proves a uniqueness result within this framework. Extensions to Maxwell fields, extremal horizons, and non-axisymmetric (type III) horizons are developed, including electromagnetic multipoles Q_n, P_n and additional geometric data in extremal cases. The work positions horizon multipoles as a robust, quasi-local toolkit for understanding black hole geometry, dynamics, and potential quantum-gravity bookkeeping, with promising applications in equations of motion, numerical relativity, and quantum gravity entropy.

Abstract

To every axi-symmetric isolated horizon we associate two sets of numbers, $M_n$ and $J_n$ with $n = 0, 1, 2, ...$, representing its mass and angular momentum multipoles. They provide a diffeomorphism invariant characterization of the horizon geometry. Physically, they can be thought of as the `source multipoles' of black holes in equilibrium. These structures have a variety of potential applications ranging from equations of motion of black holes and numerical relativity to quantum gravity.