Almost-Killing equation: Stability, hyperbolicity, and black hole Gauss law

TL;DR

This paper analyzes the stability and hyperbolicity of the almost-Killing equation (AKE), revealing conditions for well-posedness, the presence of ghosts, and a Gauss law for black hole systems in vacuum spacetimes.

Contribution

It provides a detailed Hamiltonian and hyperbolicity analysis of the AKE, identifying parameter regimes for stability and establishing a Gauss law in black hole spacetimes.

Findings

Most parameter choices lead to unbounded Hamiltonians.

The AKE is only strongly hyperbolic in the presence of ghosts.

A Gauss law for black holes in vacuum spacetimes is derived.

Abstract

We examine the Hamiltonian formulation and hyperbolicity of the almost-Killing equation (AKE). We find that for all but one parameter choice, the Hamiltonian is unbounded, and in some cases, the AKE has ghost degrees of freedom. We also show the AKE is only strongly hyperbolic for one parameter choice, which corresponds to a case in which the AKE has ghosts. Fortunately, one finds that the AKE reduces to the homogeneous Maxwell equation in a vacuum, so that with the addition of the divergence-free constraint (a "Lorenz gauge"), one can still obtain a well-posed problem that is stable in the sense that the corresponding Hamiltonian is positive definite. An analysis of the resulting Komar currents reveals an exact Gauss law for a system of black holes in vacuum spacetimes and suggests a possible measure of matter content in asymptotically flat spacetimes.