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We examine the Hamiltonian formulation and hyperbolicity of thealmost-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 offreedom. We also show the AKE is only strongly hyperbolic for one parameterchoice, 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 thecorresponding Hamiltonian is positive definite. An analysis of the resultingKomar currents reveals an exact Gauss law for a system of black holes in vacuumspacetimes and suggests a possible measure of matter content in asymptoticallyflat spacetimes.

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