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.
Feng, J. C., Gasperin, E., & Williams, J. (2019). Almost-Killing equation: Stability, hyperbolicity, and black hole Gauss law. Physical Review D, 100(12), [124034 ]. https://doi.org/10.1103/PhysRevD.100.124034