Divergence equations and uniqueness theorem of static black holes

Abstract

Equations of divergence type in static spacetimes play a significant role in the proof of uniqueness theorems of black holes. We generalize the divergence equation originally discovered by Robinson in four dimensional vacuum spacetimes into several directions. We find that the deviation from spherical symmetry is encoded in a symmetric trace-free tensor $H_{ij}$ on a static timeslice. This tensor is the crux for the construction of the desired divergence equation, which allows us to conclude the uniqueness of the Schwarzschild black hole without using Smarr's integration mass formula. In Einstein-Maxwell(-dilaton) theory, we apply the maximal principle for a number of divergence equations to prove the uniqueness theorem of static black holes. In higher $(n\ge 5)$ dimensional vacuum spacetimes, a central obstruction for applicability of the current proof is the integration of the $(n-2)$-dimensional scalar curvature over the horizon cross-section, which has been evaluated to be a topological constant by the Gauss-Bonnet theorem for $n=4$. Nevertheless, it turns out that the $(n-1)$-dimensional symmetric and traceless tensor $H_{ij}$ is still instrumental for the modification of the uniqueness proof based upon the positive mass theorem, as well as for the derivation of the Penrose-type inequality.