IDEAL characterization of higher dimensional spherically symmetric black holes

Abstract

In general relativity, an IDEAL (Intrinsic, Deductive, Explicit, ALgorithmic) characterization of a reference spacetime metric $g_0$ consists of a set of tensorial equations $T[g]=0$, constructed covariantly out of the metric $g$, its Riemann curvature and their derivatives, that are satisfied if and only if $g$ is locally isometric to the reference spacetime metric $g_0$. We give the first IDEAL characterization of generalized Schwarzschild-Tangherlini spacetimes, which consist of $\Lambda$-vacuum extensions of higher dimensional spherically symmetric black holes, as well as their versions where spheres are replaced by flat or hyperbolic spaces. The standard Schwarzschild black hole has been previously characterized in the work of Ferrando and S\'aez, but using methods highly specific to $4$ dimensions. Specialized to $4$ dimensions, our result provides an independent, alternative characterization. We also give a proof of a version of Birkhoff's theorem that is applicable also on neighborhoods of horizon and horizon bifurcation points, which is necessary for our arguments.