Uniqueness and non-uniqueness of static vacuum black holes in higher dimensions

TL;DR

This work addresses the uniqueness of static vacuum black holes in higher dimensions, proving that asymptotically flat solutions are uniquely given by the Schwarzschild–Tangherlini metric, while dropping asymptotic flatness yields non-unique static black holes with the same spherical horizon topology. The proof adapts a conformal/positive-mass framework to reduce to a Laplace equation on flat space and solves a Dirichlet boundary-value problem to establish uniqueness for connected horizons. It also constructs infinite families of non–AF static black holes by using Bohm Einstein metrics on the horizon cross-section, arguing these have lower entropy and are likely unstable, thereby delineating the boundary between AF uniqueness and non-AF non-uniqueness. The results illuminate how horizon geometry and asymptotic conditions shape black-hole uniqueness in higher dimensions and suggest links to horizon thermodynamics and holography.

Abstract

We prove the uniqueness theorem for asymptotically flat static vacuum black hole solutions in higher dimensional space-times. We also construct infinitely many non-asymptotically flat regular static black holes on the same spacetime manifold with the same spherical topology.