Uniqueness of Static Black Holes in Lovelock Gravity

Abstract

Black hole uniqueness is one of the fundamental predictions of general relativity: the spherically symmetric Schwarzschild--Tangherlini spacetime is the unique regular, asymptotically flat, static vacuum black hole solution of Einstein's equations. We show that an analogous result holds perturbatively in Lovelock gravity. Assuming the appropriate positive-mass rigidity statement, any regular, asymptotically flat, static vacuum black hole solution of Lovelock gravity on a generic branch admitting a regular Taylor expansion about general relativity is uniquely determined, at fixed ADM mass, by the corresponding spherically symmetric Lovelock black hole solution. In particular, Einstein--Gauss--Bonnet gravity admits no distinct nonspherical static branch perturbatively connected to Schwarzschild--Tangherlini, and the unique solution is Boulware--Deser. The same recursive mechanism extends to the full Lovelock class. Our result shows that static black hole uniqueness can extend beyond general relativity and is not a property peculiar to the Einstein equations.