Uniqueness of static vacuum asymptotically flat black holes and equipotential photon surfaces in $n+1$ dimensions \`a la Robinson

TL;DR

This paper extends the uniqueness theorems for static vacuum black holes and photon surfaces to higher dimensions, providing geometric inequalities and relating to existing results in the field.

Contribution

It generalizes known four-dimensional uniqueness proofs to higher dimensions and establishes new geometric inequalities for static vacuum spacetimes with photon surfaces.

Findings

Proves uniqueness of higher-dimensional static vacuum black holes.

Derives geometric inequalities involving scalar curvature.

Extends classical results to n+1 dimensions.

Abstract

In this paper, we combine and generalize to higher dimensions the approaches to proving the uniqueness of connected (3+1)-dimensional static vacuum asymptotically flat black hole spacetimes by M\"uller zum Hagen--Robinson--Seifert and by Robinson. Applying these techniques, we prove and/or reprove geometric inequalities for connected (n + 1)-dimensional static vacuum asymptotically flat spacetimes with either black hole or equipotential photon surface or specifically photon sphere inner boundary. In particular, assuming a natural upper bound on the total scalar curvature of the boundary, we recover and extend the well-known uniqueness results for such black hole and equipotential photon surface spacetimes. We also relate our results and proofs to existing results, in particular to those by Agostiniani--Mazzieri and by Nozawa--Shiromizu--Izumi--Yamada.