Photon surfaces with equipotential time-slices

TL;DR

This paper characterizes photon surfaces in static, spherically symmetric spacetimes and proves a uniqueness theorem for spacetimes with equipotential photon surfaces, extending understanding of geometric structures in general relativity.

Contribution

It provides a local classification of photon surfaces in certain symmetric spacetimes and establishes a uniqueness result linking equipotential photon surfaces to Schwarzschild spacetime.

Findings

Photon surfaces are characterized in static, spherically symmetric spacetimes.

Any static, vacuum, asymptotically isotropic spacetime with an equipotential photon surface is Schwarzschild.

The classification includes well-known solutions like Schwarzschild and Reissner--Nordström.

Abstract

Photon surfaces are timelike, totally umbilic hypersurfaces of Lorentzian spacetimes. In the first part of this paper, we locally characterize all possible photon surfaces in a class of static, spherically symmetric spacetimes that includes Schwarzschild, Reissner--Nordstr\"om, Schwarzschild-anti de Sitter, etc., in $n+1$ dimensions. In the second part, we prove that any static, vacuum, "asymptotically isotropic" $n+1$-dimensional spacetime that possesses what we call an "equipotential" and "outward directed" photon surface is isometric to the Schwarzschild spacetime of the same (necessarily positive) mass, using a uniqueness result by the first named author.