On equipotential photon surfaces in (electro-)static spacetimes of arbitrary dimension

TL;DR

This paper characterizes photon surfaces in higher-dimensional static spacetimes, providing new geometric insights and a uniqueness theorem for electrostatic spacetimes with equipotential photon surfaces, extending existing results.

Contribution

It offers a complete characterization of photon surfaces in spherically symmetric spacetimes and establishes a uniqueness theorem for electrostatic spacetimes with equipotential photon surfaces.

Findings

Photon surfaces characterized in higher-dimensional static spacetimes.

New geometric insights into equipotential photon surfaces.

Uniqueness theorem for electrostatic spacetimes with photon surfaces.

Abstract

We study timelike, totally umbilic hypersurfaces -- called photon surfaces -- in $n+1$-dimensional static, asymptotically flat spacetimes, for $n+1\geq4$. First, we give a complete characterization of photon surfaces in a class of spherically symmetric spacetimes containing the (exterior) subextremal Reissner--Nordstr\"om spacetimes, and hence in particular the (exterior) positive mass Schwarzschild spacetimes. Next, we give new insights into the spacetime geometry near equipotential photon surfaces and provide a new characterization of photon spheres (not appealing to any field equations). We furthermore show that any asymptotically flat electrostatic electro-vacuum spacetime with inner boundary consisting of equipotential, (quasi-locally) subextremal photon surfaces and/or non-degenerate black hole horizons must be isometric to a suitable piece of the necessarily subextremal Reissner--Norstr\"om spacetime of the same mass and charge. Our uniqueness result applies work by Jahns and extends and complements several existing uniqueness theorems. Its proof fundamentally relies on the lower regularity rigidity case of the Riemannian Positive Mass Theorem.