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 (exterior) Schwarzschild, Reissner–Nordström, and Schwarzschild–anti de Sitter in n + 1 dimensions. In the second part, we prove that any static, vacuum, and “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 obtained by the first author.
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