The geometry of photon surfaces

TL;DR

This paper generalizes the concept of photon spheres to arbitrary space-times, proving their relation to black holes and singularities, and derives evolution equations for photon surface areas.

Contribution

It introduces a generalized definition of photon surfaces in arbitrary space-times and establishes their connection to black holes and matter distributions under energy conditions.

Findings

Black holes in static spherically symmetric space-times are surrounded by photon spheres.

Photon spheres are linked to naked singularities and matter distributions under energy conditions.

An evolution equation for photon surface area in non-static spherically symmetric space-times is derived.

Abstract

The photon sphere concept in Schwarzschild space-time is generalized to a definition of a photon surface in an arbitrary space-time. A photon sphere is then defined as an SO(3)xR-invariant photon surface in a static spherically symmetric space-time. It is proved, subject to an energy condition, that a black hole in any such space-time must be surrounded by a photon sphere. Conversely, subject to an energy condition, any photon sphere must surround a black hole, a naked singularity or more than a certain amount of matter. A second order evolution equation is obtained for the area of an SO(3)-invariant photon surface in a general non-static spherically symmetric space-time. Many examples are provided.