The geometry of photon surfaces

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 $\text{SO(3)×}\mathbb{R}$ -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 $\text{SO(3)}$ -invariant photon surface in a general nonstatic spherically symmetric space–time. Many examples are provided.