Uniqueness of static photon surfaces: Perturbative approach

A photon surface S is defined as a three-dimensional timelike hypersurface such that any null geodesic initially tangent to S continues to be included in S, like $r=3M$ of the Schwarzschild spacetime. Using analytic solutions to static perturbations of a Schwarzschild spacetime, we examine whether a nonspherical spacetime can possess a distorted static photon surface. It is shown that if the region outside of $r=3M$ is vacuum, no distorted photon surface can be present. Therefore, we establish the perturbative uniqueness for an asymptotically flat vacuum spacetime with a static photon surface. It is also pointed out that if matter is present in the outside region, there is a possibility that a distorted photon surface could form.