Uniqueness of static photon surfaces: Perturbative approach

TL;DR

This paper investigates whether nonspherical static photon surfaces can exist in vacuum spacetimes, concluding that they cannot, thus establishing a perturbative uniqueness result for such surfaces in Schwarzschild-like spacetimes.

Contribution

The paper proves the perturbative uniqueness of static photon surfaces in vacuum spacetimes and explores conditions for their possible existence with matter.

Findings

No distorted photon surface exists outside the Schwarzschild radius in vacuum.

Perturbative analysis confirms uniqueness of static photon surfaces in asymptotically flat vacuum spacetimes.

Presence of matter outside the photon surface could allow for distorted photon surfaces.

Abstract

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.