Uniqueness of the static Einstein-Maxwell spacetimes with a photon sphere

TL;DR

This paper proves that under certain conditions, static asymptotically flat Einstein-Maxwell spacetimes with a photon sphere are uniquely described by the Reissner-Nordström solution, extending the understanding of spacetime uniqueness in general relativity.

Contribution

It establishes a new uniqueness theorem for Einstein-Maxwell spacetimes with a photon sphere, characterizing them as Reissner-Nordström solutions with specific charge-to-mass ratios.

Findings

Photon sphere has constant mean and scalar curvature.

Spacetimes with a photon sphere are isometric to Reissner-Nordström solutions.

Charge-to-mass ratio constraint: Q^2/M^2 ≤ 9/8.

Abstract

We consider the problem of uniqueness of static and asymptotically flat Einstein-Maxwell spacetimes with a photon sphere $P^3$. We are using a naturally modified definition of a photon sphere for electrically charged spacetimes with the additional property that the one-form $\iota_{\xi}F$ is normal to the photon sphere. For simplicity we are restricting ourselves to the case of zero magnetic charge and assume that the lapse function regularly foliates the spacetime outside the photon sphere. With this information we prove that $P^3$ has constant mean curvature and constant scalar curvature. We also derive a few equations which we later use to prove the main uniqueness theorem, i. e. the static asymptotically flat Einstein-Maxwell spacetimes with a non-extremal photon sphere are isometric to the Reissner-Nordstr\"om one with mass $M$ and electric charge $Q$ subject to $\frac{Q^2}{M^2}\le \frac{9}{8}$.