Classification of the static and asymptotically flat Einstein-Maxwell-dilaton spacetimes with a photon sphere

TL;DR

This paper classifies static, asymptotically flat Einstein-Maxwell-dilaton spacetimes with photon spheres, proving properties of photon spheres and explicitly constructing all solutions with non-extremal photon spheres.

Contribution

It provides a complete classification of Einstein-Maxwell-dilaton spacetimes with photon spheres, including explicit solutions and new relations between geometric and physical properties.

Findings

Photon spheres have constant mean and scalar curvature.

Lapse function, electrostatic potential, and dilaton field are functionally dependent.

All solutions with non-extremal photon spheres are explicitly constructed.

Abstract

We consider the problem for the classification of static and asymptotically flat Einstein-Maxwell-dilaton spacetimes with a photon sphere. It is first proven that the photon spheres in Einstein-Maxwell-dilaton gravity have constant mean and constant scalar curvature. Then we derive some relations between the mean curvature and the physical characteristics of the photon spheres. Using further the symmetries of the dimensionally reduced Einstein-Maxwell-dilaton field equations we show that the lapse function, the electrostatic potential and the dilaton field are functionally dependent in the presence of a photon sphere. Using all this we prove the main classification theorem by explicitly constructing all Einstein-Maxwell-dilaton solutions possessing a non-extremal photon sphere.