Uniqueness of the static spacetimes with a photon sphere in Einstein-scalar field theory

Understand

In the present paper we prove a uniqueness theorem for the static and asymptotically flat solutions to the Einstein-scalar field equations which possess a photon sphere.

• We show that such solutions are uniquely specified by their mass $M$ and scalar charge $q$ and that they are isometric to the Janis-Newman-Winicour solution with the same mass and scalar charge subject to the inequality $\frac{q^2}{M^2}<3$.