Upper bound on the radii of black-hole photonspheres

Understand

One of the most remarkable predictions of the general theory of relativity is the existence of black-hole "photonspheres", compact null hypersurfaces on which massless particles can orbit the central black hole.

• We prove that every spherically-symmetric asymptotically flat black-hole spacetime is characterized by a photonsphere whose radius is bounded from above by $r_{\gamma} \leq 3M$, where $M$ is the total ADM mass of the black-hole spacetime.
• It is shown that hairy black-hole configurations conform to this upper bound.
• In particular, the null circular geodesic of the (bald) Schwarzschild black-hole spacetime saturates the bound.