# Upper bound on the radii of black-hole photonspheres

## TL;DR

This paper establishes an upper limit on the radius of photonspheres in spherically symmetric black-hole spacetimes, showing that it cannot exceed three times the black hole's mass, with implications for hairy black holes.

## Contribution

It proves a universal upper bound on photonsphere radii in spherically symmetric asymptotically flat black holes, including hairy black holes, extending previous results.

## Key findings

- Photonsphere radius is bounded by 3 times the black hole mass.
- The Schwarzschild black hole saturates this bound.
- Hairy black holes also conform to this upper limit.

## Abstract

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.