The existence and upper bound for stable photon spheres in static spherically symmetric black holes

TL;DR

This paper establishes conditions for the existence of stable photon spheres around static spherically symmetric black holes and derives a universal upper bound on their radius, impacting astrophysical observations and black hole physics.

Contribution

It provides the first general proof of existence conditions and a universal upper bound for stable photon spheres outside black holes with matter fields.

Findings

Stable photon spheres exist under specific matter conditions.

A strict upper bound of r_sps < 6M is derived for certain black holes.

The results apply broadly to hairy black holes and external matter configurations.

Abstract

In this work, we establish the existence conditions and a universal upper bound on the radius of stable photon spheres (SPS) outside the event horizons of static, spherically symmetric, asymptotically flat black holes surrounded by matter fields. We prove that stable photon spheres exist if the external matter satisfies specific conditions. Furthermore, under the additional assumption of a monotonically decreasing mass-radius ratio $m(r)/r^3$ outside the horizon, we derive a strict upper bound on the radius $r_{\mathrm{sps}}$ of any stable photon sphere: $r_{\mathrm{sps}}<6M$, where $M$ is the asymptotic mass of the black hole. This bound is independent of specific black hole solutions and broadly applies to hairy black holes and other configurations with external matter fields meeting the stated energy conditions. Our results resolve fundamental questions regarding the existence and spatial constraints on stable photon orbits, with implications for gravitational lensing, accretion disk dynamics (e.g., the Aschenbach effect), and black hole shadow observations.