The Existence and Distribution of Photon Spheres Near Spherically Symmetric Black Holes -- A Geometric Analysis

TL;DR

This paper provides a geometric proof of the relation between stable and unstable photon spheres near spherically symmetric black holes, demonstrating their existence and distribution using intrinsic curvature analysis.

Contribution

It introduces a new geometric proof of photon sphere distribution and existence near black holes, applicable to various asymptotic behaviors and black hole types.

Findings

Existence of photon spheres near general black holes

Alternating stable and unstable photon spheres distribution

Validation of the relation n_stable - n_unstable = -1

Abstract

Photon sphere has attracted significant attention since the capture of black hole shadow images by Event Horizon Telescope. Recently, a number of studies have highlighted that the number of photon spheres and their distributions near black holes are strongly constrained by black hole properties. Specifically, for black holes with event horizons and proper asymptotic behaviors, the number of stable and unstable photon spheres satisfies the relation $n_{\text{stable}} - n_{\text{unstable}} = -1$. In this study, we provide a new proof on this relation using a geometric analysis, which is carried out using intrinsic curvatures in the optical geometry of black hole spacetimes. Firstly, we demonstrate the existence of photon spheres near black holes assuming most general asymptotic behaviors (asymptotically flat black holes, asymptotically de-Sitter and anti-de-Sitter black holes). Subsequently, we prove that the stable and unstable photon spheres near black holes must be one-to-one alternatively separated from each other, such that each unstable photon sphere is sandwiched between two stable photon spheres (and each stable photon sphere is sandwiched between two unstable photon spheres). Our analysis is applicable to any spherically symmetric black hole spacetimes.