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

TL;DR

This paper develops a purely geometric framework based on optical geometry to analyze photon spheres around static, spherically symmetric black holes. Photon spheres are located where the geodesic curvature in the 2D optical geometry vanishes, $\kappa_g(r_{ph})=0$, while stability is governed by the sign of the Gaussian curvature $\mathcal{K}(r)$ (with $\mathcal{K}>0$ indicating stability and $\mathcal{K}<0$ instability); this approach is shown to be equivalent to the conventional effective potential criterion. Using asymptotic analysis and the Gauss-Bonnet theorem, the authors prove the existence of photon spheres for asymptotically flat, de Sitter, and anti-de Sitter black holes and demonstrate that stable and unstable photon spheres must be alternately distributed, yielding $n_{\text{stable}} - n_{\text{unstable}} = -1$. The method is metric-agnostic within the assumed symmetry and provides a geometric generalization of prior topological results, with potential extensions to rotating spacetimes and other ultra-compact objects, and relevance to black hole shadow observations such as those by the Event Horizon Telescope.

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.

Figures (11)

• Figure 1: The variation of geodesic curvature $\kappa_{g}(r)$ in 2-dimensional optical geometry with respect to radial coordinate $r$. The geodesic curvature in the near horizon limit is related to the surface gravity of black holes, via $\lim_{r \to r_{H}}\kappa_{g}(r)=-\kappa_{\text{surface}} < 0$. If the geodesic curvature in the infinite distance limit satisfies $\lim_{r \to \infty}\kappa_{g}(r) > 0$, the equation $\kappa_{g}(r)=0$ must have at least one solution.
• Figure 2: The stable and unstable photon spheres in black hole spacetimes are one-to-one alternatively separated from each other.
• Figure 3: The choice of region $D$ in 2-dimensional optical geometry in the Gauss-Bonnet theorem to demonstrate that stable and unstable photon spheres in the vicinity of black hole are one-to-one alternatively separated from each other.
• Figure 4: This figure summarizes the four different cases in which the surface integral of Gaussian curvature vanishes in region $D$ (namely $\int_{D}\mathcal{K}\cdot dS = 0$). Case I: the inner boundary photon sphere is stable and the outer boundary photon sphere is unstable, with $\mathcal{K}(r_{i}) > 0$ and $\mathcal{K}(r_{i+1}) < 0$. Case II: the inner boundary photon sphere is unstable and the outer boundary photon sphere is stable, with $\mathcal{K}(r_{i}) < 0$ and $\mathcal{K}(r_{i+1}) > 0$. Case III: both the inner and outer boundary photon spheres are stable, with $\mathcal{K}(r_{i}) > 0$ and $\mathcal{K}(r_{i+1}) > 0$. Case IV: both the inner and outer boundary photon spheres are unstable, with $\mathcal{K}(r_{i}) < 0$ and $\mathcal{K}(r_{i+1}) < 0$. In the last two cases, the inner and outer boundary photon spheres at $r=r_{i}$ and $r_{i+1}$ cannot be adjacent, and an additional photon sphere must exist between $r_{i}$ and $r_{i+1}$. The factor $\sqrt{\tilde{g}^{\text{OP-2d}}}$ in surface integral is always positive, so the sign of Gaussian curvature can be easily observed in this figure. This figure corresponds to the situations where Gaussian curvature is continuous in region $D$.
• Figure 5: This figure summarizes the four different cases in which the surface integral of Gaussian curvature vanishes in region $D$ (namely $\int_{D}\mathcal{K}\cdot dS = 0$). Case I: the inner boundary photon sphere is stable and the outer boundary photon sphere is unstable, with $\mathcal{K}(r_{i}) > 0$ and $\mathcal{K}(r_{i+1}) < 0$. Case II: the inner boundary photon sphere is unstable and the outer boundary photon sphere is stable, with $\mathcal{K}(r_{i}) < 0$ and $\mathcal{K}(r_{i+1}) > 0$. Case III: both the inner and outer boundary photon spheres are stable, with $\mathcal{K}(r_{i}) > 0$ and $\mathcal{K}(r_{i+1}) > 0$. Case IV: both the inner and outer boundary photon spheres are unstable, with $\mathcal{K}(r_{i}) < 0$ and $\mathcal{K}(r_{i+1}) < 0$. In the last two cases, the inner and outer boundary photon spheres at $r_{i}$ and $r_{i+1}$ cannot be adjacent, and an additional photon sphere must exist between $r_{i}$ and $r_{i+1}$. The factor $\sqrt{\tilde{g}^{\text{OP-2d}}}$ in surface integral is always positive, so the sign of Gaussian curvature can be easily observed in this figure. This figure corresponds to the situations where Gaussian curvature admits finite number of discontinuous points with respect to radial coordinate $r$.