The universal topological charge of black hole photon spheres in higher dimensions

TL;DR

This paper extends a topological approach to analyze photon spheres around higher-dimensional black holes, revealing a universal topological charge of -1 that guarantees the existence of at least one photon sphere outside the event horizon.

Contribution

It demonstrates that the topological charge of photon spheres is a dimensionally invariant property, applicable to various higher-dimensional black hole solutions, thus providing a universal characterization.

Findings

Topological charge of photon spheres is always -1 across dimensions.

Existence of at least one unstable photon sphere outside the event horizon is guaranteed.

The topological charge remains invariant for different black hole solutions.

Abstract

A recently developed topological approach offers novel insights into photon spheres, which are fundamental to the formation of black hole shadows. In this study, we extend this topological analysis to higher-dimensional, static, spherically symmetric, and asymptotically flat black holes. By examining the asymptotic properties of the vector field associated with the photon spheres, we demonstrate that their topological charge is consistently -1. This result is a dimensionally independent invariant, guaranteeing the existence of at least one standard (unstable) photon sphere outside the event horizon. We further explore this conclusion by analyzing two distinct regular black hole solutions derived from pure gravity theory, confirming that the topological charge remains -1 irrespective of the spacetime dimension. These results provide a robust and universal characterization of photon spheres in higher-dimensional spacetimes.