Bounds on the photon sphere radius for spherically symmetric black holes in n-dimensional Einstein gravity

TL;DR

This paper derives bounds on the photon sphere radius for higher-dimensional black holes in Einstein gravity, linking it to the black hole mass and horizon radius under specific energy conditions.

Contribution

It provides the first derivation of upper and lower bounds on photon sphere radii in n-dimensional Einstein gravity with anisotropic matter fields.

Findings

Upper bound: $r_\gamma \le [(n-1)M]^{1/(n-3)}$

Lower bound: $r_\gamma \ge (\frac{n-1}{2})^{1/(n-3)} r_H$

Bounds generalize four-dimensional results to higher dimensions.

Abstract

The photon sphere, a hypersurface of null circular geodesics, plays a fundamental role in characterizing black hole spacetimes, influencing phenomena such as black hole shadows, gravitational lensing, and quasinormal modes. In this work, we derive both upper and lower bounds on the photon sphere radius for static, spherically symmetric, asymptotically flat black holes within $n$-dimensional Einstein gravity ($n\ge 4$), assuming an anisotropic matter field satisfying the weak energy condition and a non-positive trace of the energy-momentum tensor. For the upper bound, we obtain $r_\gamma\le [(n-1)M]^{\frac{1}{n-3}}$, where $M$ is the ADM mass. In the four-dimensional case ($n=4$), this reduces to $r_\gamma\le 3M$, in agreement with previous results. For the lower bound, under the additional assumption that $|r^{n-1}p_r(r)|$ is monotonically decreasing, we prove $r_\gamma\ge (\frac{n-1}{2})^{1/(n-3)}r_H$, where $r_H$ is the radius of the outer event horizon; for $n=4$ this gives $r_\gamma\ge \frac{3}{2}r_H$, also consistent with previous four-dimensional result. These results provide dimension-dependent geometric constraints that generalize well-known four-dimensional bounds to a specific class of higher-dimensional black holes (described by a Tangherlini-type metric) and deepen our understanding of spacetime structure in higher-dimensional gravitational theories.