A universal lower bound on the photon sphere radius in higher-dimensional black holes

TL;DR

This paper establishes a universal lower bound on the photon sphere radius for higher-dimensional static, spherically symmetric black holes, extending previous four-dimensional results under specific energy conditions.

Contribution

It derives a new lower bound on the photon sphere radius applicable to black holes in any dimension $n \\ge 4$, generalizing Hod's theorem to higher dimensions.

Findings

The photon sphere radius satisfies $r_\gamma \ge (\frac{n-1}{2})^{1/(n-3)} r_H$.

For $n=4$, the bound reduces to $r_\gamma \ge \frac{3}{2} r_H$.

The result applies under weak energy and monotonicity conditions.

Abstract

The photon sphere, a hypersurface of circular null geodesics, plays a fundamental role in characterizing black hole spacetimes, influencing phenomena such as black hole shadows, gravitational lensing, and quasinormal modes. While universal upper bounds on the photon sphere radius have been established for both four-dimensional and higher-dimensional black holes, the question of a corresponding lower bound in higher-dimensional black holes remains less explored. In this work, we derive a universal lower bound for the photon sphere radius in static, spherically symmetric, asymptotically flat black hole spacetimes of arbitrary dimension $n\ge 4$. Under the assumptions of the weak energy condition, a non-positive trace of the energy-momentum tensor, and a monotonicity condition on the radial pressure function $|r^{n-1}p_r(r)|$, we prove that the photon sphere radius $r_\gamma$ satisfies $r_\gamma\ge (\frac{n-1}{2})^{1/(n-3)}r_H$, where $r_H$ is the radius of the outer event horizon. For $n=4$, this reduces to the known result $r_\gamma\ge \frac{3}{2}r_H$. Our result generalizes Hod's four-dimensional theorem to higher dimensions, and provides a new geometric constraint on the structure of black holes in extended theories of gravity.