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

TL;DR

This work proves a universal lower bound on the photon-sphere radius for static, spherically symmetric, asymptotically flat black holes in $n\ge 4$ dimensions, under the weak energy condition, a non-positive energy-momentum trace, and a monotonic radial pressure profile. By relating the exterior mass, horizon radius, and the photon-sphere condition, the authors derive $r_\gamma \ge (\frac{n-1}{2})^{1/(n-3)} r_H$, generalizing Hod's four-dimensional result and imposing a geometric constraint on higher-dimensional black holes. The bound tightens to the known $r_\gamma \ge \tfrac{3}{2} r_H$ in $n=4$ and becomes weaker with increasing dimension, approaching unity in the large-$n$ limit. These results have implications for the structure of black holes in extended theories of gravity and for interpreting observational signatures like shadows and lensing in higher dimensions.

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 $r_γ$ satisfies $r_γ\ge (\frac{n-1}{2})^{1/(n-3)}r_H$, where $r_H$ is the radius of the outer event horizon. For $n=4$, this reduces to the known result $r_γ\ge \frac{3}{2}r_H$. 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.