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

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.

9 pages, no figures, Matched with the version published in Eur. Phys. J. C

References in corpus (6)

• First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole
• First Sagittarius A* Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole in the Center of the Milky Way
• Light ring stability in ultra-compact objects
• Upper bound on the radii of black-hole photonspheres
• On the Size of a Black Hole: The Schwarzschild is the Biggest
• Lower bound on the radii of black-hole photonspheres