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

Abstract: In this work, we derive a universal upper bound for the photon sphere radius in static, spherically symmetric, asymptotically flat black hole spacetimes of arbitrary dimension $n\ge 4$, in the presence of an anisotropic matter field satisfying the weak energy condition and a non-positive trace of the energy-momentum tensor. Using the effective potential method, we obtain the bound $r_γ\le [(n-1)M]^{\frac{1}{n-3}}$, where $M$ is the ADM mass of the black hole. This bound reduces to $r_γ\le 3M$ in four dimensions, consistent with the known result in the literature. Our result provides a dimension-dependent upper bound for photon spheres and deepens the understanding of spacetime structure in higher-dimensional gravitational theories.