Bounds on the radius of outermost photon sphere and black hole shadow in $n$-dimensional Einstein gravity

Abstract

The photon sphere, which determines the contour of the black hole shadow, provides a direct probe of strong-field gravity. In this work, we derive model-independent bounds on both the outermost photon sphere radius $r_{γ,\mathrm{out}}$ and the shadow radius $r_{\mathrm{sh}}$ for static, spherically symmetric, asymptotically flat black holes in $n$-dimensional ($n\ge 4$) Einstein gravity, supported by an anisotropic matter field. We first establish a rigorous upper bound on $r_{γ,\mathrm{out}}$ under the Weak Energy Condition (WEC), the Strong Energy Condition (SEC), and an asymptotic decay condition on the matter fields, proving $r_{γ,\mathrm{out}} \le \bigl[(n-1)M\bigr]^{\frac{1}{n-3}}$, where $M$ is the ADM mass, with the bound saturated by the vacuum Schwarzschild-Tangherlini black hole. We further discuss the lower bound on $r_{γ,\mathrm{out}}$, clarifying that a conditional bound can be obtained from the innermost photon sphere under an extra monotonicity assumption. Turning to the black hole shadow, we derive both upper and lower bounds on $r_{\mathrm{sh}}$. Making use of the photon sphere upper bound and the fact that, under the WEC and SEC, the effective potential for null geodesics outside the outermost photon sphere is bounded below by that of the vacuum solution with the same ADM mass, we obtain $r_{\mathrm{sh}}\le\sqrt{\frac{n-1}{n-3}}\,\bigl[(n-1)M\bigr]^{\frac{1}{n-3}}$, while for the lower bound, using only the WEC, we prove $r_{\mathrm{sh}} \ge \left( \frac{n-1}{2} \right)^{\frac{1}{n-3}} \sqrt{\frac{n-1}{n-3}}r_H$, where $r_H$ is the horizon radius. Our results provide clear geometric constraints on the observable signatures of higher-dimensional black holes and underline the special role of the shadow as a robust observable.