Universal Bounds on Horizons, Photon Spheres, and Shadows: The Role of Energy Conditions in Spherically Symmetric Black Holes

TL;DR

This paper establishes universal bounds on the sizes of horizons, photon spheres, and shadows of spherically symmetric black holes, showing Schwarzschild black holes maximize these features under the weak energy condition.

Contribution

It provides rigorous bounds on black hole geometric features and analyzes extremal cases, linking these bounds to energy conditions and the asymptotic structure of spacetime.

Findings

Schwarzschild black holes set the maximum size bounds for horizons and shadows.

Extremal black holes have bounds on horizon locations depending on the lapse function.

Outer horizon pressure is non-negative, preventing violation of the strong energy condition outside black holes.

Abstract

In this work, we derive rigorous and universal bounds on the geometric characteristics of black holes in asymptotically flat spacetimes under assumptions that weak energy condition is satisfied. We prove that the event horizon radius, the photon sphere , and the shadow ones take their maximal values in the Schwarzschild black hole case. Any additional matter distribution satisfying the weak energy condition necessarily decreases these radii relative to their Schwarzschild counterparts. Thus, the Schwarzschild solution provides an absolute upper bound on observable size characteristics of static, spherically symmetric black holes. We further analyze configurations possessing two distinct horizons and investigate their extremal regime, in which the inner and outer horizons merge. For extremal black holes, we establish both lower and upper bounds on the extremal horizon location. These bounds depend on the asymptotic structure of the lapse function, in particular on the presence or absence of a $1/r^2$ term in its asymptotic expansion. We derive explicit conditions on the lapse function determining when the extremal Reissner-Nordstrom radius provides a lower bound and when it instead serves as an upper bound. In addition, we prove that in asymptotically flat spacetimes the pressure at the outer event horizon is always either positive or equal to zero. As a consequence, the strong energy condition can not be violated outside the black hole, even in models of regular black holes where it may be violated in the interior region to avoid singularity formation.