Universal bounds on the size of a black hole

TL;DR

This paper establishes universal bounds relating the sizes of photon spheres, shadows, and horizons in static black holes within Einstein gravity, supported by proofs in symmetric cases and evidence in general static spacetimes.

Contribution

It proposes and partially proves universal inequalities linking key black hole size parameters, extending understanding of black hole geometry.

Findings

Derived bounds on photon sphere and shadow areas in terms of black hole mass

Proved inequalities in spherically symmetric cases

Presented evidence supporting bounds in general static spacetimes

Abstract

For static black holes in Einstein gravity, if matter fields satisfy a few general conditions, we conjecture that three characteristic parameters about the spatial size of black holes, namely the outermost photon sphere area $A_{\mathrm{ph,out}}$, the corresponding shadow area $A_{\mathrm{sh,out}}$ and the horizon area $A_{\mathcal{H}}$ satisfy a series of universal inequalities $9A_{\mathcal{H}}/4\leq A_{\mathrm{ph,out}}\leq A_{\mathrm{sh,out}}/3\leq 36\pi M^2$, where $M$ is the ADM mass. We present a complete proof in the spherically symmetric case and some pieces of evidence to support it in general static cases. We also discuss the properties of the photon spheres in general static spacetimes and show that, similar to horizon, photon spheres are also conformal invariant structures of the spacetimes.