On the Size of a Black Hole: The Schwarzschild is the Biggest

Abstract

We consider static black holes in Einstein gravity and study parameters characterizing the black hole size, namely the radii of the horizon $R_+$, photon sphere $R_{\rm ph}$ and black hole shadow $R_{\rm sh}$. We find a sequence of inequalities $\frac32 R_+\, \le\, R_{\rm ph}\, \le \,\frac{1}{\sqrt3} R_{\rm sh}\,\le\, 3 M$, where $M$ is the black hole mass. These are consistent with and beyond the previously known upper bounds in literature. The Schwarzschild black hole saturates all the inequalities, making it the biggest of all for given mass. The inequalities include an upper bound of entropy for any quantum system with given energy. We also point out that some black holes satisfying the dominant energy condition can trap photons to form a stable photon shield outside the event horizon, but the shadow hides it from an observer at infinity.