Universal bounds on the size of a black hole

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 $$A_{\mathrm {ph,out}}$$ A ph , out , the corresponding shadow area $$A_{\mathrm {sh,out}}$$ A sh , out and the horizon area $$A_{{\mathcal {H}}}$$ A H satisfy a series of universal inequalities $$9A_{{\mathcal {H}}}/4\le A_{\mathrm {ph,out}}\le A_{\mathrm {sh,out}}/3\le 36\pi M^2$$ 9 A H / 4 ≤ A ph , out ≤ A sh , out / 3 ≤ 36 π 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.