Universal bounds on the size of a black hole
Run-Qiu Yang, H. Lü
Abstract
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 $$A_{\mathrm {ph,out}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>ph</mml:mi> <mml:mo>,</mml:mo> <mml:mi>out</mml:mi> </mml:mrow> </mml:msub> </mml:math> , the corresponding shadow area $$A_{\mathrm {sh,out}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>sh</mml:mi> <mml:mo>,</mml:mo> <mml:mi>out</mml:mi> </mml:mrow> </mml:msub> </mml:math> and the horizon area $$A_{{\mathcal {H}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>H</mml:mi> </mml:msub> </mml:math> satisfy a series of universal inequalities $$9A_{{\mathcal {H}}}/4\le A_{\mathrm {ph,out}}\le A_{\mathrm {sh,out}}/3\le 36\pi M^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>9</mml:mn> <mml:msub> <mml:mi>A</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:mo>/</mml:mo> <mml:mn>4</mml:mn> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>ph</mml:mi> <mml:mo>,</mml:mo> <mml:mi>out</mml:mi> </mml:mrow> </mml:msub> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>A</mml:mi> <mml:mrow> <mml:mi>sh</mml:mi> <mml:mo>,</mml:mo> <mml:mi>out</mml:mi> </mml:mrow> </mml:msub> <mml:mo>/</mml:mo> <mml:mn>3</mml:mn> <mml:mo>≤</mml:mo> <mml:mn>36</mml:mn> <mml:mi>π</mml:mi> <mml:msup> <mml:mi>M</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:math> , 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.