Area bound for a surface in a strong gravity region

TL;DR

This paper establishes an upper bound on the area of certain surfaces in strong gravity regions of asymptotically flat spacetimes, linking geometric properties to the total mass using inverse mean curvature flow.

Contribution

It proves a new area bound for surfaces with positive mean curvature derivatives in non-negative Ricci scalar hypersurfaces, identifying the photon sphere as the extremal case.

Findings

The area bound is $A_0 \,\leq\, 4\pi (3Gm)^2$.

Equality holds for the photon sphere in Schwarzschild spacetime.

The result applies to surfaces with positive mean curvature and derivatives in specific hypersurfaces.

Abstract

For asymptotically flat spacetimes, using the inverse mean curvature flow, we show that any compact $2$-surface, $S_0$, whose mean curvature and its derivative for outward direction are positive in spacelike hypersurface with non-negative Ricci scalar satisfies the inequality $A_0 \leq 4 \pi (3Gm)^2$, where $A_0$ is the area of $S_0$ and $m$ is the total mass. The upper bound is realized when $S_0$ is the photon sphere in a hypersurface isometric to $t=$const. slice of the Schwarzschild spacetime.