Area bound for surfaces in generic gravitational field

TL;DR

This paper establishes an area bound for certain surfaces in gravitational fields, generalizing the Penrose inequality to asymptotically flat and AdS spaces using inverse mean curvature flow methods.

Contribution

It introduces the concept of attractive gravity probe surfaces and proves an area inequality applicable to multiple components, extending previous results in gravitational geometry.

Findings

Proved area inequality for AGPS in asymptotically flat spaces.

Derived similar inequality for AdS spaces.

Applicable to spaces with multiple surface components.

Abstract

We define an attractive gravity probe surface (AGPS) as a compact 2-surface $S_\alpha$ with positive mean curvature $k$ satisfying $r^a D_a k / k^2 \ge \alpha$ (for a constant $\alpha>-1/2$) in the local inverse mean curvature flow, where $r^a D_a k$ is the derivative of $k$ in the outward unit normal direction. For asymptotically flat spaces, any AGPS is proved to satisfy the areal inequality $A_\alpha \le 4\pi [ ( 3+4\alpha)/(1+2\alpha) ]^2(Gm)^2$, where $A_{\alpha}$ is the area of $S_\alpha$ and $m$ is the Arnowitt-Deser-Misner (ADM) mass. Equality is realized when the space is isometric to the $t=$constant hypersurface of the Schwarzschild spacetime and $S_\alpha$ is an $r=\mathrm{constant}$ surface with $r^a D_a k / k^2 = \alpha$. We adapt the two methods of the inverse mean curvature flow and the conformal flow. Therefore, our result is applicable to the case where $S_\alpha$ has multiple components. For anti-de Sitter (AdS) spaces, a similar inequality is derived, but the proof is performed only by using the inverse mean curvature flow. We also discuss the cases with asymptotically locally AdS spaces.