Area bound for surfaces in generic gravitational field

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.

28 pages, 8 figures. This version has been accepted by PTEP

References in corpus (2)

• Transversely trapping surfaces: Dynamical version
• Formation of dynamically transversely trapping surfaces and the stretched hoop conjecture

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