Area bound for surfaces in generic gravitational field. (14th July 2021)

• Record Type: Journal Article
• Authors: Izumi, Keisuke; Tomikawa, Yoshimune; Shiromizu, Tetsuya; Yoshino, Hirotaka
• Is Part Of: Progress of theoretical and experimental physics. Volume 2021:Number 8(2021)
• Publication Date: 2021-07-14
• Subjects: E01; Physics -- Periodicals; Mathematical physics -- Periodicals; 530.05
• DOI: 10.1093/ptep/ptab089
• Languages: English
• ISSNs: 2050-3911

Abstract

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 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.