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