Attractive gravity probe surfaces in higher dimensions

Abstract

A generalization of the Riemannian Penrose inequality in $n$-dimensional space ($3\le n<8$) is done. We introduce a parameter $\alpha$ ($-\frac{1}{n-1}<\alpha < \infty$) indicating the strength of the gravitational field, and define a refined attractive gravity probe surface (refined AGPS) with $\alpha$. Then, we show the area inequality for a refined AGPS, $A \le \omega_{n-1} \left[ (n+2(n-1)\alpha)Gm /(1+(n-1)\alpha) \right]^{\frac{n-1}{n-2}}$, where $A$ is the area of the refined AGPS, $\omega_{n-1}$ is the area of the standard unit $(n-1)$-sphere, $G$ is Newton's gravitational constant and $m$ is the Arnowitt-Deser-Misner mass. The obtained inequality is applicable not only to surfaces in strong gravity regions such as a minimal surface (corresponding to the limit $\alpha \to \infty$), but also to those in weak gravity existing near infinity (corresponding to the limit $\alpha \to -\frac{1}{n-1}$).