Attractive gravity probe surfaces in higher dimensions

Abstract

A generalization of the Riemannian Penrose inequality in n-dimensional space (3 ≤ n < 8) is done. We introduce a parameter α ($-\frac{1}{n-1}<\alpha < \infty$) indicating the strength of the gravitational field, and define a refined attractive gravity probe surface (refined AGPS) with α. 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, ωn − 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 α → ∞), but also to those in weak gravity existing near infinity (corresponding to the limit $\alpha \rightarrow -\frac{1}{n-1}$) .