A Penrose-type inequality for static spacetimes

TL;DR

This paper proves a Penrose-type inequality for static spacetimes, providing a lower bound on total mass related to boundary mean curvature, extending previous inequalities and characterizing Schwarzschild slices.

Contribution

It establishes a new lower bound on mass for static spacetimes under the timelike convergence condition, extending Minkowski-type inequalities to higher dimensions.

Findings

Proves a Penrose-type inequality for static spacetimes.

Characterizes equality cases as Schwarzschild slices.

Extends Riemannian Penrose inequality to all dimensions.

Abstract

We establish a lower bound on the total mass of the time slices of (n + 1)-dimensional asymptotically flat standard static spacetimes under the timelike convergence condition. The inequality can be viewed equivalently as a Minkowski-type inequality in these spaces, i.e. as a lower bound on the total mean curvature of the boundary, and thus extends inequalities from [3], [22], [19], and [10]. Equality is achieved only by slices of Schwarzschild space and is related to the characterization of quasi-spherical static vacuum metrics from [10]. As a notable special case of the main inequality, we obtain the Riemannian Penrose inequality in all dimensions for static spaces under the TCC.