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A Penrose-type inequality for static spacetimes

Brian Harvie

TL;DR

This work proves a Penrose-type lower bound on the mass of asymptotically flat static triples under the timelike convergence condition, namely $m \ge \frac{1}{2} \left( \frac{|\Sigma|}{w_{n-1}} \right)^{\frac{n-2}{n-1}} - \frac{1}{2(n-1) w_{n-1}} \int_{\Sigma} V H\,d\sigma$, with equality characterizing a Schwarzschild region. The authors introduce a monotone quantity $Q(t)$ along inverse mean curvature flow, show its non-increase under both smooth and weak IMCF, and derive a flux bound for $V$ from the mass formula, all under the timelike convergence condition. The results extend Minkowski-type and Penrose-type inequalities to higher-dimensional static spacetimes and yield a rigidity statement: equality forces the geometry to be Schwarzschild; this also provides a Riemannian Penrose-type inequality for static spaces in all dimensions under TCC. Overall, the paper provides a robust geometric-analytic framework linking mass, horizon geometry, and static vacuum rigidity in higher-dimensional general relativity.

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.

A Penrose-type inequality for static spacetimes

TL;DR

This work proves a Penrose-type lower bound on the mass of asymptotically flat static triples under the timelike convergence condition, namely , with equality characterizing a Schwarzschild region. The authors introduce a monotone quantity along inverse mean curvature flow, show its non-increase under both smooth and weak IMCF, and derive a flux bound for from the mass formula, all under the timelike convergence condition. The results extend Minkowski-type and Penrose-type inequalities to higher-dimensional static spacetimes and yield a rigidity statement: equality forces the geometry to be Schwarzschild; this also provides a Riemannian Penrose-type inequality for static spaces in all dimensions under TCC. Overall, the paper provides a robust geometric-analytic framework linking mass, horizon geometry, and static vacuum rigidity in higher-dimensional general relativity.

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.

Paper Structure

This paper contains 6 sections, 14 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. The timelike convergence condition and the ADM Mass
  3. A Monotone Quantity for the Inverse Mean Curvature Flow
  4. Monotonicity for Weak Inverse Mean Curvature Flow
  5. The Equality Case
  6. Proof of Proposition \ref{['horizon']}

Key Result

Theorem 1.1

Let $(M^{n},g,V)$ be a static triple which is asymptotically flat in the sense of Definition af and which satisfies the TCC dec. Let $\Sigma^{n-1} \subset \partial M^{n}$ be an outer-minimizing boundary component of $M$. Assume that Then we have the following lower bound on the mass $m$ of $(M^{n},g)$: where $w_{n-1}$ is the area of the unit sphere and $|\Sigma|$ is the $(n-1)$-dimensional area

Theorems & Definitions (25)

  • Theorem 1.1: Penrose-Type Inequality for Static Spaces
  • Corollary 1.2: Riemannian Penrose Inequality for Static Spaces
  • Remark 1.3
  • Proposition 2.1: The TCC on Time Slices
  • proof
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Lemma 3.1
  • Proposition 3.2
  • ...and 15 more