A geometric theory of zero area singularities in general relativity

TL;DR

This paper develops a geometric framework for zero area singularities in Riemannian manifolds, generalizing known cases like negative mass Schwarzschild spacetime, and establishes a mass inequality involving these singularities.

Contribution

It introduces a theory of zero area singularities with nontrivial topology and defines their mass, extending the Riemannian Penrose Inequality to include such singularities.

Findings

Lower bound on ADM mass in terms of singularity masses

Equality case achieved by negative mass Schwarzschild metric

Corollary extends Positive Mass Theorem to incomplete metrics

Abstract

The Schwarzschild spacetime metric of negative mass is well-known to contain a naked singularity. In a spacelike slice, this singularity of the metric is characterized by the property that nearby surfaces have arbitrarily small area. We develop a theory of such "zero area singularities" in Riemannian manifolds, generalizing far beyond the Schwarzschild case (for example, allowing the singularities to have nontrivial topology). We also define the mass of such singularities. The main result of this paper is a lower bound on the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the masses of its singularities, assuming a certain conjecture in conformal geometry. The proof relies on the Riemannian Penrose Inequality. Equality is attained in the inequality by the Schwarzschild metric of negative mass. An immediate corollary is a version of the Positive Mass Theorem that allows for certain types of incomplete metrics.