Proof of the Riemannian Penrose Conjecture Using the Positive Mass Theorem

Abstract

We prove the Riemannian Penrose conjecture, an important case of a conjecture made by Roger Penrose in 1973, by defining a new flow of metrics. This flow of metrics stays inside the class of asymptotically flat Riemannian 3-manifolds with nonnegative scalar curvature which contain minimal spheres. In particular, if we consider a Riemannian 3-manifold as a totally geodesic submanifold of a space-time in the context of general relativity, then outermost minimal spheres with total area $A$ correspond to apparent horizons of black holes contributing a mass $\sqrt{A/16\pi }$, scalar curvature corresponds to local energy density at each point, and the rate at which the metric becomes flat at infinity corresponds to total mass. The Riemannian Penrose conjecture then states that the total mass of an asymptotically flat 3-manifold with nonnegative scalar curvature is greater than or equal to the mass contributed by the black holes. The flow of metrics we define continuously evolves the original 3-metric to a Schwarzschild 3-metric, which represents a spherically symmetric black hole in vacuum. We define the flow such that the area of the minimal spheres (which flow outward) and hence the mass contributed by the black holes in each of the metrics in the flow is constant, and then use the positive mass theorem to show that the total mass of the metrics is nonincreasing. Then since the total mass equals the mass of the black holes in a Schwarzschild metric, the Riemannian Penrose conjecture follows. This result improves upon the beautiful work of Huisken and Ilmanen, who used inverse mean curvature flows of surfaces to show that the total mass is at least the mass contributed by the largest black hole.

arXiv commentPreliminary version, 78 pages, 8 figures

References in corpus (1)

• Collapsing Shells and the Isoperimetric Inequality for Black Holes

Cited by in corpus (22)

• Initial data engineering
• Towards the classification of static vacuum spacetimes with negative cosmological constant
• Mass and Angular Momentum in General Relativity
• From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity
• Area-charge inequality for black holes
• General Bounds on Holographic Complexity
• Proof of the Riemannian Penrose Inequality with Charge for Multiple Black Holes
• Positive Mass Theorem on Manifolds admitting Corners along a Hypersurface
• Boundary value problems for Dirac--type equations, with applications
• Geometric inequalities for black holes
• The mass of asymptotically hyperbolic Riemannian manifolds
• Bekenstein Bounds, Penrose Inequalities, and Black Hole Formation
• On the Riemannian Penrose inequality with charge and the cosmic censorship conjecture
• Reduction Arguments for Geometric Inequalities Associated With Asymptotically Hyperboloidal Slices
• Anti-gravity à la Carlotto-Schoen
• Recent results in mathematical relativity
• Some Recent Progress in Classical General Relativity
• Variational Effect of Boundary Mean Curvature on ADM Mass in General Relativity
• The Penrose Inequality
• The Einstein-Klein-Gordon Equations, Wave Dark Matter, and the Tully-Fisher Relation
• Conformal Einstein evolution
• Remark on the Limit Case of Positive Mass Theorem for Manifolds with Inner Boundary