arXiv subjects

Hubert L. Bray

Publications and source records attributed to Hubert L. Bray.

20 records · Page 2Linked to original sources

arXiv2003

Black holes and the Penrose inequality in general relativity

In a paper \cite{P} in 1973, R. Penrose made a physical argument that the total mass of a spacetime which contains black holes with event horizons of total area $A$ should be at least $\sqrt{A/16π}$. An important special case of this physical statement translates into a very beautiful mathematical inequality in Riemannian geometry known as the Riemannian Penrose inequality. One particularly geometric aspect of this problem is the fact that apparent horizons of black holes in this setting correspond to minimal surfaces in Riemannian 3-manifolds. The Riemannian Penrose inequality was first proved by G. Huisken and T. Ilmanen in 1997 for a single black hole \cite{HI} and then by the author in 1999 for any number of black holes \cite{Bray}. The two approaches use two different geometric flow techniques. The most general version of the Penrose inequality is still open. In this talk we will sketch the author's proof by flowing Riemannian manifolds inside the class of asymptotically flat 3-manifolds (asymptotic to $\real^3$ at infinity) which have nonnegative scalar curvature and contain minimal spheres. This new flow of metrics has very special properties and simulates an initial physical situation in which all of the matter falls into the black holes which merge into a single, spherically symmetric black hole given by the Schwarzschild metric. Since the Schwarzschild metric gives equality in the Penrose inequality and the flow decreases the total mass while preserving the area of the horizons of the black holes, the Penrose inequality follows. We will also discuss how these techniques can be generalized in higher dimensions.

math.DG↗

arXiv1999

Proof of the Riemannian Penrose Conjecture Using the Positive Mass Theorem

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 $\sqrt{A/16π}$, 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.

math.DG↗