Black Holes, Geometric Flows, and the Penrose Inequality in General Relativity, Volume 49, Number 11

1372 NOTICES OF THE AMS VOLUME 49, NUMBER 11 I n 1973 R. Penrose [13] made a physical argument that the total mass of a spacetime containing black holes with event horizons of total area A should be at least √ 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. The Riemannian Penrose inequality was first proved by G. Huisken and T. Ilmanen in 1997 for a single black hole [8] and then by the author in 1999 for any number of black holes [1]. The two approaches use two different geometric flow techniques. The most general version of the Penrose inequality is still open. A natural interpretation of the Penrose inequality is that the mass contributed by a collection of black holes is (at least) √ A/16π . More generally, the question, How much matter is in a given region of a spacetime? is still very much an open problem [6]. In this paper we will discuss some of the qualitative aspects of mass in general relativity, look at some informative examples, and describe the two very geometric proofs of the Riemannian Penrose inequality. Total Mass in General Relativity Two notions of mass which are well understood in general relativity are local energy density at a point and the total mass of an asymptotically flat spacetime. However, defining the mass of a region larger than a point but smaller than the entire universe is not at all well understood. Suppose (M3, g) is a Riemannian 3-manifold isometrically embedded in a (3+1) dimensional Lorentzian spacetime N4. Suppose that M3 has zero second fundamental form in the spacetime. This is a simplifying assumption which allows us to think of (M3, g) as a “t = 0” slice of the spacetime. (The second fundamental form is a measure of how much M3 curves inside N4; M3 is also sometimes called “totally geodesic” since geodesics of N4 which are tangent to M3 at a point stay inside M3 forever.) The Penrose inequality (which allows for M3 to have general second fundamental form) is known as the Riemannian Penrose inequality when the second fundamental form is set to zero. We also want to consider only (M3, g) that are asymptotically flat at infinity, which means that for some compact set K, the “end” M3 \K is diffeomorphic to R3 \ B1(0), where the metric g is asymptotically approaching (with certain decay conditions) the standard flat metric δij on R3 at infinity. The simplest example of an asymptotically flat manifold is (R3, δij ) itself. Other good examples are the conformal metrics (R3, u(x)δij ), where u(x) approaches a constant sufficiently rapidly at Hubert L. Bray is assistant professor of mathematics at the Massachusetts Institute of Technology. His email address is [email protected].