A Continuing Investigation of the Penrose Conjecture in General Relativity

Massachusetts Institute Of Technology, Cambridge MA

Investigators

Abstract

Abstract Award: DMS-9971960 Principal Investigator: Hubert L. Bray The goal of this project is to better understand the Penrose Conjecture in General Relativity, which is closely related to the Positive Mass Theorem. Both statements can be thought of as attempts to describe the relationship between the local energy density of a space-time N (whose metric has signature 3,1) and the total mass of N. In physical terms, the Positive Mass Theorem states that an isolated gravitational system with nonnegative local energy density must have nonnegative total energy. The idea is that nonnegative energy densities must "add up" to something nonnegative. The Penrose Conjecture, on the other hand, states that if an isolated gravitational system with nonnegative local energy density contains black holes contributing a mass m, then the total energy of the system must be at least m. These compelling physical statements translate into highly nontrivial geometric statements about asymptotically flat manifolds with nonnegative scalar curvature. The Positive Mass Theorem was not proven until 1979 by Schoen and Yau, and the most general version of the Penrose Conjecture is still open. The investigator recently proved the Riemannian Penrose Conjecture, which is the Penrose Conjecture for a 3-dimensional space-like hypersurface M of N with zero second fundamental form. The proof uses a new approach to the problem (which came out of research supported by the NSF), and the theorem is the strongest version of the Penrose Conjecture proved to date. This research aims to extend these results to prove the most general version of the Penrose Conjecture, where the hypersurface M is not required to have zero second fundamental form. It is also hoped that these new techniques can be used to understand higher dimensional cases and the behavior of quasi-local mass in General Relativity. Einstein's Theory of General Relativity is one of two primary theories (along with Quantum Mechanics) thought to best describe the laws of physics. In the long run, it is hoped that a better understanding of the laws of physics will lead to advances which could lead to a better standard of living for people. Already, fundamental advances in understanding the laws of physics have made modern technology possible. The research in this project goes to the heart of the behavior of matter in General Relativity and attempts to answer fundamental questions about the additivity of energy and momentum in space-time. Even so, achieving these goals would only represent a small step forward in understanding the implications and intricacies of General Relativity