A Continuing Investigation of the Penrose Conjecture in General Relativity

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AbstractAward: DMS-9971960Principal Investigator: Hubert L. BrayThe goal of this project is to better understand the PenroseConjecture in General Relativity, which is closely related to thePositive Mass Theorem. Both statements can be thought of asattempts to describe the relationship between the local energydensity of a space-time N (whose metric has signature 3,1) andthe total mass of N. In physical terms, the Positive MassTheorem states that an isolated gravitational system withnonnegative local energy density must have nonnegative totalenergy. The idea is that nonnegative energy densities must "addup" to something nonnegative. The Penrose Conjecture, on theother hand, states that if an isolated gravitational system withnonnegative local energy density contains black holescontributing a mass m, then the total energy of the system mustbe at least m. These compelling physical statements translateinto highly nontrivial geometric statements about asymptoticallyflat manifolds with nonnegative scalar curvature. The PositiveMass Theorem was not proven until 1979 by Schoen and Yau, and themost general version of the Penrose Conjecture is still open.The investigator recently proved the Riemannian PenroseConjecture, which is the Penrose Conjecture for a 3-dimensionalspace-like hypersurface M of N with zero second fundamental form.The proof uses a new approach to the problem (which came out ofresearch supported by the NSF), and the theorem is the strongestversion of the Penrose Conjecture proved to date. This researchaims to extend these results to prove the most general version ofthe Penrose Conjecture, where the hypersurface M is not requiredto have zero second fundamental form. It is also hoped thatthese new techniques can be used to understand higher dimensionalcases and the behavior of quasi-local mass in General Relativity.Einstein's Theory of General Relativity is one of two primarytheories (along with Quantum Mechanics) thought to best describethe laws of physics. In the long run, it is hoped that a betterunderstanding of the laws of physics will lead to advances whichcould lead to a better standard of living for people. Already,fundamental advances in understanding the laws of physics havemade modern technology possible. The research in this projectgoes to the heart of the behavior of matter in General Relativityand attempts to answer fundamental questions about the additivityof energy and momentum in space-time. Even so, achieving thesegoals would only represent a small step forward in understandingthe implications and intricacies of General Relativity