A Continuing Investigation of the Penrose Conjecture in General Relativity

项目编号：dms -9971960项目负责人：Hubert L. bray该项目的目的是为了更好地理解广义相对论中与正质量定理密切相关的penrosecjecture。这两种说法都可以被认为是试图描述时空N的局部能量密度（其度规具有签名3,1）与N的总质量之间的关系。在物理术语中，正质量定理表明，具有非负局部能量密度的孤立引力系统必须具有非负的总能量。这个想法是，非负能量密度必须“加”成非负的东西。另一方面，彭罗斯猜想指出，如果一个具有非负局部能量密度的孤立引力系统包含贡献质量为m的黑洞，那么该系统的总能量必须至少为m。这些引人注目的物理陈述转化为关于具有非负标量曲率的渐近平坦流形的高度非平凡的几何陈述。正质量定理直到1979年才被舍恩和丘证明，而彭罗斯猜想的最一般版本仍然是开放的。研究者最近证明了黎曼彭罗斯猜想，即具有零秒基本形式的三维类空间超曲面M (N)的彭罗斯猜想。这个证明使用了一种新的方法来解决这个问题（这是由美国国家科学基金会支持的研究得出的），而且这个定理是迄今为止证明的彭罗斯猜想的最强版本。本研究旨在扩展这些结果，以证明彭罗斯猜想的最一般版本，其中超曲面M不需要具有零秒基本形式。我们也希望这些新技术可以用来理解广义相对论中更高维度的情况和准局部质量的行为。爱因斯坦的广义相对论被认为是最能描述物理定律的两个主要理论之一（另一个是量子力学）。从长远来看，人们希望对物理定律的更好理解将带来进步，从而提高人们的生活水平。在理解物理定律方面取得的基本进展已经使现代技术成为可能。这个项目的研究进入了广义相对论中物质行为的核心，并试图回答关于时空中能量和动量的可加性的基本问题。即便如此，实现这些目标也只是在理解广义相对论的含义和复杂性方面向前迈出的一小步

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