Applications of geometric analysis to general relativity and geometric flows

几何分析在广义相对论和几何流中的应用

• 批准号: 1405152
• 负责人: Mu-Tao Wang
• 金额: $ 21.5万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-08-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405152&HistoricalAwards=false
• 关键词: Applications; geometric; analysis; general; relativity

AbstractAward: DMS 1405152, Principal Investigator: Mu-Tao WangThe principal investigator proposes to study the notions of mass, energy, angular momentum, and center of mass in general relativity. These notions are of most fundamental importance in any branch of physics. However, since Einstein's time, there has been great difficulty to find physically acceptable definitions of these concepts for gravitation. The solutions of many unsolved problems such as how a black hole forms and how black holes collide rely essentially on these notions. Recently, the principal investigator and his collaborators successfully discovered definitions for both isolated systems (e.g. when the observer is very far away from a star) and non-isolated systems (e.g. when the observer is at close range with two stars rotating about each other ). The definitions they found satisfy many highly desirable properties, including the first precise dynamical description of Einstein's equation.This proposal describe plans to further explore these new definitions and their applications. The results obtained in this project are expected to be crucial steps towards deeper understanding of the universe in a large scale. The principal investigator also proposes to study geometric flows. These are differential equations that model how a geometric shape deforms and evolves to an optimal form in the most efficient way. The proposed research is expected to have applications in general relativity and mathematical physics. Another line of investigation will study applications of quasi-local mass and quasi-local conserved quantities such as angular momentum and center of mass, which he recently discovered with his collaborators, aiming to anchor and resolve several problems in classical general relativity. Immediate goals of this proposal include resolving the invariant mass conjecture in general relativity, justification of the definitions of conserved quantities at both the quasi-local and total levels, and applications in the study of the dynamics of the Einstein equation. The principal investigator will also continue his research on inverse mean curvature flows and mean curvature flows. Immediate goals include the proof of a Gibbons-Penrose inequality in Schwarzschild spacetime and the dynamical stability of the mean curvature flow.

摘要奖项：DMS 1405152，首席研究员：王木涛首席研究员提议研究广义相对论中的质量、能量、角动量和质心的概念。这些概念在物理学的任何分支中都具有最基本的重要性。然而，自爱因斯坦时代以来，要找到这些引力概念的物理上可接受的定义一直非常困难。许多未解决问题的解决，例如黑洞如何形成以及黑洞如何碰撞，本质上依赖于这些概念。最近，首席研究员和他的合作者成功地发现了孤立系统（例如，当观察者距离恒星很远时）和非孤立系统（例如，当观察者距离很近，两颗恒星围绕彼此旋转时）的定义。他们发现的定义满足了许多非常理想的性质，包括爱因斯坦方程的第一个精确的动力学描述。该提案描述了进一步探索这些新定义及其应用的计划。该项目获得的结果预计将成为更深入地了解大范围宇宙的关键步骤。 首席研究员还提议研究几何流。这些微分方程模拟几何形状如何变形并以最有效的方式演化为最佳形式。拟议的研究预计将在广义相对论和数学物理中得到应用。 另一条研究方向将研究他最近与合作者发现的准局域质量和准局域守恒量（例如角动量和质心）的应用，旨在锚定和解决经典广义相对论中的几个问题。 该提案的直接目标包括解决广义相对论中的不变质量猜想、准局域和总体水平上守恒量定义的合理性以及在爱因斯坦方程动力学研究中的应用。首席研究员还将继续进行反平均曲率流和平均曲率流的研究。 近期目标包括证明史瓦西时空中的吉本斯-彭罗斯不等式以及平均曲率流的动态稳定性。