Problems in Mathematical General Relativity

该项目的主要目标是应用几何分析的方法来研究广义相对论（GR）引起的问题。在项目的第一部分，PI计划研究GR中的准局域质量的性质。研究将基于PI与王慕涛和丘成桐之前的工作。目的是研究拟局域质量的单调性和变分性质。对于该项目的第二部分，PI计划研究GR中的其他准局部守恒量。PI希望在准局部质量研究中开发的方法将有助于锚定准局部角动量和质心的定义。 在该项目的第三部分，PI计划调查几何不等式所产生的GR. PI预计，准本地质量和角动量的一个很好的概念将是重要的彭罗斯不等式和质量角动量不等式的研究。 最后，PI计划研究引力辐射。这项研究将基于PI与Lydia Bieri和Shing-Tung Yau先前关于记忆效应的工作，其中引力能的辐射与测试粒子位移的测量有关。在广义相对论中，孤立系统的总能量的概念是重要的，而在有限区域内的质量、动量或角动量的测量则是广义相对论中许多基本问题的关键。由于引力的非局部性质，这一点特别重要。拟议的研究评估宇宙任何区域所包含的能量、动量或角动量。这使得在引力场较强的区域进行广义相对论的研究成为可能。因此，拟议的研究将有助于更好地了解黑洞的形成和过程中的引力辐射。

The main goal of this project is to apply methods from geometric analysis to study problems arising from general relativity (GR). In the first part of the project, the PI plans to investigate properties of quasi-local mass in GR. The research will be based on the PI's previous work with Mu-Tao Wang and Shing-Tung Yau. The goal is to study the monotonicity and the variational properties of quasi-local mass. For the second part of the project, the PI plans to investigate other quasi-local conserved quantities in GR. The PI expects that the method developed in the study of quasi-local mass will help anchoring the definition of quasi-local angular momentum and center of mass. In the third part of the project, the PI plans to investigate geometric inequalities arising from GR. The PI expects that a good notion of quasi-local mass and angular momentum will be important in the study of the Penrose inequality and the mass-angular-momentum inequality. Finally, the PI plans to study gravitational radiation. The research will be based on PI's previous work with Lydia Bieri and Shing-Tung Yau on the memory effect where the radiation of gravitational energy is related to measurement of displacements of test particles. The PI plans to investigate gravitational radiation using quasi-local mass and to study the role of other conserved quantities in gravitational radiation.While the concept of total energy of isolated systems is important in general relativity, the measurement of mass, momentum or angular momentum contained in a finite region is essential in many fundamental problems in general relativity. This is particularly important due to the non-local nature of gravitation. The proposed research evaluates the energy, momentum or angular momentum contained in any region of the universe. This allows the study of general relativity in regions where the gravitational field is strong. As a result, the proposed research will lead to a better understanding of formation of black holes and the gravitational radiation during the process.