Problems in Mathematical General Relativity

这个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中准局域质量的性质。这项研究将基于PI之前与王慕涛和邱成栋的工作。目的是研究准局部质量的单调性和变分性质。在项目的第二部分，PI计划研究GR中的其他准局部守恒量。PI期望在准局域质量研究中发展的方法将有助于锚定准局域角动量和质心的定义。在项目的第三部分，PI计划调查由GR引起的几何不平等。PI预计，准局域质量和角动量的良好概念将在彭罗斯不等式和质量-角动量不等式的研究中发挥重要作用。最后，PI计划研究引力辐射。这项研究将基于Pi之前与Lydia Bieri和游成栋关于记忆效应的工作，在记忆效应中，引力能量的辐射与测试粒子的位移测量有关。PI计划利用准局域质量研究引力辐射，并研究其他守恒量在引力辐射中的作用。虽然孤立系统的总能量概念在广义相对论中很重要，但有限区域中包含的质量、动量或角动量的测量在广义相对论的许多基本问题中都是必不可少的。由于万有引力的非局部性质，这一点尤其重要。这项拟议的研究评估了宇宙中任何区域包含的能量、动量或角动量。这使得在引力场较强的地区研究广义相对论成为可能。因此，拟议的研究将有助于更好地理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.