Problems in Mathematical General Relativity

本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中的问题。在项目的第一部分，PI计划研究GR中准局部质量的性质。该研究将基于PI先前与王慕涛和丘成东的工作。目的是研究准局部质量的单调性和变分性质。对于项目的第二部分，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。在项目的第三部分，PI计划研究由GR引起的几何不等式。PI期望准局部质量和角动量的良好概念将在Penrose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计划研究重力辐射。这项研究将基于PI之前与Lydia Bieri和Yau Shing-Tung关于记忆效应的工作，其中引力能的辐射与测试粒子的位移测量有关。PI计划利用准局部质量来研究引力辐射，并研究其他守恒量在引力辐射中的作用。在广义相对论中，孤立系统的总能量的概念是重要的，而在广义相对论的许多基本问题中，有限区域内的质量、动量或角动量的测量是必不可少的。由于引力的非局域性，这一点尤为重要。提出的研究评估能量，动量或角动量包含在宇宙的任何区域。这使得在引力场强的区域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.