Problems in Mathematical General Relativity

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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.