MATHEMATICAL PROBLEMS IN GENERAL RELATIVITY

该项目旨在研究爱因斯坦方程的各个方面，该方程在广义相对论中控制着时空几何的动力学。PI特别感兴趣的是研究引力非常强的解决方案，包括存在时空奇点的情况，以及物质场（如恒星和气体）与引力之间存在强烈相互作用的情况。更广泛的影响包括与学生的合作，该项目提出了一些问题，据信这些问题已经为研究生的研究做好了准备。PI试图在这个项目中回答一系列问题。一般黑洞的奇异边界是什么样的？当物质场存在时，奇异边界会发生什么变化？在奇异边界附近的物质场本身又会发生什么变化？物质场的自引力“零壳层”是如何传播和相互作用的？激波是如何在自重力气体中形成的？在解决这些问题的同时，PI将开发出对广义相对论之外的双曲型偏微分方程有用的数学工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The project aims at studying various aspects of the Einstein equations, which govern the dynamics of spacetime geometry in general relativity. The PI is particularly interested in studying the solution regimes where gravity is very strong, including in the presence of spacetime singularities, as well as settings where there is a strong interaction between matter fields (such as stars and gases) and gravity. Broader impacts include work with students, and the project presents some questions that are believed to be ready for research by graduate students.THe PI seeks to answer a series of questions in this project. What does the singular boundary of generic black holes look like? What happens to the singular boundary when matter fields are present, and what happens to the matter fields themselves near the singular boundary? How do self-gravitating "null shells of matter fields propagate and interact? How does shock form in self-gravitating gas? While solving these problems, the PI will develop mathematical tools useful for hyperbolic partial differential equations beyond the application in general relativity.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.