MATHEMATICAL PROBLEMS IN GENERAL RELATIVITY

该项目旨在研究爱因斯坦方程的各个方面，这些方程支配着广义相对论中时空几何的动力学。PI对研究重力非常强的解决方案特别感兴趣，包括在存在时空奇点的情况下，以及在物质场(如恒星和气体)与重力之间存在强烈相互作用的环境中。更广泛的影响包括与学生的合作，该项目提出了一些问题，据信这些问题已经准备好供研究生研究。PI试图回答该项目中的一系列问题。一般黑洞的奇异边界是什么样子的？当物质场存在时，奇异边界会发生什么，以及在奇异边界附近，物质场本身会发生什么？自引力物质场的零壳层如何传播和相互作用？如何在自引力气体中形成激波？在解决这些问题时，PI将开发出适用于一般相对论之外的双曲型偏微分方程组的数学工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.