Singularities in General Relativity

这项研究项目研究爱因斯坦真空方程解的时空奇异性，爱因斯坦真空方程是一组描述无物质位置的重力的非线性偏微分方程组。这组方程的解可以从规则的初始数据产生奇点；一些这样的奇点位于黑洞的视界之内。一个基本的数学问题涉及到理解这种奇点是如何从通用的初始数据中产生的。特别是，这个项目探索的问题包括：黑洞内部的边界是什么样子的？它们总是单数吗？有没有不在黑洞区域的奇点？这些都是广义相对论中的强和弱宇宙审查猜想试图解决的问题--对这些猜想的研究是这个项目的核心动机。该项目的一个特别重点是了解黑洞内部的奇异边界，并了解形成囚禁表面(与黑洞密切相关的结构)的准确标准。研究人员和合作者最近的工作解决了球对称爱因斯坦-麦克斯韦(实)标量场的强宇宙审查猜想，表明由一般数据产生的解具有奇异的黑洞内部，从这个意义上说，时空度规不能以适当的规则方式扩展。这个项目的目的是将这些结果推广到没有对称性假设的环境中，特别是给出由显式Kerr解的扰动引起的黑洞边界的完整描述。在一个不同的方向上，这个项目还旨在建立保证被困表面形成的标准，作为理解广义相对论中弱宇宙审查猜想的一步。

This research project studies spacetime singularities in the solutions to the Einstein vacuum equations, a system of nonlinear partial differential equations that describes gravity in locations devoid of matter. Solutions to this system of equations can develop singularities from regular initial data; some such singularities are inside a black hole event horizon. A fundamental mathematical question concerns understanding how such singularities arise from generic initial data. In particular, this project explores questions that include: What does the boundary of the interior of black holes look like? Are they always singular? Are there singularities that are not in the black hole regions? These are questions that the strong and weak cosmic censorship conjectures in general relativity seek to address -- study of those conjectures is the central motivation for this project.A particular focus of the project is to understand the singular boundary of the interior of black holes and also to understand sharp criteria under which trapped surfaces (structures closely related to black holes) may form. Recent work of the investigator and collaborators provides a resolution of the strong cosmic censorship conjecture for the Einstein-Maxwell (real) scalar field in spherical symmetry, showing that solutions arising from generic data have singular black hole interiors, in the sense that the spacetime metric cannot be extended in a suitably regular manner. This project aims to extend these results to settings without symmetry assumptions, in particular to give a complete description of the boundary of black holes that arise from perturbations of the explicit Kerr solutions. In a different direction, this project also aims to establish criteria that guarantee that trapped surfaces form, as a step towards understanding the weak cosmic censorship conjecture in general relativity.