The Nonlinear Stability of Black Holes and the Structure of Spacetime Singularities in General Relativity

这个项目致力于回答关于黑洞和时空结构的基本问题。这项研究是对爱因斯坦广义相对论方程解的性质的数学研究，其重点是时空奇点。该项目旨在推进严谨的数学，以加深对这一卓越的引力理论所描述的物理世界各个方面的理解，包括来自遥远星系的光的引力透镜，全球定位系统所需的卫星时钟的修正，以及从双黑洞碰撞中观察到的引力波。这项研究旨在从数学上证明黑洞的非线性稳定性，并回答广义相对论中有关时空奇点性质的基本问题。本项目的第一个主要目标是为克尔度量的非线性稳定性猜想提供数学证明。这项研究的另外两个主要目标是了解具有奇点的各种时空的结构。一系列项目将集中于证明黑洞中确实存在奇点，但它们并不像之前预期的那样“凶猛”。另一类具有奇点的时空的出现类似于最初在纯数学背景下完成的构造，也引起了理论物理学的兴趣。第二个系列项目将集中于在广义相对论中实现这一结构，并分析由此产生的奇点的精确性质。

This project strives to answer fundamental questions about black holes and the structure of spacetime. The research is a mathematical investigation of properties of solutions to Einstein's equations of general relativity that focuses on spacetime singularities. The project aims to advance rigorous mathematics to deepen understanding of the aspects of the physical world that are described by this remarkable theory of gravitation, which include the gravitational lensing of light from distant galaxies, the corrections to satellite clocks necessary for global positioning systems, and the gravitational waves observed from binary black hole collisions.This research aims to mathematically prove the nonlinear stability of black holes and answer fundamental questions concerning the nature of spacetime singularities in general relativity. The first major objective in this project is to provide a mathematical proof of the nonlinear stability conjecture for the Kerr metric. The other two major objectives of this research concern understanding the structure of various spacetimes with singularities. A series of projects will focus on showing that singularities do indeed exist in black holes, but they are not as "ferocious" as previously expected. Another class of spacetimes with singularities arise in analogy to a construction first done in the context of pure mathematics that has also attracted interest from theoretical physics. A second series of projects will concentrate on implementation of this construction in general relativity and analyze the precise nature of the resulting singularities.