Mathematical Problems in General Relativity

广义相对论中的数学问题

• 批准号: 2005435
• 负责人: Jonathan Luk
• 金额: $ 33万
• 依托单位: Stanford University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005435&HistoricalAwards=false
• 关键词: Mathematical; Problems; General; Relativity

This project studies the Einstein's remarkable theory of general relativity, described by the Einstein equations. One fascinating prediction of Einstein's theory is the ubiquity of singularities. The research aims at giving a precise mathematical description of these singularities. For instance, we study the structure of singularities inside black holes, and also the effects of highly singular and highly oscillatory gravitational waves. In addition, the PI will also undertake training of students (he currently has a PhD student) and continue disseminating his work in both the mathematics and physics communities. Some of the work will be communicated to a wider public audience; the PI's work has already been featured in a popular science magazine.We describe a sample of questions that we seek to answer. What does the singular boundary of generic black holes look like? How does the situation change in the special case of extremal black holes? Are there other types of singularities? What does the interaction of multiple impulsive gravitational waves look like? Can one describe appropriate limits of highly oscillatory solutions?Finally, we also propose some questions to understand the dynamics of the Einstein equations in the presence of matter fields. While we consider a wide range of questions, they are unified by the underlying analytic and geometric techniques. Through solving these problems we will also 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.

这个项目研究爱因斯坦的非凡的广义相对论，由爱因斯坦方程描述。爱因斯坦理论中一个令人着迷的预言是奇点无处不在。该研究旨在对这些奇点进行精确的数学描述。例如，我们研究黑洞内部奇点的结构，以及高度奇异和高度振荡的引力波的影响。此外，PI还将对学生进行培训（他目前有一名博士生），并继续在数学和物理界传播他的工作。其中一些工作将被传达给更广泛的公众受众; PI的工作已经在一本科普杂志上进行了专题报道。我们描述了一个我们试图回答的问题样本。一般黑洞的奇异边界是什么样子的？在极端黑洞的特殊情况下，情况会如何变化？还有其他类型的奇点吗？多重脉冲引力波的相互作用是什么样子的？能否描述高振荡解的适当极限？最后，我们还提出了一些问题，以了解爱因斯坦方程的动力学存在的物质场。虽然我们考虑了广泛的问题，但它们是由基本的分析和几何技术统一的。通过解决这些问题，我们还将开发出适用于广义相对论之外的双曲型偏微分方程的数学工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Phase mixing for solutions to 1D transport equation in a confining potential

• DOI: 10.3934/krm.2022002
• 发表时间: 2021-09
• 期刊: Kinetic & Related Models
• 影响因子: 1
• 作者: S. Chaturvedi;J. Luk

A Scattering Theory Approach to Cauchy Horizon Instability and Applications to Mass Inflation

柯西视界不稳定性的散射理论方法及其在大规模通货膨胀中的应用

• DOI: 10.1007/s00023-022-01216-7
• 发表时间: 2023
• 期刊: Annales Henri Poincaré
• 作者: Luk, Jonathan;Oh, Sung-Jin;Shlapentokh-Rothman, Yakov
• 通讯作者: Shlapentokh-Rothman, Yakov

Global Nonlinear Stability of Large Dispersive Solutions to the Einstein Equations

• DOI: 10.1007/s00023-021-01148-8
• 发表时间: 2021-08
• 期刊: Annales Henri Poincaré
• 作者: J. Luk;Sung-Jin Oh