On the Mathematical Theory of Black Holes

论黑洞的数学理论

• 批准号: 2201031
• 负责人: Sergiu Klainerman
• 金额: $ 47.21万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2022
• 资助国家: 美国
• 起止时间: 2022-07-01 至 2025-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2201031&HistoricalAwards=false
• 关键词: Mathematical; Theory; Black; Holes

This project deals with problems connected to the mathematical theory of black holes which, if solved, would contribute greatly to our understanding of these most fascinating physical objects of the physical Universe. The project will not only advance the science of black holes, but, by developing new mathematical techniques of relevance to other fields of mathematical physics, it promotes the progress of Mathematics and Science in the broadest possible sense. The Principal Investigator (PI) has a strong track record in targeting specific problems with broad appeal in various areas of Partial Differential Equations (PDE). In that sense, the problems considered as part of this project, though specific to the subject at hand, can also be reformulated in other important problems of mathematical physics. The project also fits well with respect to the long term goal of the PI to help create a vibrant scientific community working on mathematical problems connected to these areas. In particular, the project provides research training opportunities for graduate students. The project focuses on some of the main open problems concerning black holes such as rigidity, stability, and collapse, with special emphasis being given to the fundamental problem of the stability of the Kerr family. This is a precise, explicit, family of solutions to the Einstein vacuum equations, depending on two parameters (the mass and the angular momentum), on which our theoretical understanding of black holes is based. Thus, the problems mentioned above are not only deep from a mathematical point of view, they also have serious implications in Astrophysics. This is particularly true about the problem of stability, since if the Kerr family were to be found unstable, black holes would be nothing more than mathematical artifacts. The PI interprets these three related problems as mathematical "tests of reality" for black holes. All three problems require a deep understanding of the dynamics of the Einstein field equations in a strong gravitational field regime. This is a tall task which can only be achieved by a systematic mathematical analysis of the underlining structure of the problems, using new geometric PDE techniques. The resolution of these problems promotes the progress of Mathematics and Science beyond the subject matter of this particular project. Indeed, it is expected, as it often happened in the past, that some of the techniques developed in connection to these problems will be relevant to other important partial differential equations of mathematical physics.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.

这个项目涉及与黑洞的数学理论有关的问题，如果解决了这些问题，将极大地有助于我们理解物理宇宙中这些最迷人的物理对象。该项目不仅将推动黑洞科学的发展，而且通过开发与数学物理其他领域相关的新的数学技术，它将在最广泛的意义上促进数学和科学的进步。首席调查员(PI)在针对偏微分方程组(PDE)各个领域具有广泛吸引力的具体问题方面有着良好的记录。在这个意义上，被认为是这个项目的一部分的问题，虽然是针对手头的主题，但也可以在其他重要的数学物理问题中重新表述。该项目也非常符合PI的长期目标，即帮助创建一个充满活力的科学社区，致力于解决与这些领域有关的数学问题。特别是，该项目为研究生提供了研究培训机会。该项目集中于一些与黑洞有关的主要公开问题，如刚性、稳定性和崩塌，特别强调科尔族的稳定性这一根本问题。这是一组精确的、显式的爱因斯坦真空方程的解，依赖于两个参数(质量和角动量)，我们对黑洞的理论理解基于这两个参数。因此，上面提到的问题不仅从数学的角度来看是深刻的，它们在天体物理学中也具有严重的影响。在稳定性问题上尤其如此，因为如果克尔家族被发现不稳定，黑洞将只不过是数学上的产物。PI将这三个相关问题解释为黑洞的数学“现实测试”。所有这三个问题都需要深入理解爱因斯坦场方程在强引力场中的动力学。这是一项艰巨的任务，只有使用新的几何偏微分方程技术，对问题的基本结构进行系统的数学分析，才能实现这一任务。这些问题的解决推动了数学和科学的进步，超出了这一特定项目的主题。事实上，正如过去经常发生的那样，预计与这些问题相关的一些技术将与其他重要的数学物理偏微分方程式相关。这一奖项反映了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。