Mathematical problems in general relativity

广义相对论中的数学问题

• 批准号: RGPIN-2017-06103
• 负责人: ARETAKIS, STEFANOS
• 金额: $ 2.19万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=644110
• 关键词: Mathematical; problems; general; relativity

General relativity is a theory proposed by Einstein in 1915 as a unified theory of space, time and gravitation and is fundamentally rooted in classical mechanics. Its main objective is the understanding of the evolution of gravitational physical systems such as planetary systems, multiple stars, black holes, (a cluster of) galaxies and ultimately the universe as a whole. What distinguishes general relativity from other physical theories is that "general relativity describes the evolution of systems in terms of differential equations (i.e.~physical laws) imposed directly on the space-time metric'', connecting thus geometry, analysis and physics. These differential equations, known as the Einstein equations, give rise to a theory of great mathematical subtlety and beauty. ******The research program will strive to resolve certain fundamental mathematical questions pertaining to the main conjectures in general relativity. Specifically, the project will investigate the stability problem for the so-called extremal and sub-extremal Kerr black holes aiming at establishing the relevance of black holes from a theoretical point of view. It will also develop a scattering theory for the Einstein equations which is expected to provide new insights into the study of the propagation of gravitational waves. Finally, the program will investigate the gluing problem for characteristic initial data of hyperbolic equations. The latter is an extension of the Riemannian gluing problem for elliptic equations and is intensively studied in geometric analysis.******The proposal, if successful, will lead to the discovery of genuinely new analytical techniques applicable to a wider spectrum of problems in geometry, analysis, partial differential equations and mathematical physics. We have previously used modern techniques of mathematical analysis to obtain new and mathematically rigorous results that led to very fruitful and productive contacts with members of the high-energy physics, numerical relativity and astrophysics communities. The program will contribute to further strengthen the ties between these communities and pure mathematics.******As part of this research program we will train undergraduate and graduate students, and attract and supervise highly qualified doctoral students and post-docs from Canada and worldwide aiming to establish the University of Toronto as a central place for research in the rapidly expanding collaborative area of mathematical general relativity.

广义相对论是爱因斯坦于1915年提出的一个理论，是空间、时间和引力的统一理论，基本上植根于经典力学。它的主要目标是了解引力物理系统的演变，如行星系统，多星，黑洞，（一个）星系团，最终是整个宇宙。广义相对论与其他物理理论的区别在于，“广义相对论用微分方程描述系统的演化（即~物理定律）直接施加在时空度规上”，从而将几何学、分析学和物理学联系起来。这些微分方程，被称为爱因斯坦方程，产生了一个伟大的数学微妙和美丽的理论。** 该研究计划将努力解决与广义相对论主要理论有关的某些基本数学问题。具体而言，该项目将研究所谓的极值和亚极值克尔黑洞的稳定性问题，旨在从理论角度建立黑洞的相关性。它还将发展爱因斯坦方程的散射理论，预计将为引力波传播的研究提供新的见解。最后，程序将研究双曲型方程特征初值的胶合问题。后者是椭圆型方程的黎曼粘合问题的扩展，在几何分析中得到了深入的研究。这项建议如果成功，将导致发现真正新的分析技术，适用于更广泛的问题，在几何，分析，偏微分方程和数学物理。我们以前使用现代数学分析技术，以获得新的和数学上严格的结果，导致非常富有成效和富有成效的接触与高能物理，数值相对论和天体物理学社区的成员。该计划将有助于进一步加强这些社区和纯数学之间的联系。作为这项研究计划的一部分，我们将培养本科生和研究生，并吸引和监督来自加拿大和世界各地的高素质的博士生和博士后，旨在建立多伦多大学作为一个中心的研究在数学广义相对论的迅速扩大的合作领域。