Global Studies of Einstein Spacetimes

爱因斯坦时空的全球研究

• 批准号: 1305766
• 负责人: Vincent Moncrief
• 金额: $ 6万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2014-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1305766&HistoricalAwards=false
• 关键词: Global; Studies; Einstein; Spacetimes

One of the main aims of this project is to continue the development of light-cone estimates for the Einstein equations that parallel those that have already been developed and successfully used to prove a global existence theorem for the Yang-Mills-Higgs equations. Though strict global existence is not valid for general relativity the Penrose cosmic censorship conjecture provides its natural, gravitational analogue and the resolution of this conjecture is considered to be one of the main open mathematical problems in classical general relativity. Recent analysis has succeeded to derive an explicit, exact integral formula for the curvature of a vacuum spacetime, at an arbitrary point. This integral formula should also admit applications to other fundamental, open mathematical problems such as the rigorous proof of black hole stability. An additional, principal aim of this project is to continue work on the development of improved numerical methods for the analysis of gravitational waves. A primary goal is to determine how best to integrate this discovery with standard, unconstrained, numerical evolution techniques. A final goal of this project is to adapt recently developed "Euclidean signature semi-classical" approximation methods to the study of the Wheeler-DeWitt equation of formal, quantum general relativity.Much of the research involved in this project is of interest to mathematicians working in geometry and partial differential equations. Techniques developed for studying, numerically, the propagation of waves that literally extend all the way to infinity could well prove to be useful in other areas of science and engineering. It is anticipated that these same methods, when applied to the study of gravitational radiation, will have a significant influence on the ongoing worldwide efforts to model, numerically, interesting sources of such waves and thereby, ultimately, to impact the experimental search efforts. The proposal to further develop "Euclidean-signature semi-classical" methods for the study of quantum field theories, especially gauge theories, should have far-reaching implications for the understanding of such systems. It is expected that enlisting the contributions of students and younger researchers in these endeavors should have significant implications for the development of human resources in the mathematical and physical sciences and this is an additional, principal aim of the project.

该项目的主要目的之一是继续发展爱因斯坦方程的光锥估计，这些估计与那些已经开发并成功用于证明杨-米尔斯-希格斯方程的全局存在定理的估计平行。尽管严格的全局存在性对广义相对论无效，但彭罗斯宇宙审查猜想提供了它的自然的、引力的类比，并且该猜想的求解被认为是经典广义相对论中主要的开放数学问题之一。最近的分析已经成功地推导出真空时空在任意一点的曲率的一个明确的、精确的积分公式。这个积分公式也可以应用于其他基本的、开放的数学问题，比如黑洞稳定性的严格证明。这个项目的另一个主要目的是继续发展改进的引力波分析的数值方法。主要目标是确定如何最好地将这一发现与标准的、不受约束的数值进化技术结合起来。该项目的最终目标是采用最近发展的“欧几里得签名半经典”近似方法来研究形式量子广义相对论的惠勒-德维特方程。这个项目中涉及的许多研究对研究几何和偏微分方程的数学家来说是很有兴趣的。从数值上研究波的传播的技术可以一直延伸到无限远，这在其他科学和工程领域也很有用。预计这些相同的方法在应用于重力辐射研究时，将对正在进行的世界范围内对这种波的有趣来源进行数值模拟的努力产生重大影响，从而最终影响实验搜索工作。进一步发展“欧几里得签名半经典”方法来研究量子场论，特别是规范理论，应该对理解这些系统具有深远的意义。预计在这些努力中征集学生和年轻研究人员的贡献将对数学和物理科学人力资源的发展产生重大影响，这是该项目的另一个主要目标。