On the Mathematical Theory of Black Holes

论黑洞的数学理论

• 批准号: 1800841
• 负责人: Sergiu Klainerman
• 金额: $ 27万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-09-01 至 2021-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1800841&HistoricalAwards=false
• 关键词: Mathematical; Theory; Black; Holes

At the most basic level Black Holes are nothing but special, explicit, solutions of the Einstein equations of General Relativity. To have physical reality they have to possess properties which make them detectable to observations. Chief among them is stability to perturbations. This translates to a very difficult, deep and beautiful mathematical conjecture which is the main focus of the proposed research. The goal is to show that any small perturbation, at a given time, of a given black hole solution remains small for all later times. Thus all the important features of a unperturbed black hole are preserved by the perturbations.To solve the problem of stability of black holes one has to overcome a large range of difficult issues. The high non-linearity of the Einstein equations makes it hard to separate them; they are all interconnected. At the heart of all these difficulties is the issue of gauge. That means, roughly, that one cannot solve the stability problem without finding a specific coordinate system in which one can establish decay of the perturbations, i.e. to show that the perturbations converge to zero for large times. In collaboration with J. Szeftel in France the PI had proposed a new method to construct these coordinates based on the full covariance properties of the Einstein equations. The main methodology of the PI is to combine this new method, called Generally Covariant Modulation theory, with the the so called vector-field method.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.

在最基本的层面上，黑洞只不过是爱因斯坦广义相对论方程的特殊的、显式的解。为了具有物理真实性，它们必须具有使它们能够被观察到的特性。其中最主要的是对扰动的稳定性。这意味着一个非常困难、深刻和美丽的数学猜想，这是拟议研究的主要焦点。我们的目标是证明，在给定时间，给定黑洞解的任何微小扰动在以后的所有时间都保持很小。因此，不受扰动的黑洞的所有重要特征都被微扰所保留。要解决黑洞的稳定性问题，人们必须克服一系列困难的问题。爱因斯坦方程的高度非线性使人们很难将它们分开；它们都是相互关联的。所有这些困难的核心是衡量标准的问题。粗略地说，这意味着如果不找到一个特定的坐标系来确定扰动的衰减性，即证明扰动在很大程度上收敛到零，就不能解决稳定性问题。在与法国的J.Szeftel的合作下，国际和平研究所提出了一种新的方法，根据爱因斯坦方程的全部协方差特性来构造这些坐标。PI的主要方法是将这种新方法，通常称为协变调制理论，与所谓的矢量场方法相结合。该奖项反映了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。