FRG: Mathematical Theory of Gravitational Collapse in General Relativity

FRG：广义相对论中引力塌缩的数学理论

• 批准号: 1065710
• 负责人: Sergiu Klainerman
• 金额: $ 147.05万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1065710&HistoricalAwards=false
• 关键词: FRG; Mathematical; Theory; Gravitational; Collapse

Gravitational collapse is a central problem in General Relativity intimately tied, mathematically, to the issue of the long time behavior of general solutions to the Einstein field equations. The problem can be neatly captured in the so called final state conjecture (FSC): generic asymptotically flat initial data sets have maximal future developments, namely solutions of the Einstein vacuum equations, which look, asymptotically, in any finite region of space, as a Kerr black hole solution. The focused research effort described in this proposal identifies four related problems whose solution will greatly advance our understanding of the FSC. The first three, 1) Uniqueness and 2) Stability of the Kerr solutions and 3) Formation of black holes, are at the heart of the theory of black holes. The last one, 4) Bounded L2 curvature conjecture and break-down criterions (the problem of evolution for very rough initial conditions and optimal sufficient conditions which insure regularity of the space-time), is a topic loosely connected with the celebrated cosmic censorship conjecture, which is itself a necessary ingredient, and a formidable intellectual obstacle, for resolving the FSC.Each of the four related problems identified above are, in themselves, very difficult and deep challenges in general relativity which have witnessed a lot of progress in recent years based on new analytical and geometric ideas. Furthermore, recent advances in numerical relativity have allowed new classes of solutions to the field equations to be obtained in highly non-trivial situations. Advances toward solving the FSC will require continued development within individual methodologies, though we plan to accelerate progress via a closer interaction amongst the different approaches. Cooperation between mathematical and numerical relativity can substantially help the former by endowing it with a powerful experimental tool and also help the latter to formulate key questions connected with the development of new numerical codes. To be successful, this will require training a new class of researcher, proficient in both the relevant formal mathematics and high performance scientific computing. We will engage graduate students and postdoctoral scholars in this effort, giving them the skills necessary to have a strong impact in their future careers, whether in academia or the broader work force. The research carried out will strengthen the foundations of general relativity and our understanding of black holes, which is of significant import to the broader nascent scientific field of gravitational wave astronomy.

引力坍缩是广义相对论中的一个中心问题，在数学上与爱因斯坦场方程的一般解的长时间行为问题密切相关。这个问题可以在所谓的终态猜想（FSC）中巧妙地捕捉到：一般的渐近平坦的初始数据集具有最大的未来发展，即爱因斯坦真空方程的解，在任何有限的空间区域中，渐近地看起来都是克尔黑洞的解。本提案中所描述的重点研究工作确定了四个相关问题，其解决方案将大大促进我们对FSC的理解。前三个，1）克尔解的唯一性和2）克尔解的稳定性和3）黑洞的形成，是黑洞理论的核心。第四部分，4）有界L2曲率猜想及其失效准则（非常粗糙的初始条件和最佳充分条件的演化问题，确保时空的规则性），是一个与著名的宇宙监督猜想松散联系的话题，这本身就是一个必要的成分，也是一个可怕的智力障碍，上述四个相关问题中的每一个本身都是广义相对论中非常困难和深刻的挑战，近年来，基于新的分析和几何思想，广义相对论取得了很大的进展。此外，最近的进展，在数值相对论允许新的类的解决方案，以获得在高度非平凡的情况下的场方程。解决FSC的进展将需要在各个方法中继续开发，尽管我们计划通过不同方法之间更密切的互动来加速进展。数学相对论和数值相对论之间的合作可以大大帮助前者，赋予它一个强大的实验工具，也可以帮助后者制定与新的数字代码的发展有关的关键问题。要取得成功，这将需要培训一批新的研究人员，精通相关的正规数学和高性能科学计算。我们将让研究生和博士后学者参与这项工作，为他们提供必要的技能，使他们在未来的职业生涯中产生强大的影响，无论是在学术界还是更广泛的劳动力。开展的研究将加强广义相对论的基础和我们对黑洞的理解，这对引力波天文学这一更广泛的新兴科学领域具有重要意义。