FRG: Mathematical Theory of Gravitational Collapse in General Relativity

引力坍缩是广义相对论的一个中心问题，在数学上与爱因斯坦场方程解的长期行为密切相关。这个问题可以巧妙地用所谓的终态猜想(FSC)来描述：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，它在空间的任何有限区域渐近地看起来是克尔黑洞解。这份提案中描述的重点研究工作确定了四个相关的问题，这些问题的解决将极大地促进我们对FSC的理解。前三个问题，1)克尔解的唯一性和2)稳定性，3)黑洞的形成，是黑洞理论的核心。最后一个问题，4)有界L2曲率猜想和分解准则(非常粗糙的初始条件和确保时空正则性的最佳充分条件的演化问题)，是一个与著名的宇宙审查猜想松散联系的话题，而宇宙审查猜想本身就是解决FSC的一个必要因素，也是一个巨大的智力障碍。上述四个相关问题中的每一个，本身就是广义相对论中非常困难和深刻的挑战，近年来，基于新的解析和几何思想，广义相对论取得了很大进展。此外，数值相对论的最新进展使我们能够在非常重要的情况下获得场方程解的新类别。虽然我们计划通过不同方法之间更紧密的互动来加快进展，但解决FSC的进展将需要在个别方法中继续发展。数学和数值相对性之间的合作可以为前者提供一个强大的实验工具，从而大大帮助前者，也有助于后者制定与新数字代码开发有关的关键问题。为了取得成功，这将需要培训一批新的研究人员，他们精通相关的形式数学和高性能科学计算。我们将吸引研究生和博士后学者参与这一努力，让他们掌握必要的技能，以便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在更广泛的劳动力大军中。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，这对更广泛的新生科学领域--引力波天文学--具有重要意义。

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.