FRG: Mathematical Theory of Gravitational Collapse in General Relativity

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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.