Singularities and Black Holes in General Relativity

广义相对论最令人震惊的预言之一是黑洞的存在；时空区域如此密集，以至于任何东西，甚至光，都无法逃脱。除了拥有迷人的物理特性外，黑洞在我们对天体物理学的现代理解中起着至关重要的作用。例如，预计任何经历引力坍缩的物体最终都会沉降到黑洞中，而且，人们认为黑洞可以保护我们免受潜在危险的引力奇点的影响。然而，这两个对我们的宇宙观如此重要的期望，都缺乏明确的数学理解。这个项目旨在促进我们对黑洞稳定性问题和“裸奇点”的潜在存在的理解，“裸奇点”是指没有隐藏在黑洞后面的奇异解。在过去十年中，在一般黑洞背景下建立模型方程的稳定性结果方面取得了巨大进展。与此同时，对于一类特殊的非旋转黑洞的真正稳定性问题的理解也取得了很大的进展。我们研究计划的一部分是将这两个研究方向结合起来，最终为旋转黑洞建立一个明确的稳定性理论。我们研究计划的另一部分是证明确实存在不隐藏在黑洞后面的奇异时空；然而，我们也将表明，在轻微的扰动下，这些特定的时空变得不稳定并发展成黑洞。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

One of the most shocking predictions of general relativity is the existence of black holes; regions of spacetime so dense that nothing, not even light, can escape. In addition to possessing fascinating physical properties, black holes play an essential role in our modern understanding of astrophysics. For example, it is expected that any object undergoing gravitational collapse eventually settles down to a black hole, and also, it is thought that black holes serve to shield us from potentially dangerous gravitational singularities. However, both of these expectations, so fundamental to our view of the universe, lack a definitive mathematical understanding. This project aims to advance our understanding of both the problems of black hole stability and the potential existence of "naked singularities", singular solutions which are not hidden away behind a black hole.In the last decade there has been a tremendous progress in establishing stability results for model equations on general black hole backgrounds. At the same time, there has been much progress in understanding the true stability problem for the special class of non-rotating black holes. One part of our research plan is to combine these two research directions and to eventually establish a definitive stability theory for rotating black holes. Another part of our research plan is to show that there do in fact exist singular spacetimes which are not hidden behind black holes; however, we will also show that upon a slight perturbation, these particular spacetimes become unstable and develop a black hole.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.