Stability of Black Holes and the Nature of Singularities in General Relativity

黑洞时空可能是爱因斯坦广义相对论中最令人震惊的预言。这些时空区域是如此密集，以至于它们产生的引力场如此强大，以至于绝对没有东西可以逃脱。尽管爱因斯坦最初怀疑它们的存在，但它们最终被理论物理界接受，2015年，来自两个黑洞合并的引力波直接(！)由LIGO合作观察(导致2017年诺贝尔奖)。尽管黑洞看似自相矛盾和可怕，但实际上它们在广义相对论中扮演着非常重要的簿记角色。也就是说，从这门学科的最初几年起，人们就知道爱因斯坦方程允许发展出严重的奇点，其中时空的意义完全被打破。然而，在这些早期的例子中，奇点总是隐藏在黑洞内部，既然没有任何东西能从黑洞中逃脱，只要我们不进入黑洞，我们就可以继续求解爱因斯坦方程，而忽略时空破裂带来的问题。在这个背景下，黑洞的出现是如此的偶然，以至于彭罗斯认为这可能不是偶然的，并做出了他著名的“宇宙审查猜想”(CCC)，即一般而言，广义相对论中形成的所有奇点总是隐藏在黑洞中。由于没有任何明显的反例，CCC最终达成了广泛的共识，并奠定了我们对经典广义相对论的所有理解的基础。然而，对CCC坚持的原因缺乏真正的理解，仍然是我们对广义相对论理解的一个明显差距，并阻碍了该理论在某些基本问题上的进展。第一批甚至给出了为什么CCC可能成立的启发式原因的著作是Christodoulou为某一模型系统撰写的一系列论文，其中Christodoulou构造了“裸奇点”(不隐藏在黑洞中的奇点)，但随后表明CCC一般适用于该模型。我的建议的第一部分以Christodoulou的工作为起点，并开始讨论爱因斯坦方程的类似结果。我们调查赤裸裸的奇点形成的例子，并理解为什么它们实际上不是通用的。克里斯托杜卢的许多技巧都与他研究的特定模型密切相关。我的建议的另一部分是开发与这些技术类似的技术，这些技术有望在爱因斯坦方程的一般背景下发挥作用。最后，我的提议的最后一部分，建立在许多其他数学家和物理学家的工作的基础上，试图在理解黑洞的稳定性属性方面取得进展。黑洞的稳定性对于充分理解它们在CCC中的作用将是至关重要的。

Black hole spacetimes are perhaps the most shocking prediction of Einstein's theory of General Relativity. These are regions of spacetime which are so dense that they produce a gravitational field so strong that absolutely nothing can escape. Though Einstein himself originally doubted their existence, they were eventually accepted by the theoretical physics community and, in 2015, gravitational waves from a merger of two black holes were directly(!) observed by the LIGO collaboration (leading to the 2017 Nobel Prize). Despite their seemingly paradoxical and scary nature, black holes actually serve a very important bookkeeping role in General Relativity. Namely, from the very first years of the subject, it has been known that the Einstein equations permit the development of severe singularities where the very meaning of spacetime completely breaks down. However, in these early examples, the singularities always turned out to be hidden inside a black hole, and since nothing can escape from the black hole, as long as we do not enter the black hole, we could continue to solve Einstein's equation and ignore the problem posed by the spacetime breakdown. The appearance of the black holes in this setting was so fortuitous that Penrose decided it was likely no accident and made his famous "Cosmic Censorship Conjecture" (CCC) that, generically, all singularities which formed in General Relativity would always be hidden inside a black hole. Due to the lack of any apparent counter-example, the CCC eventually achieved widespread consensus and came to underlie all of our understanding of classical General Relativity. However, the lack of a true understanding of why the CCC holds has remained a glaring gap in our understanding of General Relativity and prevents progress on certain foundational problems with the theory. The first works which gave even a heuristic reason as to why the CCC may hold were a sequence of papers written by Christodoulou for a certain model system, where Christodoulou constructed "naked singularities" (singularities which are not hidden inside black holes) but then showed that generically the CCC holds for this model. The first part of my proposal takes Christodoulou's work as a starting point, and begins to address the analogous results for the Einstein equations. We investigate examples of naked singularity formation and understand why they are, in fact, not generic. Many of Christodoulou's techniques are intimately related to the specific model he studied. Another part of my proposal develops analogues of these techniques that can be expected to work in the general context of the Einstein equations. Finally, the last part of my proposal, building on the work of many other mathematicians and physicists, seeks to make progress on understanding stability properties of black holes. The stability of black holes will be of fundamental importance to fully understanding their role in the CCC.