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

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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.