This project strives to better understand two basic predictions of Einstein's theory of gravity: black holes and spacetime singularities. The first major objective to be addressed is to resolve the question of whether the simplest examples of black holes are indeed stable objects as observed from the outside. If they are not, then much of our astrophysical world-picture will need to be revised. The second major objective concerns understanding what generic black holes look like from the inside. It has long been expected that observers entering black hole regions eventually encounter "ferocious singularities", where the very structure of spacetime breaks down. The project will strive to show, based in part on previous work of the proposer, that a portion of this singularity is not as "ferocious" as previously thought. Both the above objectives can be formulated as conjectures in pure mathematics concerning the governing equations of general relativity, the celebrated Einstein equations, and thus can be addressed by modern techniques of mathematical analysis. This project will thus contribute to this long tradition of rigorous mathematics making fundamental statements about our physical world. The subject of black holes and spacetime singularities has inspired the popular imagination well beyond the confines of the scientific community, and work on this proposal will strengthen the ties between scientific research and the general public.
At a more technical level, the first objective of the proposal concerns the dynamic stability of the Kerr metrics, a two parameter family of explicit solutions to the celebrated Einstein equations that represent the simplest black hole solutions. A series of projects is outlined, of increasing difficulty, culminating in a complete resolution of the Kerr stability conjecture in the fully nonlinear setting, the statement that small perturbations of Kerr initial states stay close to the Kerr family in the black hole exterior and asymptote back to it as time goes to infinity. The second major objective of this proposal concerns the analysis of black hole interiors. The original expectation was that generic black holes would terminate in a spacelike singular boundary where the geometry of spacetime is completely destroyed. Previous work of the proposer in a spherically symmetric toy problem, itself based on previous numerical work in the physics literature, showed that singularities do indeed occur, but that (at least) a portion of the singular boundary is null, not spacelike, and far less "singular" than expected. Through a series of projects of increasing complexity, this proposal aims to definitively resolve this question, now not for a toy model, but for the vacuum Einstein equations themselves, without any symmetry assumptions. In addition to resolving well-known fundamental conjectures about this venerable set of equations, this project strives to introduce new techniques of mathematical analysis which have the potential to influence more general developments in the hyperbolic partial differential equations governing classical physics.
