Caltech/UCLA Joint Analysis Seminar
The celebrated theorems of Hawking and Penrose show that under appropriate assumptions on the matter model, a large, open set of initial data for Einstein's equations lead to geodesically incomplete solutions. However, these theorems are "soft" in that they do not yield any information about the nature of the incompleteness, leaving open the possibilities that i) it is tied to the blowup of some invariant quantity (such as curvature) or ii) it is due to a more sinister phenomenon, such as incompleteness due to lack of information for how to uniquely continue the solution (this is roughly known as the formation of a Cauchy horizon). Despite the "general ambiguity," in the mathematical physics literature, there are heuristic results, going back 50 years, suggesting that whenever a certain "sub-criticality" condition holds, the Hawking–Penrose incompleteness is caused by the formation of a Big Bang singularity, that is, curvature blowup along an entire spacelike hypersurface. In recent joint work with I. Rodnianski and G. Fournodavlos, we have given a rigorous proof of the heuristics. More precisely, our results apply to Sobolev-class perturbations – without symmetry – of generalized Kasner solutions whose exponents satisfy the sub-criticality condition. Our main theorem shows that – like the generalized Kasner solutions – the perturbed solutions develop Big Bang singularities. In this talk, I will provide an overview of our result and explain how it is tied to some of the main themes of investigation by the mathematical general relativity community, including the remarkable work of Dafermos–Luk on the stability of Kerr Cauchy horizons. I will also discuss the new gauge that we used in our work, as well as intriguing connections to other problems concerning stable singularity formation.