Session Q01: Progress and Challenges in Analytic Gravity

Strong Cosmic Censorship and the Stability of AVTD Behavior in Cosmological Space-Times

The Strong Cosmic Censorship Conjecture (SCCC) suggests that the “singularities” in cosmological space-times are generically characterized by unbounded curvature so that consequently these space-times generically cannot be extended beyond their maximal domains of dependence. One of the most effective tools for proving SCCC holds for a family of cosmological solutions of Einstein’s equations is to verify that these solutions all exhibit Asymptotically Velocity Term Dominated (AVTD) behavior in a neighborhood of their singularities. After presenting some background history of known results and conjectures concerning the behavior of the gravitational field near the singularity in cosmological space-times, we discuss recent work done with collaborators, as well as other recent results by others which show that AVTD behavior does occur in an open set of solutions within certain families in a neighborhood of the Kasner solutions.   

Emergence of Spacetime from the Algebra of Modular Hamiltonians

Modular flow, generated by a modular Hamiltonian, is an operation in quantum field theory that preserves a given subregion.  In a holographic setting modular flow on the boundary preserves a dual subregion of the bulk space-time.  Modular flow leaves the boundary of this subregion fixed, thus a point in the bulk corresponds to a family of modular Hamiltonians whose fixed surfaces all intersect at the bulk point.  This family defines a maximal subalgebra within the algebra Η generated by all modular Hamiltonians.  This suggests an algebraic approach to bulk reconstruction, in the spirit of non-commutative geometry: the bulk spacetime can be recovered as the space of maximal subalgebras of Η.   

Stable Big Bang Formation in General Relativity: The Sub-Critical Regime

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. Despite the uncertainty, 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 cosmological Big Bang singularity, that is, curvature blowup along an entire spacelike hypersurface. In recent joint work with G. Fournodavlos and I. Rodnianski, we have given a rigorous proof of this phenomenon, i.e., a proof of stable monotonic curvature blowup along a spacelike hypersurface in the complete regime where the heuristics suggest it might occur. 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 works of Dafermos-Luk on the stability of Kerr Cauchy horizons and of Van de Moortel on the structure of singularities inside black holes. 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.