Localized formation of quiescent big bang singularities

TL;DR

This paper proves a localized big bang formation result for Einstein-scalar field equations, introducing a new foliation method that allows for matter-independent analysis and detailed asymptotic descriptions.

Contribution

It presents a novel localized big bang formation theorem using a new foliation approach, independent of matter models, and provides geometric initial data on the singularity.

Findings

Established conditions for localized big bang formation in Einstein-scalar field equations.

Introduced a new foliation by level sets of a time function satisfying a second order differential equation.

Achieved a symmetric hyperbolic formulation that synchronizes the singularity and describes asymptotics.

Abstract

We prove a localized big bang formation result, which does not require proximity of the initial data to any background solution. Suppose that we are given initial data for the Einstein--nonlinear scalar field equations on an open set $U \subset \mathbb{R}^3$. We identify a general condition on the initial data such that if the condition is satisfied in a large enough neighborhood of $x \in U$, then the corresponding maximal globally hyperbolic development has a local quiescent big bang singularity with curvature blow up to the past of $x$. We achieve the localization by introducing a new kind of foliation by spacelike hypersurfaces, given by the level sets of a time function satisfying a certain second order differential equation. This time function allows us to synchronize the singularity while at the same time yielding a symmetric hyperbolic formulation of Einstein's equations. Our new formulation also has two key advantages over previous localized big bang stability results. First, it is independent of the matter model, so it is possible that it could be used to prove big bang formation with matter models different from a scalar field. And second, it allows us to conclude that our solutions induce geometric initial data on the singularity, thus giving a complete description of the asymptotics towards the big bang.