Quiescent Big Bang Singularities

QC 2026-10-22

Abstract

This thesis is about the rigorous mathematical study of solutions to Einstein's equations of general relativity that exhibit Big Bang singularities. We focus on quiescent singularities. Roughly speaking, this refers to solutions whose asymptotics are convergent towards the singularity. In order to study them, we take two different perspectives. The first one is the singular initial value problem. This means that we start by prescribing geometric initial data at the Big Bang, and then construct a corresponding solution. The second one is the dynamical formation of singularities. That is, starting from initial data on a Cauchy surface, we look for conditions that ensure that the corresponding solution exhibits Big Bang formation. An important aspect of our results is that we do not make any symmetry assumptions, and we do not assume proximity to any fixed background solutions.

Paper A is not directly concerned with the study of Big Bang singularities, but it served as a training ground for some of the recurring ideas of the thesis. In it, we study a special class of systems of linear wave equations on cosmological spacetimes, called silent equations. We obtain asymptotics of all orders for the solutions, and show that solutions are uniquely determined by the asymptotic data. Then we apply the results to study the source free Maxwell's equations on Kasner spacetimes near the initial singularity.

In Paper B we prove that the Einstein-nonlinear scalar field equations have a geometric singular initial value problem formulation. That is, given geometric initial data on a Big Bang singularity, we show the existence of a solution to the equations that realizes the given initial data. Moreover, we show that this solution is unique within the class of spacetimes under consideration.

In Paper C we go in the opposite direction. We identify a general condition on initial data on a Cauchy surface that ensures that the corresponding development has a quiescent Big Bang, and that it induces geometric initial data on the singularity.

An issue with the results of Paper C is that they cannot be localized in space in a straightforward way. In Paper D, we solve this issue by introducing a new gauge for Einstein's equations, which allows us to prove a localized version of the results of Paper C. Specifically, we identify a general condition on initial data that ensures that the corresponding solution has a quiescent Big Bang to the past of a point on the initial hypersurface.

Link to DiVA