Blow-up of waves on singular spacetimes with generic spatial metrics
Abstract
We study the asymptotic behaviour of solutions to the linear wave equation on cosmological spacetimes with Big Bang singularities and show that appropriately rescaled waves converge against a blow-up profile. Our class of spacetimes includes Friedman-Lema\^itre-Robertson-Walker (FLRW) spacetimes with negative sectional curvature that solve the Einstein equations in presence of a perfect irrotational fluid with . As such, these results are closely related to the still open problem of past nonlinear stability of such FLRW spacetimes within the Einstein scalar field equations. In contrast to earlier works, our results hold for spatial metrics of arbitrary geometry, hence indicating that the matter blow-up in the aforementioned problem is not dependent on spatial geometry. Additionally, we use the energy estimates derived in the proof in order to formulate open conditions on the initial data that ensure a non-trivial blow-up profile, for initial data sufficiently close to the Big Bang singularity and with less harsh assumptions for .
Keywords
Cite
@article{arxiv.2110.14999,
title = {Blow-up of waves on singular spacetimes with generic spatial metrics},
author = {David Fajman and Liam Urban},
journal= {arXiv preprint arXiv:2110.14999},
year = {2022}
}
Comments
24 pages. Two paragraphs in the introduction have been updated to more accurately reflect the relationship of our results and approach to the surrounding literature, in particular to [14] and [15]. As in the previous version, the paper also includes a pointwise blow-up condition for $\gamma=2$ and we discuss its strengths and weaknesses compared to the other conditions