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-Lemaître-Robertson-Walker (FLRW) spacetimes with negative sectional curvature that solve the Einstein equations in presence of a perfect irrotational fluid with $p=(\gamma -1)\rho$. 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 $\gamma <2$.

arXiv comment24 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

References in corpus (8)

• A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation
• Stable Big Bang formation for Einstein's equations: The complete sub-critical regime
• The maximal development of near-FLRW data for the Einstein-scalar field system with spatial topology ${\mathbb{S}}^3$
• Wave equations on silent big bang backgrounds
• A unified approach to the Klein-Gordon equation on Bianchi backgrounds
• Solutions of the wave equation bounded at the Big Bang
• Wave asymptotics at a cosmological time-singularity: classical and quantum scalar fields
• Asymptotics of linearized cosmological perturbations

Cited by in corpus (2)

• Decay of solutions of the wave equation in cosmological spacetimes -- a numerical analysis
• Cosmic Censorship near FLRW spacetimes with negative spatial curvature