The wave equation near flat Friedmann–Lemaître–Robertson–Walker and Kasner Big Bang singularities

Abstract

We consider the wave equation, g ψ = 0, in fixed flat Friedmann-Lemaître-Robertson-Walker and Kasner spacetimes with topology R + × T 3 . We obtain generic blow up results for solutions to the wave equation towards the Big Bang singularity in both backgrounds. In particular, we characterize open sets of initial data prescribed at a spacelike hypersurface close to the singularity, which give rise to solutions that blow up in an open set of the Big Bang hypersurface {t = 0}. The initial data sets are characterized by the condition that the Neumann data should dominate, in an appropriate L 2 -sense, up to two spatial derivatives of the Dirichlet data. For these initial configurations, the L 2 (T 3 ) norms of the solutions blow up towards the Big Bang hypersurfaces of FLRW and Kasner with inverse polynomial and logarithmic rates respectively. Our method is based on deriving suitably weighted energy estimates in physical space. No symmetries of solutions are assumed.

Contracting asymptotics of the linearized lapse-scalar field sub-system of the Einstein-scalar field equations

“…(3.17) and (3.18) for asymptotic data (ϕ (0) , ϕ (1) ) = Ψ −1 (u * , ϕ * ). On the other hand, it states that for any choice of asymptotic data (ϕ (0) , ϕ (1) ) there exists a solution of the Cauchy problem for Eqs. (1) ) which realizes these asymptotic data via Eqs.…”

“…It can therefore be interpreted as the asymptotic representation of the full degrees of freedom of the solution set as it contains both the first asymptotic datum, ϕ (0) , associated with the t-power 0, and the second asymptotic datum, ϕ (1) , associated with the t-power 2A 2 . Hence it does make sense to call (ϕ (0) , ϕ (1) ) asymptotic data. Observe that these two quantities are associated with the following limits of an arbitrary solution:…”

“…First we hope to be able to choose the background fieldsφ(t, x),α(t, x),π(t, x) in Eqs. (2.10) -(2.12) to carry the main singular behavior of solutions of the lapse-scalar field system in the sense that the functions f (1)…”

“…(4.14), we can find a unique solution v 2 of the σ 2 -version such that Eq. (4.2) holds by matching both v (0) and v (1) in Eq. (4.15) (one can prove that this can be done in this case).…”

“…(1.2) of the lapse-scalar field equations. This feature is not seen, for example in the study of wave equations on cosmological background spacetimes in [1], due to the lack of a coupled elliptic equation in that setting. We expect that subsystem Eqs.…”