Asymptotically Kasner-like singularities

Abstract

We prove existence, uniqueness and regularity of solutions to the Einstein vacuum equations taking the formx , where a ij (t, x) and p i (x) are regular functions without symmetry or analyticity assumptions. These metrics are singular and asymptotically Kasner-like as t → 0 + . These solutions are expected to be highly non-generic, and our construction can be viewed as solving a singular initial value problem with Fuchsian-type analysis where the data are posed on the "singular hypersurface" {t = 0}. This is the first such result without imposing symmetry or analyticity.To carry out the analysis, we study the problem in a synchronized coordinate system. In particular, we introduce a novel way to perform (weighted) energy estimates in such a coordinate system based on estimating the second fundamental forms of the constant-t hypersurfaces.

Wave equations on silent big bang backgrounds

“…Assume, additionally, that there are constants δ q and q > 0 such that [q(•, t) − (n − 1)] C 0 ( M ) ≤ δ q e q τ (t) (4. 19) for all t ≤ t 0 . Let KG := min{ q , ε Sp } and u be a solution to (4.18).…”

“…In [4,15], results are derived in these contexts in the class of real analytic solutions, using Fuchsian techniques. Two more recent results on specifying data on the singularity are [3,19]. The results of [19] (cf.…”

“…Two more recent results on specifying data on the singularity are [3,19]. The results of [19] (cf. also [27]) are of particular importance, in that they apply to the Einstein vacuum equations in 3 + 1-dimensions in the absence of symmetries.…”

“…the companion article [47] for a discussion. On the other hand, the results of [19] are in the C ∞ -setting.…”

“…This discrepancy is related to a methodological issue we expect to be of importance. In the results, such as [4,15,19], where the authors specify the asymptotics, the directions corresponding to maximal/minimal asymptotic contraction are given a priori. Knowing these directions is of central importance when proving the existence of solutions.…”