Quasilinear Hyperbolic Fuchsian Systems and AVTD Behavior in T 2-Symmetric Vacuum Spacetimes

Abstract

We set up the singular initial value problem for quasilinear hyperbolic Fuchsian systems of first order and establish an existence and uniqueness theory for this problem with smooth data and smooth coefficients (and with even lower regularity). We apply this theory in order to show the existence of smooth (generally not analytic) T 2 -symmetric solutions to the vacuum Einstein equations, which exhibit AVTD (asymptotically velocity term dominated) behavior in the neighborhood of their singularities and are polarized or half-polarized.

On the initial geometry of a vacuum cosmological spacetime

“…On the other hand, if we assume that A ⋆ is constant in θ, the half-polarized condition, then e 4U a 2 A 2 θ ∼ e −2τ and (2.66) holds. This is consistent with the finding in [1] that the half-polarized condition is needed for the Fuchsian method to work.…”

“…Then for some N ∈ Z + , Λ > 1, ǫ > 0, α ∈ (0, 1), U ⊂ X and x ′ ∈ U, there is a sequence of times {t i } ∞ i=1 with lim i→∞ t i = 0 so that the proposition fails for t = t i . In particular, for each i, the rescaled pointed flow E t i is not ǫ-close to a Lorentzian cone as described in case (1).…”

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

“…The Fuchsian method has been used in a number of cases [2,3,15,20,25,26,29] to iden-tify asymptotic degrees of freedom -asymptotic data -(often determined heuristically by formal expansions at the singularity) and then for rigorously validating this identification (by proving the existence of a solution of the equation which realizes each choice of asymptotic data in some well-defined sense). It typically yields continuous maps from asymptotic data to Cauchy data.…”

Second-order hyperbolic Fuchsian systems: Asymptotic behavior of geodesics in Gowdy spacetimes

“…(11), below] reduce to such an equation (in the socalled polarized case Q ¼ 0; see below). For any real positive constant , let…”

“…[11]) has been found to apply to a variety of problems arising in physics and, especially, in general relativity. In earlier work, we considered Gowdy spacetimes with spatial toroidal topology, and we applied the theory to the construction of the so-called asymptotically velocity-dominated solutions [12,13] of the Gowdy equations [14][15][16].…”