Quasilinear hyperbolic Fuchsian systems and AVTD behavior in T2-symmetric vacuum spacetimes

TL;DR

This paper develops a mathematical framework for solving certain hyperbolic systems near singularities and applies it to prove the existence of specific vacuum solutions in Einstein's theory that display AVTD behavior near singularities.

Contribution

It establishes a new existence and uniqueness theory for quasilinear hyperbolic Fuchsian systems and applies it to construct non-analytic T2-symmetric vacuum solutions with AVTD behavior.

Findings

Existence of smooth T2-symmetric vacuum solutions with AVTD behavior.

Solutions exhibit polarized or half-polarized singularities.

The theory applies to systems with low regularity data.

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) T2-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.