Stability of AVTD Behavior within the Polarized $T^2$-symmetric vacuum spacetimes

TL;DR

This paper proves the nonlinear stability of certain Kasner solutions in polarized $T^2$-symmetric vacuum spacetimes, showing that small perturbations lead to AVTD behavior and curvature blow-up in the contracting universe.

Contribution

It establishes the nonlinear stability of Kasner solutions with specific parameters within polarized $T^2$-symmetric vacuum spacetimes, extending understanding of their asymptotic behavior.

Findings

Kasner solutions with |K-1|>2 are non-linearly stable.

Small perturbations exhibit AVTD behavior.

Kretschmann scalar blows up near singularity.

Abstract

We prove stability of the family of Kasner solutions within the class of polarized $T^2$-symmetric solutions of the vacuum Einstein equations in the contracting time direction with respect to an areal time foliation. All Kasner solutions for which the asymptotic velocity parameter $K$ satisfies $|K-1|>2$ are non-linearly stable, and all sufficiently small perturbations exhibit asymptotically velocity term dominated (AVTD) behavior and blow-up of the Kretschmann scalar.