Stability of Asymptotic Behavior Within Polarised $T^2$-Symmetric Vacuum Solutions with Cosmological Constant

TL;DR

This paper proves the nonlinear stability of asymptotic behaviors of perturbed Kasner solutions within polarised $T^2$-symmetric vacuum spacetimes with a cosmological constant, extending previous results to include non-zero $\Lambda$ and broader perturbations.

Contribution

It generalizes stability results for Kasner solutions to include arbitrary cosmological constants and a wider class of perturbations within polarised $T^2$-symmetric vacuum solutions.

Findings

Areal time coordinate covers the entire interval $(0, T_0]$ for certain solutions.

Stability of asymptotic behavior is established for perturbations with $\Lambda eq 0$.

Results extend previous $\Lambda=0$ stability results to more general settings.

Abstract

We prove the nonlinear stability of the asymptotic behavior of perturbations of subfamilies of Kasner solutions in the contracting time direction within the class of polarised $T^2$-symmetric solutions of the vacuum Einstein equations with arbitrary cosmological constant $\Lambda$. This stability result generalizes the results proven in [3], which focus on the $\Lambda=0$ case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for $\Lambda=0$, the results established here apply to a wider class of perturbations of Kasner solutions within the family of polarised $T^2$-symmetric vacuum solutions than those considered in [3] and [26]. Our results establish that the areal time coordinate takes all values in $(0, T_0]$ for some $T_0 > 0$, for certain families of polarised $T^2$-symmetric solutions with cosmological constant.