# Stability Within $T^2$-Symmetric Expanding Spacetimes

## TL;DR

This paper extends the understanding of the stability and asymptotic behavior of $T^2$-symmetric Einstein solutions by proving a nonpolarised analogue with broader applicability, supported by numerical simulations.

## Contribution

It introduces a nonpolarised analysis of $T^2$-symmetric Einstein flows, generalizing previous polarised results and revealing the instability of polarised asymptotics.

## Key findings

- Similar decay rates for normalized energy in the broader class
- Existence of a locally attractive set outside the main theorem's scope
- Polarised asymptotics are shown to be unstable

## Abstract

We prove a nonpolarised analogue of the asymptotic characterization of $T^2$-symmetric Einstein Flow solutions completed recently by LeFloch and Smulevici. In this work, we impose a condition weaker than polarisation and so our result applies to a larger class. We obtain similar rates of decay for the normalized energy and associated quantities for this class. We describe numerical simulations which indicate that there is a locally attractive set for $T^2$-symmetric solutions not covered by our main theorem. This local attractor is distinct from the local attractor in our main theorem, thereby indicating that the polarised asymptotics are unstable.