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

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.

28 pages. Fixed minor typos and made several clarifications. Agrees with published version

References in corpus (6)

• Analytic description of singularities in Gowdy spacetimes
• Asymptotic Behavior in Polarized {\bf T}${}^2$-symmetric Vacuum Spacetimes
• Stabilizing relativistic fluids on spacetimes with non-accelerated expansion
• Future global stability for relativistic perfect fluids with linear equations of state $p=K\rho$ where $1/3<K<1/2$
• Areal Foliation and AVTD Behavior in T^2 Symmetric Spacetimes with Positive Cosmological Constant
• Kasner-like regions near crushing singularities

Cited by in corpus (6)

• Stable Big Bang formation for Einstein's equations: The complete sub-critical regime
• Localized big bang stability for the Einstein-scalar field equations
• Stress and Geometry for Isotropic Singularities
• Stability of Asymptotic Behavior Within Polarised $T^2$-Symmetric Vacuum Solutions with Cosmological Constant
• Well-posedness of anisotropic and homogeneous solutions to the Einstein-Boltzmann system with a conformal-gauge singularity
• Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability