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Stability of AVTD behavior Within the polarized \(\mathbb{T}^2\)-symmetric vacuum spacetimes. (English) Zbl 1498.83002

Summary: We prove stability of the family of Kasner solutions within the class of polarized \(\mathbb{T}^2\)-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.

MSC:

83C05 Einstein’s equations (general structure, canonical formalism, Cauchy problems)
35Q75 PDEs in connection with relativity and gravitational theory
83F05 Relativistic cosmology
53C12 Foliations (differential geometric aspects)
35B44 Blow-up in context of PDEs

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