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具有宇宙学常数的极化$T^2$ -对称真空解的渐近稳定性

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

Ellery Ames, Florian Beyer, James Isenberg, Todd Oliynyk

备注：
20 pages
发表信息：
Phil. Trans. R. Soc. A 380: 20210173 (2022)
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摘要：

本文证明了具有任意宇宙学常数$Λ$的真空Einstein方程的极化$T^2$对称解类中Kasner解的子族在收缩时间方向上扰动的渐近行为的非线性稳定性。这个稳定性结果推广了[3]中证明的结果，[3]中的结果集中在$Λ=0$的情况下，并且在那篇文章中，证明依赖于面积时间叶理和Fuchsian技术。甚至对于$Λ=0$，这里建立的结果也适用于极化$T^2$对称真空解族中Kasner解的更广泛的扰动类，而不是[3]和[26]中考虑的那些。我们的研究结果表明，对于某些$T_0 > 0$，对于某些具有宇宙学常数的极化$T^2$对称解族，面积时间坐标取$(0, T_0]$中的所有值.

英文摘要：

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 $Λ$. This stability result generalizes the results proven in [3], which focus on the $Λ=0$ case, and as in that article, the proof relies on an areal time foliation and Fuchsian techniques. Even for $Λ=0$, 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.

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