爱因斯坦-斯卡拉场方程的局部大爆炸稳定性

IF 2.6 1区 数学 Q1 MATHEMATICS, APPLIED
Florian Beyer, Todd A. Oliynyk
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引用次数: 0

摘要

我们证明了爱因斯坦尺度场方程的弗里德曼-勒梅特-罗伯逊-沃克(Friedmann-Lemaître-Robertson-Walker,FLRW)解在(n\ge 3\) 时空中的收缩方向上的非线性稳定性,这些解定义在形式为\((0,t_0]\times \mathbb {T}{}^{n-1}\),\(t_0>0\) 的时空流形上。稳定性是在初始数据同步的假设下建立的,这意味着在初始超曲面 \(\Sigma = \{t_0\}\times \mathbb {T}{}^{n-1}\) 上,标量场 \(\tau = \exp \bigl (\sqrt{frac{2(n-2)}{n-1}}\phi \bigr )\)是常数,也就是说,(\Sigma =\tau ^{-1}(\{t_0\})\).由于我们证明了所有与 FRLW 足够接近的初始数据集都可以通过爱因斯坦-标量场方程演化成同步的新的初始数据集,所以这一假设并没有失去一般性。通过使用 \(\tau\) 作为时间坐标,我们确定了扰动 FLRW 时空流形的形式为 \(M = \bigcup _{t\in (0,t_0]}\tau ^{-1}(\{t\})\cong (0,t_0]\times \mathbb {T}{}^{n-1}\)、扰动的FLRW解在\(\tau \searrow 0\) 时是渐近点的卡斯纳(Kasner),而在\(\tau =0\)时会出现一个大爆炸奇点,其特征是标量曲率的膨胀。我们过去的稳定性证明的一个重要方面是,我们使用了爱因斯坦-标量场方程的双曲规减。因此,稳定性证明中使用的所有估计都是局部的,我们利用这一特性为FLRW解建立了相应的局部过去稳定性结果。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Localized Big Bang Stability for the Einstein-Scalar Field Equations

We prove the nonlinear stability in the contracting direction of Friedmann–Lemaître–Robertson–Walker (FLRW) solutions to the Einstein-scalar field equations in \(n\ge 3\) spacetime dimensions that are defined on spacetime manifolds of the form \((0,t_0]\times \mathbb {T}{}^{n-1}\), \(t_0>0\). Stability is established under the assumption that the initial data is synchronized, which means that on the initial hypersurface \(\Sigma = \{t_0\}\times \mathbb {T}{}^{n-1}\), the scalar field \(\tau = \exp \bigl (\sqrt{\frac{2(n-2)}{n-1}}\phi \bigr ) \) is constant, that is, \(\Sigma =\tau ^{-1}(\{t_0\})\). As we show that all initial data sets that are sufficiently close to FRLW ones can be evolved via the Einstein-scalar field equation into new initial data sets that are synchronized, no generality is lost by this assumption. By using \(\tau \) as a time coordinate, we establish that the perturbed FLRW spacetime manifolds are of the form \(M = \bigcup _{t\in (0,t_0]}\tau ^{-1}(\{t\})\cong (0,t_0]\times \mathbb {T}{}^{n-1}\), the perturbed FLRW solutions are asymptotically pointwise Kasner as \(\tau \searrow 0\), and a big bang singularity, characterised by the blow up of the scalar curvature, occurs at \(\tau =0\). An important aspect of our past stability proof is that we use a hyperbolic gauge reduction of the Einstein-scalar field equations. As a consequence, all of the estimates used in the stability proof can be localized and we employ this property to establish a corresponding localized past stability result for the FLRW solutions.

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来源期刊
CiteScore
5.10
自引率
8.00%
发文量
98
审稿时长
4-8 weeks
期刊介绍: The Archive for Rational Mechanics and Analysis nourishes the discipline of mechanics as a deductive, mathematical science in the classical tradition and promotes analysis, particularly in the context of application. Its purpose is to give rapid and full publication to research of exceptional moment, depth and permanence.
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