Einstein-标量场系统的线性稳定性及其在非线性大爆炸形成中的应用
A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation
摘要：
我们线性化的爱因斯坦-标量场方程，表示相对于恒定中曲率（CMC）传输的空间坐标规范，周围的成员著名的家庭Kasner解决方案$(0,\infty) \times \mathbb{T}^3$。卡斯纳解模拟了一个空间均匀的标量场在一个（典型的）空间各向异性时空中的演化，这个时空向未来扩展，并且在$\lbrace t = 0 \rbrace$有一个“大爆炸”奇点。我们将线性化系统的初始数据沿$\lbrace t = 1 \rbrace \simeq \mathbb{T}^3$放置，并研究线性解在$t \downarrow 0$坍缩方向上的行为。我们的第一个主要结果是证明了线性解的近似$L^2$单调性恒等式。使用它，我们证明了一个线性稳定性的结果，持有时，背景Kasner解决方案是足够接近的Friedmann-Lematre-Robertson-Walker（FLRW）解决方案。特别是，我们表明，作为$t \downarrow 0$，各种时间重新缩放组件的线性解收敛到沿$\lbrace t = 0 \rbrace$定义的正则函数。此外，我们激励的首选方向的近似单调性表明，CMC运输的空间坐标规范可以被看作是一个限制版本的一个家庭的抛物线规范的推移变量;近似单调性的身份和相应的线性稳定性的结果也保持在抛物线规范，但相应的抛物偏微分方程是局部适定的方向$t \downarrow 0$。最后，基于线性稳定性的结果，我们概述了以下结果的证明，其完整的证明将出现在其他地方：FLRW解决方案是全局非线性稳定的崩溃方向$t \downarrow 0$在小扰动下，其数据在$\lbrace t = 1 \rbrace$。
英文摘要：
We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on $(0,\infty) \times \mathbb{T}^3$. The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a "Big Bang" singularity at $\lbrace t = 0 \rbrace$. We place initial data for the linearized system along $\lbrace t = 1 \rbrace \simeq \mathbb{T}^3$ and study the linear solution's behavior in the collapsing direction $t \downarrow 0$. Our first main result is the proof of an approximate $L^2$ monotonicity identity for the linear solutions. Using it, we prove a linear stability result that holds when the background Kasner solution is sufficiently close to the Friedmann-Lema\^ıtre-Robertson-Walker (FLRW) solution. In particular, we show that as $t \downarrow 0$, various time-rescaled components of the linear solution converge to regular functions defined along $\lbrace t = 0 \rbrace$. In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMC-transported spatial coordinates gauge can be viewed as a limiting version of a family of parabolic gauges for the lapse variable; an approximate monotonicity identity and corresponding linear stability results also hold in the parabolic gauges, but the corresponding parabolic PDEs are locally well-posed only in the direction $t \downarrow 0$. Finally, based on the linear stability results, we outline a proof of the following result, whose complete proof will appear elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing direction $t \downarrow 0$ under small perturbations of its data at $\lbrace t = 1 \rbrace$.
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