论文标题
爱因斯坦方程的稳定大爆炸形成：完整的亚临界体制
Stable Big Bang formation for Einstein's equations: The complete sub-critical regime
论文作者
论文摘要
For $(t,x) \in (0,\infty)\times\mathbb{T}^D$, the generalized Kasner solutions are a family of explicit solutions to various Einstein-matter systems that start out smooth but then develop a Big Bang singularity as $t \downarrow 0$, i.e., curvature blowup along a spacelike hypersurface.家族由Kasner指数$ \ widetilde {q} _1，\ cdots，\ widetilde {q} _d \ in \ mathbb {r} $进行参数化，这满足了两个代数约束。数学物理学文献中有启发式方法，可以追溯到50多年来，这表明在Kasner初始数据的扰动下，大爆炸形成应该是稳定的，只要指数在以下意义上是“亚临界”： $ \ mathop {\ max_ {i，j，b = 1，\ cdots，d}} _ {i <j} \ {\ widetilde {q} _i+\ widetilde {q} _jj- _ _j- \ widetilde {q}以前的作品已经显示了在更强的假设下奇异性的稳定性：1）Einstein-Scalar场系统具有$ d = 3 $和$ \ \ \ \ \ \\\\\\\\\\\ \ ai {q} _1 \ of wideTilde {q} _2 _2 _2 \ of bout \ bout \ widetilde {q} _3 \ _3 \ y egaum 39 $ with $ \ max_ {i = 1，\ cdots，d} | \ widetilde {q} _i | <1/6 $。我们证明，对于\ emph {all}亚临界卡斯纳指数，Kasner的奇异性在动态上是稳定的，从而在完整的策略中证明了稳定的单调单调型曲率爆炸的启发式。我们以$ D \ geq3 $和$ 1+d $ dimensional einstein-vacuum方程为$ d \ geq $ d $ d $ d $ d $ - 维度级赛车系统。此外，对于$ 1+3 $尺寸的爱因斯坦 - 维库姆方程式，一般而言，我们证明所有单数Kasner解决方案在两极分化的$ u（1）$ - 对称的初始数据下都具有稳定的大爆炸。除了我们在偏光$ u（1）$ - 对称解决方案上的工作外，我们的结果适用于无对称的Sobolev空间中的开放式初始数据。
For $(t,x) \in (0,\infty)\times\mathbb{T}^D$, the generalized Kasner solutions are a family of explicit solutions to various Einstein-matter systems that start out smooth but then develop a Big Bang singularity as $t \downarrow 0$, i.e., curvature blowup along a spacelike hypersurface. The family is parameterized by the Kasner exponents $\widetilde{q}_1,\cdots,\widetilde{q}_D \in \mathbb{R}$, which satisfy two algebraic constraints. There are heuristics in the mathematical physics literature, going back more than 50 years, suggesting that the Big Bang formation should be stable under perturbations of the Kasner initial data, given say at $\lbrace t = 1 \rbrace$, as long as the exponents are "sub-critical" in the following sense: $\mathop{\max_{I,J,B=1,\cdots,D}}_{I < J} \{\widetilde{q}_I+\widetilde{q}_J-\widetilde{q}_B\}<1$. Previous works have shown the stability of the singularity under stronger assumptions: 1) the Einstein-scalar field system with $D = 3$ and $\widetilde{q}_1 \approx \widetilde{q}_2 \approx \widetilde{q}_3 \approx 1/3$ or 2) the Einstein-vacuum equations for $D \geq 39$ with $\max_{I=1,\cdots,D} |\widetilde{q}_I| < 1/6$. We prove that the Kasner singularity is dynamically stable for \emph{all} sub-critical Kasner exponents, thereby justifying the heuristics in the full regime where stable monotonic-type curvature blowup is expected. We treat the $1+D$-dimensional Einstein-scalar field system for $D \geq3$ and the $1+D$ dimensional Einstein-vacuum equations for $D \geq 10$. Moreover, for the Einstein-vacuum equations in $1+3$ dimensions, where instabilities are in general expected, we prove that all singular Kasner solutions have stable Big Bangs under polarized $U(1)$-symmetric perturbations of their initial data. Our results hold for open sets of initial data in Sobolev spaces without symmetry, apart from our work on polarized $U(1)$-symmetric solutions.
