The Maximal Development of Near-FLRW Data for the Einstein-Scalar Field System with Spatial Topology S3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddside
The Maximal Development of Near-FLRW Data for the Einstein-Scalar Field System with Spatial Topology S3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddside
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具有空间拓扑的爱因斯坦标量场系统的近FLRW数据的最大发展 S3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{
DOI：
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发表时间：
2018
影响因子：
2.4
通讯作者：
Jared Speck
中科院分区：
文献类型：
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作者：
Jared Speck
具有空间拓扑的爱因斯坦标量场系统的friedman - lematrer - robertson - walker （FLRW）解[12pt]{最小}usepackageamsfonts{ useppackageamsfonts }useppackageamsfonts{ useppackageamssyb} useppackageamssfs{ setlengthoddsidemargin}-69pt{ }egindocument{}{}{}{}{}{}$${{mathbb{S}}^3}$$ enddocument{模拟了一个从奇异类空间超表面（大爆炸）发出的宇宙}，沿着这个超表面各种时空曲率不变量爆炸，只是为了在未来以对称的方式重新坍缩（大紧缩）。在本文中，我们给出了在FLRW演变的时间中点上FLRW数据扰动的最大发展的完整描述。我们证明了扰动解也表现出沿一对类空间超表面的曲率爆炸，这表明了大爆炸和大紧缩的动态稳定性。此外，我们对解的渐近行为给出了一个清晰的描述，特别是在奇异超表面上，各种时间尺度的解变量收敛到与相应的时间尺度FLRW张量场接近的正则张量场。我们的证明关键依赖于L2documentclass[12pt]minimal {uspackageamsmath} uspackagewasysym{ uspackageamsfonts} uspackageamssyb {uspackageamssyb} uspackageamssfs {uspackageupgreek} setlengththoddsidemargin{ -69pt} egindocument {}{}{}{}{}{}$${L^2}$$ enddocument{类型}的近似单调性恒恒性，其{精神与我们在}与I. Rodnianski的联合工作中使用的恒恒性一致。其中，我们用空间拓扑T3documentclass[12pt]证明了接近空间平面的解决方案的类似结果：最小化useppackageamsmath {useppackageamsfonts} useppackageamssymb{ useppackageamssyb} useppackageamsfs{ useppackageupgreek }setlengthoddsidemargin{-}69pt egindocument {}{}{}{}{}{}$${{mathbb{T}}^3}$$ enddocument{。在}本文中，我们依靠{新的}成分{来处理在S3documentclass[12pt]minimal上的近圆空间}度量{usepackageamsmath useppackagewasysym useppackageamsfonts useppackageamssymb }useppackageamssy{ useppackageamsfs} useppackageupgreek{ setlengthoddsidemargin}-{69pt egindocument}{}{}{}{}{}$${{mathbb{S}}^3}$$ enddocument{，其曲率在}初始数据超{曲面}附近{是}顺序统一的。特别地，我们的证明{依赖于(i)}构造一个全局定义的空间矢量场框架，该框架适应于圆形度量在S3documentclass[12pt]上的对称性。最小的useppackageamsmath。useppackageamsfonts。useppackageamssymb。useppackageamssyb。{useppackageamsfs}。{useppackageupgreek}。{setlengthoddsidemargin}-{69pt egindocument}{}{}{}{}$${{mathbb{S}}^3}$$ enddocument{；（}ii）各种几何量相对于框架元素的李氏导数的估计；（iii）对FLRW解的尺度因子在奇异超曲面附近的渐近行为的尖锐估计。
The Friedmann–Lemaître–Robertson–Walker (FLRW) solution to the Einstein-scalar field system with spatial topology S3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${{mathbb{S}}^3}$$end{document} models a universe that emanates from a singular spacelike hypersurface (the Big Bang), along which various spacetime curvature invariants blow up, only to re-collapse in a symmetric fashion in the future (the Big Crunch). In this article, we give a complete description of the maximal developments of perturbations of the FLRW data at the chronological midpoint of the FLRW evolution. We show that the perturbed solutions also exhibit curvature blowup along a pair of spacelike hypersurfaces, signifying the dynamic stability of the Big Bang and the Big Crunch. Moreover, we provide a sharp description of the asymptotic behavior of the solution up to the singularities, showing in particular that various time-rescaled solution variables converge to regular tensorfields on the singular hypersurfaces that are close to the corresponding time-rescaled FLRW tensorfields. Our proof crucially relies on L2documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${L^2}$$end{document}-type approximate monotonicity identities in the spirit of the ones we used in our joint works with I. Rodnianski, in which we proved similar results for nearly spatially flat solutions with spatial topology T3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${{mathbb{T}}^3}$$end{document}. In the present article, we rely on new ingredients to handle nearly round spatial metrics on S3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${{mathbb{S}}^3}$$end{document}, whose curvatures are order-unity near the initial data hypersurface. In particular, our proof relies on (i) the construction of a globally defined spatial vectorfield frame adapted to the symmetries of a round metric on S3documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$${{mathbb{S}}^3}$$end{document}; (ii) estimates for the Lie derivatives of various geometric quantities with respect to the elements of the frame; and (iii) sharp estimates for the asymptotic behavior of the FLRW solution’s scale factor near the singular hypersurfaces.