{"title":"爱因斯坦-斯卡拉场方程的局部大爆炸稳定性","authors":"Florian Beyer, Todd A. Oliynyk","doi":"10.1007/s00205-023-01939-9","DOIUrl":null,"url":null,"abstract":"<div><p>We prove the nonlinear stability in the contracting direction of Friedmann–Lemaître–Robertson–Walker (FLRW) solutions to the Einstein-scalar field equations in <span>\\(n\\ge 3\\)</span> spacetime dimensions that are defined on spacetime manifolds of the form <span>\\((0,t_0]\\times \\mathbb {T}{}^{n-1}\\)</span>, <span>\\(t_0>0\\)</span>. Stability is established under the assumption that the initial data is <i>synchronized</i>, which means that on the initial hypersurface <span>\\(\\Sigma = \\{t_0\\}\\times \\mathbb {T}{}^{n-1}\\)</span>, the scalar field <span>\\(\\tau = \\exp \\bigl (\\sqrt{\\frac{2(n-2)}{n-1}}\\phi \\bigr ) \\)</span> is constant, that is, <span>\\(\\Sigma =\\tau ^{-1}(\\{t_0\\})\\)</span>. 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 <i>synchronized</i>, no generality is lost by this assumption. By using <span>\\(\\tau \\)</span> as a time coordinate, we establish that the perturbed FLRW spacetime manifolds are of the form <span>\\(M = \\bigcup _{t\\in (0,t_0]}\\tau ^{-1}(\\{t\\})\\cong (0,t_0]\\times \\mathbb {T}{}^{n-1}\\)</span>, the perturbed FLRW solutions are asymptotically pointwise Kasner as <span>\\(\\tau \\searrow 0\\)</span>, and a big bang singularity, characterised by the blow up of the scalar curvature, occurs at <span>\\(\\tau =0\\)</span>. 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.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 1","pages":""},"PeriodicalIF":2.6000,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Localized Big Bang Stability for the Einstein-Scalar Field Equations\",\"authors\":\"Florian Beyer, Todd A. Oliynyk\",\"doi\":\"10.1007/s00205-023-01939-9\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>We prove the nonlinear stability in the contracting direction of Friedmann–Lemaître–Robertson–Walker (FLRW) solutions to the Einstein-scalar field equations in <span>\\\\(n\\\\ge 3\\\\)</span> spacetime dimensions that are defined on spacetime manifolds of the form <span>\\\\((0,t_0]\\\\times \\\\mathbb {T}{}^{n-1}\\\\)</span>, <span>\\\\(t_0>0\\\\)</span>. Stability is established under the assumption that the initial data is <i>synchronized</i>, which means that on the initial hypersurface <span>\\\\(\\\\Sigma = \\\\{t_0\\\\}\\\\times \\\\mathbb {T}{}^{n-1}\\\\)</span>, the scalar field <span>\\\\(\\\\tau = \\\\exp \\\\bigl (\\\\sqrt{\\\\frac{2(n-2)}{n-1}}\\\\phi \\\\bigr ) \\\\)</span> is constant, that is, <span>\\\\(\\\\Sigma =\\\\tau ^{-1}(\\\\{t_0\\\\})\\\\)</span>. 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 <i>synchronized</i>, no generality is lost by this assumption. By using <span>\\\\(\\\\tau \\\\)</span> as a time coordinate, we establish that the perturbed FLRW spacetime manifolds are of the form <span>\\\\(M = \\\\bigcup _{t\\\\in (0,t_0]}\\\\tau ^{-1}(\\\\{t\\\\})\\\\cong (0,t_0]\\\\times \\\\mathbb {T}{}^{n-1}\\\\)</span>, the perturbed FLRW solutions are asymptotically pointwise Kasner as <span>\\\\(\\\\tau \\\\searrow 0\\\\)</span>, and a big bang singularity, characterised by the blow up of the scalar curvature, occurs at <span>\\\\(\\\\tau =0\\\\)</span>. 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.</p></div>\",\"PeriodicalId\":55484,\"journal\":{\"name\":\"Archive for Rational Mechanics and Analysis\",\"volume\":\"248 1\",\"pages\":\"\"},\"PeriodicalIF\":2.6000,\"publicationDate\":\"2023-12-08\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Archive for Rational Mechanics and Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://link.springer.com/article/10.1007/s00205-023-01939-9\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Archive for Rational Mechanics and Analysis","FirstCategoryId":"100","ListUrlMain":"https://link.springer.com/article/10.1007/s00205-023-01939-9","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
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.
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.