Localized big bang stability for the Einstein-scalar field equations
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866929209008455680 |
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| author | Beyer, Florian Oliynyk, Todd A. |
| author_facet | Beyer, Florian Oliynyk, Todd A. |
| contents | We prove the nonlinear stability in the contracting direction of Friedmann-Lemaître-Robertson-Walker (FLRW) solutions to the Einstein-scalar field equations in $n\geq 3$ 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 \textit{synchronized}, which means that on the initial hypersurface $Σ= \{t_0\}\times \mathbb{T}^{n-1}$ the scalar field $τ= \exp\bigl(\sqrt{\frac{2(n-2)}{n-1}}ϕ\bigr) $ is constant, that is, $Σ=τ^{-1}(\{t_0\})$. 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 \textit{synchronized}, no generality is lost by this assumption. By using $τ$ as a time coordinate, we establish that the perturbed FLRW spacetime manifolds are of the form $M = \bigcup_{t\in (0,t_0]}τ^{-1}(\{t\})\cong (0,t_0]\times \mathbb{T}^{n-1}$, the perturbed FLRW solutions are asymptotically pointwise Kasner as $τ\searrow 0$, and a big bang singularity, characterised by the blow up of the scalar curvature, occurs at $τ=0$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2112_07730 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Localized big bang stability for the Einstein-scalar field equations Beyer, Florian Oliynyk, Todd A. General Relativity and Quantum Cosmology Analysis of PDEs We prove the nonlinear stability in the contracting direction of Friedmann-Lemaître-Robertson-Walker (FLRW) solutions to the Einstein-scalar field equations in $n\geq 3$ 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 \textit{synchronized}, which means that on the initial hypersurface $Σ= \{t_0\}\times \mathbb{T}^{n-1}$ the scalar field $τ= \exp\bigl(\sqrt{\frac{2(n-2)}{n-1}}ϕ\bigr) $ is constant, that is, $Σ=τ^{-1}(\{t_0\})$. 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 \textit{synchronized}, no generality is lost by this assumption. By using $τ$ as a time coordinate, we establish that the perturbed FLRW spacetime manifolds are of the form $M = \bigcup_{t\in (0,t_0]}τ^{-1}(\{t\})\cong (0,t_0]\times \mathbb{T}^{n-1}$, the perturbed FLRW solutions are asymptotically pointwise Kasner as $τ\searrow 0$, and a big bang singularity, characterised by the blow up of the scalar curvature, occurs at $τ=0$. 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. |
| title | Localized big bang stability for the Einstein-scalar field equations |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs |
| url | https://arxiv.org/abs/2112.07730 |