A localized construction of Kasner-like singularities
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866909438026186752 |
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| author | Athanasiou, Nikolaos Fournodavlos, Grigorios |
| author_facet | Athanasiou, Nikolaos Fournodavlos, Grigorios |
| contents | We construct local, in spacetime, singular solutions to the Einstein vacuum equations that exhibit Kasner-like behavior in their past boundary. Our result can be viewed as a localization (in space) of the construction in \cite{FL}. We also prove a refined uniqueness statement and give a simple argument that generates general asymptotic data for Kasner-like singularities, enjoying all expected degrees of freedom, albeit only locally in space. The key difference of the present work with \cite{FL} is our use of a first order symmetric hyperbolic formulation of the Einstein vacuum equations, relative to the connection coefficients of a parallelly propagated orthonormal frame which is adapted to the Gaussian time foliation. This makes it easier to localize the construction, since elliptic estimates are no longer required to complete the energy argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_16630 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A localized construction of Kasner-like singularities Athanasiou, Nikolaos Fournodavlos, Grigorios Analysis of PDEs General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry 83C75 (Primary) 35A21, 35Q76, 83C05, 83F05 (Secondary) We construct local, in spacetime, singular solutions to the Einstein vacuum equations that exhibit Kasner-like behavior in their past boundary. Our result can be viewed as a localization (in space) of the construction in \cite{FL}. We also prove a refined uniqueness statement and give a simple argument that generates general asymptotic data for Kasner-like singularities, enjoying all expected degrees of freedom, albeit only locally in space. The key difference of the present work with \cite{FL} is our use of a first order symmetric hyperbolic formulation of the Einstein vacuum equations, relative to the connection coefficients of a parallelly propagated orthonormal frame which is adapted to the Gaussian time foliation. This makes it easier to localize the construction, since elliptic estimates are no longer required to complete the energy argument. |
| title | A localized construction of Kasner-like singularities |
| topic | Analysis of PDEs General Relativity and Quantum Cosmology Mathematical Physics Differential Geometry 83C75 (Primary) 35A21, 35Q76, 83C05, 83F05 (Secondary) |
| url | https://arxiv.org/abs/2412.16630 |