Static solutions to the spherically symmetric Einstein-Vlasov system: a particle-number-Casimir approach
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866914680431181824 |
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| author | Andréasson, Håkan Kunze, Markus |
| author_facet | Andréasson, Håkan Kunze, Markus |
| contents | Existence of spherically symmetric solutions to the Einstein-Vlasov system is well-known. However, it is an open problem whether or not static solutions arise as minimizers of a variational problem. Apart from being of interest in its own right, it is the connection to non-linear stability that gives this topic its importance. This problem was considered in \cite{Wol}, but as has been pointed out in \cite{AK}, the paper \cite{Wol} contained serious flaws. In this work we construct static solutions by solving the Euler-Lagrange equation for the energy density $ρ$ as a fixed point problem. The Euler-Lagrange equation originates from the particle number-Casimir functional introduced in \cite{Wol}. We then define a density function $f$ on phase space which induces the energy density $ρ$ and we show that it constitutes a static solution of the Einstein-Vlasov system. Hence we settle rigorously parts of what the author of \cite{Wol} attempted to prove. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2202_01835 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Static solutions to the spherically symmetric Einstein-Vlasov system: a particle-number-Casimir approach Andréasson, Håkan Kunze, Markus General Relativity and Quantum Cosmology Mathematical Physics Existence of spherically symmetric solutions to the Einstein-Vlasov system is well-known. However, it is an open problem whether or not static solutions arise as minimizers of a variational problem. Apart from being of interest in its own right, it is the connection to non-linear stability that gives this topic its importance. This problem was considered in \cite{Wol}, but as has been pointed out in \cite{AK}, the paper \cite{Wol} contained serious flaws. In this work we construct static solutions by solving the Euler-Lagrange equation for the energy density $ρ$ as a fixed point problem. The Euler-Lagrange equation originates from the particle number-Casimir functional introduced in \cite{Wol}. We then define a density function $f$ on phase space which induces the energy density $ρ$ and we show that it constitutes a static solution of the Einstein-Vlasov system. Hence we settle rigorously parts of what the author of \cite{Wol} attempted to prove. |
| title | Static solutions to the spherically symmetric Einstein-Vlasov system: a particle-number-Casimir approach |
| topic | General Relativity and Quantum Cosmology Mathematical Physics |
| url | https://arxiv.org/abs/2202.01835 |