The transition to phenomenological behaviour of static solutions of the Einstein-Dirac system for an increasing number of fermions
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| Format: | Preprint |
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2025
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| _version_ | 1866915445458599936 |
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| author | Andréasson, Håkan Blomqvist, Joakim |
| author_facet | Andréasson, Håkan Blomqvist, Joakim |
| contents | Static spherically symmetric solutions to the Einstein-Dirac system were constructed numerically for the first time in 1999 by Finster, Smoller and Yau \cite{FSY1} in the case of two fermions. In 2020 this result was generalized by Leith, Hooley, Horne and Dritschel \cite{LHHD} to a system consisting of an even number $κ$ of fermions. They constructed solutions for $2\leqκ\leq 90$. The purpose of the present investigation is to compare the properties of static solutions of the Einstein-Dirac system with static solutions of the Einstein,-Vlasov system as the number of fermions increases, that is, for $2\leqκ\leq 180$. Since the Einstein-Vlasov system is a fully classical physical model, whereas the Einstein-Dirac system is semiclassical and thus has a quantum signature, this framework provides an excellent opportunity to study the transition from quantum to classical behaviour. It turns out that even for a comparatively small number of particles, the features of the solutions are remarkably similar. For both systems, we find highly relativistic solutions having a multi-peak structure with strikingly similar characteristics. We also investigate the maximum compactness ratio $\sup 2m/r$ of the solutions. The solutions of both systems share the fundamental properties regarding the maximum compactness ratio and obey the inequality derived in \cite{A2}. Furthermore, we investigate the sign of the pressure components of solutions of the Einstein-Dirac system. For small values of $κ$, there are regions where the radial pressure is negative. These regions disappear as $κ$ increases. This supports the interpretation we make as a transition from quantum to classical behaviour as the number of fermions increases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_15995 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The transition to phenomenological behaviour of static solutions of the Einstein-Dirac system for an increasing number of fermions Andréasson, Håkan Blomqvist, Joakim General Relativity and Quantum Cosmology Static spherically symmetric solutions to the Einstein-Dirac system were constructed numerically for the first time in 1999 by Finster, Smoller and Yau \cite{FSY1} in the case of two fermions. In 2020 this result was generalized by Leith, Hooley, Horne and Dritschel \cite{LHHD} to a system consisting of an even number $κ$ of fermions. They constructed solutions for $2\leqκ\leq 90$. The purpose of the present investigation is to compare the properties of static solutions of the Einstein-Dirac system with static solutions of the Einstein,-Vlasov system as the number of fermions increases, that is, for $2\leqκ\leq 180$. Since the Einstein-Vlasov system is a fully classical physical model, whereas the Einstein-Dirac system is semiclassical and thus has a quantum signature, this framework provides an excellent opportunity to study the transition from quantum to classical behaviour. It turns out that even for a comparatively small number of particles, the features of the solutions are remarkably similar. For both systems, we find highly relativistic solutions having a multi-peak structure with strikingly similar characteristics. We also investigate the maximum compactness ratio $\sup 2m/r$ of the solutions. The solutions of both systems share the fundamental properties regarding the maximum compactness ratio and obey the inequality derived in \cite{A2}. Furthermore, we investigate the sign of the pressure components of solutions of the Einstein-Dirac system. For small values of $κ$, there are regions where the radial pressure is negative. These regions disappear as $κ$ increases. This supports the interpretation we make as a transition from quantum to classical behaviour as the number of fermions increases. |
| title | The transition to phenomenological behaviour of static solutions of the Einstein-Dirac system for an increasing number of fermions |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2503.15995 |