Caustics in the spherically symmetric Einstein-dust system
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866911308640681984 |
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| author | Bick, David |
| author_facet | Bick, David |
| contents | Caustics-envelopes formed by the trajectories of fluid particles-arise in proposed dynamical extensions for shell-crossing singularities occurring in the Einstein-dust system. In this study, a local existence result is established, describing the dynamics in a neighbourhood of such caustics. Specifically, we obtain spherically symmetric spacetimes $(M,g_{μν})$ containing a caustic $\mathcal{C}$, which, in the quotient $M/SO(3)$, is a timelike curve forming a singular boundary between a 2-dust region and a vacuum region. The spacetimes are constructed from solutions to a PDE problem posed with a spacelike direction of evolution. Curvature invariants and energy densities diverge as the caustic is approached. Consequently the metric has limited regularity $g\in C^{1,1/2}$ and is shown to satisfy Einstein's equation weakly. On the complement of the caustic, the metric is smooth and satisfies Einstein's equation classically. A (degenerate) coordinate system is identified in which the dynamical variables are smooth with extension to the caustic. Finally, a novel family of static, spherically symmetric spacetimes is identified, complementing the local construction above. Each spacetime contains an eternal annular 2-dust region bounded by a pair of caustics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_07812 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Caustics in the spherically symmetric Einstein-dust system Bick, David General Relativity and Quantum Cosmology Mathematical Physics Analysis of PDEs Differential Geometry Caustics-envelopes formed by the trajectories of fluid particles-arise in proposed dynamical extensions for shell-crossing singularities occurring in the Einstein-dust system. In this study, a local existence result is established, describing the dynamics in a neighbourhood of such caustics. Specifically, we obtain spherically symmetric spacetimes $(M,g_{μν})$ containing a caustic $\mathcal{C}$, which, in the quotient $M/SO(3)$, is a timelike curve forming a singular boundary between a 2-dust region and a vacuum region. The spacetimes are constructed from solutions to a PDE problem posed with a spacelike direction of evolution. Curvature invariants and energy densities diverge as the caustic is approached. Consequently the metric has limited regularity $g\in C^{1,1/2}$ and is shown to satisfy Einstein's equation weakly. On the complement of the caustic, the metric is smooth and satisfies Einstein's equation classically. A (degenerate) coordinate system is identified in which the dynamical variables are smooth with extension to the caustic. Finally, a novel family of static, spherically symmetric spacetimes is identified, complementing the local construction above. Each spacetime contains an eternal annular 2-dust region bounded by a pair of caustics. |
| title | Caustics in the spherically symmetric Einstein-dust system |
| topic | General Relativity and Quantum Cosmology Mathematical Physics Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2512.07812 |