Solving the Einstein Constraints Numerically on Compact Three-Manifolds Using Hyperbolic Relaxation
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917602334343168 |
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| author | Zhang, Fan Lindblom, Lee |
| author_facet | Zhang, Fan Lindblom, Lee |
| contents | The effectiveness of the hyperbolic relaxation method for solving the Einstein constraint equations numerically is studied here on a variety of compact orientable three-manifolds. Convergent numerical solutions are found using this method on manifolds admitting negative Ricci scalar curvature metrics, i.e. those from the $H^3$ and the $H^2\times S^1$ geometrization classes. The method fails to produce solutions, however, on all the manifolds examined here admitting non-negative Ricci scalar curvatures, i.e. those from the $S^3$, $S^2\times S^1$, and the $E^3$ classes. This study also finds that the accuracy of the convergent solutions produced by hyperbolic relaxation can be increased significantly by performing fairly low-cost standard elliptic solves using the hyperbolic relaxation solutions as initial guesses. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_08880 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Solving the Einstein Constraints Numerically on Compact Three-Manifolds Using Hyperbolic Relaxation Zhang, Fan Lindblom, Lee General Relativity and Quantum Cosmology The effectiveness of the hyperbolic relaxation method for solving the Einstein constraint equations numerically is studied here on a variety of compact orientable three-manifolds. Convergent numerical solutions are found using this method on manifolds admitting negative Ricci scalar curvature metrics, i.e. those from the $H^3$ and the $H^2\times S^1$ geometrization classes. The method fails to produce solutions, however, on all the manifolds examined here admitting non-negative Ricci scalar curvatures, i.e. those from the $S^3$, $S^2\times S^1$, and the $E^3$ classes. This study also finds that the accuracy of the convergent solutions produced by hyperbolic relaxation can be increased significantly by performing fairly low-cost standard elliptic solves using the hyperbolic relaxation solutions as initial guesses. |
| title | Solving the Einstein Constraints Numerically on Compact Three-Manifolds Using Hyperbolic Relaxation |
| topic | General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2402.08880 |