The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866917762715090944 |
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| author | Graf, Melanie Grant, James D. E. Kunzinger, Michael Steinbauer, Roland |
| author_facet | Graf, Melanie Grant, James D. E. Kunzinger, Michael Steinbauer, Roland |
| contents | We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$-metrics, and of $C^0$-trapped submanifolds. By regularisation, we show that, under these weak conditions, causal geodesics necessarily become non-maximising. This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1706_08426 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics Graf, Melanie Grant, James D. E. Kunzinger, Michael Steinbauer, Roland Mathematical Physics General Relativity and Quantum Cosmology Differential Geometry 83C75, 53B30 We show that the Hawking--Penrose singularity theorem, and the generalisation of this theorem due to Galloway and Senovilla, continue to hold for Lorentzian metrics that are of $C^{1, 1}$-regularity. We formulate appropriate weak versions of the strong energy condition and genericity condition for $C^{1,1}$-metrics, and of $C^0$-trapped submanifolds. By regularisation, we show that, under these weak conditions, causal geodesics necessarily become non-maximising. This requires a detailed analysis of the matrix Riccati equation for the approximating metrics, which may be of independent interest. |
| title | The Hawking-Penrose singularity theorem for $C^{1,1}$-Lorentzian metrics |
| topic | Mathematical Physics General Relativity and Quantum Cosmology Differential Geometry 83C75, 53B30 |
| url | https://arxiv.org/abs/1706.08426 |