The $C^3$-null gluing problem: linear and nonlinear analysis
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866914917835079680 |
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| author | Sansom, Robert |
| author_facet | Sansom, Robert |
| contents | In this paper, we investigate the $C^3$-null gluing problem for the Einstein vacuum equations, that is, we consider the null gluing of up to and including third-order derivatives of the metric. In the regime where the characteristic data is close to Minkowski data, we show that this $C^3$-null gluing problem is solvable up to a $20$-dimensional space of obstructions. The obstructions correspond to $20$ linearly conserved quantities: $10$ of which are already present in the $C^2$-null gluing problem analysed by Aretakis, Czimek and Rodnianski, and $10$ are novel obstructions inherent to the $C^3$-null gluing problem. The $10$ novel obstructions are linearly conserved charges calculated from third-order derivatives of the metric. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2408_10859 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $C^3$-null gluing problem: linear and nonlinear analysis Sansom, Robert General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry In this paper, we investigate the $C^3$-null gluing problem for the Einstein vacuum equations, that is, we consider the null gluing of up to and including third-order derivatives of the metric. In the regime where the characteristic data is close to Minkowski data, we show that this $C^3$-null gluing problem is solvable up to a $20$-dimensional space of obstructions. The obstructions correspond to $20$ linearly conserved quantities: $10$ of which are already present in the $C^2$-null gluing problem analysed by Aretakis, Czimek and Rodnianski, and $10$ are novel obstructions inherent to the $C^3$-null gluing problem. The $10$ novel obstructions are linearly conserved charges calculated from third-order derivatives of the metric. |
| title | The $C^3$-null gluing problem: linear and nonlinear analysis |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2408.10859 |