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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.03373 |
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| _version_ | 1866917697982300160 |
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| author | Câmara, M. Cristina Cardoso, Gabriel Lopes |
| author_facet | Câmara, M. Cristina Cardoso, Gabriel Lopes |
| contents | The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix $M$ depending on the Weyl coordinates $ρ$ and $v$, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the $(ρ,v)$ plane on which some elements of $M(ρ,v)$ tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_03373 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Riemann-Hilbert problems, Toeplitz operators and ergosurfaces Câmara, M. Cristina Cardoso, Gabriel Lopes Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Analysis of PDEs Functional Analysis The Riemann-Hilbert approach, in conjunction with the canonical Wiener-Hopf factorisation of certain matrix functions called monodromy matrices, enables one to obtain explicit solutions to the non-linear field equations of some gravitational theories. These solutions are encoded in the elements of a matrix $M$ depending on the Weyl coordinates $ρ$ and $v$, determined by that factorisation. We address here, for the first time, the underlying question of what happens when a canonical Wiener-Hopf factorisation does not exist, using the close connection of Wiener-Hopf factorisation with Toeplitz operators to study this question. For the case of rational monodromy matrices, we prove that the non-existence of a canonical Wiener-Hopf factorisation determines curves in the $(ρ,v)$ plane on which some elements of $M(ρ,v)$ tend to infinity, but where the space-time metric may still be well behaved. In the case of uncharged rotating black holes in four space-time dimensions and, for certain choices of coordinates, in five space-time dimensions, we show that these curves correspond to their ergosurfaces. |
| title | Riemann-Hilbert problems, Toeplitz operators and ergosurfaces |
| topic | Mathematical Physics General Relativity and Quantum Cosmology High Energy Physics - Theory Analysis of PDEs Functional Analysis |
| url | https://arxiv.org/abs/2404.03373 |