Intrinsic rigidity of extremal horizons
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917239800725504 |
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| author | Dunajski, Maciej Lucietti, James |
| author_facet | Dunajski, Maciej Lucietti, James |
| contents | We prove that the intrinsic geometry of compact cross-sections of any vacuum extremal horizon must admit a Killing vector field. If the cross-sections are two-dimensional spheres, this implies that the most general solution is the extremal Kerr horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant. Furthermore, we also deduce that any non-trivial vacuum near-horizon geometry, with a non-positive cosmological constant, must have a Lie algebra of Killing vector fields that contains $\mathfrak{sl}(2)\times \mathfrak{u}(1)$ in all dimensions under no symmetry assumptions. We also show that, if the cross-sections are two-dimensional, the horizon Einstein equation is equivalent to a single fourth order PDE for the Kähler potential, and that this equation is explicitly solvable on the sphere if the corresponding metric admits a Killing vector. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_17512 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Intrinsic rigidity of extremal horizons Dunajski, Maciej Lucietti, James General Relativity and Quantum Cosmology High Energy Physics - Theory Differential Geometry We prove that the intrinsic geometry of compact cross-sections of any vacuum extremal horizon must admit a Killing vector field. If the cross-sections are two-dimensional spheres, this implies that the most general solution is the extremal Kerr horizon and completes the classification of the associated near-horizon geometries. The same results hold with a cosmological constant. Furthermore, we also deduce that any non-trivial vacuum near-horizon geometry, with a non-positive cosmological constant, must have a Lie algebra of Killing vector fields that contains $\mathfrak{sl}(2)\times \mathfrak{u}(1)$ in all dimensions under no symmetry assumptions. We also show that, if the cross-sections are two-dimensional, the horizon Einstein equation is equivalent to a single fourth order PDE for the Kähler potential, and that this equation is explicitly solvable on the sphere if the corresponding metric admits a Killing vector. |
| title | Intrinsic rigidity of extremal horizons |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Differential Geometry |
| url | https://arxiv.org/abs/2306.17512 |