Uniqueness of the extremal Schwarzschild de Sitter spacetime
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910302037082112 |
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| author | Katona, David Lucietti, James |
| author_facet | Katona, David Lucietti, James |
| contents | We prove that any analytic vacuum spacetime with a positive cosmological constant in four and higher dimensions, that contains a static extremal Killing horizon with a maximally symmetric compact cross-section, must be locally isometric to either the extremal Schwarzschild de Sitter solution or its near-horizon geometry (the Nariai solution). In four-dimensions, this implies these solutions are the only analytic vacuum spacetimes that contain a static extremal horizon with compact cross-sections (up to identifications). We also consider the analogous uniqueness problem for the four-dimensional extremal hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a spectral problem for the laplacian on compact hyperbolic surfaces, if a cohomological obstruction to the uniqueness of infinitesimal transverse deformations of the horizon is absent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_04238 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uniqueness of the extremal Schwarzschild de Sitter spacetime Katona, David Lucietti, James General Relativity and Quantum Cosmology High Energy Physics - Theory Differential Geometry 83C57 (primary) 83C05 (secondary) We prove that any analytic vacuum spacetime with a positive cosmological constant in four and higher dimensions, that contains a static extremal Killing horizon with a maximally symmetric compact cross-section, must be locally isometric to either the extremal Schwarzschild de Sitter solution or its near-horizon geometry (the Nariai solution). In four-dimensions, this implies these solutions are the only analytic vacuum spacetimes that contain a static extremal horizon with compact cross-sections (up to identifications). We also consider the analogous uniqueness problem for the four-dimensional extremal hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a spectral problem for the laplacian on compact hyperbolic surfaces, if a cohomological obstruction to the uniqueness of infinitesimal transverse deformations of the horizon is absent. |
| title | Uniqueness of the extremal Schwarzschild de Sitter spacetime |
| topic | General Relativity and Quantum Cosmology High Energy Physics - Theory Differential Geometry 83C57 (primary) 83C05 (secondary) |
| url | https://arxiv.org/abs/2309.04238 |