Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866916790944137216 |
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| author | Graf, Olivier Holzegel, Gustav |
| author_facet | Graf, Olivier Holzegel, Gustav |
| contents | We prove that there are no non-stationary (with respect to the Hawking vectorfield) real mode solutions to the Teukolsky equations on all $(3+1)$-dimensional subextremal Kerr-anti-de Sitter spacetimes. We further prove that stationary solutions do not exist if the black hole parameters satisfy the Hawking-Reall bound and $\left|a\sqrt{-Λ}\right|<\frac{\sqrt{3}}{20}$. We conclude with the statement of mode stability which preludes boundedness and decay estimates for general solutions which will be proven in a separate paper. Our boundary conditions are the standard ones which follow from fixing the conformal class of the metric at infinity and lead to a coupling of the two Teukolsky equations. The proof relies on combining the Teukolsky-Starobinsky identities with the coupled boundary conditions. In the stationary case the proof exploits elliptic estimates which fail if the Hawking-Reall bound is violated. This is consistent with the superradiant instabilities expected in that regime. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_02801 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes Graf, Olivier Holzegel, Gustav General Relativity and Quantum Cosmology Analysis of PDEs We prove that there are no non-stationary (with respect to the Hawking vectorfield) real mode solutions to the Teukolsky equations on all $(3+1)$-dimensional subextremal Kerr-anti-de Sitter spacetimes. We further prove that stationary solutions do not exist if the black hole parameters satisfy the Hawking-Reall bound and $\left|a\sqrt{-Λ}\right|<\frac{\sqrt{3}}{20}$. We conclude with the statement of mode stability which preludes boundedness and decay estimates for general solutions which will be proven in a separate paper. Our boundary conditions are the standard ones which follow from fixing the conformal class of the metric at infinity and lead to a coupling of the two Teukolsky equations. The proof relies on combining the Teukolsky-Starobinsky identities with the coupled boundary conditions. In the stationary case the proof exploits elliptic estimates which fail if the Hawking-Reall bound is violated. This is consistent with the superradiant instabilities expected in that regime. |
| title | Mode stability for the Teukolsky equations on Kerr-anti-de Sitter spacetimes |
| topic | General Relativity and Quantum Cosmology Analysis of PDEs |
| url | https://arxiv.org/abs/2205.02801 |