$\mathrm{SL}(2,\mathbb{R})$ families of Kerr black holes
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908387831185408 |
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| author | Penna, Robert |
| author_facet | Penna, Robert |
| contents | The stationary, axisymmetric sector of vacuum general relativity (with zero cosmological constant) enjoys an $\mathrm{SL}(2,\mathbb{R})$ symmetry called the Matzner-Misner group. We study the action of the Matzner-Misner group on the Kerr black hole. We show that the group acts naturally on a three parameter generalization of the usual two parameter Kerr solution. The new parameter represents a large diffeomorphism which gives the spacetime an asymptotic angular velocity. We explain how the $\mathrm{SL}(2,\mathbb{R})$ symmetry organizes the space of three parameter Kerr solutions into the classical analogue of principal series representations. We show that the $\mathrm{SL}(2,\mathbb{R})$ Casimir operator is the Bekenstein-Hawking entropy. The Matzner-Misner group sits inside a much larger Kac-Moody symmetry called the Geroch group. We show that the Kac-Moody level of the Kerr black hole is the Bekenstein-Hawking entropy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_00184 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathrm{SL}(2,\mathbb{R})$ families of Kerr black holes Penna, Robert High Energy Physics - Theory General Relativity and Quantum Cosmology The stationary, axisymmetric sector of vacuum general relativity (with zero cosmological constant) enjoys an $\mathrm{SL}(2,\mathbb{R})$ symmetry called the Matzner-Misner group. We study the action of the Matzner-Misner group on the Kerr black hole. We show that the group acts naturally on a three parameter generalization of the usual two parameter Kerr solution. The new parameter represents a large diffeomorphism which gives the spacetime an asymptotic angular velocity. We explain how the $\mathrm{SL}(2,\mathbb{R})$ symmetry organizes the space of three parameter Kerr solutions into the classical analogue of principal series representations. We show that the $\mathrm{SL}(2,\mathbb{R})$ Casimir operator is the Bekenstein-Hawking entropy. The Matzner-Misner group sits inside a much larger Kac-Moody symmetry called the Geroch group. We show that the Kac-Moody level of the Kerr black hole is the Bekenstein-Hawking entropy. |
| title | $\mathrm{SL}(2,\mathbb{R})$ families of Kerr black holes |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2506.00184 |