Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes
Abstract Ricci flow is a natural gradient flow of the Einstein-Hilbert action. Here we consider the analog for the Einstein-Maxwell action, which gives Ricci flow with a stress tensor contribution coupled to a Yang-Mills flow for the Maxwell field. We argue that this flow is well-posed for static sp...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2023-03-01
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| Series: | Journal of High Energy Physics |
| Subjects: | |
| Online Access: | https://doi.org/10.1007/JHEP03(2023)074 |
| _version_ | 1797795921093722112 |
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| author | Davide De Biasio Julian Freigang Dieter Lüst Toby Wiseman |
| author_facet | Davide De Biasio Julian Freigang Dieter Lüst Toby Wiseman |
| author_sort | Davide De Biasio |
| collection | DOAJ |
| description | Abstract Ricci flow is a natural gradient flow of the Einstein-Hilbert action. Here we consider the analog for the Einstein-Maxwell action, which gives Ricci flow with a stress tensor contribution coupled to a Yang-Mills flow for the Maxwell field. We argue that this flow is well-posed for static spacetimes with pure electric or magnetic potentials and show it preserves both non-extremal and extremal black hole horizons. In the latter case we find the flow of the near horizon geometry decouples from that of the exterior. The Schwarzschild black hole is an unstable fixed point of Ricci flow for static spacetimes. Here we consider flows of the Reissner-Nordström (RN) fixed point. The magnetic RN solution becomes a stable fixed point of the flow for sufficient charge. However we find that the electric RN black hole is always unstable. Numerically solving the flow starting with a spherically symmetric perturbation of a non-extremal RN solution, we find similar behaviour in the electric case to the Ricci flows of perturbed Schwarzschild, namely the horizon shrinks to a singularity in finite time or expands forever. In the magnetic case, a perturbed unstable RN solution has a similar expanding behaviour, but a perturbation that decreases the horizon size flows to a stable black hole solution rather than a singularity. For extremal RN we solve the near horizon flow for spherical symmetry exactly, and see in the electric case two unstable directions which flow to singularities in finite flow time. However, even turning these off, and fixing the near horizon geometry to be that of RN, we numerically show that the flows appear to become singular in the vicinity of its horizon. |
| first_indexed | 2024-03-13T03:25:20Z |
| format | Article |
| id | doaj.art-8709851cb72a43af8eb4f751fcbffa6c |
| institution | Directory Open Access Journal |
| issn | 1029-8479 |
| language | English |
| last_indexed | 2024-03-13T03:25:20Z |
| publishDate | 2023-03-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of High Energy Physics |
| spelling | doaj.art-8709851cb72a43af8eb4f751fcbffa6c2023-06-25T11:06:47ZengSpringerOpenJournal of High Energy Physics1029-84792023-03-012023315110.1007/JHEP03(2023)074Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holesDavide De Biasio0Julian Freigang1Dieter Lüst2Toby Wiseman3Max-Planck-Institut für Physik, Werner-Heisenberg-InstitutMax-Planck-Institut für Physik, Werner-Heisenberg-InstitutMax-Planck-Institut für Physik, Werner-Heisenberg-InstitutTheoretical Physics Group, Blackett Laboratory, Imperial College LondonAbstract Ricci flow is a natural gradient flow of the Einstein-Hilbert action. Here we consider the analog for the Einstein-Maxwell action, which gives Ricci flow with a stress tensor contribution coupled to a Yang-Mills flow for the Maxwell field. We argue that this flow is well-posed for static spacetimes with pure electric or magnetic potentials and show it preserves both non-extremal and extremal black hole horizons. In the latter case we find the flow of the near horizon geometry decouples from that of the exterior. The Schwarzschild black hole is an unstable fixed point of Ricci flow for static spacetimes. Here we consider flows of the Reissner-Nordström (RN) fixed point. The magnetic RN solution becomes a stable fixed point of the flow for sufficient charge. However we find that the electric RN black hole is always unstable. Numerically solving the flow starting with a spherically symmetric perturbation of a non-extremal RN solution, we find similar behaviour in the electric case to the Ricci flows of perturbed Schwarzschild, namely the horizon shrinks to a singularity in finite time or expands forever. In the magnetic case, a perturbed unstable RN solution has a similar expanding behaviour, but a perturbation that decreases the horizon size flows to a stable black hole solution rather than a singularity. For extremal RN we solve the near horizon flow for spherical symmetry exactly, and see in the electric case two unstable directions which flow to singularities in finite flow time. However, even turning these off, and fixing the near horizon geometry to be that of RN, we numerically show that the flows appear to become singular in the vicinity of its horizon.https://doi.org/10.1007/JHEP03(2023)074Black HolesBlack Holes in String TheoryClassical Theories of GravityModels of Quantum Gravity |
| spellingShingle | Davide De Biasio Julian Freigang Dieter Lüst Toby Wiseman Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes Journal of High Energy Physics Black Holes Black Holes in String Theory Classical Theories of Gravity Models of Quantum Gravity |
| title | Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes |
| title_full | Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes |
| title_fullStr | Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes |
| title_full_unstemmed | Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes |
| title_short | Gradient flow of Einstein-Maxwell theory and Reissner-Nordström black holes |
| title_sort | gradient flow of einstein maxwell theory and reissner nordstrom black holes |
| topic | Black Holes Black Holes in String Theory Classical Theories of Gravity Models of Quantum Gravity |
| url | https://doi.org/10.1007/JHEP03(2023)074 |
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