On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes
The lack of a positive-definite and conserved energy is a serious obstacle in the black hole stability problem. In this work, we will show that there exists a positive-definite and conserved Hamiltonian energy for axially symmetric linear perturbations of the exterior of Kerr black hole spacetimes....
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2025 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2025
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/213522 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes. Nishanth Gudapati. SIGMA 21 (2025), 054, 72 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862634503882670080 |
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| author | Gudapati, Nishanth |
| author_facet | Gudapati, Nishanth |
| citation_txt | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes. Nishanth Gudapati. SIGMA 21 (2025), 054, 72 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | The lack of a positive-definite and conserved energy is a serious obstacle in the black hole stability problem. In this work, we will show that there exists a positive-definite and conserved Hamiltonian energy for axially symmetric linear perturbations of the exterior of Kerr black hole spacetimes. In the first part, based on the Hamiltonian dimensional reduction of 3 + 1 axially symmetric, Ricci-flat Lorentzian spacetimes to a 2 + 1 Einstein-wave map system with the negatively curved hyperbolic 2-plane target, we construct a positive-definite, spacetime gauge-invariant energy functional for linear axially symmetric perturbations in the exterior of Kerr black holes, in a manner that is also gauge-independent on the target manifold. In the construction of the positive-definite energy, various dynamical terms at the boundary of the orbit space occur critically. In the second part, after setting up the initial value problem in harmonic coordinates, we prove that the positive energy for the axially symmetric linear perturbative theory of Kerr black holes is strictly conserved in time by establishing that all the boundary terms dynamically vanish for all times. This result implies a form of dynamical linear stability of the exterior of Kerr black hole spacetimes.
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| first_indexed | 2026-03-14T22:05:08Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-213522 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-14T22:05:08Z |
| publishDate | 2025 |
| publisher | Інститут математики НАН України |
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| spelling | Gudapati, Nishanth 2026-02-18T11:24:08Z 2025 On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes. Nishanth Gudapati. SIGMA 21 (2025), 054, 72 pages 1815-0659 2020 Mathematics Subject Classification: 83C57; 58E20; 58E30; 35C15; 35Q75 arXiv:2507.08326 https://nasplib.isofts.kiev.ua/handle/123456789/213522 https://doi.org/10.3842/SIGMA.2025.054 The lack of a positive-definite and conserved energy is a serious obstacle in the black hole stability problem. In this work, we will show that there exists a positive-definite and conserved Hamiltonian energy for axially symmetric linear perturbations of the exterior of Kerr black hole spacetimes. In the first part, based on the Hamiltonian dimensional reduction of 3 + 1 axially symmetric, Ricci-flat Lorentzian spacetimes to a 2 + 1 Einstein-wave map system with the negatively curved hyperbolic 2-plane target, we construct a positive-definite, spacetime gauge-invariant energy functional for linear axially symmetric perturbations in the exterior of Kerr black holes, in a manner that is also gauge-independent on the target manifold. In the construction of the positive-definite energy, various dynamical terms at the boundary of the orbit space occur critically. In the second part, after setting up the initial value problem in harmonic coordinates, we prove that the positive energy for the axially symmetric linear perturbative theory of Kerr black holes is strictly conserved in time by establishing that all the boundary terms dynamically vanish for all times. This result implies a form of dynamical linear stability of the exterior of Kerr black hole spacetimes. I acknowledge the gracious hospitality of Institut des Hautes Études Scientifiques (IHES) at Bures-sur-Yvette in Fall 2016. In a previous work [61], the coauthor Vincent Moncrief had fittingly paid tribute to A. Taub, J. Marsden, and S. Dain for their influence on him and for making fundamental contributions in this direction. I take this opportunity to pay tribute to his own outstanding contributions to general relativity and Hamiltonian methods. On an individual note, for his mentorship, encouragement, and enjoyable interactions, I am indebted to him. This project has spanned several years, and my gratitude is also due to all the colleagues who have continually supported me and to the institutions that hosted me. Special thanks are due to my host Hermann Nicolai at the Albert Einstein Institute at Golm (DFG grant no. GU 1513/2-1), where significant aspects of this work were completed. A part of this work was done while I was a Postdoc at the Department of Mathematics, Yale University, where I benefited from conducive working conditions. Finally, I thank the referees for the feedback. This work was supported by the German Research Foundation (Grant numbers GU 1513/1-1 and GU 1513/2-1). en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes Article published earlier |
| spellingShingle | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes Gudapati, Nishanth |
| title | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_full | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_fullStr | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_full_unstemmed | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_short | On Axially Symmetric Perturbations of Kerr Black Hole Spacetimes |
| title_sort | on axially symmetric perturbations of kerr black hole spacetimes |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/213522 |
| work_keys_str_mv | AT gudapatinishanth onaxiallysymmetricperturbationsofkerrblackholespacetimes |