Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations
We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form...
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| Veröffentlicht in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Datum: | 2022 |
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| Format: | Artikel |
| Sprache: | Englisch |
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Інститут математики НАН України
2022
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| Online Zugang: | https://nasplib.isofts.kiev.ua/handle/123456789/211534 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations. Igor Khavkine. SIGMA 18 (2022), 011, 57 pages |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862710640970301440 |
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| author | Khavkine, Igor |
| author_facet | Khavkine, Igor |
| citation_txt | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations. Igor Khavkine. SIGMA 18 (2022), 011, 57 pages |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into a sparse triangular form, where the diagonal blocks consist of spin- scalar Regge-Wheeler equations (for spins = 0,1,2). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous, more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges.
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| first_indexed | 2026-03-19T12:55:46Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-211534 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-19T12:55:46Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Khavkine, Igor 2026-01-05T12:28:37Z 2022 Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations. Igor Khavkine. SIGMA 18 (2022), 011, 57 pages 1815-0659 2020 Mathematics Subject Classification: 35Q75; 34L99; 34L05; 68W30 arXiv:2004.09651 https://nasplib.isofts.kiev.ua/handle/123456789/211534 https://doi.org/10.3842/SIGMA.2022.011 We consider the vector and the Lichnerowicz wave equations on the Schwarzschild spacetime, which correspond to the Maxwell and linearized Einstein equations in harmonic gauges (or, respectively, in Lorenz and de Donder gauges). After a complete separation of variables, the radial mode equations form complicated systems of coupled linear ODEs. We outline a precise abstract strategy to decouple these systems into a sparse triangular form, where the diagonal blocks consist of spin- scalar Regge-Wheeler equations (for spins = 0,1,2). Building on the example of the vector wave equation, which we have treated previously, we complete a successful implementation of our strategy for the Lichnerowicz wave equation. Our results go a step further than previous, more ad-hoc attempts in the literature by presenting a full and maximally simplified final triangular form. These results have important applications to the quantum field theory of and the classical stability analysis of electromagnetic and gravitational perturbations of the Schwarzschild black hole in harmonic gauges. Research of the author's research was partially supported by the Praemium Academiae of M. Markl, GACR project GA18-07776S, and RVO: 67985840. Early stages of this work were completed while the author was affiliated with the University of Rome II (Tor Vergata) and with the University of Milan (Statale). Thanks to Claudio Dappiaggi, Felix Finster, Christian Gerard, Dietrich Hafner, Peter Hintz, Dmitry Jakobson, Niky Kamran, Vesselin Petkov, Iosif Polterovich, Ko Sanders, Werner Seiler, and Artur Sergyeyev for interesting discussions. Thanks to Stefanos Aretakis and Mihalis Dafermos for their hospitality during a visit to Princeton University, where an early version of this work was first presented in the Spring of 2016. The author also thanks Francesco Bussola for checking the mode decompositions (5.6c) and (5.6d) as part of his MSc thesis [13]. And final thanks go to Roman O. Popovych and to anonymous referees, whose comments greatly improved the presentation of the manuscript. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations Article published earlier |
| spellingShingle | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations Khavkine, Igor |
| title | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_full | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_fullStr | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_full_unstemmed | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_short | Explicit Triangular Decoupling of the Separated Lichnerowicz Tensor Wave Equation on Schwarzschild into Scalar Regge-Wheeler Equations |
| title_sort | explicit triangular decoupling of the separated lichnerowicz tensor wave equation on schwarzschild into scalar regge-wheeler equations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211534 |
| work_keys_str_mv | AT khavkineigor explicittriangulardecouplingoftheseparatedlichnerowicztensorwaveequationonschwarzschildintoscalarreggewheelerequations |