Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities
Abstract In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild metrics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively correspond to the closed a...
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| Format: | Article |
| Language: | English |
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Springer International Publishing
2021
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| Online Access: | https://hdl.handle.net/1721.1/131466 |
| _version_ | 1811087639976083456 |
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| author | Hung, Pei-Ken Keller, Jordan Wang, Mu-Tao |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Hung, Pei-Ken Keller, Jordan Wang, Mu-Tao |
| author_sort | Hung, Pei-Ken |
| collection | MIT |
| description | Abstract
In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild metrics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively correspond to the closed and co-closed, or polar and axial, solutions in the case of four spacetime dimensions, while the two-tensor portion is a new feature in the higher-dimensional setting. Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge decomposition, we produce decoupled gauge-invariant master quantities satisfying Regge-Wheeler type wave equations in each of the three portions. The scalar and co-vector quantities respectively generalize the Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four spacetime dimensions; beyond these quantities, we discover a higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the co-vector portion. In addition, our work provides the first verification that the scalar master quantity satisfies its putative Regge-Wheeler equation. In the analysis of the master quantities, we strengthen the mode stability result of Kodama-Ishibashi to a uniform boundedness estimate in all dimensions; further, we prove decay estimates in the case of six or fewer spacetime dimensions. In the case of more than six spacetime dimensions, we discover an obstruction to Morawetz type estimates arising from negative potential terms growing quadratically in spacetime dimension. Finally, we provide a rigorous argument that linearized solutions of low angular frequency are decomposable as a sum of pure gauge solution and linearized Myers-Perry solution, the latter solutions generalizing the linearized Kerr solutions in four spacetime dimensions. |
| first_indexed | 2024-09-23T13:49:40Z |
| format | Article |
| id | mit-1721.1/131466 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:49:40Z |
| publishDate | 2021 |
| publisher | Springer International Publishing |
| record_format | dspace |
| spelling | mit-1721.1/1314662024-01-02T18:39:26Z Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities Hung, Pei-Ken Keller, Jordan Wang, Mu-Tao Massachusetts Institute of Technology. Department of Mathematics Abstract In this paper, we study solutions to the linearized vacuum Einstein equations centered at higher-dimensional Schwarzschild metrics. We employ Hodge decomposition to split solutions into scalar, co-vector, and two-tensor pieces; the first two portions respectively correspond to the closed and co-closed, or polar and axial, solutions in the case of four spacetime dimensions, while the two-tensor portion is a new feature in the higher-dimensional setting. Rephrasing earlier work of Kodama-Ishibashi-Seto in the language of our Hodge decomposition, we produce decoupled gauge-invariant master quantities satisfying Regge-Wheeler type wave equations in each of the three portions. The scalar and co-vector quantities respectively generalize the Moncrief-Zerilli and Regge-Wheeler quantities found in the setting of four spacetime dimensions; beyond these quantities, we discover a higher-dimensional analog of the Cunningham-Moncrief-Price quantity in the co-vector portion. In addition, our work provides the first verification that the scalar master quantity satisfies its putative Regge-Wheeler equation. In the analysis of the master quantities, we strengthen the mode stability result of Kodama-Ishibashi to a uniform boundedness estimate in all dimensions; further, we prove decay estimates in the case of six or fewer spacetime dimensions. In the case of more than six spacetime dimensions, we discover an obstruction to Morawetz type estimates arising from negative potential terms growing quadratically in spacetime dimension. Finally, we provide a rigorous argument that linearized solutions of low angular frequency are decomposable as a sum of pure gauge solution and linearized Myers-Perry solution, the latter solutions generalizing the linearized Kerr solutions in four spacetime dimensions. 2021-09-20T17:17:11Z 2021-09-20T17:17:11Z 2020-06-27 2020-09-24T21:17:55Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/131466 Annals of PDE. 2020 Jun 27;6(2):7 en https://doi.org/10.1007/s40818-020-00083-x Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. Springer Nature Switzerland AG application/pdf Springer International Publishing Springer International Publishing |
| spellingShingle | Hung, Pei-Ken Keller, Jordan Wang, Mu-Tao Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities |
| title | Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities |
| title_full | Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities |
| title_fullStr | Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities |
| title_full_unstemmed | Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities |
| title_short | Linear Stability of Higher Dimensional Schwarzschild Spacetimes: Decay of Master Quantities |
| title_sort | linear stability of higher dimensional schwarzschild spacetimes decay of master quantities |
| url | https://hdl.handle.net/1721.1/131466 |
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