Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes
For any sub-extremal Kerr spacetime with non-zero angular momentum, we find an open family of non-zero masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein–Gordon equation. If desired, for any non-zero integer m, an exponentially growing...
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| Format: | Article |
| Language: | English |
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Springer Berlin Heidelberg
2016
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| Online Access: | http://hdl.handle.net/1721.1/104927 |
| _version_ | 1826212188156592128 |
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| author | Shlapentokh-Rothman, Yakov Mordechai |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Shlapentokh-Rothman, Yakov Mordechai |
| author_sort | Shlapentokh-Rothman, Yakov Mordechai |
| collection | MIT |
| description | For any sub-extremal Kerr spacetime with non-zero angular momentum, we find an open family of non-zero masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein–Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to |am|/2Mr+. In addition to its direct relevance for the stability of Kerr as a solution to the Einstein–Klein–Gordon system, our result provides the first rigorous construction of a superradiant instability. Finally, we note that this linear instability for the Klein–Gordon equation contrasts strongly with recent work establishing linear stability for the wave equation. |
| first_indexed | 2024-09-23T15:17:45Z |
| format | Article |
| id | mit-1721.1/104927 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:17:45Z |
| publishDate | 2016 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1049272022-09-29T14:01:01Z Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes Shlapentokh-Rothman, Yakov Mordechai Massachusetts Institute of Technology. Department of Mathematics Shlapentokh-Rothman, Yakov Mordechai For any sub-extremal Kerr spacetime with non-zero angular momentum, we find an open family of non-zero masses for which there exist smooth, finite energy, and exponentially growing solutions to the corresponding Klein–Gordon equation. If desired, for any non-zero integer m, an exponentially growing solution can be found with mass arbitrarily close to |am|/2Mr+. In addition to its direct relevance for the stability of Kerr as a solution to the Einstein–Klein–Gordon system, our result provides the first rigorous construction of a superradiant instability. Finally, we note that this linear instability for the Klein–Gordon equation contrasts strongly with recent work establishing linear stability for the wave equation. National Science Foundation (U.S.) (Grant DMS-0943787) 2016-10-21T21:40:16Z 2016-10-21T21:40:16Z 2014-04 2016-08-18T15:24:07Z Article http://purl.org/eprint/type/JournalArticle 0010-3616 1432-0916 http://hdl.handle.net/1721.1/104927 Zhang, ShunRong et al. “Ionospheric Longitudinal Variations at Midlatitudes: Incoherent Scatter Radar Observation at Millstone Hill.” Science China Technological Sciences 55.5 (2012): 1153–1160. en http://dx.doi.org/10.1007/s00220-014-2033-x Communications in Mathematical Physics 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-Verlag Berlin Heidelberg application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Shlapentokh-Rothman, Yakov Mordechai Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes |
| title | Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes |
| title_full | Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes |
| title_fullStr | Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes |
| title_full_unstemmed | Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes |
| title_short | Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes |
| title_sort | exponentially growing finite energy solutions for the klein gordon equation on sub extremal kerr spacetimes |
| url | http://hdl.handle.net/1721.1/104927 |
| work_keys_str_mv | AT shlapentokhrothmanyakovmordechai exponentiallygrowingfiniteenergysolutionsforthekleingordonequationonsubextremalkerrspacetimes |