Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes
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Author(s)
Shlapentokh-Rothman, Yakov Mordechai
Date Issued
April 2014
Journal
Communications in Mathematical Physics
Publisher
Springer Berlin Heidelberg
Citation
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.
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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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.
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DOI of Published Version
https://doi.org/10.1007/s00220-014-2033-x
