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 \({\frac{\left|am\right|}{2Mr_+}}\). 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.
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Communicated by P. T. Chruściel
This work was partially supported by NSF grant DMS-0943787.
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Shlapentokh-Rothman, Y. Exponentially Growing Finite Energy Solutions for the Klein–Gordon Equation on Sub-Extremal Kerr Spacetimes. Commun. Math. Phys. 329, 859–891 (2014). https://doi.org/10.1007/s00220-014-2033-x
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DOI: https://doi.org/10.1007/s00220-014-2033-x