Exponentially Growing Finite Energy Solutions for the Klein-Gordon Equation on Sub-Extremal Kerr Spacetimes

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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 (Formula presented). 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.

Original languageEnglish (US)
Pages (from-to)859-891
Number of pages33
JournalCommunications In Mathematical Physics
Volume329
Issue number3
DOIs
StatePublished - Aug 2014

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

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