TY - JOUR

T1 - Exponentially Growing Finite Energy Solutions for the Klein-Gordon Equation on Sub-Extremal Kerr Spacetimes

AU - Shlapentokh-Rothman, Yakov

N1 - Funding Information:
This work was partially supported by NSF grant DMS-0943787.

PY - 2014/8

Y1 - 2014/8

N2 - 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.

AB - 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.

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U2 - 10.1007/s00220-014-2033-x

DO - 10.1007/s00220-014-2033-x

M3 - Article

AN - SCOPUS:84901819637

SN - 0010-3616

VL - 329

SP - 859

EP - 891

JO - Communications In Mathematical Physics

JF - Communications In Mathematical Physics

IS - 3

ER -