  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.
