  Aretakis has proved the existence of an instability of a massless scalar
field at the horizon of an extreme Kerr or Reissner-Nordstrom black hole: for
generic initial data, a transverse derivative of the scalar field at the
horizon does not decay, and higher transverse derivatives blow up. We show that
a similar instability occurs for linearized gravitational, and electromagnetic,
perturbations of an extreme Kerr black hole. We show also that the massless
scalar field instability occurs for extreme black hole solutions of a large
class of theories in various spacetime dimensions.
