  In this paper, we provide an elementary, unified treatment of two distinct
blue-shift instabilities for the scalar wave equation on a fixed Kerr black
hole background: the celebrated blue-shift at the Cauchy horizon (familiar from
the strong cosmic censorship conjecture) and the time-reversed red-shift at the
event horizon (relevant in classical scattering theory).
  Our first theorem concerns the latter and constructs solutions to the wave
equation on Kerr spacetimes such that the radiation field along the future
event horizon vanishes and the radiation field along future null infinity
decays at an arbitrarily fast polynomial rate, yet, the local energy of the
solution is infinite near any point on the future event horizon. Our second
theorem constructs solutions to the wave equation on rotating Kerr spacetimes
such that the radiation field along the past event horizon (extended into the
black hole) vanishes and the radiation field along past null infinity decays at
an arbitrarily fast polynomial rate, yet, the local energy of the solution is
infinite near any point on the Cauchy horizon.
  The results make essential use of the scattering theory developed in [M.
Dafermos, I. Rodnianski and Y. Shlapentokh-Rothman, A scattering theory for the
wave equation on Kerr black hole exteriors, preprint (2014) available at
\url{http://arxiv.org/abs/1412.8379}] and exploit directly the time-translation
invariance of the scattering map and the non-triviality of the transmission
map.
