  The geometry very near the horizon of a near-extreme Reissner-Nordstrom black
hole is described by the direct product of a near-$AdS_2$ spacetime with a
two-sphere. While near-$AdS_2$ is locally diffeomorphic to $AdS_2$ the two
connect differently with the asymptotically flat part of the geometry of
(near-)extreme Reissner-Nordstrom. In previous work, we solved analytically the
coupled gravitational and electromagnetic perturbation equations of
$AdS_2\times S^2$ and the associated connection problem with extreme
Reissner-Nordstrom. In this paper, we give the solution for perturbations of
near-$AdS_2\times S^2$ and make the connection with near-extreme
Reissner-Nordstrom. Our results here may also be thought of as computing the
classical scattering matrix for gravitational and electromagnetic waves which
probe the region very near the horizon of a highly charged spherically
symmetric black hole.
