Near-$$AdS_2$$ perturbations and the connection with near-extreme Reissner–Nordstrom

Abstract

Abstract The geometry very near the horizon of a near-extreme Reissner–Nordstrom black hole is described by the direct product of a near-$$AdS_2$$AdS2 spacetime with a two-sphere. While near-$$AdS_2$$AdS2 is locally diffeomorphic to $$AdS_2$$AdS2 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 $$AdS_2\times S^2$$AdS2×S2 and the associated connection problem with extreme Reissner–Nordstrom. In this paper, we give the solution for perturbations of near-$$AdS_2\times S^2$$AdS2×S2 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.