Scattering of gravitational and electromagnetic waves off $AdS_2\times S^2$ in extreme Reissner-Nordstrom

TL;DR

This paper analytically solves coupled gravitational and electromagnetic perturbation equations on the $AdS_2\times S^2$ background, connecting near-horizon and asymptotic regions of the extreme Reissner-Nordstrom black hole to compute scattering properties.

Contribution

It provides the first analytical solutions to perturbation equations on $AdS_2\times S^2$ and connects these to the full ERN black hole perturbations using matched asymptotic expansions.

Findings

Analytical solutions for perturbations on $AdS_2\times S^2$

Connection of near-horizon and asymptotic solutions

Calculation of classical scattering matrix for waves

Abstract

The direct product of two-dimensional anti-de Sitter spacetime with a two-sphere is an exact solution of four-dimensional Einstein-Maxwell theory without a cosmological constant. In this paper, we analytically solve the coupled gravitational and electromagnetic perturbation equations of $AdS_2\times S^2$ in Einstein-Maxwell theory. On the other hand, $AdS_2\times S^2$ also describes the near-horizon region of the extreme Reissner-Nordstrom (ERN) black hole. We therefore also solve the connection problem: we show how the $AdS_2\times S^2$ perturbation equations arise from an appropriate near-horizon approximation of the corresponding equations for the ERN and then, using matched asymptotic expansions, we analytically extend the $AdS_2\times S^2$ solutions away from the near-horizon region connecting them with solutions in the far asymptotically flat region. From the point of view of ERN our results may be thought of as computing the classical scattering matrix for gravitational and electromagnetic waves which probe the near-horizon region of the black hole.