Evolution of perturbations inside a charged black hole: Linear scalar field

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Abstract
We present a simple method for the analysis of perturbations inside a charged black hole, to which we refer as the late-time expansion. This method has the advantage that it can be straightforwardly applied to nonlinear perturbations as well. It can also be generalized to (linear and nonlinear) metric perturbations in Kerr spacetime, allowing the investigation of the singularity at the inner horizon of a generically perturbed spinning black hole. In this paper we describe, in full detail, the application of our method to a linear scalar field on a Reissner-Nordström background. In particular, we analyze the asymptotic behavior near the inner horizon. The scalar field itself is found to be finite at the inner horizon, but its gradient diverges there. We compare our results to those obtained previously by Gursel and co-workers and by Chandrasekhar and Hartle (by a different method) and find full consistency.