Behaviour of scalar perturbations of a Reissner-Nordström black hole inside the event horizon

Abstract
This paper considers general scalar perturbations of a Reissner-Nordstrdöm black hole and examines the qualitative behaviour of these perturbations in the region between and on the inner and outer horizons ( r - ≼ r ≼ r + ). Initial data are specified in terms of the ingoing radiation crossing the outer (event) horizon. The only essential restriction on these data is that the radiation should not die away too slowly on this horizon. The resultant perturbations are shown to be bounded and continuous. It is also shown that if ũ is any retarded null coordinate such that ũ = 0 on the event horizon, then the perturbations tend to zero along lines of constant radius as ũ ↓ 0. In particular, all these properties hold for pertur­bations on the inner horizon. For certain types of scalar field (including the zero rest mass scalar field) perturbations vanish at the crossover point on the inner horizon.

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