The interior structure of rotating black holes 3. Charged black holes

Abstract

This paper extends to the case of charged rotating black holes the conformally stationary, axisymmetric, conformally separable solutions presented for uncharged rotating black holes in a companion paper. In the present paper, the collisionless fluid accreted by the black hole may be charged. The charge of the black hole is determined self-consistently by the charge accretion rate. As in the uncharged case, hyper-relativistc counter-streaming between ingoing and outgoing streams drives inflation at (just above) the inner horizon, followed by collapse. If both ingoing and outgoing streams are charged, then conformal separability holds during early inflation, but fails as inflation develops. If conformal separability is imposed throughout inflation and collapse, then only one of the ingoing and outgoing streams can be charged: the other must be neutral. Conformal separability prescribes a hierarchy of boundary conditions on the ingoing and outgoing streams incident on the inner horizon. The dominant radial boundary conditions require that the incident ingoing and outgoing number densities be uniform with latitude, but the charge per particle must vary with latitude such that the incident charge densities vary in proportion to the radial electric field. The sub-dominant angular boundary conditions require specific forms of the incident number- and charge-weighted angular motions. If the streams fall freely from outside the horizon, then the prescribed angular conditions can be achieved by the charged stream, but not by the neutral stream. Thus, as in the case of an uncharged black hole, the neutral stream must be considered to be delivered ad hoc to just above the inner horizon.