Before discussing the astrophysical implications of our result, we must address the three major limitations of our approach. First, we have made no attempt to include relativistic effects (beyond our simple treatment of the radius of marginal stability) on the dynamics or electrodynamics of the disk/field system. Our model is an adequate representation for slowly spinning black holes (where the radius of marginal stability is rather large and in a comparatively low-gravity region of spacetime), but we acknowledge that a full relativistic electrodynamic treatment is required to robustly treat the case of rapidly rotating black holes. While the same basic phenomenon of magnetic flux trapping by the plunge region should be at work around rapidly rotating black holes, the geometry of the system (i.e., the fact that the radius of marginal stability is much closer to the event horizon) might be unfavorable for producing dramatic enhancements in the black hole-threading magnetic field. On the other hand, an ergospheric wind (Punsly & Coroniti 1990; Punsly 1991) could aid in the production of a strong poloidal field (through the inertial effects of the outflowing plasma) as well as the inward dragging of field (through the removal of angular momentum from the accretion flow).
Second, we assume the existence of a pre-existing large scale magnetic field. The origin of such a field depends upon the system under consideration. For the accreting black hole at the heart of a Gamma-Ray Burst (GRB) collapsar, such a field may arise naturally from the collapsed stellar envelope. In the case of AGN, the field corresponds to that of the accreting interstellar medium. For GBHBs, the presence of a large scale field probably depends on the mode of accretion, with wind-accretors likely possessing a much stronger and better organized large scale field than Roche-lobe overflow accretors.
Third, we assume axisymmetric large scale fields with a disk magnetosphere consisting of force-free and purely poloidal field. As mentioned in Section 2, a more physical treatment would entail matching an MHD wind solution to the disk-plane flux function. With such an approach, one could capture the inertial effects of a disk outflow on the field structure, the hoop stresses resulting from any toroidal fields present, and the angular momentum losses in the disk due to the wind. These could have competing effects on the ultimate ability of the disk to drag the field into the plunge region. The inertial effects will tend to bend the field lines outwards, increasing the field-line curvature at the disk plane and hence increasing outward diffusion of the field. The loss of disk angular momentum to the wind, on the other hand, would lead to an increase in the radial velocity of the accretion flow but no change in the magnetic diffusivity. This, in turn, increases the inwards advection of the magnetic field. Clearly, more detailed calculations of this scenario are warranted. As for the axisymmetric assumption, we note that Spruit & Uzdensky (2005) have recently examined the dragging of a large scale magnetic field by an accretion disk under the assumption that the MHD turbulence in the disk concentrates the field into small bundles (giving rise to the accretion disk equivalent of Sun spots). Through an analysis of the dynamics of these bundles, they conclude that this is a generally favorable scenario for accumulating a large amount of magnetic flux in the central regions of the disk. Thus, in at least one specific model, an extreme deviation from axisymmetry aids in the inward dragging of magnetic flux.
We reiterate that the principal result of this paper is that the existence of a plunge region together with magnetic field dragging in the accretion disk can significantly enhance the black hole-threading magnetic field and hence the BZ effect. Furthermore, the enhancement becomes increasing effective for thicker disks or higher magnetic Prandtl numbers. However, we acknowledge that the caveats given above, together with the dependence of the enhancement on the size of the dead zone, prevents us from further quantifying the enhancement.