Force-Free Black Hole Magnetospheres

It is argued that a force-free degenerate electrodynamic (FFDE) magnetosphere of a Kerr black hole with 0 $\lt$ $\Omega _{\rm F}$ $\lt$ ${\Omega_{\rm H}}$ consists of the outer classical and inner general-relativistic domains. This is described by a simple DC dual-circuit model, with dissipative membranes as two loads at a “force-free infinity surface” (S $_{{\rm ff}\infty }$ ) with $\omega$ $=$ 0 and at a “force-free horizon surface” (S $_{\rm ffH}$ ) with $\omega$ $=$ ${\Omega_{\rm H}}$ , where $\omega$ , ${\Omega_{\rm H}}$ , and $\Omega _{\rm F}$ are the frame-dragging, the horizon and the field line angular frequencies. It is beneath upper null surface, S $_{\rm N}$ , at $\omega$ $=$ $\Omega _{\rm F}$ between the two domains that dual unipolar batteries (double EMF’s) exist back-to-back, oppositely directed, with a pair-creation gap between. The total energy flux ${\boldsymbol{S}}_E$ is a linear sum of the two fluxes: the hole’s outward spin-down energy flux ${\boldsymbol{S}}_{\rm SD}$ originating at S $_{\rm ffH}$ and the Poynting flux $\boldsymbol{S}_{\rm EM}$ emitted at S $_{\rm N}$ in both the outward and inward directions, with ${\boldsymbol{S}}_E$ being proportional to $\Omega _{\rm F}$ , ${\boldsymbol{S}}_{\rm SD}$ to $\omega$ and $\boldsymbol{S}_{\rm EM}$ to ( $\Omega _{\rm F}$ $-$ $\omega$ ) along each field line. Applying a perturbation method for a split-monopolar field with a spin-parameter $h$ $\ll$ 1, the analytic solution of the stream equation is given, and the double eigenvalue problem due to the `criticality condition’ at the outer/inner fast surfaces S $_{\rm oF}$ /S $_{\rm iF}$ and the `boundary condition’ at S $_{\rm N}$ is solved to yield the final eigenvalue $\Omega _{\rm F}$ , in terms of ${\Omega_{\rm H}}$ and $f_{\rm H}$ $=$ 0.5676. The ratio of the output power reaching S $_{{\rm ff}\infty }$ to the dissipation on S $_{\rm ffH}$ is $\epsilon$ $=$ 1 $+$ (4 $/$ 5)(1 $-$ $f_{\rm H}$ ) $h^2$ .