The evolution of a black hole's force-free magnetosphere

This paper elucidates the structure of a stationary axisymmetric force-free magnetosphere of a Kerr black hole, and the hole's evolution due to extraction of rotational energy by the Blandford–Znajek process. It is argued that there is an ‘effective’ ergoregion inside the static-limit surface where the densities of ‘energy at infinity’ and angular momentum of the field are negative. In the outer half of the charged magnetosphere, a pulsar-type centrifugal slingshot wind blows outwards, whereas in the inner half, a similar centrifugal slingshot wind blows inwards. The critical conditions for both winds at the fast magnetosonic points near the ‘stretched’ horizon and the infinity horizon, yield the two expressions for the angular momentum flux per unit flux tube I(Ψ) as a function of stream function Ψ in terms of $$({\Omega}_\text{H}-{\Omega}_\text{F})$$ and Ω_F and respective magnetic fluxes, where Ω_H is the hole's angular velocity and Ω_F is the angular velocity of field line Ψ. Matching the two I(Ψ)s at both ends of each field line yields $${\Omega}_\text{F}={\Omega}_\text{H}/(1+\zeta ) \,\text{and} \,I(\Psi)=\left[\zeta /2(1+\zeta ) \right]{\Omega}_{\text H} (B_P\omega^2)_\rm H$$, where $$\zeta(\Psi)\equiv(B_P\omega^2)_\text L/(B_P\omega^2)_\rm H$$ is a parameter expressing degree of collimation of field lines and suffix L denotes the boundary to the ‘load region’. For $$\zeta \rightarrow 0$$, the extraction process is almost adiabatic, and the hole with angular momentum J_H loses its rotational energy, following the path of constant entropy with infinite long time in the $${J}_\text{H}-{\Omega}_\text{H}\,\text{plane}. \,\text{For}\,\zeta\rightarrow \infty $$, all of the rotational energy is wasted by the surface Joule dissipation and the hole with mass M_H moves on a curve of constant M_H in the $${J}_\text{H}-{\Omega}_\text{H}$$ plane. The effects of variable ζ upon a hole's magnetospheric structure, wind system and evolution are discussed in some detail.