Note on bunching of field lines in black hole magnetospheres

Numerical simulations of Blandford-Znajek energy extraction at high spin have revealed that field lines tend to bunch near the poles of the event horizon. We show that this behavior can be derived analytically from the assumption of fixed functional dependence of current and field line rotation on magnetic flux. The argument relies crucially on the existence of the Znajek condition, which offers nontrivial information about the fields on the horizon without requiring a full force-free solution. We also provide some new analytic expressions for the parabolic field configuration.

Figure 1

Bunching of field lines for radial, parabolic, and hyperbolic geometries. Diagrams of the field line geometry appear in the upper right. On the left, the dimensionless current $\mathcal{I}$ (black line) and angular velocity $\mathcal{O}$ (gray line) as a function of magnetic flux $\psi /{\psi }_{*}$ on the horizon $\psi <{\psi }_{*}$, which we assume to hold at all spin. On the right, the radial magnetic field $B^r={\partial }_{\theta }\psi /\sqrt{-g}$ as a function of angle $\theta <\pi /2$ on the horizon for different values of the spin $a=0.1$, 0.5, 0.9, 0.99, 0.999, 0.9999 (upwards on the left).