Black hole electrodynamics: How does unipolar induction work in Kerr black holes?

Abstract

Abstract It is argued that through magnetohydrodynamic couplings with the field-line angular velocity/the cross-field potential gradient ΩF, the frame-dragging angular velocity ω acquires a new role of the gravito-electric potential gradient, which produces the along-field gradient of the potential gradient. With use of the field-line angular velocity, ΩFω ≡ ΩF − ω, measured by fiducial observers living in “absolute space” circulating with ω, it is then shown that the hole's spin rate ΩH also has another side of the along-field difference of the potential gradient between the surface-at-infinity S∞ and the horizon surface SH, i.e., ΩH = ΔΩFω ≡ (ΩFω)∞ − (ΩFω)H, which gives rise to a voltage drop ΔV ∝ ΔΩFω = ΩH at the interface SN with (ΩFω)N = 0 between the outer and inner force-free domains, leading to formation of a magnetized gap under SN. The EMFs due to a pair of unipolar induction batteries, ${\cal E}_{\rm out}$ and ${\cal E}_{\rm in}$, drive currents to a pair of circuits in the two domains, and satisfy relation ${\cal E}_{\rm out}-{\cal E}_{\rm in}=\Delta V$. The eigenvalue ΩF is determined by the criticality-boundary condition in MHD wind theory in terms of ΩH, i.e., ≈ (1/2)ΩH. The present gap model with a pair of batteries and a strong voltage drop is fundamentally different from any existing pulsar gap models.