Electromagnetic Extraction of Energy from Kerr Black Holes

Abstract

Abstract The energy extraction process from Kerr black holes is elucidated compared with that from pulsars in force-free degenerate electrodynamics (FFDE). It is argued for the force-free magnetosphere of a Kerr black hole in a steady axisymmetric state that the function of unipolar inductors in the presence of global magnetic fluxes threading the horizon is equipped by a frame-dragging effect in the Kerr spacetime metric to the upper null surface, $\mathrm{S}_{\mathrm{N}}$, with the axial distance, $\varpi_{\mathrm{N}}$, where the local value of the angular velocity of the frame-dragging, $\omega$, is equal to the rotational frequency of each field line, i.e., $\omega(\varpi_{\mathrm{N}}) = \Omega_{\mathrm{F}}$, and the rotational velocity of each field line, $\boldsymbol{v}_{\mathrm{F}}$, measured by “zero-angular-momentum-observers” changes in sign. There must be a pair-creation gap at $\mathrm{S}_{\mathrm{N}}$, from which the paired winds, ingoing and outgoing, are initiated. The poloidal electric current, $I$, for each wind is determined by solving the eigenvalue problem with the “criticality condition” imposed at the fast surfaces nearly at the horizon and at infinity, and the “current-closure condition” at $\mathrm{S}_{\mathrm{N}}$ determines the ultimate eigenvalues of $\Omega_{\mathrm{F}}$ and $I$ in terms of the hole’s angular velocity, $\Omega_{\mathrm{H}} = \omega(r_{\mathrm{H}})$, and the distribution of the magnetic flux at the horizon. It is not in the ordinary ergosphere, but in the effective ergosphere in the region of $\Omega_{\mathrm{H}} \gt \omega \gt \Omega_{\mathrm{F}}$ and $\boldsymbol{v}_{\mathrm{F}}<0$ that general-relativistic effects play the most significant role for electromagnetic extraction of the hole’s energy. The flow there must be regarded as an ingoing magneto-centrifugal wind, rather than an accretion flow. In a kind of extreme state that “force-free” field lines be frozen in “massless” particles in FFDE, the “membrane paradigm” combined with the DC circuit model is useful in black hole as well as pulsar magnetospheres.