Black-hole electrodynamics: an absolute-space/universal-time formulation*

Abstract

This paper reformulates and extends the Blandford–Znajek theory of a stationary, axisymmetric magnetosphere anchored in a black hole and in its accretion disc. Such a magnetosphere should transfer much of the rotational energy of the hole and orbital energy of the disc into an intense flux of electromagnetic energy – which in turn might be the energizer for quasars and active galactic nuclei. Our reformulation of the theory attempts to make it accessible to plasma astrophysicists who have little experience with general relativity. This is done by replacing the relativist's ‘unified spacetime’ viewpoint with an equivalent Galilean-type ‘absolute-space-plus-universal-time’ viewpoint, and by replacing the electromagnetic field tensor |$F_{\mu\nu}$| with electric and magnetic fields E and B that reside in the absolute space outside the black hole. The resulting formalism resembles the theory of axisymmetric pulsar magnetospheres; it will, we hope, permit a fairly easy transfer of physical intuition and results from the pulsar problem to the black-hole problem. The Blandford–Znajek theory focused primarily on force-free regions of the magnetosphere. This paper, in addition to recasting the force-free theory in new language, extends it to encompass regions that are degenerate (E · B = 0) but not force-free, and regions that are neither degenerate nor force-free. Blandford & Znajek showed that the magnetospheric structure in the force-free region is determined by a general relativistic ‘stream differential equation’. This paper presents an action principle for the stream equation, it elucidates the boundary conditions that one must pose on the stream function ψ, and it shows that ψ and the poloidal magnetic field distribute themselves over the hole's horizon in such a manner as to extremize the horizon's electromagnetic surface energy. This paper also constructs a general relativistic version of DC electronic circuit theory and uses it to elucidate the flows of electric current and of electromagnetic power in the magnetosphere. The circuit-theory analysis, and independently a torque-balance analysis, suggests that those magnetic field lines which thread the hole will be dragged into rotation with roughly half the angular velocity of the hole – and, consequently, that the hole will deliver to the magnetosphere the maximum electromagnetic power permitted by the horizon strengths of the magnetic fields.