Timelike Geodesic Currents in the Stationary, Axisymmetric, Force-free   Magnetosphere of a Kerr Black Hole
  The structural properties of geodesic currents in an ambient Kerr background is studied from an analytical point of view. The geodesics in the congruence correspond to charged particles that carry energy and angular momentum from the black hole through the Blandford-Znajek mechanism. It is shown that the resulting magnetosphere naturally satisfies the Znajek regularity condition. Particular attention is paid here to the energy extracted by matter currents rather than by electromagnetic Poynting fluxes. 
Govind Menon, Charles Dermer