  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.
