High resolution radio imaging of AGN have revealed that some sources present motion of superluminal knots and transverse stratification of their jet. Recent observational projects have provided new observational constraints on the central region of rotating black holes in AGN, suggesting that there is an inner- or spine-jet surrounded by a disk wind. This relativistic spine-jet is likely to be composed of electron - positron pairs extracting energy from the black hole. In this article we present an extension and generalization to relativistic jets in Kerr metric of the meridional self similar mechanism. We aim at modeling the inner spine-jet of AGN as the relativistic light outflow emerging from a spherical corona surrounding a Kerr black hole. The model is built by expanding the metric and the forces with colatitude to first order in the magnetic flux function. Conversely to previous models, effects of the light cylinder are not neglected. Solutions with high Lorentz factor are obtained and provide spine-jet models up to the polar axis. As in previous publications, we calculate the magnetic collimation efficiency parameter, which measures the variation of the available energy across the field lines. This collimation efficiency is an integral of the model, generalizing to Kerr metric the classical magnetic rotator efficiency criterion. We study the variation of the magnetic efficiency and acceleration with the spin of the black hole and show their high sensitivity to this integral. These new solutions model collimated or radial, relativistic or ultra-relativistic outflows. We discuss the relevance of our solutions to model the M87 spine-jet. We study the efficiency of the central black hole spin to collimate a spine-jet and show that the jet power is of the same order with that determined by numerical simulations.
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