We study the accretion of relativistic Vlasov gas onto a Kerr black hole,
regarding the particles as distributed throughout all the space, other than
just in the equatorial plane. We solve the relativistic Liouville equation in
the full 3+1 dimensional framework of Kerr geometry. For the flow that is
stationary and axial symmetric, we prove that the distribution function is
independent of the conjugate coordinates. For an explicit distribution that can
approximate to Maxwell-J\"{u}ttner distribution, we further calculate the
particle current density, the stress energy momentum tensor and the unit
accretion rates of mass, energy and angular momentum. The analytic results at
large distance are shown to be consistent with the limits of the numerical ones
computed at finite distance. Especially, we show that the unit mass accretion
rate agrees with the Schwarzschild result in the case of low temperature limit.
Furthermore, we find from the numerical results that the three unit accretion
rates vary with the angle in Kerr metric and the accretion of Vlasov gas would
slow down the Kerr black hole. The closer to the equator, the faster it slows
down the black hole.Comment: 28 pages, 11 figures, accepted for publication in Physical Review