Near-horizon behavior of nonequatorial accelerated particle motion and high energy particle collisions

We consider the motion of a particle in the background of a stationary axially symmetric generic black hole. A particle experiences the action of a force of unspecified nature. We require the force to remain finite in a comoving frame. The result is expressed in terms of several integers characterizing the Taylor expansion of the metric coefficients near the horizon. We show that the polar component of the four-velocity remains finite. As a result, the scenarios of high energy particle collisions, found previously in the context of the BSW effect for equatorial motion, do not change qualitatively in the nonequatorial case. The fact that the polar component is finite, also enables us to fill some gaps in the description of the BSW effect in previous works. This includes (i) classification of particle trajectories for nonequatorial motion, (ii) more coherent derivation of kinematics of the BSW effect, and (iii) discussion of bands in which the BSW effect takes place. Our results have a quite general character and can be used not only in description of high energy particle collisions but also in diverse astrophysical problems for which motion is not constrained by the equatorial plane.