Kinematic restrictions on particle collisions near extremal black holes: A unified picture

In 2009, Banados, Silk and West (BSW) pointed out the possibility of having an unbounded limit of centre-of-mass collision energy for test particles in the field of an extremal Kerr black hole, if one of them has fine-tuned parameters and the collision point is approaching the horizon. The possibility of this "BSW effect" attracted much attention: it was generalised to arbitrary ("dirty") rotating black holes and an analogy was found for collisions of charged particles in the field of non-rotating charged black holes. Our work considers the unification of these two mechanisms, which have so far been studied only separately. Exploring the enlarged parameter space, we find kinematic restrictions that may prevent the fine-tuned particles from reaching the limiting collision point. These restrictions are first presented in a general form, which can be used with an arbitrary black-hole model, and then visualised for the Kerr-Newman solution by plotting the "admissible region" in the parameter space of critical particles, reproducing some known results and obtaining a number of new ones. For example, we find that (marginally) bounded critical particles with enormous values of angular momentum can, curiously enough, approach the degenerate horizon, if the charge of the black hole is very small. Such "mega-BSW" behaviour is excluded in the case of a vacuum black hole, or a black hole with large charge. It may be interesting in connection with the small "Wald charge" induced on rotating black holes in external magnetic fields.