Abstract
Bañados et al. (BSW) found that Kerr black holes can act as particle accelerators with collisions at arbitrarily high center-of-mass energies. Recently, collisions of particles with spin around some rotating black holes have been discussed. In this paper, we study the BSW mechanism for spinning particles by using a metric ansatz which describes a general rotating black hole. We notice that there are two inequivalent definitions of center-of-mass (CM) energy for spinning particles. We mainly discuss the CM energy defined in terms of the worldline of the particle. We show that there exists an energy-angular momentum relation e=Omega _h j that causes collisions with arbitrarily high energy near extremal black holes. We also provide a simple but rigorous proof that the BSW mechanism breaks down for nonextremal black holes. For the alternative definition of the CM energy, some authors find a new critical spin relation that also causes the divergence of the CM mass. However, by checking the timelike constraint, we show that particles with this critical spin cannot reach the horizon of the black hole. Further numerical calculation suggests that such particles cannot exist anywhere outside the horizon. Our results are universal, independent of the underlying theories of gravity.
Highlights
The aim of our paper is to explore this issue in a modelindependent way
For Kerr black holes Q = 0, Eq (63) further reduces to e2. This lower bound imposed on the energy of the particle per unit mass is a necessary condition for the BSW mechanism, which has been ignored in the literature
One can check that the divergence condition X = 0 violates the constraint (33) at the horizon, i.e., the divergent CM energy cannot be obtained for the nonextremal black hole through the collision on the horizon when the particle satisfied the critical relations
Summary
The aim of our paper is to explore this issue in a modelindependent way. A general stationary axisymmetric black hole can be described by the metric ansatz ds2 = α(r, θ )2 − (r, θ )dt2 + dr 2 (r, θ ). We derive the general critical relation j = he, where j and e are the angular momentum and energy per unit mass of the particle, and h is the angular velocity of the horizon. This relation leads to the divergence of the CM energy around extremal black holes. Definition 2 leads to a critical spin, causing divergence of the CM energy in Kerr–Sen black holes [21]. 3, we derive the divergence condition of the CM energy associated with Definition 1 for extremal black holes and show that the BSW mechanism breaks down for nonextremal black holes.
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