Bañados, Silk and West (BSW) [1] demonstrated some years ago that two particles colliding near the degenerate horizon of an extreme Kerr black hole could create a large center of mass (CM) energy if one of the particles has a critical angular momentum; thus, extreme Kerr black holes can act as natural particle accelerators. Nowadays, this process is known as the BSW mechanism, which was found for the first time by Piran, Shaham and Katz in 1975 [2,3,4]. Then, based on the infinite acceleration being able to occur not only for extremal black holes but also for non-extremal ones [5], it was shown that an infinite energy in the CM frame of colliding particles is a universal property of rotating black holes as long as the angular momentum of one of the colliding particles approaches the critical value [6]. The requirement that the black hole be rotating seems to be an essential ingredient in obtaining ultra-high CM energy; however, it was shown that the similar effect exists for non-rotating charged black holes [7]. Moreover, the extension of the BSW mechanism to non-extremal backgrounds shows that particles cannot collide with arbitrarily high energies at the outer horizon and that ultra-energetic collisions can only occur near the Cauchy horizon of a Kerr black hole with any spin parameter [8]. The non-extremal Kerr-de Sitter black holes could also act as particle accelerators with arbitrarily high CM energy if one of the colliding particles has the critical angular momentum [9].
The BSW mechanism has been studied for different black hole geometries; for instance, rotating charged cylindrical black holes were studied in Ref. [10] and an extreme rotating black holes in Horava–Lifshitz gravity in Ref. [11]. However, the fundamental parameter of Horava–Lifshitz gravity avoids an infinite value of the CM energy being obtained. Higher-dimensional rotating black holes have been studied in Refs. [12,13,14] and lower-dimensional black holes in Refs. [15,16,17,18,19]. The collision of two neutral particles on the vicinity of the extremal Kerr black hole horizon considered as a complete vacuum spacetime in its own right was studied in [20]. Particle collision in the strong gravitational field of a rotating black hole in a Randall–Sundrum brane with a nonvanishing cosmological constant was studied in [21]. The particle collisions near the cosmological horizon of non-extremal Reissner–Nordstrom de Sitter black holes were studied in Ref. [22], charged dilatonic black holes in Ref. [23], non-rotating and rotating regular black holes in Refs. [24,25,26,27], and extremal modified Hayward and Bardeen rotating black holes in Ref. [28]. On the other hand, charged particles in general stationary charged black holes was considered in Ref. [29], and on string black holes in Ref. [30]. The collisions of spinning particles on rotating black holes in Refs. [31, 32], and on Schwarzschild black holes was considered in Ref. [33]; however, the unavoidable appearance of superluminal motion and the change of trajectories from timelike to spacelike can be avoided as the energy in the CM frame can grow unbounded provided that one of the particles is not exactly critical but slightly deviates from the critical trajectory [34]. On the other hand, the formation of black holes through the BSW mechanism was investigated in [35].
The aim of this work is to consider the special class of well-known three-dimensional warped anti-de Sitter black holes solutions [36,37,38,39] and to study, via the BSW mechanism, the possibility of obtaining unbounded energy in the CM frame of two colliding particles and to analyze the effect of the warped parameter that controls the stretching deformation on this. It is worth noting that warped AdS3AdS3 black holes can be viewed as discrete quotients of warped AdS3AdS3 spacetime just like the BTZ black holes as discrete quotients of the AdS3AdS3. Also, when the warped parameter ν=1ν=1, the metric reduces to the metric of a BTZ black hole in a rotating frame. An important feature of these black holes is that the Killing vector ∂t∂t is spacelike everywhere in spacetime and consequently its ergoregion extends to infinity. Also, it is known that two particles in the ergosphere lead to infinity growth of the energy of the CM frame, provided the angular momentum of one of the two particles has a large negative angular momentum and a fixed energy at infinity for the Kerr black holes [5], which was subsequently proven to be a universal property of the ergosphere [40].
It worth mentioning that the collision of two particles near the horizon of a BTZ black hole was studied in Refs. [16,17,18]. In Refs. [16, 17] the authors found that the particle with the critical angular momentum could exist inside the outer horizon of the BTZ black hole regardless of the particle energy with the BSW process being possible on the inner horizon for the non-extremal BTZ black hole. Also, the BSW process could also happen for the extremal BTZ black hole, where the particle with the critical angular momentum could only exist on the degenerate horizon. On the other hand, in Ref. [18], the authors studied the collision of two particles on an event horizon and outside of the BTZ black hole, and they showed that although in principle the CM energy of two ingoing particles can be arbitrarily large on an event horizon, if either of the two particles has a critical angular momentum and the other has a non-critical angular momentum, the critical particles never reach the event horizon. However, the motion of a particle with a subcritical angular momentum is allowed near an extremal rotating BTZ black hole and that a CM energy for a tail-on collision at a point can be arbitrarily large in a critical angular momentum limit.
On the other hand, the warped AdS3AdS3 space has non-AdS asymptotics and is not a solution of pure three-dimensional gravity, and it appears as a submanifold, at fixed polar coordinate, of the near horizon extreme Kerr black hole [41, 42]. So, it is interesting to study if the BSW effect, which was first noted for the extreme Kerr black holes, also occurs in the warped AdS3AdS3 black hole in analogy with the extremal Kerr geometry. In fact, we will show that the BSW effect is possible on the outer horizon in the extremal warped AdS3AdS3 black hole, and the particle with critical angular momentum can reach the degenerate horizon when a condition on its energy is fulfilled, which resembles to what occurs in the extremal Kerr-AdS black hole; however, in the extremal Kerr-AdS black hole two conditions must be fulfilled [9]; besides, this effect is also possible on the inner horizon for the non-extremal warped AdS3AdS3 black hole. In addition, we describe the kinds of orbits in this background for timelike and null geodesics by analyzing the effective potential. It is worth mentioning that the study of null geodesics in backgrounds with and without horizon are very important in studies about reconstruction of scalar field in the bulk from data defined on the boundary. It is known that for an asymptotically AdS spacetime, the reconstruction of a scalar field in the bulk from data defined on the boundary fails if there exist null geodesic originating in the bulk that do not reach the boundary [43,44,45], which is analogous to what occurs in flat spacetime, where a necessary condition for a continuous bulk reconstruction of a field from data defined on a boundary is that every null geodesic originating in the bulk intersects the boundary [46]. We will show that, similar to the BTZ black hole, not all the outgoing null geodesics outside the warped AdS3AdS3 black hole can reach infinity. On the other hand, the warped AdS3AdS3/CFT22 correspondence was proposed in [38], the asymptotical symmetry group analysis was studied in [47,48,49,50], and some studies of quasinormal modes and real-time correlators were carried out in [51, 52].
The manuscript is organized as follows: In Sect. 2 we give a brief review of the three-dimensional warped AdS black hole. Then, we study the particle’s motion in the three-dimensional warped black hole background in Sect. 3. In Sect. 4 we obtain the CM energy of two colliding particles, and in Sect. 5 we study the radial motion of a particle with critical angular momentum and we investigate the possibility that the black hole acts as a particle accelerator. Finally, our conclusions are in Sect. 6.
https://link.springer.com/article/10.1140%2Fepjc%2Fs10052-018-5814-0
P. A. González, Ramón Bécar, Yerko Vásquez
