>K9m 2013, TOM 144, bmii. 1 (7), rap. 92 96

© 2013



M. Sharif* Nida Haider**

Department of Mathematics, University of the Punjab Lahore-54590, Pakistan

Received February 7, 2013

We study the center-of-mass energy of the particles colliding in the vicinity of acceleration and event horizons of the Plebanski and Demianski class of black holes. We calculate the collision energy of uncharged particles in the center-of-mass frame that are freely falling along the equatorial plane of a charged accelerating and rotating black hole with an NUT parameter. This energy turns out to be infinite in the nonextremal case, while in the extremal case, it becomes infinitely large near the event horizon only if the particle has the critical angular momentum. We conclude that the center-of-mass energy depends on the rotation and the NUT parameter.

DOI: 10.7868/S0044451013070092

Black holes (BHs) arc the most important prediction of general relativity and are studied by detecting their effects 011 the nearby matter. One of the interesting features of BHs is that they can behave like a particle accelerator (accelerating charged particles to high speed). Recently, the physics of ultra-high energies in the context of particle accelerators is receiving much attention. Therefore, the study of naturally occurring processes in the vicinity of astrophysical objects is of great significance. Collision energies up to 10 TeV can be observed by the largest terrestrial accelerators like the Tevatron and Large Hadron Collider. In this regard, the center-of-mass energy (CME) provides the collision energy required for the production of new particles. Black holes can accelerate and collide particles with an unlimited CME. This infinite increase in energy in the center-of-mass (CM) frame is due to the blue-shifting of particles near the horizon.

The possibility of obtaining infinite growth of energy in the CM frame due to particles colliding near the horizon of a BH was discussed by Banados, Silk, and West (the BSW effect) [1]. They showed that the rotating BH can behave as a particle accelerator and observed high CMEs for particles propelling along the equatorial plane in the locality of the Kerr (extremal) BH. In [2], it was pointed out that the CME is finite

E-mail: msharif.math'&pu.edu.pk

** E-mail: nida.haiderl2(&gmail.<'om

for the Kerr (nonextremal) BH. An infinite CME due to particle scattering in the Kerr (nonextremal) BH was found in [3]; an infinite CME for particles colliding in the Kerr de Sitter (extremal) BH was observed in [4]. The infinite CME at the cosmological horizon of the Reissner Nordstrom (RN) de Sitter BH was studied in [5]. The infinite CME for the critical particles (with fine-tuned angular momentum) colliding along the equatorial plane of the Sen (extremal and nonextremal) BH and the Kerr Newman BH were discussed in [61.

The BSW effect near the event horizon of the Kerr Taub NUT BH was investigated in [7]. In [8], the authors discussed the collision of particles around the four-dimensional Kaluza Klein (extremal) BH and found the infinitely large CME near the horizon in both rotating and nonrotating cases. Joshi and Patil [9] found that the CME turns out to be high in the naked singularity of the RN and Kerr BHs. The same authors [10] proved that a high CME can also be seen in regular BHs for particular values of the parameters m and q. The BSW effect for the Ayon Beato Garcia Bronnikov BH, the Einstein Maxwell dilaton axion BH, and the Banados Teitelboim Zanelli BH was discussed in [11]. In [12], the CME was generalized for charged particles moving in an electromagnetic field and braneworld BHs were discussed. It was proved in [13] that a nonrotating but charged RN (extremal or nonextremal) BH can also serve as an accelerator with an arbitrarily high CME of charged particles colliding near the horizon.

The effect of acceleration on the CME for particles involved in nonequatorial motion and colliding in the Kerr Newman BH was discussed in [14]. Arbitrarily high CME for colliding particles with nonequatorial motion near the horizon of an (extremal) Kerr Newman BH were obtained in [15]. In [16], nonequatorial motion of particles colliding in dirty BHs was discussed and the CME was found to grow without bound. This generalizes the results of the equatorial motion. Collisions of the innermost orbit particle in a nonequatorial plane of an (extremal) Kerr BH were studied in [17], and the CME was found to be unboundedly high. The CME of a Plebanski Demianski (PD) (extremal) BH with a zero NUT parameter near the acceleration and event horizons was studied in [18].

In a recent paper, we have studied the CME of a PD (nonextremal) BH with a zero NUT parameter near the event horizon [19]. In this paper, we explore the CME for charged accelerating and rotating (extremal and nonextremal) BHs with an NUT parameter near the event and acceleration horizons. The particles are assumed to be colliding in the equatorial plane. In general, the NUT parameter is associated with the twisting property of the BH. Plebanski and Demianski presented a class of type-D BHs known as the family of PD BHs [20]. These are described by the metric

ds2 = —f(r,0)dt2

, dr2 - 2H(r. 0) dt dd) -i + S(r,0)<202 + K{r,0)d<p2

where /, g, H, E, and K are functions that describe different BHs in this class.

We consider a PD BH with a NUT parameter, described by the metric [21]

(ls2 = J Q_

n2 (P

dt — ( u sin2 0 + 41 sin2 ^ ) d<t>


—Ydi'2--5- [a dt — (r2 + (u + I)2) d<t>] '

tjf P"

C sin2 ßdß2L (1)

Q =

(uj2k ■

2 Mr-

+ r)

cj2k i2 - /2



a (a + I)



a (a — I)


P = sin2 8{\ — «3 cos (9 — «4 cos2 (9) = Psin20, , a2ul,

« au , , a3 = 2—M



"(uTÄ- ■


«4 =

2 2 -a* a


(uj2k -


Here, M and a respectively represent the IIlclSS and rotation of BH, and the parameters e and g are the electric and magnetic charges. Moreover, a is the acceleration of a BH and / is the NUT parameter. The rotation parameter u; in terms of a and k is given by




k = 1




3a /"-


/ 2 1 2 (e +g

It is interesting to mention here that all the parameters a, M, e, g, and a vary independently, but u; depends on the rotation and NUT parameters. For a = 0, the metric reduces to the Kerr Newman BH with an NUT parameter. I11 the absence of an NUT parameter, it reduces to a charged accelerating and rotating BH. Further, the limit a = 0 leads to the Kerr Newman BH, and a = 0 yields the RN BH. I11 addition, if e = 0 = g, then we have a Schwarzschild BH, while the limit / = 0 = a leads to the C-metric.

The horizons of BH (1) can be found by setting g(r,9) = 0, which yields

(uj2k -





2 Mr-

^ 2 n -TK'I- = 0,

which is quadratic in r with the roots

r± =

(LjH- + e2 + g2)—-M



. a/

(iSk + e2 + g2)— - AT



a2 - I-

(uj2k + e2 + g2


where r± respectively represent the outer (event) and inner horizons. For the existence of horizons, the condition is



Q = 1 —- (/ + «cos(9) r. p" = r" + (I + acosQ)-


Huo2k + e2 + g2)^^Mf >




ai a(u + I)' a2 a(u — l)

arc acceleration horizons. The angular velocity at the outer horizon is

HH =

9ht> .900

which, in our case, takes the form


Vh =

4 + (a + /)2


We consider a particle exhibiting the geodesic motion in the PD BH. Let

ua = (Ul,UrM°,U*)

be the four-velocity of the particle, which is restricted to equatorial motion (9 = tt/2), leading to

U° = 0.

We can define the energy and angular momentum of the particle as

E = -gab(d,)aUh = -g„U1 -g^V*, L = gah(dlP)aUh = gllPUt +glPlPU*.

These are conserved throughout the motion, termed as constants of motion. With Eq. (1), these quantities become

E =


L =

ny2 [Q

1 r 7

n y2 I1

1 [


[Q - Pa2]U' +

[Pa(r2 + (a + If) - Q( a + 21)] U^ (4) [Pa(r2 + (a + If) ^ Q(a + 21)}U' +



[(nr2 + (« + Iff - Q(a + 21 f)] u*. (5)

These lead to the four-velocity components 1

UL =

-{E(P(r2 + (u+lff^Q(u+2lf

PQ(r2 + /2

- L(Pa(r2 + (a + If) - Q( a + 2/))}, (6)

U0 =


PQ(r'2 + P

-{E(Pu(r2 + (u + lf

^Q(u + 2l)) + L(Q^Pu2)}. (7) Using the normalization condition,

gahUaUh = -1,

we find the radial component of the velocity as

Tjr _ ,__Q__|__1_ x

L (r2+/2)n2 P(r2+/2)2n2

x [E2[P(r2 + (a + iff ^ Q(a + 21 f] -- L2(Q^Pu2)^2EL(Pu(r2+(u+lf)^Q(u+2I))]


where ± correspond to the radially ingoing and outgoing particles.

We introduce the effective potential as


Vcff(r) =

Ur~ + V.ffir) = 0,



(r2 +/2)Q2 P(r2 +/2)2Q2

x {E2[P{r2 + (a + Iff - Q(a + 2 if] -—L2(Q—Pu2) —2EL [Pu(r2 + (u+lf)^Q(u+2I)]}.

The conditions for a circular orbit are

,,ff(r) = 0. ¿iksa_„.

Because the timelike component of the four-velocity is greater than zero (causally connected), Eq. (C) leads to

E(P(r2+ (a + lff ^Q(a + 2lf)>

> L [Pa(r2 + (a + If) - Q( a + 21)] , (9)

which reduces (at the horizon) to


E >

(a +1)2

Furthermore, the angular velocity of the BH (at r =

= r+) is


(u + I)2

which yields

E > flHL.


We now discuss the C^IE for two colliding particles with rest masses mi and m2 moving in the equatorial plane. In terms of the four-momentum

pi = m-il'l, i = 1,2, a = t,r,9,(f), the CME of two particles is [7]

^cm = Pi Pa i'

which yields

E';m = 'lni\in>

(mi^m'2)2 -d-wrr;:

'im i m>

The CME of these particles turns out to be


' - r in

'(mi - m2)2 M(r) - N(r)

\/2ii) i nt > where

'hit i nt >




M(r) = PQp2 + {(EiL-2 + E2Li) x

x [Q(a + 21)^Pa(r2 + (a + lf)} + +E1E2(P(r2 + (u+I)2)2^Q(u+2I)2)^L1L2(Q^Pu2)},

AT(r) = \/ni(r)n2(r)

(see [18]). For the CME near the event horizon, the term in the right side of Eq. (12) becomes undetermined. Using l'Hospital rule, we then find

'(mi — tn-2 )2 M'ir) -N'(r)

^/'2iii\in-> with

'hit i m >



M'(r) |P=P+ = E1E2[4Pr+(rl + (a+lf)^Ql(a+2I)y -LxLiQ' - (EiL2 + E2Li)(2Par+ — Q'(a + 21)),



' + 2\fri\ (r+ )n2(r+)

x {!%[ (r+ )

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