A Slowly Rotating Black Hole in Horava-Lifshitz Gravity and a 3+1 Dimensional Topological Black Hole: Motion of Particles and BSW Mechanism
aa r X i v : . [ g r- q c ] N ov A Slowly Rotating Black Hole in Horava-Lifshitz Gravity and a3+1 Dimensional Topological Black Hole: Motion of Particles andBSW Mechanism
Ibrar Hussain, ∗ Mubasher Jamil, † and Bushra Majeed ‡ School of Electrical Engineering and Computer ScienceNational University of Sciences and TechnologyH-12, Islamabad, Pakistan. School of Natural Sciences, National University of Sciences and Technology,H-12, Islamabad, Pakistan.
The motion of a neutral particle in the vicinity of a slowly rotating black hole in theHorava-Lifshitz theory of gravity and 3+1 dimensional topological Lifshitz black holeis investigated. Geodesics for radial motion of the particles are also plotted. Somedifferent cases of the orbital motion of the particle are discussed where maximumand minimum values of the effective potential are calculated. Further the Ba˜nados,Silk and West (BSW) mechanism is studied for these black holes. It is shown thatthe centre-of-mass energy (CME) of two colliding uncharged particles at the horizonof these black holes remains finite. Thus the BSW effect cannot be seen in thesecases.Key words: Motion of particles; Black holes; Horava-Lifshitz gravity, 3+1 dimen-sional topological Lifshitz black hole; BSW mechanism.
PACS numbers: 04.70-s, 04.50.Kd ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: bushra˙[email protected]
I. INTRODUCTION
To combine the Einstein’s theory of relativity with quantum mechanics or to obtain atheory of quantum gravity (QG) is an unsolved problem in physics. In this regard severalattempts have been made (for detail see [1]). In 2009 Peter Horava proposed a candidatetheory for QG in 4 spacetime dimensions which is based on the explicit violation of localLorentz invariance[2]. This is a power counting renormalizable, higher order gravity modeland reduces to General Relativity (GR) at large distances. It may provide a candidate ofultraviolet completion of GR [2].In the past few years some static and slowly rotating black hole spacetimes have beenproposed in Horava-Lifshitz (HL) gravity [3–9]. In this regard Lu-Mei-Pope have studieda spherically symmetric black hole with some dynamical parameter λ [9]. Cai et. al haveobtained a topological black hole solution[5]. Another black hole solution which is asymptot-ically flat has been found by Kehagias and Sfetsos by taking λ = 1 [3]. Besides, some slowlyrotating black hole solutions in HL gravity have been investigated in literature [4, 6–8].Recently researchers have given a great attention to the Lifshitz spacetimes for findingthe gravity duals of Lifshitz fixed points due to the AdS/CFT correspondence for condensedmatter physics [10]. From the perspective of quantum field theory, there are many scaleinvariant theories for studying such critical points. Lifshitz field theory is a toy model whichexhibits this scale invariance. Such theories exhibit the anisotropic scale invariance expressedas t → λ z t , and x → λx , with z = 1, where z is the relative scale dimension of time andspace. The Lifshitz spacetimes are given by ds = r z ℓ z dt − ℓ r dr − r ℓ d~x , (1)where ~x is a D − D denotes the spacetime dimension and ℓ represents the length scale in the geometry. As mentioned, this spacetime is interestingdue to its invariance under anisotropic scale transformation and represents the gravitationaldual of strange metals [11]. When z = 1, then (1) is the usual anti-de Sitter metric inPoincar´e coordinates. The metric of the Lifshitz black hole is asymptotically similar to(1). Topological Lifshitz black hole with critical exponent z = 2 in 3 + 1 dimensions wasproposed by Mann [12]. Particles motion on topological black hole in 3 + 1 dimensions hasbeen studied in [13].Particle dynamics around black holes is a topic of interest for physicist for the last fewdecades. In this direction several papers have been appeared in the literature. Particlemotion in the vicinity of a magnetized Schwarzschild black hole was studied in [14]. Particledynamics in the Riessner-Nordstrom, Kerr and Kerr-Newman black holes was investigated byPugliese et. al. [15–18]. An analysis of motion of particles in higher dimensional black holeshas appeared in [19]. Vasudevan et. al. have investigated particle motion in Myers-Perryblack hole in all dimensions [20]. The motion of particles around near-extreme BraneworldKerr black holes has analyzed in [21]. In a regular black hole spacetime, motion of particleshas discussed by Garcia et. al [22]. Particles motion in the background of a black hole inBranward geometry has investigated in [23]. Enolskii et. al. considered motion of particles inthe Kehagias and Sfetsos black hole [24]. They have shown that neither massless nor massiveparticles with non-vanishing angular momentum can reach the singularity. The dynamicsof particles around Kehagias and Sfetsos black hole immersed in an external magnetic fieldhas studied in [25]. In [26] motion of charged particles in a rotating black hole immersedin a uniform magnetic field has considered. The influence of quintessence on the motionof particles in the vicinity of Schwarzschild black hole has discussed by Fernando [27] andLi and Zhang [28]. The tunneling radiation characteristic of charged particles from theReissner-Nordstrm-anti de Sitter black hole has studied in [29].Ba˜nados, Silk and West (BSW) proposed that rotating black holes may act as particleaccelerators [30]. In the vicinity of extremal Kerr black hole, they have found that infinitecentre-of-mass energy (CME) can be achieved during the collision of particles. The BSWeffect was then studied for different black hole spacetimes [31–50]. For a brief review onecan see [46] and references therein. BSW effect for particles in the vicinity of a rotatingblack hole in HL gravity has studied by Abdujabbarov and Ahmedov in [25]. They havefound some limitation on the CME of the accelerating particles. For another slowly rotatingblack hole in HL gravity the BSW mechanism has investigated by Sadeghi and Pourhassan[51]. They have obtained an infinite CME for colliding particles arbitrarily close to thehorizon of the slowly rotating balck hole. Consequently in this paper we consider a slowlyrotating black hole solution in HL theory of gravity given by Barausse et. al [7], and the3 + 1 dimensional topological Lifshitz black hole, to study motion of particles in the vicinityof the black holes. We also investigate the BSW process for these black hole to look at theCME of the colliding neutral particles.The plan of the paper is as follows. In Section 2 we formulate the equations of motionfor a particle in the field of the slowly rotating black hole in the HL gravity. In subsectionsgeodesics of the radially moving particle are plotted. We also analyze the behavior of theeffective potential to look at its maximum and minimum values. Expression for CME of thecolliding particles is obtained. In Section 3 we study the motion of particles in the vicinityof 3 + 1 dimensional topological Lifshitz black hole. Behavior of the effective potential andCME of the colliding particles around this black hole are studied in the following subsections.We summarize our discussion in the last Section. Throughout this paper we use G = c = 1. II. MOTION OF PARTICLES IN THE VICINITY OF SLOWLY ROTATINGBLACK HOLE IN HORAVA-LIFSHITZ GRAVITY
The line element for a slowly rotating black hole in HL theory of gravity is given by [7] ds = f ( r ) dt − B ( r ) f ( r ) dr − r ( dθ + sin θdφ ) + ǫr sin θ Ω( r, θ ) dtdφ + O ( ǫ ) , (2)where f ( r ) and B ( r ) are given as a series expansion for the spherically symmetric staticasymptotically flat solution in terms of the inverse radial coordinates x = αr (for detail onecan see [8]). For simplicity, we consider only those terms which are linear in x and neglectthe higher order terms. Therefore f ( r ) = 1 + αr , B ( r ) = 1 , (3)where α is an arbitrary constant with dimensions of length. In (2) ǫ is known as the book-keeping parameter of the expansion in the rotation. The motion of a neutral particle in thevicinity of a black hole is determined by the following Lagrangian L = 12 g µν ˙ x µ ˙ x ν , (4)where g µν is the metric tensor and over-dot denotes the derivative with respect to the geodeticparameter. Here µ, ν = 0 , , ,
3. Using (2) in (4), we get L = 12 h f ( r ) ˙ t − B ( r ) f ( r ) ˙ r − r ˙ θ − r sin ˙ φ + ǫr sin θ Ω( r, θ ) ˙ t ˙ φ + O ( ǫ ) i . (5)The time coordinate basis ∂/∂t and azimuthal angular coordinate basis ∂/∂φ are Killing vec-tors of the metric (2) we considered here, therefore conserved quantities along the geodesicsof the particles are the energy E , of the particle and the angular momentum L , of the particlerespectively. The Euler -Lagrange equations give the following relations upon integration˜ E ≡ − Em = ∂ L ∂ ˙ t = f ( r ) ˙ t + ǫ r sin θ )Ω ˙ φ, (6)˜ L ≡ − Lm = ∂ L ∂ ˙ φ = − r sin θ ˙ φ + ǫ r sin θ )Ω ˙ t, (7)here m is mass of particle and terms of order O ( ǫ ) are retained while higher order terms areignored in whole calculations. With-out loss of generality, the motion of the particles canbe confined to the equatorial plane, θ = π/
2. For θ = π/ t from (6) and(7), we get ˙ φ = Lmr − ǫ (cid:16) E Ω2 mf (cid:17) . (8)Further substituting (8) in (6), we get˙ t = − mf (cid:16) E + ǫ L (cid:17) , (9)To find ˙ r we use the normalization condition: u µ u µ = 1 , (10)here u µ denotes the velocity components of the moving particles. For the metric underconsideration, (10) yields f ˙ t − ˙ r f − r ˙ φ + 2( ǫr Ω) ˙ t ˙ φ = 1 . (11)Making use of (8) and (9) in (11), we get˙ r = f (cid:16) E m f − L m r − (cid:17) , (12)or drdτ = ˙ r = ± s f (cid:16) E m f − L m r − (cid:17) , (13)here ± sign denotes the radial velocity of the outgoing and ingoing particles respectively. A. Geodesics of a Radially Moving Particle
If the particle is moving radially towards black hole we can get the geodesics of suchparticle. Using (9) and (13) together for zero angular momentum, L = 0, we obtain drdt ≡ ˙ r ˙ t = ∓ p E − f ( r ) E f − ( r ) , (14) FIG. 1: Geodesics for a radially ingoing particle coming from infinity with some initial velocity,reaching r = r b (dashed line), and going back to infinity. We chose α = 1, E = 1 . where E ≡ −
E/m . Changing position of a radially moving particle with the passage of timecould be obtained from (14), where positive root gives the path of the particle going awayfrom black hole, and negative root gives the path of an ingoing particle. Note that (14) isdefined only if E > f ( r ), or r > α/ ( E − E = 1 .
5, then there is a boundary, r b , r b ≡ r = α . , (15)beyond which the particle can not go. Geodesics defined in (14) could be understood betterby plotting in ( r, t ) coordinates, Fig(1). B. Circular Orbits and Behavior of Effective Potential
Here we analyze the effective potential to look at the stable and unstable circular orbitsof the moving particles. It is known that the possible range for the motion of the particleis given by E ≥ V (see for example [28]). Further the orbits of massive particles can becircular if the V eff attains it maximum and minimum values. It means that for the minimumvalue of V eff the circular orbits are stable and for the maximum value of V eff the circularorbits are unstable. For inflection point V ′′ = 0, yields the marginally stable orbit,We can write (12) as below ˙ r = 1 m B ( E − V eff ) , (16)where V eff = f ( L r + m ) , (17)is the effective potential and V ′ eff ≡ dV eff dr = − m r α − L (2 r + 3 α ) r . (18)Solving V ′ eff = 0, for r we get r ± = − L ± √ L − L m α m α . (19)For real values of r , L ≥ m α is required. Also V ′′ eff ≡ d V eff dr = 2( m r α + 3 L ( r + 2 α )) r . (20)Now we check the behavior of V eff at the above obtained radii given by (19). Note that at L = 3 m α , particle has a circular orbit, of radius | r o | = 3 α, and V ′′ eff | r = r o = 0 . Behavior of V eff with changing value of radius, r , is demonstrated in Fig. (2).If L > m α then V ′′ eff | r = r + = 2 L m α ( − L + 3 m α + √ L − L m α )( − L + √ L − L m α ) , (21)and V ′′ eff | r = r − = 2 L m α ( L − m α + √ L − L m α )( L + √ L − L m α ) . (22)Let L − α m = β the above expressions (21) and (22) reduce to: V ′′ eff | r = r + = 2 L m α ( − β + βL )( − L + βL ) , (23)and V ′′ eff | r = r − = 2 L m α ( β + βL )( L + βL ) , (24) FIG. 2: Evolution of effective potential versus radius r , parameters values are set as, L = 3, α = 1, m = 1. At r = r o = 3 there is a circular orbit of the particle. We see that quantity in the R.H.S. of (23) is greater than zero if βL > β , βL > L , or if β > βL, L > Lβ, but both conditions lead to the contradiction: β > L > β. (25)Hence V eff does not attain minimum value at r = r + . The V eff has maximum at r + if β < L (which gives 3 α m >
0, a physically true constraint) therefore the circular orbits areunstable.Now there are two possible cases for V ′′ eff | r = r − > β + βL and L + βL are positive quantities which give β > − L or 3 m α < β + βL < L + βL <
0. This implies β < − L or 3 m α >
0, which isphysically true. So V eff | r = r − > , if β < − L . Hence V eff is minimum at r = r − if β < − L and the circular orbits are stable. C. Center of Mass Energy
Energy in the center of mass frame is defined as [30] E cm = m √ q − g µν u µ u ν , (26)where u µi ≡ dx µ dτ , i = 1 , E cm = A h − E m f + 1 m f n E + fr ( L + L )( m f − E ) − f m (2 E − f m ) + L L f r o / + L L m r i / , (28)here A = m √
2. We observe that near the horizon (i.e. at f ( r ) = 0), E cm in (28) becomesundefined. However expanding the numerator in (28), about the horizon yields E cm = 2 m r m α ( L + L ) . (29)The CME in (29) could be infinite, if the angular momentum of one of the particles hasinfinite value, for which the particle could not reach the horizon of the black hole. Thus theCME in (29) can not be unlimited. III. MOTION OF THE PARTICLES IN THE VICINITY OF 3+1DIMENSIONAL TOPOLOGICAL BLACK HOLE
The metric of four dimensional Lifshitz black hole is given by [12] ds = r f ( r ) ℓ dt − dr f ( r ) − r ( dθ + sinh θdφ ) , (30)where f ( r ) = r ℓ − . (31) f ( r ) = 0, gives the event horizon at r h = ℓ/ √
2, where ℓ denotes the length scale in thegeometry. To find out the CME of the colliding particles in the vicinity of Lifshitz black0hole, we first find out the geodesics structure of the particles. Using the standard Lagrangianprocedure we find the conserved quantities of the moving particles of mass m ( m = m = m ). For each particle the velocity components are given by˙ t = − r Em ℓ f r , (32)and ˙ φ = Lmr . (33)The normalization condition given in (10), used for the metric defined in (30) gives˙ r = ℓ mr ( E − V eff ) , (34)or ˙ r = ± ℓr √ m p E − V eff , (35)the ± signs stand for radial velocities of outgoing and ingoing particles respectively, and V eff is defined as V eff = f r ℓ [ m + L mr ] . (36)This expression for V eff is already derived in literature [13], we rederive it using dimensionallycorrect equations (32) and (33). Geodesics of a radially moving particle for this black holeare also already plotted in [13]. A. Behavior of Effective Potential
Differentiating (36) with respect to r , we have V ′ eff = r (2 L − m ( ℓ − r )) ℓ m . (37)Solving V ′ eff = 0, for r we get r = 0 , r ± = ± √− L + ℓ m m . (38)Furthermore, d V eff dr ≡ V ′′ eff = 2 L − m ( ℓ − r ) ℓ m . (39)We evaluate (39) at above obtained radii given in (38), to check the nature of V eff . At r = r ± ,(39) becomes d V eff dr ≡ V ′′ eff = 4 L mℓ + 2 mℓ . V ′′ eff > m ℓ > L . It shows that V eff is minimum at r = r ± , for L < ℓ / ℓ = 10 as in [13], we get −√ < L < √ . If the particles approachthe black hole from opposite directions then their angular momentums would have oppositesign.
B. Center of Mass Energy
For two neutral particles falling freely from rest at infinity the CME in the vicinity of3 + 1 dimensional topological Lifshitz black hole is calculated by using (32), (33) and (35)(in (35) we choose negative sign only, and its reason was explained before in section 2) inequation of CME, we get E cm = m √ h ( L + L )( − r m + f mEℓ ) + L L m r (1 + f L L mEℓ ) + f r m Eℓ i / . (40)Expanding the numerator in (40) at the horizon of the black we obtain E cm = √ ℓ ( L − L ) . (41)Note that for the CME to be positive L > L . The CME in (41) goes to infinity if L attain infinite value, for which the particle could not reach the black hole horizon, hence inthis case CME will remains finite. IV. CONCLUSIONS
In the background of a slowly rotating black hole in the HL theory of gravity and 3 + 1dimensional topological Lifshitz black hole we have analyzed the motion of massive (neu-tral) particles, falling freely from infinity with zero velocity there. Analytical expressions foreffective potentials and the CME of neutral particles have obtained by using the standardLagrangian methods. Geodesics of the particle coming from infinity, with some initial ve-locity, approaching black hole radially, has also plotted in ( r, t ) coordinates. It has observedthat in the case of slowly rotating black hole in HL gravity the particle coming from infinity2approaches the black hole and a smooth curve has obtained which ends at the boundary r = r b = 2. Then we have combined that curve with the curve obtained for the particlemoving away from the black hole, Fig. (1) has obtained.We have examined the behavior of effective potentials in both cases of black holes. Cal-culations have shown that in the case of slowly rotating black hole in the HL gravity, V eff in (21), does not attain minimum value at r = r + given by (19) and is maximum if β < L is satisfied. It shows that the unstable circular orbits of the particle exist there. While V eff in (22) is minimum at r = r − if β < − L . In this case there exists stable circular orbits ofthe moving particles. For the 3 + 1 dimensional topological Lifshitz black hole V eff in (36)is minimum at both r = r ± given by (38), for −√ < L < √
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