Particle acceleration in Horava-Lifshitz black holes
aa r X i v : . [ h e p - t h ] A ug Particle acceleration in Horava-Lifshitz blackholes
J. Sadeghi a,b ∗ and B. Pourhassan a,b † a Sciences Faculty, Department of Physics, Mazandaran University,P .O .Box 47416-95447, Babolsar, Iran b Institute for Studies in Theoretical Physics and Mathematics (IPM),P.O.Box 19395-5531, Tehran, Iran
August 26, 2018
Abstract
In this paper we calculate the center-of-mass energy of two colliding test particles nearthe rotating and non-rotating Horava-Lifshitz black hole. For the case of slowly rotat-ing KS solution of Horava-Lifshitz black hole we compare our results with the case ofKerr black holes. We confirm the limited value of the center-of-mass energy for thestatic black holes and unlimited value of the center-of-mass energy for the rotatingblack holes. Numerically, we discuss temperature dependence of the center-of-mass en-ergy on the black hole horizon. We obtain the critical angular momentum of particles.In this limit the center-of-mass energy of two colliding particles in the neighborhood ofthe rotating Horava-Lifshitz black hole could be arbitrarily high. We found appropri-ate conditions where the critical angular momentum could have an orbit outside thehorizon. Finally, we obtain center-of-mass energy corresponding to this circle orbit.
Keywords:
Particle acceleration, Center-of-mass energy, Horava-Lifshitz black hole. ∗ Email: [email protected] † Email: [email protected] ontents Introduction
Recently, Banados, Silk and West (BSW) [1] shown that free particles falling from rest atinfinity outside a Kerr black holes may collide with arbitrarily high center-of-mass (CM) en-ergy and hence the maximally rotating black hole might be regarded as a Planck-energy-scalecollider. They proposed that this might lead to signals from ultra high energy collisions suchas dark matter particles. The spinning Kerr black holes as particle accelerators discussed bythe Ref. [2], they found that the ultra-energetic collisions cannot occur near black holes innature. In the Ref. [3] elastic and inelastic scattering of particles in the gravitational field ofstatic and rotating Kerr black holes was considered and found that the CM energy is limitedfor the static and can be unlimited for the rotating black holes. In the Ref. [4] new resultsfor the CM energy of particles for the non-extremal black holes are obtained and found thatthe CM energy has limited value, but if one takes into account multiple scattering the CMenergy becomes infinite [5]. These results extended to the charged spinning black hole asKerr-Newman [6] and non-rotating charged black holes as Reissner-Nordstrom backgrounds[7]. In near-extremal Reissner-Nordstrom black hole it is found that there always exists afinite upper bound of CM energy, which decreases with the black hole charge. From abovestudies we find that having infinite CM energy of colliding particles is a generic property ofa rotating black holes [8]. This universal property helps us to understand unknown channelsof reaction between elementary particles. Also there are similar studies in other kinds ofblack holes [9-15].In this paper we take advantages from above studies and investigate the particle accelerationprocess in the Horava-Lifshitz (HL) black holes [16]. The HL gravity is an interesting quan-tum gravity theory,which has stimulated an developed study on cosmology and black holesolutions [17-20]. Because it is power counting renormalizable non-relativistic gravity theorygiving up the Lorentz invariance. It is expected that the HL black hole solutions asymptoti-cally become Einstein gravity solutions. Significant reason for consideration of HL black holeas particle accelerator is that the slowly rotating Kerr black hole is recovered by the slowlyrotating black hole solutions in the HL gravity in the IR limit [21]. Slowly rotating blackholes means that one considers the first order of rotating parameter, it may be interpretedas a black hole arising from the breaking of spherical symmetry to axial symmetry. So, itis interesting to calculate the CM energy of two colliding particles in the rotating HL blackhole and compare it with the case of Kerr black holes. Also one can investigate universalityof unlimited CM energy on the horizon of the rotating HL black holes, which proposed inthe Ref. [8]. Therefore our motivation is to examine if the BSW effect remains valid in thecase of rotating black hole solutions of the modified gravity models such as HL gravity.This paper organized as follows. In the section 2 we review static HL black holes and discussabout horizon structure of several kinds of HL black holes. In the section 3 we calculate theCM energy of two colliding particles in the HL black holes and obtain finite value of CMenergy in the non-rotating black hole. In order to investigate universality of CM energy werecall rotating HL black hole in the section 4 and calculate the CM energy of the rotatingHL black holes in the section 5. Also we compare our result for slowly rotating KS black holewith the case of Kerr black holes. Finally we give conclusion and summarize our results.3
Horava-Lifshitz black holes
In this section we recall HL black hole and discuss about horizon structure of these kindof black holes. Such discussions will be useful in the study of particle acceleration. Thefour-dimensional gravity action of HL theory is given by the following expression, S = Z dtdx √ g ¯ N [ ˜ L + L + L ] , (1)where ˜ L = 2 k ( K ij K ij − λK ) L = − k ω C ij C ij + + k µ ω ǫ ijk R (3) il ∇ j R (3) lk − k µ R (3) ij R (3) ij L = k µ − λ ) (cid:18) − λ R (3) ) + Λ W R (3) − W (cid:19) + µ R (3) , (2)where k , λ and µ are constant parameters, and the Cotton tensor is defined as the following, C ij = ǫ ijk ∇ k ( R jl − Rδ jl ) , (3)also the extrinsic curvature is defined as, K ij = 12 ¯ N ( g ij − ∇ i N j − ∇ j N i ) , (4)Furthermore, ¯ N and N i are the lapse and shift functions respectively, which are used ingeneral relativity in order to split the space-time dimensions. Indeed, we considered theprojectable version of HL gravity with detailed balanced principle, and an IR modificationterm µ R (3) .As we know the vacuum metric of the HL black holes is given by [22], ds = f ( r ) dt − dr f ( r ) − r ( dθ + sin θdφ ) , (5)where we used the natural units ( c = G = 1), and f ( r ) is, f ( r ) = 1 + ( ω − Λ W ) r − (cid:0) r [ ω ( ω − W ) r + β ] (cid:1) . (6)where, β is an integration constant, Λ W and ω are real constant parameters. There are twospecial cases of the HL black holes. In the first case one consider β = 4 ωM and Λ W = 0which is called the Kehagias-Sfetsos (KS) black hole solution [23]. In this case f ( r ) is, f KS ( r ) = 1 + ωr − ωr r Mωr , (7)4n the second case β = − α Λ W , and ω = 0, which is called the Lu-Mei-Pope (LMP) black holesolution [24]. In this case f ( r ) is, f LMP ( r ) = 1 − Λ W r − α √− Λ W √ r. (8)We note here that the LMP solutions are spherically symmetric space-times, but the KSsolutions are asymptotically flat space-times. In order to study equations (7) and (8) wedraw the f ( r ) function with respect to r which are shown by Fig. 1 and Fig. 2.Figure 1: Horizon structure of the KS solution (7) for M = 1. The solid line represents ω = 0 .
5, where two horizons are coincide. ω = 1 and ω = 2 . ω = µ k , it becomes Schwarzchild black hole for r ≫ ( M/ω ) , in other word for large r , or large ω ( ω → ∞ limit) in fixed r we have Schwarzchildblack hole. The constant ωM is dimensionless parameter and the mass M is transformedto k M to make mass dimension. There exist two event horizon for ωk M > / r ± = k M ± r − ωk M ! . (9)Two horizons of the equation (9) are coincide for ωk M = 1 /
2. We have the naked singu-larity for ωk M < /
2. These situations drawn in the Fig. 1 for M = 1 and different valuesof ω . In the Ref. [25] the motion of massless and massive test particles in the space-timeof the KS black hole solution studied. Also the Horizon structure of the LMP solution (8)drawn in the Fig. 2. By choosing Λ W = − α = 1 . W = −
1. The dashed line drawnfor α = 2. And the solid line drawn for α = 1 .
75, where two horizons are coincide.Also we have naked singularities for the cases of 0 < α < . f ( r ) = 0, which yields to the following equation,Λ W r + 2( ω − Λ W ) r − βr + 1 = 0 . (10)The equation (10) may be obtained by adding the following equations,1Λ W r − r = 2( ω − Λ W )Λ W , r − W r = − β Λ W . (11)The first relation of (11) yields to the following answer, r ± = s ω − Λ W W ± s W ( ω − Λ W ) ! . (12)We can find that r − is related to the inner horizon, while r + is related to the outer horizon.We can also solve ω and β from the relations in the (11), and find, ω = 12 r (1 + 3Λ W r )(1 − Λ W r ) ,β = 2 r (1 − Λ W r )(1 + Λ W r ) . (13)6n order to have positive β it should be r ≤ / Λ W , but in order to have positive ω it shouldbe r ≤ / Λ W . If we assume that ω is positive for r = 1 / Λ W , then the upper limits of β ispositive if Λ W ≥
1. On the other hand, if we assume that β is positive for r = 1 / Λ W , thenthe upper limits of ω is positive if Λ W ≥
1. It means that both cases constrained by thesome condition Λ W ≥ In order to obtain the CM energy, we should calculate the 4-velocity of the particles. Weassume that the motion of particles is on the equatorial plane. In that case we should set θ = π/ t = − Ef ( r ) , ˙ r = s f ( r ) (cid:18) E f ( r ) − L r (cid:19) , ˙ θ = 0 , ˙ φ = − Lr , (14)where E denotes the test particle energy per unit mass and L denotes the angular momentumparallel to the symmetry axis per unit mass. Also the dot denotes a derivative with respectto an affine parameter λ , which can be related to the proper time by τ = ηλ , and η satisfiesthe normalization condition η = g µν ˙ x µ ˙ x ν with η = 1 for time-like geodesics and η = 0for light-like geodesics. Here, we should note that we used sign convention (+ , − , − , − ),therefore sign of normalization condition is different with the Ref. [15] where sign conventionused as ( − , + , + , +). We use nonzero 4-velocity components (14) to obtain CM energy ofthe two-particle collision in the background of HL black holes (5). We assume that twoparticles have the angular momentum per unit mass L , L and energy per unit mass E , E , respectively. Also we take m as the rest mass of both particles. By using the relation E CM = √ m p g µν u µ u ν , one can find the CM energy of two-particle collision as thefollowing ( u = ( ˙ t, ˙ r, , ˙ φ )),¯ E CM = 1 f ( r ) r (cid:0) f ( r ) r + E E r − L L f ( r ) − H H (cid:1) , (15)where, H i = q f ( r ) r + E i r − f ( r ) L i , (16)7here i = 1 ,
2, and we re-scaled the CM energy as ¯ E CM ≡ E CM m . We interest to find ¯ E CM when the particles collide on the black hole horizon. Therefore we should set f ( r ) = 0 inthe relation (15). It is clear that the denominator of right hand side of the relation (15) iszero. So, if E E >
0, then the numerator will be zero and the value of ¯ E CM on the horizonwill be undetermined. But when E E <
0, the numerator will be negative finite value and¯ E CM on the horizon will be negative infinity, which is not physical solution.In order to obtain behavior of the CM energy of two colliding particles on the horizon weexpand the equation (15) at r + . So, the lowest order term yields to the following expression,¯ E CM ( r → r + ) = A E E r , (17)where, A ≡ r ( E + E ) − ( E L − E L ) . (18)The equation (17) is just the limiting value of the relation (15) at r → r + . In that case wecan see that if L i → E L = E L , then ¯ E CM is independent of the horizon radius.We found that the value of CM energy in the horizon is finite for E i = 0, as expected fornon-rotating black holes. Therefore, this case is not interesting in the present work.In order to compare our result with the case of Kerr black holes and investigate universalityof having infinite energy in the center of mass frame of colliding particles we should considerrotating HL black holes which is subject of the next sections. In this section, we consider rotating black hole in the HL gravity, which described by thefollowing metric [21], ds = f ( r ) dt − dr f ( r ) − r dθ − r sin θ ( dφ − aN dt ) , (19)where a = J/M is the rotation parameter, and J is spin angular momentum, and M isthe black hole mass. In order to obtain a slowly rotating black hole solution one can keepequations of motion up to the linear order of a . In that case the metric (19) reduced to thefollowing [21], ds = f ( r ) dt − dr f ( r ) − r dθ − r sin θ ( dφ − aN dtdφ ) . (20)where, N = 2 Mr . (21)Also it is found that f ( r ) in the equation (20) is KS solution which is given by the relation(7). So, the slowly rotating KS black hole solution of HL gravity is given by, ds slowKS = f KS ( r ) dt − dr f KS ( r ) − r dθ − r sin θ ( dφ − Jr dtdφ ) . (22)8he Hawking temperature of the slowly rotating KS black hole (22) is given by the followingexpression, T H = 2 ωr − πr + ( ωr + 1) , (23)where the outer horizon r + is given by the equation (9). Also the angular momentum ofslowly rotating black hole is given by [21], J = a (cid:20) r + − − ( √ ωr + )4 √ ω (cid:21) . (24)It is found that in the ω → ∞ limit the slowly rotating KS black hole solution (22) leads tothe slowly rotating Kerr solution. In the next section we calculate CM energy of two collidingparticles in the rotating HL black holes. Finally we compare results of slowly rotating KSblack hole with Kerr black holes. In this section, similar to the section 3, we calculate the CM energy of two colliding particlesin the rotating HL black hole (19). In that case 4-velocity of the particles extended to thefollowing, ˙ t = aN L − Ef ( r ) , ˙ r = s f ( r ) (cid:18) E f ( r ) + a N r − f ( r ) f ( r ) r L − aN ELf ( r ) (cid:19) , ˙ θ = 0 , ˙ φ = aN ( aN L − E ) r − f ( r ) Lf ( r ) r . (25)The radial component ˙ r gives us the effective potential via 2 V eff + ˙ r = E . The structureof the effective potential of the KS solution of the static HL black hole have been discussedin the Ref. [26]. Now we present the effective potential of both KS and LMP solutions forrotating HL black hole, V eff ( KS ) = ( L ω − L a N − r − ωr + L + ( r − L ) ωr q Mωr r ,V eff ( LM P ) = ( L ω − L a N − r − r + L + ( r − L ) α √ r r . (26)The effective potential of the KS solution (26) in limit of a → r = 0 to r = ∞ . As we expected the effective potential vanishes at r → ∞ and goes to infinity at r = 0. Also the effective potentials of the KS black hole at ω → ∞ and α → V eff = 0 yields to the twosolutions r and r with the condition r < r + < r . So, V eff ( r < r ) and V eff ( r > r ) arepositive, and V eff ( r + ) is always negative. It means that the horizon of the HL black holesare attractive, but V eff ( r < r ) and V eff ( r > r ) are repulsive. Because of the repulsive forceof the V eff ( r < r ), particles cannot fall to the center of black hole. Instead, two particlescould collide on the horizon.Figure 3: Plots of effective potential in terms of r by choosing M = 1, L = 10 and a = 0 . ω = 0 . , .
75 and 1 represented by solid, dotted and dashed linesrespectively. Right: For LMP solution, α = 1 . , . E CM = 1 f ( r ) r (cid:0) f ( r ) r + E E r − L L ( f ( r ) − a N r ) − aN r ( E L + E L ) − H H (cid:1) , (27)where, H i = q f ( r ) r + E i r − ( f ( r ) − a N r ) L i − aN r E i L i , (28)with i = 1 ,
2. The denominator of the CM energy (27) on the black hole horizon is zero, andthe numerator of it reduced to K K − p K p K , where, K i = r ( aN L i − E i ) , i = 1 , , (29)when K K >
0, the numerator of (27) will be zero and the value of the ¯ E CM on the horizonwill be undetermined. On the other hand, if K K <
0, then ¯ E CM on the horizon will be10egative infinity which is not physical solution. This result agree with the result of particleacceleration in Kerr-(anti) de Sitter black hole background [15]. It is easy to check that thisresult with a = 0 reduced to the results of the section 3. In the Fig. 4 we draw ¯ E CM fromthe relation (27) for KS solution with small and large ω separately. We find that at large ω limit our result agree with the Ref. [1] where the Kerr black holes living in a Minkowskispace-time with a zero cosmological constant considered.Figure 4: Plots of ¯ E CM in terms of r by choosing E = E = 2, M = 1 and L = − ω and set L = 0 .
5. We can read r + ≈ . E CM with the critical angular momentum. Right: We take large ω , this behavessimilar ¯ E CM near horizon ( r + = 1) of the Kerr black hole with a zero cosmological constant.On the other hand the LMP black hole solution with Λ W = − E CM in terms of r for LMP solution.It shows that the CM energy has large value on the horizon ( r + = 1).Now, in order to obtain behavior of the ¯ E CM on the horizon we consider slowly rotatingHL black hole and assume that a is small parameter, then expand ¯ E CM at the horizon andfind, ¯ E CM ( r → r + ) = A K ( r + ) K ( r + ) , (30)where, A ≡ ( K ( r + ) + K ( r + )) − ( E L − E L ) . (31)In order to study the critical angular momentum per unit mass we assume the followingcondition. If we consider K i ( r + ) = 0 ( aN L i = E i ), then A reduced to K (1 + E a N r ), inthat case we will obtain ( ¯ E CM ) | K i =0 → ∞ on the horizon. It means that, if K i ( r + ) = 0 then11igure 5: Plot of ¯ E CM in terms of r by choosing E = E = 2, L = 0 . L = − α = 2 forLMP solution. We can read r + = 1 from the Fig. 2.the CM energy of two colliding particles on the horizon of HL black hole could be arbitraryhigh (see Fig. 4). These lead us to obtain the critical angular momentum per unit massfrom the equation (21) as the following expression, L ci = E i aN = E i r J , i = 1 , , (32)It means that the particles with the critical angular momentum L ci can collide with arbitraryhigh CM energy at the horizon. This result may provides an effective way to probe theplank-scale physics in the background of a rotating HL black hole. We can see the criticalangular momentum depends on r which is differs from the case of Kerr black hole, wherethe L ci obtained depend on r [15]. However, by using the equation (24) and taking ω → ∞ limit, we can rewrite the equation (32) as L ci ∝ r , which agree with the results of [15].If we take r = − l , we will have same critical angular momentum for two black holes( L ci ( HL ) = L ci ( Kerr )), where l is related to the cosmological constant Λ by l − = − Λ / E CM ( r + ) in terms of the black hole temperature. In that case we useequations (23) and (30), and obtain the relation between T and ¯ E CM ( r + ). So, numericallywe find that ¯ E CM ( r + ) increases with temperature. It diverges at critical angular momentumand yields to the negative value for L i < L ci . The situation with L i > L ci illustrated in theFig. 6.The particle with the critical angular momentum may have an orbit outside the outerhorizon. This will be happen if, dR ( r ) dr | r = r + > , (33)12igure 6: Plot of ¯ E CM ( r + ) of KS solution in terms of the Hawking temperature for ω = 1, E i = 2, L = − a = 2. Solid, dashed and dotted lines represent L = 1 , . L c = 0 .
5, where ¯ E CM ( r + ) → ∞ .For the cases of L c < . R ( r ) ≡ ˙ r and ˙ r is given by the relation (25). This condition comes fromstability of orbital motion where dV eff /dr should be positive at the balance point. By usingthe relation (7) and (25) one can obtain the following equation, dR ( r ) dr | r = r + = ( a − E r ) Wa , (34)where, W = 2 ωr + − s Mωr ! + 6 Mr q Mωr . (35)First, we assume that a > E r . In that case, by using the relation (9) we draw W in termsof ω for KS solution of HL background. It show that the value of W is positive for ω > . M = 1 (see Fig. 7). The case of ω = 0 .
5, which yields to W = 0, is the boundary case.Already we found that the ω < . ω ≥ .
5, hence there is only a ≥ E r condition. a = E r is the boundary case which is similar to the case of W = 0. In summary with ω ≥ . a > E r , the particle with the critical angular momentum can have an orbit outsidethe outer horizon of the HL black hole. If this orbit be a circle, then the radial 4-velocitycomponent of the particle must be zero, ˙ r = 0.13igure 7: Plot of W in terms of ω for M = 1.By using the relation (25) one can obtain the angular momentum per unit mass on acircle orbit as the following, L co = aN Er + r p f ( r ) p E + a N r − f ( r ) a N r − f ( r ) ,L co = aN Er − r p f ( r ) p E + a N r − f ( r ) a N r − f ( r ) . (36)In order to obtain real solution we should have E ≥ f ( r ) − a N r . If E = f ( r ) − a N r ,then L co = L co = aNr E , and there is no circle orbit. In order to have the circle orbit, theangular momentum must be in the range L co < L < L co . We assume that L co = L , and L co = L − δ , where 0 ≤ δ ≤ L co − L co . It means that the first particle is a target andthe second particle on the circle orbit collide with the target. δ is the small parameter andinterpreted as the drift of the second particle from the circle orbit. In that case, one canobtain ¯ E CM at the circle orbit as the following,¯ E CM = 1 + E E r + Q Q r ( a N r − f ( r )) − p f ( r ) Q f ( r ) r δ + O ( δ ) , (37)where, Q i = r q E i + a N r − f ( r ) . (38)Here, δ = 0 yields to maximum of ¯ E CM and δ = L co − L co yields to minimum of ¯ E CM , so14e find, ¯ E CMmin = 1 + E E r − Q Q r ( a N r − f ( r )) , ¯ E CMmax = 1 + E E r + Q Q r ( a N r − f ( r )) , (39)and the CM energy has the rang ¯ E CMmin < ¯ E CM < ¯ E CMmax . On the other hand, if wechange position of two particles ( L co = L , and L co = L − δ ) then find the similar resultsas (39). If f ( r ) = a N r , then the maximum value of ¯ E CM goes to infinity and CM energycan reach the Planck energy. For the KS solution of HL gravity, the ¯ E CMmax correspondingto two colliding particles on the circle orbit with radius r = 2 a will be infinity, where weassume that M = 1 and ω is large. Extension of above discussions to the case of slowlyrotating is straightforward and obtained by setting a = 0 in the relations (36) and (39). Several kinds of Kerr black holes already considered as particle accelerators. In this paper,we investigated the possibility that the Horava-Lifshitz black holes may serve as particleaccelerators. Our motivation for this study is that the slowly rotating KS solution of HLblack holes reduced to slowly rotating Kerr black holes. Therefore it is interesting to calculateCM energy of two colliding test particles in the neighborhood of the rotating HL black holes.In agreement with the case of non-rotating Kerr black holes we obtained finite CM energyof two colliding particles on the horizon of the static HL black holes. In this case, wefound that if the angular momentum of particles vanished then the value of CM energy isindependent of horizon. Then we reviewed the rotating HL black hole and discussed thatthe ω → ∞ limit of the slowly rotating KS black hole solution leads to the slowly rotatingKerr solution. Therefore, we calculated CM energy of two colliding particles in the rotatingHL black hole and found that the CM energy on the horizon will be infinity. This resultconfirms the claim of the Ref. [8] where the arbitrary high CM energy of two collidingparticles at the horizon serves as general properties of rotating black holes. So, we foundthe critical angular momentum per unit mass where the CM energy goes to infinity. Weobtained special conditions where the particle with the critical angular momentum mayhave an orbit outside the outer horizon. We have shown that this orbit may be a circle whenthe angular momentum limited in the special range. Hence we calculated the CM energycorresponding to this circle orbit and found radius of circle orbit at large ω . As mentionedin the section 2 we considered projectable version of HL black hole. It is also interesting toconsider non-projectable version of HL gravity [27, 28, 29, 30] and investigate BSW effect.This is currently under investigation. 15 eferenceseferences