Holographic drag force in 5d Kerr-AdS black hole

We consider the 5d Kerr-AdS black hole as a gravity dual to rotating quark-gluon plasma. In the holographic prescription we calculate a drag force acting on a heavy quark. According to the holographic approach a heavy quark can be considered through the string in the gravity dual. We study the dynamics of the string for the Kerr-AdS backgrounds with one non-zero rotational parameter and two non-zero rotational parameters that are equal in magnitude. For the case of one non-zero rotational parameter we find good agreement with the prediction from the 4d case considered by Atmaja and Schalm.

energy physics and condensed matter phenomena [16]- [18], [35]- [41]. Among these questions applications for processes in the formation and evolution of quark-gluon plasma formed in heavy-ion collision are of particular interest. The QGP is a dense strongly coupled QCD matter (strongly coupled fluid) in the phase of deconfinement. Moreover, at high temperatures the lattice results show quasi-conformal behaviour. Therefore the holographic approach for QGP studies seems to be viable. It is also known that in non-central heavyion collisions there is a nonzero total angular momentum that is related to colliding nuclei [6], [42]- [43]. The major part of this angular momentum is taken away by the spectator nucleons, however some amount of the angular momentum remains in the QGP and is approximately conserved in time [43,44]. So this rotation in QGP may affect on a number of observables.
In [36] the authors considered the 4d Kerr-AdS black hole as a holographic dual of 3d strongly coupled quark-gluon plasma, that has an anisotropy introduced by the presence of non-zero angular momentum. It was explored how such anisotropy affects on a quark motion in the medium. Following the holographic prescription a drag force on a heavy quark was discussed. Holographically, the dynamics of a heavy quark moving in a plasma is related with a string motion with one endpoint attached at the boundary and stretched down into the black hole [20]- [21]. In [36] it was proposed that the drag force will be proportional to a pressure gradient of the dual rotating fluid. For the case of the 4d Kerr-AdS black hole, that gives predictions to a 2 + 1-dimensional theory, the assumption was verified.
In the present paper we extend the analysis of [36] to study the holographic drag force in the 5d Kerr-AdS background. We follow-up [20,21] and represent a heavy quark as a string with the end attached to the boundary of the Kerr-AdS background and stretched down to the black hole horizon. The dynamics of the string is described by the Nambu-Goto action. Calculating the corresponding conjugate momenta we can find one a drag force with which the string exerts on the quark.
A feature of 5d Kerr-AdS metrics, which is also common with higher dimensional Myers-Perry black holes, is that due to SO(4)-symmetry, 5d Kerr-AdS black holes are characterized by two rotation parameters that are associated with the number of Casimirs for SO (4) and are preserved independently. So, there are at least three cases for consideration: a) both rotational parameters have non-zero values which are different, b) rotational parameters are non-zero and equal in the absolute value; c) one of rotational parameters vanishes. In this work we focus on the cases b) and c).
The paper is organized as follows. In Section 2 we briefly review different cases of the 5d Kerr-AdS solutions both in Boyer-Lindquist and global coordinates. In Section 3, we calculate the Nambu-Goto action for the string in the Kerr-AdS background with one nonzero rotational parameter in Boyer-Lindquist coordinates. Assuming that the rotational parameter is small we find the conjugate momenta and the leading term for the drag force. In Section 4 we discuss the case with two rotational parameters that are equal in magnitude. We perform computations in Boyer-Lindquist and global AdS coordinates. In Section 5 we conclude with summary of the presented results as well with outline of future directions mainly related with NICA. In the appendix, we present calculations using the hydrodynamical approach and give useful formulae for the string conjugate momenta in the Kerr-AdS metric written in global AdS coordinates through the coordinate transformation.

D = 5 Kerr-AdS black hole
Our starting point is the five-dimensional gravity model with the negative cosmological term where G 5 is the five-dimensional Newton constant and the cosmological constant is Λ = −6l 2 . The Einstein equations following from (2.1) are given by Rotating black holes with an AdS aymptotics solve eqs. (2.2). It is known that a rotating black hole in five dimensions is characterized by the mass and two angular parameters related to Casimir invariants of SO(4). The generic five-dimensional Kerr-AdS metric with two non-zero rotational parameters in the Boyer-Lindquist coordinates is where 0 ≤ φ, ψ ≤ 2π, 0 ≤ θ ≤ π/2, and the parameter M is associated with the mass, a, b are related to the angular momentum and we also have We note that we use Hopf coordinates for the spherical part of the metric (2.3). The horizon position is defined as a largest root r + to the equation ∆ r = 0. The rotational parameters a and b are constrained such that and the angular momenta [34] are given by The Hawking temperature is defined as The metric on the boundary for (2.3) is We note that in the Boyer-Lindquist coordinates, the metric is asymptotic to AdS 5 in a rotating frame, with angular velocities Let us consider the case of the Kerr-AdS solutions with a = b, then the metric (2.3) takes the form where for (2.5) we have ∆ r = 1 r 2 (r 2 + a 2 ) 2 (1 + r 2 2 ) − 2M, (2.11) The angular momentum has the form The 5d Kerr-AdS solution with a single rotational parameter (a = 0, b = 0) reads (2.14) with ∆ r = (r 2 + a 2 )(1 + l 2 r 2 ) − 2M, The corresponding angular momentum is (2.16) The case of single rotational parameter can this solutions can be constructed from 4d case as the stationary asymptotically flat higher dimensional black holes from the work by Myers and Perry [45]. The transformations that convert (2.3) from Boyer-Lindquist coordinates to asymptotically AdS coordinates a = b = 0 [29] are Ξ a y 2 sin 2 Θ = (r 2 + a 2 ) sin 2 θ, (2.17) It should be noted the coordinates (2.17) are difficult for direct representation of the 5d Kerr-AdS metrics, except the case when we have the two non-zero rotational parameters which are equal by its magnitude: .
The position of the horizon in these coordinates reads However with M = 0 the 5d Kerr-AdS solutions come to the following form ds 2 gAdS = −(1 + y 2 2 )dT 2 + y 2 (dΘ 2 + sin 2 ΘdΦ 2 + cos 2 ΘdΨ 2 ) + , (2.20) that is a well known form of the global representation of the AdS solution.
From (2.20) it is easy to see that the 4d conformal boundary of 5d Kerr-AdS black hole is 4d R × S 3 [31,34], which is reached with y → ∞: (2.21) The boundary metrics ( 3 Holographic drag force in 5d Kerr-AdS background

Setup
The drag force is a force acting opposite to the relative motion of the heavy quark moving with respect to a surrounding quark-gluon plasma. Following the dictionary of the gauge/gravity duality the heavy quark is represented by a string suspended from the boundary of the Kerr-AdS background into the interior. A string can be described by the Nambu-Goto action where g = det g αβ is the determinant of the induced metric which is defined though the 5d spacetime metric G µν by

2)
X µ are the embedding functions of the string worldheet in the spacetime, we also assume that X µ = X µ (σ). The equations of motions have the form The conserved currnents are defined as variational derivatives on Note, that this current is related to the translational invariance. In (3.1)-(3.2) we define σ α with α = 0, 1 as the string worldsheet coordinates. So for the conjugated momenta π α µ one can write The corresponding charge reads The associated conserved charge is the total momentum in the µ-direction where Σ α is a cross-sectional surface on the worldsheet.
the time-independent force on the string is (3.10)

Straight string solution in global AdS
Let us consider a string motion in the 5d Kerr-AdS background written in the AdS coordinates. The general form of the metric in these coordinates is complicated. However, under the assumption M = 0 that corresponds to absence of the quark-gluon plasma the Kerr-AdS metric comes to the global AdS solution (2.20), that can be represented as We use the physical gauge with (σ 0 , σ 1 ) = (T, y).
For the embedding we have X µ = X µ (σ), so The induced metric is where the indices run as I, J = (Θ, Φ, Ψ) and˙= d dT , = d dy . The Nambo-Goto action reads S N G = dT dy |g|, (3.16) with the determinant of the induced metric: assuming that fluctuationsẊ I , X I are small we can write The equations of motion that follow from (3.16) with (3.18) take the following form where Θ 0 , Φ 0 and Ψ 0 are some constants corresponding to a massive quark at rest. We note that plugging the solution for the straight static string (3.21) into (3.17), we see that −g is not positively defined. This fact was mentioned in [21]. The corresponding "time-dependent" solution in the Boyer-Lindquist coordinates with where

Curved string in the 5d Kerr-AdS background with one rotational parameter
Now we turn to discussion of a curved string in the 5d Kerr-AdS background with one nonzero rotational parameter (a = 0, b = 0). We use the form of the metric in Boyer-Lindquist coordinates (2.14)-(2.15) 1 . The string worldsheet is parametrized as (σ 0 , σ 1 ) = (t, r). The embedding is characterized by Then non-zero components of the the induced metric g αβ look like where G µν are components of the 5d Kerr-AdS metric (2.14)-(2.15) and we define˙= d dt A, = d dr A. The Nambu-Goto action reads where the determinant of the induced metric built on (3.25)-(3.27) is where ∆ r , ∆ θ , Ξ a , ρ are defined as (2.15). Taking into account the solution for the straight string (3.22) we choose the following ansatz for the curved string solution We note that with β 2 = 0 we come to the case with fixed ψ, i.e. ψ(t, r) = Ψ 0 . Owing to (3.30) the components of the induced metric can be represented in the following way Correspondingly, the induced metric (3.29) takes the form From the Nambu-Goto action with (3.34) we can find out the first integrals expanded in series by a and the following relations hold with some conserved quantities P and Q and h(r) is which is actually the blackening factor for the 5d AdS-Schwarzschild black hole. It worth to be noted that ψ 1 = 0 in the case β 2 = 0. Substituting (4.10) into the Nambu-Goto action (3.28) with (3.34) we can derive the equation of motion for θ 1 expanded in series by a in the following form (3.37) Taking into account the dimensions of the quantities r we get that the dimension of the LHS of the expression (3.37) is 1 r 4 . The derivative of θ 1 can be represented as For the 5d AdS-Schwarzschild black hole with the spherical horizon and the blackening function given by (3.36), the horizon and the Hawking temperature are given by . Then we can represent (3.38) in terms of r H as follows where we also take into account that Expanding near the boundary r → +∞ we get the following relations for the conjugate momenta where we use log r+r H r−r H ≈ 2 r H r and tan −1 The components of the drag force can be found owing to (3.10) and (3.44)-(3.46) as The component dp θ dt (3.47) has a linearly divergent term −(β 2 1 − β 2 2 + 2β 1 ) 2 r with r → ∞. This term can be associated to the infinite mass of the heavy quark [21]. One can renormalize it introducing a cut-off. The forth term in (3.47) manifests the dependence on the temperature. Owning to (3.39) it yields Here the sign "−" shows that the drag force is opposite the quark movement. With respect to values of parameters β 1 and β 2 we have the following special cases This case corresponds to fixed ψ, the conjugate momentum in the ψ-direction is equals to 0. Eq. (3.51) covers the result of [36], i.e. the leading term of the drag force in the θ-direction is 3 2 r. Taking r c = 2πα m rest as a cut-off it comes to the form 6πα m rest 2 , that is exactly matches with the leading term for the drag force from [36]. The dependence of dp θ dt on the temperature is that as in the general case ∼ T H .
Here the relations seems to have the same form comparing to the generic case (3.47)-(3.49) except the coefficients. The dependence on the temperature changes to the inverse one.
4 Drag force from the 5d Kerr-AdS metric with two rotational parameters
The first integrals for φ 1 and ψ 1 can be found from the Nambu-Goto action with (4.9), in lower order by a, and we can write down where we also use a rescaling for the conserved quantities as P = P is given by (3.36). The equation of motion for θ , that follows from the Lagrangian with (4.9) after substitution (4.10), reads Owing to (3.40)-(3.41) we can represent θ 1 as follows 3 The corresponding conjugate momenta can be calculated using (3.4). On the boundary 3 The constantC1 differs from C1 by to including an imaginary constant.
r → +∞ they take the following form , (4.14) π r ψ = Q cos(Θ 0 ) 2 a + O(a 2 ). (4.15) Thanks to (3.10) the drag force can be found as As in the case with one non-zero rotational parameter, the component dp θ dt has a divergent term (β 2 1 − β 2 2 + 2(β 1 − β 2 )) 2 r which can be related to the infinite mass of the heavy quark. The forth term (4.40) represents the dependence on the temperature. It also coincides with the one rotational parameter case (3.50).

Calculations in global AdS coordinates
Now we are going to consider the string in the 5d Kerr-AdS background with a = b written in global AdS coordinates (2.18) 4 . For the worldsheet coordinates we set We suppose that the embedding is given by Θ = Θ(T, y), Φ = Ψ(T, y), Ψ = Ψ(T, y). (4.25) The string action has the form where the components of the induced metric are We take the following expansion for the transversal coordinates Plugging (4.30)-(4.32) into (4.27)-(4.29) we calculate the Nambu-Goto action (4.26) with the following determinant of the induced metric |g| = 2 y 2 (β 2 1 sin 2 ΘΦ 1 + β 2 2 cos 2 ΘΨ 1 ) + (4.33) with Ξ given by (2.12).
As the in Boyer-Lindquist coordinates the variables Φ 1 and Ψ 1 are cyclic, so we can find the first integrals for Φ 1 and Ψ 1 doing an expansion of (4.26) with (4.33) in series by a and the following relations take place Substituting (4.34) into (4.33) we derive the equation of motion for Θ 1 (4.35) The solution to (4.35) leads to the following relation Using (3.4), (4.34), (4.36) we calculate the corresponding conjugate momenta π y ψ = (y 2 + 2 y 4 − 2M )Ψ 1 a + O(a 2 ). (4.39) Finally, taking into account (3.10) we can write down the drag force components Q cos(Θ 0 ) 2 a + O(a 2 ). (4.42) Comparing the result with this for two rotational parameter a = b performed in Boyer-Lindquist coordinates, we find that the relations for dp θ dt have the common dependence on the radial coordinates (y and r, correspondingly) and the horizons (y H , r H ). However the coefficients for these terms are different. One can reach an exactly same answer if we choose the parameters for the string dynamics as β 1 = β 2 . Namely, we have Here we can also observe a degenerate case if the parameters P = Q and the constant C 1 = 0. Then the components of the drag force are equal to zero. It worth to be noted that the calculations of the drag force from hydrodynamical approach predict this result, see Appendix A (A.13).

Conclusions
In this paper we have studied a drag force acting on a heavy quark moving in the rotating quark gluon plasma within the context of the holographic duality. We have considered a 5d Kerr-AdS black hole as a holographic dual of 4d strongly coupled rotating QGP. We have focused on cases when the black hole solution has one non-zero rotational parameter and two rotational parameters, that are equal in magnitude. These cases are related to the presence of one and two Casimir invariants for SO(4), correspondingly. Following the holographic prescription we have associated a heavy quark with an end of a string suspended on the boundary of the Kerr-AdS black hole into its interior. We have solved the string equations of motion order by order in a. Then we have found the corresponding conjugate momenta, which is related to components of the drag force. In the case of one rotational parameter we performed calculations in Boyer-Lindquist coordinates and have established that the result is in agreement with the prediction of the work [36] where it was considered a lower dimensional holographic duality, namely, 4d Kerr-AdS black hole for the description of 3d rotating quark-gluon plasma. It is actually not surprising, since 4d rotating black holes have just one Casimir invariant corresponding to SO(3). In the case of two equal rotational parameters we have calculated the drag force both in Boyer-Lindquist and global AdS coordinates. It can be considered as a degenerated case when the drag force vanishes. This result matches with calculations from the hydrodynamical approach on 4-dimensional sphere.
A straightforward problem for future study is to investigate the drag force in the 5d Kerr-AdS black hole with two non-equal rotational parameters.
One interesting problem for the future work is the study of the drag forces in the rotating charged Kerr-AdS metric [47], that has interest in the context of effects related with non-zero chemical potential [48,49]. This would be relevant in the context of NICA, since it is expected to have hyperon polarization at NICA energies [50,51]. Note also that vortical effects in high-energy nuclear collision are closed related with chiral magnetic effects [52] that are also admit investigations in the holographic approach.

A Hydrodynamical calculations
Here we consider rotating fluid in 4 dimensional spacetime R × S 3 that has the metric The metric has the following non-zero Christoffel symbols The stress-energy tensor in hydrostationary equilibrium reads where g AB are components of the metric (A.1),A, B = 1, . . . , 4, ρ -the density, P -the pressure, the velocity field is Correspondingly, taking into account u A = g AB u B we also have The corresponding conservation law is One can rewrite the conservation law projecting (A.6) onto the direction orthogonal to the velocity field (g BC + u B u C ) ∇ A T AC = 0, (A.7) that can be rewritten as (g BC + u B u C )∇ C P + (ρ + P )u C ∇ C u B = 0. (A.8) Owing to rotational symmetry we have the dependence only on Θ for ρ and P : ρ = ρ(Θ), P = P (Θ), (A.9) that reduces to the following equation that is in agreement with our result for one rotational parameter and Schalm's prediction.
If Ω a = Ω b we get ∂ Θ P = 0. It matches with the holographic calculations for the case with two equal rotational parameters a = b in Boyer-Lindquist and global AdS coordinates if we choose β 1 = β 2 , C 1 = 0, P = Q. Taking into account ρ + P = sT. (A.14) The entropy density is where the Newton's constant equals G 5 = π (2N 2 c 3 ) . A relativistic pressure gradient force is given by dp θ dt = −3m rest ∂ θ P sT . (A.16) So we can write dp θ dt = −3m rest 2 π 2 T 4 N 2 c ∂ θ P, (A.17) that has a good agreement with (3.51),where we take the leading term 3 2 r and choose a cut-off as r c = 2πα m rest .
B Dependence on the horizon for the case of two equal rotational parameters.