Holograghic Brownian motion in three dimensional G\"{o}del black hole

By using the AdS/CFT correspondence and G\"{o}del black hole background, we study the dynamics of heavy quark under a rotating plasma. In that case we follow Atmaja (JHEP 1304, 021, (2013)) about Brownian motion in BTZ black hole. In this paper we receive some new results for the case of $\alpha^{2}l^{2}\neq1$. This case, we must redefine the angular velocity of string fluctuation. We obtain the time evolution of displacement square that angular velocity and show that it behaves as a Brownian particle in non-relativistic limit. In this plasma, it seems that relating the Brownian motion with physical observables is rather a difficult work. But our results match with Atmaja work in the limit $\alpha^{2}l^{2}\rightarrow1$.


Introduction
In the last several years, the holographic AdS/CFT [1]- [4] has been exploited to study of strongly coupled systems, in particular quark gluon plasmas [5]- [7]. The quark gluon plasma (QGP) is produced, when two heavy ions collide with each other at very high temperature. A relatively heavy particle, e,g. a heavy quark, immerses in a soup of quarks and gluons with small fluctuations due to its interaction with constituent of QGP. The random motion of this particle well-known as Brownian motion [8,9]. The Brownian motion is a universal phenomenon in finite temperature systems and any particle immersed in a fluid at finite temperature undergoes Brownian motion. The Brownian motion opens a wide view from microscopic nature and studying it by using the AdS/CFT correspondence. It offers a better understanding of the microscopic origin of thermodynamics of black holes, Therefore, it is a natural step to study Brownian motion using the AdS/CFT correspondence. Particulary, the AdS/CFT correspondence can be utilized to investigate the Brownian motion for a quark in the quark gluon plasma.
In the field theory or boundary side of AdS/CFT story, a mathematical description of Brownian motion is given by the Langevin equation which phenomenologically describe the force acting on Bownain particles [9]- [11] which is given by, where p is momentum of Browniam particle and γ 0 is the friction coefficient. These forces originate from losing energy to medium due to friction term (first term) and getting a random kick from the thermal bath(second term). One can learn about the microscopic interaction between the Brownian particle and the fluid constituents, if measures these forces be clear. By assuming < mẋ 2 >= T , the time evolution of displacement square is given as follows [9], where D = T γ 0 m is diffusion constant , T is the tempreture and m is the mass of Brownian particle. At early time, t ≪ 1 γ 0 (balistic regime), the Brownian particle moves with constant velocityẋ ∼ T m , while at the late time, t ≫ 1 γ 0 (diffusive regime), the particle undergoes a random walk. In the gravity or bulk side of AdS/CFT version for Brownian motion, we need a gravitational analog of a quark immersed in QGP. This is acheived by introducing a bulk fundamental string stretching between the boundary at infinity and event horizon of an asymptotically AdS black hole background [12]- [16]. The dual statement of a quark in QGP on the boundary corresponds to the black hole environment that excites the modes of string. In the context of this duality, the end of string at the boundary corresponds to the quark which shows Brownian motion and its dynamics is formulated by Langevin equation. In the formulation of AdS/CFT correspondence, fields of gravitational theory would be related to the corresponding boundary theory operators [3,4], such as their boundary value should couple to the operators in a way consist with symmetries of problem. In this way, instead of using the boundary field theory to obtain the correlation function of quantum operators, we can determine these correlators by the thermal physics of black holes and use them to compute the correlation functions. In [13]- [16], the Brownian motion has been studied in holographic setting and the time evolution of displacement square. If we consider different gravity theories, we know that strings will be live in a black hole background and excitation of the modes are done by Hawking radiation of black hole. So, different theories of gravity can be associated with various plasma in the boundary. In this paper we follow different works to investigate Brownian motion of a particle in rotating plasmas. We try to consider the motion of a particle in two dimensional rotating plasma which its gravity dual is described by three dimensional Gödel metric background. In this case, we will see that for the α parameter in the Gödel metric, new conditions for Brownian motion will be provided. As we know, the Brownian motion of a particle in two dimensional rotating plasma with the corresponding gravity of BTZ black hole has been studied in [15]. The Gödel metric background in the special case receives to the BTZ black hole [17], so the comparison of our result with what obtained in the [15], gives us motivation to understanding about the Brownian motion in rotating plasmas in general form of background as Gödel black hole. This paper is arranged as follows. In section 2, we give some review of three dimensional Gödel black hole and derive string action from this metric background. Section 3 is devoted to look for a holographic realization of Brownian motion and obtain the solution for equation of motion of string in Gödel black hole geometry. We study the Hawking radiation of the transverse modes near the outer horizon of Gödel black hole to describe the random motion of the external quark in section 4. In section 5, we make some comments about our results and close with conclusions.

Background and String action 2.1 Gödel black hole
Three dimensional Gödel spacetime is an exact solution of Einstein-Maxwell theory with a negative cosmological constant and a Chern-Simons term [18]. When the electromagnetic field acquire a topological mass α Maxwell equation will be modified by an addition term. In that case, we receive to Einstein-Maxwell-Chern-Simons system, and the geometry is the Gödel space time [19]. This theory can be viewed as a lower dimensional toy model for the bosonic part of five dimensional supergravity theory, so it can be advantage in development of string theory. Three dimensional Gödel black holes are alike their higher dimensional counterparts in special properties . The action of Einstein-Maxwell-Chern-Simons theory in three dimensions is given by [20], A general spherically symmetric static solution to the above action in various cases for the α parameter, can be written by [21], where p, q and h are functions of r as, The gauge potential is given by, with The parameters µ and J, are mass and angular momentum. The arbitrary constant ε, is a pure guage. We can rewrite metric (4) in the ADM form as follows, where and The Hawking temperature that gives the temperature of the plasma, is [22] Here r − is the inner horizon and r + is the outer horizon. In the sector α 2 l 2 > 1, we have real solution only for µ negative. In this regime, there are Gödel particles and theory supports time-like constants fields. When α 2 l 2 < 1, µ has positive values. This is the case that black hole will be constructed and theory supports space-like constants fields. For α 2 l 2 = 1 , the metric (4) reduces to BTZ metric as can be explicitly seen by transforming to the standard frame that is non-rotating at infinity with respect to anti-de Sitter space, In the standard frame, energy and angular momentum become M = µ − αJ and J, instead of µ and J in rotating frame.

String action
In the general case for a d + 2 dimensional black hole metric background is, Here x µ = r, t stands for the string worldsheet coordinates and X I = X I (x) (I,J=1,...,d-2) for the spacetime coordinates. If we stretch a string along the r direction and consider small fluctuation in the transverse direction X I , the dynamics of this string follows from the Nambu-Goto action [13], If the scalars X I do not fluctuate too far from their equilibrium values (X I = 0), we can expand the above action up to quadratic order in X I , In fact, this quadratic fluctuation lagrangian can be interpreted as taking the non-relativistic limit, so we must use the dual Langevin dynamics on boundary in the non-relativistic case.

Strings in Gödel black hole
As we said in the introduction, an external quark is dual to an open string that extends from the boundary to the horizon of the black hole [23]. We can obtain the dynamics of this string in a three dimensional Gödel black with the metric background (9) by the Nambu-Goto action (15) in the following form, We have obtained the above relation in the standard frame. The equation of motion for φ derived from (17) is

Trivial solution
The Nambu-Goto action up to quadratic terms after subsisting the small fluctuation φ → C + φ under Gödel metric background in the standard frame is given by By changing coordinate to s = x−x + (where x = r 2 2 + 2GJ α ) and defining φ(t, s) = e −iωt f ω (s), one can write the equation of motion as following, where The solution for this equation of motion is the trivial solution for the relation (18). In general, solution of this equation is very complicated. However for the extremal case µ = J(1+α 2 l 2 ) αl 2 , we can find an analytical solution as, where Y (s) is obtained from the following relation where χ = −1 ± D and z = 2x + l 2 . The complete solution of Y(s) is derived in the Appendix. With similar argument as in the ref [15], this solution is not acceptable, because it dosn't have oscillatory modes in radial coordinates and also for this trivial constant solution, one can investigate that the square root determinant of the worldsheet metric is not real everywhere, therefore it is not physical solution. Due to nonphysical motivation about the mentioned solution, we have to consider other approach, which is linear solution.

Linear solution
We can take the linear ansatz for the small fluctuation in the transverse direction φ to achieve a non-trivial solution, so we expand it as, where w is a constant angular velocity. By replacing this relation into equation (18), the solution for η is obtained as following, where π φ is a constant which has a concept as the total force to keep string moving with linear angular velocity w, and also is related to momentum conjugate of φ in r direction. At r = r N H , the numerator becomes zero, so the denominator should also vanish there, because the string solution (24) must be real everywhere along the worldsheet. For w = 0 and α 2 l 2 = 1, r N H is given by When w = 0 and α 2 l 2 = 1 , we receive the excepted relation for BTZ black hole [15]. For w = 0 and α 2 l 2 = 1, we obtain According to [15] we set dominator to zero, so we have, The external force F ext , can be obtained by considering the rotation and the topological mass of black hole which is given by, After extracting this external force, we can derive the friction coefficient γ 0 for non-zero w, by considering the relation p φ = m 0 w , as With α 2 l 2 = 1 this coefficient reduce to the excepted value γ 0 = r 2 NH 2πάm 0 for BTZ black hole [15].
The Nambu-Goto action, with the small fluctuation, φ → wt + η(r) + φ under the Gödel background becomes, The equation of motion from the above Nambu-Goto action is given by, Solving this equation is quite complicated for more values of w. However, one can find that there are some value like where this makes it possible to solve the equation of motion. In derivation the r.h side of the above relation we use r − r + = 4GJ α . The special radius r N H approaches the outer horizon of the Gödel black hole for this value of angular velocity w, where ∆(r + ) = 0( and π φ = 0), then the steady state solution is the case that r N H = r + . From the relation (33) for the angular velocity, it is evident that we can receive to w = r − r + for BTZ black hole (α 2 l 2 = 1). Furthermore for (α 2 l 2 > 1), we can check that w 2 < α 2 , but there must be some condition on µ and J to have w 2 < α 2 for (α 2 l 2 < 1). We can write the equation of motion for this terminal angular velocity with changing coordinate to s = x − x + as, where and As before, we take φ(t, s) = e −iωt f ω (s), so the equation (34) reduce to, Consequently, the independent linear solutions to the above equation, are obtained as bellow f ± ω (s) = (λζs) ±iϑ 2 F 1 (±iϑ , where or with ξ = 2ζ = r 2 .By considering the following relation for hypergeometric functions, the equation (38) reduces as, which gives oscillation modes. We have the following asymptotic behavior from the solutions near the outer horizon (s → 0) and the boundary (s → ∞),

Displacement Square
So far, we have succeeded to drive oscillation modes for a string moving in the Gödel black hole background. In the following, we follow the same procedure as in [13,15] to compute the displacement square for Brownian motion. In order to achieve this, we write the solutions for bulk equation of motion as a linear combination of f ± ω , A and B are constants. By exerting the Neumann boundary condition near the boundary, ∂ s f s (ω) = 0 with s = s c ≫ 0, to put the UV-cutoff, we obtain, (1 + 2iϑ) Note that the constant B is a pure phase, so by using the equation (42)in the near horizon we can write, Φ(t, s) = f ω (s)e iωt ∼ e −iω(t−s⋆) + e iθω e −iω(t+s⋆) .
To regulate the theory, we implement another cutoff near the outer horizon at s h = ǫ, ǫ ≪ 1 , which called IR-cutoff, we obtain, If we take B in the terms of ω by the relation (43) only, then B has continues values, since the ω can have any value. Using the relation (45) for B will satisfy our requirements to have discrete values in ǫ ≪ 1. In this case, the discreteness is [13,15], where in terms of ω, it is given by (48) Following the above processes and using IR cutoff to discrete the continues spectrum , makes it easy to find normalized bases of modes and to quantize φ(t, r) by extending in this modes.

Brownian particle Location
In this section we are going to use quantized modes of the string near the outer horizon of Gödel black hole to describe the Brownian motion of an external quark. Therefore we consider the Nambu-Goto action for certain amount of terminal angular velocity, near the outer horizon (s → 0) , where Φ ≡ 8GJ r − √ 2πλά φ . Thus, according to the same procedure for standard scalar fields, we introduce the following mode expansions, with Now, by considering the above quantum modes on the probe string in the bulk, we want to work out the dynamics of the endpoint which corresponds to a external quark. We investigate the wave-functions of the world-sheet fields in the two interesting regions: (i) near the black hole horizon and (ii) close to the boundary. From (42), near the horizon (S ∼ 0), the expansion (50) becomes, We used S = 2s = r 2 − r 2 + . On the other hand, the expansion (50) at S = R ( the location of the regulated boundary) is given by, (53) One can see that there are two modes in the solutions. The outgoing modes (ω > 0) that are excited because of Hawking radiation [24,25] and incoming modes (ω < 0) which fall into black hole. The outgoing mode correlators are determined by the thermal density matrix, and the expectation value of occupation number is given by the Bose-Einstein distribution, with β = 1 T . Using the knowledge of relation (54) about outgoing modes correlators in the bulk, we can investigate the motion of the endpoint of the string at S = R ≫ 1. We can also determine the behavior of the Brownian motion, by computing displacement square, as came in (2). So we can predict the nature of Brownian motion of external particle on the boundary. For this purpose, we compute the modes correlators at S = R ≫ 1 as, By utilizing (48), we can write above relation in the integral form. We observe that the integral is diverge. So we regularize it by normal ordering the a ,a † oscillators : a ω a † ω ≡ a † ω a ω , then we have, and the displacement square becomes, with and we have defined The evaluation of these integrals and their behavior for R ≫ 1 and a ≫ 1 can be found in Appendix B of [13]. From the relation (60), we can see that when R ≫ 1, we have a ∝ 1 c . Thus in general case for a, we use the following relations for the integrals (59), However, for R ≫ 1 and c ≪ 1 (α 2 l 2 → 1), then a ≫ 1, one can utilize the following relation for I 1 and I 2 , . (63)

Conclusion
In this paper, we used Ref. [15] and obtained the time evolution of the displacement square for realization of the Brownian motion of an external quark in a plasma which corresponds to Gödel black hole. We showed to have a acceptable solution with oscillatory modes, we had to change the definition of terminal angular velocity. We found that in general case (α 2 l 2 = 1) our results for displacement square are different in the comparison with Ref. [15]. However, in α 2 l 2 = 1 limit, We confirmed that our results are agree with the work of Atmaja [15]. We derived the physical mass, but we found that relating the displacement square with physical observables is a difficult work. This is the problem that we would like to consider in future work. Also we would like to investigate the Brownian motion of external quarks in different environments, in particular plasmas which correspond to metric backgrounds as Lifshitz geometry [27] and metric backgrounds with Hyperscaling violation [28,29].