Holographic model with a NS-NS field

Abstract

We consider a holographic model constructed through using the D4/D8- brane configuration with a NS-NS background field. We study some properties of the effective field theory in this intersecting brane construction, and calculate the effects of this NS-NS background field on some underlying dynamics. We also discuss some other general brane configurations.

June 2009

## 1 Introduction

The AdS/CFT correspondence proposed by Maldacena is one realization of the holographic principle [1]. It means IIB string theory on the gravity background is dual to the supersymmetric Yang-Mills field theory in the boundary [2]-[5]. In order to use the AdS/CFT method to study more realistic physics, one need to consider some necessary properties, such as less supersymmetries, confinement and chiral symmetry breaking in constructing models. It was firstly given some investigations in the references [4] and [7]. In [8], Karch et al. introduce fundamental flavors into the holographic models by adding some probe flavor D-branes. In the quenching approximation, the supergravity method is a useful way to study some strong coupling dynamics of the boundary field theory. Follow this proposal, many holographic models are constructed to study the strong coupling physics by the supergravity approximation [9]-[14]. If the number of flavor branes is almost equal to the color brane number (), it means the flavor backreaction becomes large, then the quenching approximation isn’t reliable [15].

Now we generalize to consider the color brane background with a NS-NS background field. In string theory, we know that a NS-NS background field produces some non-commutative effects in the field theory on the D-brane world-volume [16], [18] and [19]. The strong and weak coupled regimes of these non-commutative effective field theory can be analyzed as the same way in [20]. After adding the flavor D-brane into such background, one can obtain some non-commutative QCD-like effective field theory on the intersecting region of the brane configuration. Such model was firstly studied in [13]. It is interesting to construct some holographic models with a NS-NS field from string theory, and investigate some properties of these models by using the gauge/gravity correspondence. Then we can understand such NS-NS background field how to affect the underlying strong coupled dynamics of the effective field theory.

In this paper, we construct a holographic model through using the brane configuration D4/D8- just like the Sakai-Sugimoto model [12]. The difference with the Sakai-Sugimoto model is now the color D4-branes gravity background with a NS-NS field (see appendix). If there only exists a constant NS-NS background, then it will be equivalent to turn on a magnetic field on the flavor brane worldvolume by the gauge invariance. But now except for this NS-NS field, the dilaton and R-R field also include this non-commutative parameter . Thus, it can’t be gauged away by add an additional gauge field on the flavor brane.

Then the effective field theory on the intersecting region of this brane configuration is four-dimensional and non-commutative along the coordinates and . And some strong coupled physics of this effective theory can be studied through investigating the dynamics of the flavor D8-brane in the D4-brane gravity background. If one don’t consider the gauge field on the flavor D8-brane, then the effective D8-brane action is same as the commutative case in [12]. The reason is the non-commutative parameter from the dilaton will cancel out with the contribution of the square root part in the D8-brane DBI action. It means that this NS-NS background field doesn’t have any effects on the dynamics of the flavor D8-brane. But after a magnetic field along the and directions is turned on, then the non-commutative parameter will combine together with this magnetic field, and can’t be canceled in the effective D8-brane action. Finally, this effective DBI action contains the contribution of the NS-NS field. And its effects on the chiral symmetry breaking and phase transition etc. can be analyzed in details through using the supergravity/Born-Infeld approximation. We also investigate a fundamental string moving in the near horizon geometry of the black D4-branes. And we calculate the drag force of moving quark and the Regge trajectory of mesons in this non-commutative quark-gluon-plasma(QGP), and analyze the NS-NS how to influence on these quantities.

This papers is organized as follows. In the section 2, we firstly give a brane construction, and study the D8-brane dynamics without and with a magnetic field in the near horizon geometry of the color D4-branes background with a NS-NS field. In the section 3, we generalize to study some other brane configurations by using the same way as in the previous section. Then in the section 4, we study the physics of a fundamental string moving through the near horizon background of the black D4-branes with a NS-NS field. The section 5 is some conclusions and discussions. And finally there is an appendix.

## 2 Brane configuration

We consider a brane configuration which is composed of D4, D8 and brane in IIA string theory. The D4 are the color branes , while the D8 and branes produce the flavor degrees of freedom. Their extending directions of these branes are as follows

 0123456789Nc D4:xxxxxNf D8,¯¯¯¯¯¯¯D8:xxxxxxxxx (2.1)

In this brane configuration, the D8 and branes are parallel each other and intersect with the D4 branes along directions . All the others are the transverse directions to the intersection region. We assume the coordinate to be compactified on a circle , so it satisfies a periodic condition . Then the fermions on the D4 brane with anti-periodic boundary condition on this circle get mass and are decoupled from the low energy effective theory. Also we assume the number of the color and flavor branes satisfies the condition . In this quenching approximation, the backreaction of the flavor branes on the color branes can be ignored. Due to the existence of the NS-NS field, the coordinates and are non-commutative

 [x1, x2]∼b, (2.2)

the low energy effective theory in the intersecting region is a non-commutative field theory. If let the non-commutative parameter vanish, the gravity background here reduces to the usual near horizon geometry of D4 branes without a NS-NS background field. So our model correspondingly goes to the Sakai-Sugimoto model [12]. This means that this holographic model is connected to Sakai-Sugimoto model through varying the non-commutative parameter . Thus, the holographic model with a NS-NS background field can be regarded as a non-commutative deformation to the Sakai-Sugimoto holographic model.

After compactifying the coordinate and using the gravity background (A.6) introduced in the appendix, we get the following gravity solution

 ds2=(uR)3/2[−dt2+h(dx21+dx22)+dx23+fdx24]+(Ru)3/2(du2f+u2dΩ24), (2.3) R3=π4gsNc,h=11+a3u3,a3≡b2R3,H=1−u3KKu3, B12=a3u3b(1+a3u3)=1b(1−h),e2ϕ=g2su3/2R3/2b2h, (2.4) C01234=g−1sh,C034=g−1sb.

In order to avoid the singularity, the coordinate will be periodic with a radius

 x4∼x4+δx4=x4+4π3R3/2u1/2KK. (2.5)

From this radius of the coordinate , we know it corresponds to a KK mass scale

 MKK=2πδx4=32u1/2KKR3/2. (2.6)

By using the solution (2.4), and doing a double Wick rotation between the coordinate and , we obtain a black hole solution as follows

 ds2=(uR)3/2[−Hdt2+h(dx21+dx22)+dx23+dx24]+(Ru)3/2(du2H+u2dΩ24), (2.7) R3=π4gsNc,h=11+a3u3,a3≡b2R3,H=1−u3Hu3, B12=a3u3b(1+a3u3)=1b(1−h),e2ϕ=g2su3/2R3/2b2h, (2.8) C01234=g−1sh,C034=g−1sb.

Now the coordinates and are all compactified. The time coordinate need to be periodic

 t∼t+δt=t+4π3R3/2u1/2H, (2.9)

then this background is well-defined. And the corresponding temperature of this black hole is

 T=34πu1/2HR3/2. (2.10)

Through a Hawking-Page phase transition at a critical point , the low temperature solution (2.4) will go to the black hole background (2.8).

### 2.1 Low temperature phase

For the low temperature phase, the dominated gravity background is the equation (2.4). Now we consider the dynamics of the probe D8- brane in this background. Assume the transverse coordinate of D8 branes is a function of the radial coordinate , i.e. . Then the induced metric on the D8 branes is

 ds2 = (uR)3/2[−dt2+h(dx21+dx22)+dx23]+(Ru)3/2u2dΩ24 (2.12) +[(Ru)3/2f−1+f(uR)3/2(∂x4∂u)2]du2,

Using the D-brane effective action

 S=SDBI+SCS, (2.13) SDBI=−Tp∫dp+1ξe−ϕ√−det(P[G+B]μν+2πFμν), (2.14) SCS=Tp∫∑iP[Ci∧eB2]∧e2πF2,

then we get the D8-brane action is 111Here we don’t turn on the gauge field on the D8 brane worldvolume, so to investigate the dynamics of D8-brane in the gravity background (2.4), it is enough to only use the effective action of one single flavor D8 brane.

 SDBI∼∫duu13/4√(Ru)3/2f−1+f(uR)3/2(∂x4∂u)2, (2.15)

and

 SCS=0. (2.16)

Except for the proportional coefficient, this action is same as the D8-brane effective in the D4-branes background without a NS-NS background field. The reason is the included in the term will be canceled out with the same factor in the square root part of the DBI action. Thus, at low temperature, the NS-NS background field doesn’t influence on the dynamics of the probe D8-brane in this non-commutative gravity background. Its dynamics is same with the usual commutative case [12] and [21].

Then one can get two different solutions from the equation of motion: one is the connected configuration of the D8- branes, the other is the separated case. The connected solution corresponds to the chiral symmetry breaking phase, while the chiral symmetry is preserved for the separated solution. But the on-shell energy difference between the connected and separated solution is always negative, so this connected configuration is dominated in the low temperature phase [21]. The chiral symmetry is always broken , and the corresponding Nambu-Goldstone boson can be found through calculating the meson spectra [12].

### 2.2 High temperature phase

In the high temperature phase, the corresponding gravity background is the black solution (2.8). We still assume the transverse coordinate of the flavor D8-brane depends on the coordinate , then the induced metric on the D8-brane is

 ds2 = (uR)3/2[−Hdt2+h(dx21+dx22)+dx23] (2.18) +((Ru)3/2H−1+(uR)3/2(∂x4∂u)2)du2+(Ru)3/2u2dΩ24.

So the effective action of the D8-brane is

 S∼∫duu13/4H1/2√(Ru)3/2H−1+(uR)3/2(∂x4∂u)2. (2.19)

Clearly, it still doesn’t depend on the NS-NS background field, and is also same as the commutative case [12] except for the proportional coefficient. Then all the other analysis of the D8- dynamics will be same. Through comparing the energy of these two solutions, one can find there exists two phase at high temperature. One is the chiral symmetry breaking phase, the other is chiral symmetry restoration phase. Between these two phases, it has a chiral phase transition at a critical temperature. Below this critical temperature the system is located at the chiral symmetry breaking phase i.e. , otherwise it will be in the chiral symmetry restoration phase.

### 2.3 Adding a magnetic field

Now we introduce a constant magnetic field along the and directions

 2πF12=B, (2.20)

on the flavor D8-brane222 Holographic models with a constant magnetic or electric field were studied in [22]-[25].. This magnetic field is equivalent to a constant NS-NS field by the gauge invariance on the D8-brane. This NS-NS field will contribute a constant transportation to the NS-NS field

 Bnew=B12+B′12=B+1−h(u)b. (2.21)

This new background is still a solution of the IIA supergravity. Since the dilation and R-R field in the gravity background all depend on the parameter , this magnetic field can’t be canceled out through a redefinition of the parameter . Then the flavor D8-brane effective action contains the contributions of the NS-NS and magnetic field together. In the following we will analyze the effect of the parameters and on the flavor brane dynamics in details.

#### 2.3.1 Low temperature

The induced metric on the flavor D8-brane is same as the equation (2.12). After a magnetic field (2.20) is turned on, the effective action of the probe D8 brane is

 S ∼ ∫duu7/4√h√A(u)√(Ru)3/2f−1+(uR)3/2f(∂x4∂u)2, (2.22)

where the is defined as

 A(u)≡u3R3+b2u3+B21+2bB. (2.23)

It is clear that the factor can’t be canceled each other with the in the action (2.22) through introducing this magnetic field. If the magnetic field vanishes , then this effective action will reduce to the action (2.15).

Now the equation of motion from the action (2.22) is

 ∂∂x4⎡⎢ ⎢ ⎢ ⎢⎣u5/2√hf√A(u)√f+f−1(Ru)3u′2⎤⎥ ⎥ ⎥ ⎥⎦=0. (2.24)

Choose a boundary condition at and perform an integration333Here the denotes ., we get a first derivative equation

 u5/2√hf√A(u)√f+f−1(uR)3u′2=u5/20√f(u0)A(u0)√h(u0). (2.25)

Then the equation is

 u′=√P(u), (2.26) P(u)≡(uR)3f2[(uu0)5h(u0)A(u)f(u)h(u)A(u0)f(u0)−1]. (2.27)

Defining , and , we can rewrite the above equation as

 P = u30R3y3f(y)2[y5h(1)A(y)f(y)h(y)A(1)f(1)−1] (2.28) = u30R3z−1f(z)2[z−5/3h(1)A(z)f(z)h(z)A(1)f(1)−1]. (2.29)

So the shape of this connected D8- brane solution is

 x4(u)=∫uu0du√P(u). (2.30)

The corresponding chiral symmetry in the effective field theory is broken because of this connected solution. Then the asymptotic distance between the D8 and reads

 L=2∫∞u0du√P(u)=2u03∫10dzz4/3√P(z). (2.31)

We plot some figures 1-3 to show this asymptotic distance how to depend on the variable , and in this low temperature phase444In this section, we alway choose in doing the numerical calculations..

These figures shows the asymptotic distance between the D8 and brane is became larger with increasing the value at arbitrary value and . For a fixed and increasing , then is decreased. However, for a fixed and increasing , the distance is also increased. So the magnetic field and non-commutative parameter have a converse contribution to the asymptotic distance .

Substituting the equation of motion (2.25) into the effective action (2.22), we obtain the on-shell energy of this connected D8- configuration

 Sconnected∼∫∞u0duu7/4√A(u)√h ⎷(Ru)3/2f−1+f(uR)3/21P(u). (2.32)

And the energy of the separated D8- brane solution is

 Sseparated∼∫∞uKKduR3/4u√A(u)√h(u)f(u). (2.33)

Then the energy difference of these two solutions is

 δS = Sconnected−Sseparated (2.34) ∼ ∫∞u0duu√A(u)√h(u)(√f−1+f(uR)31P(u)−f−1/2) (2.36) −∫u0uKKduu√A(u)√h(u)f(u) ∼ (2.38) −∫1yKKdyy√A(y)√h(y)f(y).

In the figures 4 and 5, it shows this energy difference varying with the parameter , and .

We find the contribution of the parameters and are similar to the energy difference . And this energy difference is always negative at arbitrary values of the parameters , and . So the connected D8- brane configuration is always dominated, and the chiral symmetry of the effective non-commutative field theory is always broken in the low temperature.

#### 2.3.2 High temperature

If the magnetic field (2.20) is included, the effective D8 brane action in the black hole background (2.8) is

 S ∼ ∫duu7/4√A√h√(Ru)3/2+H(u)(uR)3/2(∂x4∂u)2. (2.39)

Now the effective action contains the parameters and through the factor and . Since the action (2.39) doesn’t explicitly depend on the coordinate , the Hamiltonian relative to the variable is conserved. So the equation of motion is

 ∂∂x4⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣u5/2H√A(u)√h(u)(H+(Ru)3u′2)⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦=0. (2.40)

As the same way used at low temperature, we choose a boundary condition as at , it corresponds to the connected D8- brane solution. Then we obtain a first derivative equation

 u5/2H√A√h(H+(Ru)3u′2)=u5/20√H(u0)A(u0)√h(u0). (2.41)

In the new variables and , the equation can be written as

 u′ = √Q(u), (2.42) Q(u) ≡ (uR)3(u5H(u)2A(u)h(u0)h(u)A(u0)H(u0)u50−H(u)) (2.43) = u30R3z−1(H(z)2A(z)h(1)z5/3h(z)A(1)H(1)−H(z)). (2.44)

So the asymptotic distance between the D8 and defines

 L=2∫∞u0du√Q(u)=2u03∫10dzz4/3√Q(z). (2.45)

Through some numerical calculations, we plot figures 6, 7 and 8 about this asymptotic distance varying with the parameter , and .

The asymptotic distance of the D8 and brane become larger with increasing the value and a fixed value . However, for a fixed , then is decreased with increasing the value . As at low temperature, the magnetic parameter and non-commutative parameter have a converse contribution to the asymptotic distance between D8 and .

For this connected D8- brane solution, after inserting the solution into the effective action (2.39), we obtain its on-shell energy

 Sconnected∼∫∞u0u7/4A(u)√h(u) ⎷(Ru)3/2+(uR)3/2H(u)Q(u). (2.46)

Similar as the low temperature case, there also exists a separated D8- solution, and its energy is

 Sseparated∼∫∞uHduu7/4A(u)√h(u)(Ru)3/4. (2.47)

So the difference of the energy is

 δS = Sconnected−Sseparated (2.48) ∼ ∫∞u0duu√A(u)√h(u)(√1+(uR)3H(u)Q(u)−1) (2.50) −∫u0uHduu√A(u)√h(u) ∼ 13∫10dz√A(z)z5/3√h(z)⎛⎜⎝ ⎷1+u30R3H(z)Q(z)z−1−1⎞⎟⎠ (2.52) −∫1yHdyy√A(y)√h(y).

We explicitly show this energy difference in the figures 10 and 10.

It is clear that the energy difference between the connected and separated solutions has critical point , i.e. critical temperature . If , this holographic model becomes the Sakai-Sugimoto model. This chiral phase transition is studied in [21]. At and a finite , the chiral phase transition is also investigated in [24]. The results got here at or is all same as in [21] and [24]. At a fixed value , the critical point value becomes smaller with increasing the value in figure 10(a). However, it become larger with increasing the value for a fixed in figure 10(b). The influences of the parameters and is very little. Just like the effects on the asymptotic distance, these two parameters also have converse contributions to the critical point of phase transition.

Below these critical points, the energy difference is negative, otherwise, it become positive. It means there exists two phases at high temperature. The phase transition is first-order between each other. Under the value , the connected solution is dominated, then the chiral symmetry in the four-dimensional field theory will be broken . Otherwise, the dominated solution is the separated D8- configuration, and the chiral symmetry will be restored.

## 3 General brane configuration

In this section, we generalize to consider some other brane configurations. Here we mainly consider the color brane still is the D4-branes, so the corresponding gravity backgrounds are the equations (2.4) and (2.8).

Firstly, let us consider the following brane constructions

 0123456789Nc D4:xxxxxNf D6,¯¯¯¯¯¯¯D6:xxxxxxxNf D4,¯¯¯¯¯¯¯D4:xxxxx (3.1)

The effective field theory on the intersecting region is some four-dimensional non-commutative theory. Follow the same method in the previous subsection, we can analyze the dynamics of these non-commutative field theory at strong coupled regime by supergravity approximation. The effective action of the flavor brane in the low temperature background (2.4) and black hole background (2.8) are as follows

 Slow∼∫duun√(Ru)3/2f−1+(uR)3/2(∂x4∂u)2, (3.2) SHigh∼∫duun√(Ru)3/2+H(u)(uR)3/2(∂x4∂u)2. (3.3)

For the flavor D6-brane, the parameter is , while for the D4-brane. The non-commutative parameter is canceled in the effective action. So the classical dynamics of flavor brane is same as the commutative cases. At low temperature, the chiral symmetry is always broken. However, there exists a chiral phase transition in the high temperature phase. Below a critical temperature, the chiral symmetry is broken, it corresponds to the connected D8- brane solution. Otherwise, the solution is a separated brane configuration, and this symmetry is restored.

Also some three-dimensional non-commutative effective field theories can be constructed through the following brane configurations

 0123456789Nc D4:xxxxxNf D6,¯¯¯¯¯¯¯D6:xxxxxxxNf D4,¯¯¯¯¯¯¯D4:xxxxx (3.4)

Now the flavor D6 and D4-brane effective actions at low and high temperature background read

 Slow∼∫duum√(Ru)3/2f−1+(uR)3/2(∂x4∂u)2, (3.5) Shigh∼∫duum√(Ru)3/2+H(u)(uR)3/2(∂x4∂u)2, (3.6)

where the parameter is equal to for the D6-brane, and is for the D4-brane. Again these effective action are same as the commutative cases.

As the same discussions in the above subsections, one can turn on a magnetic field along the directions and on the worldvolume of the flavor branes. Then in the flavor D-brane effective action there will exist some contributions of the and through and . Here we list these effective actions: (a) for brane configurations (3.1), the corresponding effective actions are

 SlowbB∼∫duun−3/2√h(u)√A(u)√(Ru)3/2f−1+(uR)3/2(∂x4∂u)2, (3.7) SHighbB∼∫duun−3/2√h(u)√A(u)√(Ru)3/2+H(u)(uR)3/2(∂x4∂u)2; (3.8)

(b) for brane configurations (3.4), the actions read

 SlowbB∼∫duum−3/2√h(u)√A(u)√(Ru)3/2f−1+(uR)3/2(∂x4∂u)2, (3.9) ShighbB∼∫duum−3/2√h(u)√A(u)√(Ru)3/2+H(u)(uR)3/2(∂x4∂u)2. (3.10)

Then we can investigate some effects of the NS-NS and magnetic field on the string coupled dynamics of the non-commutative effective field theory through using gauge/gravity correspondence. One will find some similar results as the previous subsection. And maybe these holographic models can be used to study some condensed matter physics, for example the quantum Hall effect [36].

Also we can generalize to consider some other color brane background with a NS-NS field, and construct the general Dq/Dp- brane configurations. Then one can study some influences of this NS-NS background field on some properties of the effective theories living on the intersecting parts of these brane configurations by the AdS/CFT correspondence.

## 4 String in non-commutative background

In this section, we consider the dynamics of a fundamental sting in the high temperature phase. We calculate the drag force of a quark moving through the QGP, and also study the Regge trajectory of a meson in this non-commutative QGP.

### 4.1 Quark in non-commutative QGP

The Nambu-Goto action of a fundamental string is

 S=−12πα′∫dτdσ√−detgαβ+12πα′∫P[B],   gαβ≡∂αXμ∂βXνGμν. (4.1)

We consider the two endpoints of a fundamental string separately attached on the black horizon and flavor D8-brane. Following the same way in [26] and [27], we parameterize the world-sheet coordinates of this fundamental string as and , and assume the endpoint (quark) on the flavor brane moving along the direction with

 x2=vt+ξ(u). (4.2)

Since there exists a rotational symmetry in the and plane, it is equivalent to let quark moving along the direction .

So the induced metric on the string worldsheet is

 gττ=−(uR)3/2(H−hv2),  gτσ=gστ=(uR)3/2hvξ′, (4.3) gσσ=(uR)3/2hξ′2+(Ru)3/2H−1, (4.4)

where the denotes . Then inserting it into the string action, we get

 S = −12πα′∫dtduL, (4.5) L ≡ √1−hH−1v2+(uR)3hHξ′2. (4.6)

The NS-NS field is a two-form along the directions and . Since in the coordinate parameterizations the coordinate is independent of the string world-sheet coordinates and , this NS-NS field doesn’t have any contributions to the string action. The string world-sheet momentum of string world-sheet is defined as

 Prx2=∂L∂(∂rx2)=Πξ, (4.7)

where the canonical momentum is . The string momentum is the energy associated by the fundamental string. So the drag force is

 −f=12πα′Πξ. (4.8)

Since the string action (4.6) doesn’t explicitly depend on the variable , the canonical momentum is conserved relative to the parameter . Then the equation of motion is derived as

 ξ′=±Πξ(Ru)3/2  ⎷1−hH−1v2(uR)3h2H2−hHΠ2ξ. (4.9)

Here we need to choose ”+” equation because the fundamental string is received a drag force. The is positive, and is 0 at the horizon and is equal to 1 at infinity. So to preserve the quantity to be positive in the square root in the equation (4.9), the canonical momentum need to satisfy

 Πξ=(ucR)3/2h(uc)v, (4.10)

where the coordinate is chosen as

 u3c=12a3(−(1−a3u3H−v2)+√(1−a3u3H−v2)2+4a3u3H). (4.11)

Thus we obtain the drag force is

 f=12πα′(ucR)3/2h(uc)v. (4.12)

In the figure 11, it shows this drag force how to depend on the non-commutative parameter .

At a fixed velocity , the drag force will be decreased with increasing the value . And at some large , the drag force will vanish. Thus, the NS-NS field can decrease the viscosity of the non-commutative QGP to a moving quark.

If the parameter is very small, then the can be expanded as

 u3c=u3H1−v2(1−v2u3H(1−v2)2a3)+O(a). (4.13)

So at the leading order, the drag force becomes

 f=12πα′(uHR)3/2v√1−v2(1−v2u3H(1−v2)2a3). (4.14)

And if let the non-commutative parameter , this drag force reduces to the commutative case.

### 4.2 Meson in non-commutative QGP

Now we consider a meson moving through the non-commutative hot QGP. Here the black hole background is the metric (2.8). Defined , then this gravity background is written as

 ds2=(uR)3/2[−Hdt2+h(dρ2+ρ2dφ2)+dx23+dx24]+(Ru)3/2H−1du2, (4.15) Bρφ=B12ρ. (4.16)

where the part is omitted. To study a meson moving through the QGP with a velocity , we instead to choose the meson to be rest, while boost a QGP wind along the direction with a constant velocity [29] and [30]. Then the final metric takes the following form