A Strongly Coupled Anisotropic Fluid From Dilaton Driven Holography

A Strongly Coupled Anisotropic Fluid From Dilaton Driven Holography

Sachin Jain, Nilay Kundu, Kallol Sen, Aninda Sinha and Sandip P. Trivedi
Department of Theoretical Physics, Tata Institute of Fundamental Research,
Colaba, Mumbai, 400005, India
Centre for High Energy Physics, Indian Institute of Science,
C.V. Raman Avenue, Bangalore 560012, India. E-mail:
sachin@theory.tifr.res.in, nilay.tifr@gmail.com, kallol@cts.iisc.ernet.in, asinha@cts.iisc.ernet.in, trivedi.sp@gmail.com
Abstract:

We consider a system consisting of dimensional gravity with a negative cosmological constant coupled to a massless scalar, the dilaton. We construct a black brane solution which arises when the dilaton satisfies linearly varying boundary conditions in the asymptotically region. The geometry of this black brane breaks rotational symmetry while preserving translational invariance and corresponds to an anisotropic phase of the system. Close to extremality, where the anisotropy is big compared to the temperature, some components of the viscosity tensor become parametrically small compared to the entropy density. We study the quasi normal modes in considerable detail and find no instability close to extremality. We also obtain the equations for fluid mechanics for an anisotropic driven system in general, working upto first order in the derivative expansion for the stress tensor, and identify additional transport coefficients which appear in the constitutive relation. For the fluid of interest we find that the parametrically small viscosity can result in a very small force of friction, when the fluid is enclosed between appropriately oriented parallel plates moving with a relative velocity.

preprint: TIFR/TH/14-16

1 Introduction

The AdS/CFT correspondence has opened the door for an interesting dialogue between the study of strongly coupled field theories, of interest for example in the study of QCD and condensed matter physics, and the study of gravity. See, for example, the reviews [1, 2, 3, 4, 5] and references therein. An important area in which there has already been a fruitful exchange is the study of transport properties, and the related behavior of fluids, which arise in strongly coupled systems, see [6] for a review. Another interesting area has been the study of black holes and branes which carry hair or have reduced symmetry. The existence of such solutions, which at first sight contradict the conventional lore on no-hair theorems, was motivated by the existence of corresponding phases on the field theory side, see [7, 8, 9, 10], for example.

Among solutions with reduced symmetry which have been found are black branes which describe homogeneous but anisotropic phases, i.e. phases which leave translational invariance, at least in some generalized sense, intact, but which break rotational symmetry. An example of such a phase on the field theory side is a spin density wave phase, well known in condensed matter physics. Such black brane solutions were systematically studied in [11, 12, 13] and it was shown that they can be classified using the Bianchi classification developed earlier for understanding homogeneous cosmologies. A natural follow up question is about the transport properties in such anisotropic phases. One would like to know whether these transport properties reveal some interesting and qualitatively new types of behavior when compared to weakly coupled theories or strongly coupled isotropic situations which have already been studied via gravity.

In this paper we turn to examining this question in one of the simplest examples of an homogeneous but anisotropic phase. This phase is obtained by considering a five dimensional system consisting of gravity coupled to a massless scalar, the dilaton, in the presence of a negative cosmological constant. The anisotropic phase arises when the dilaton is subjected to a linearly varying boundary condition along a spatial direction in the asymptotically region. More precisely, a non-normalisable mode for the dilaton is turned on in the asymptotically region which is linearly varying along one of the boundary spatial directions. The spatial direction along which the dilaton varies breaks isotropy. The resulting solution is characterized by two parameters, , which is proportional to the dilaton gradient and is a measure of the breaking of isotropy, and , the temperature. By taking the dimensionless ratio to be large or small we can consider the highly anisotropic case or the mildly anisotropic one respectively.

Next, we analyze the transport properties of this system, and studying its stability by both examining the thermodynamics and the spectrum of quasi normal modes in some detail. We also set up the fluid mechanics which arises in anisotropic phases in some generality and apply these general considerations to study the system at hand.

When the anisotropy is small its effects can be incorporated systematically, starting with the well known rotationally invariant black brane in , in a perturbative expansion in . The resulting changes in thermodynamics and transport properties are small, as expected. It is the region where the anisotropy is big, with , that is truly interesting, and where qualitatively new phenomenon or behavior could arise. It is this region of parameter space that we mostly explore in this paper.

Before proceeding it is worth mentioning that the linearly varying dilaton actually breaks translational invariance along with rotational invariance. Thus, it would seem that this system is quite different from the ones mentioned above which describe homogeneous but anisotropic phases. In fact the similarity is closer than one might expect because for the more standard situation, where the dilaton takes a constant asymptotic value, it is well known that the thermodynamics of the dual CFT is independent of this asymptotic value, in the gravity approximation. As a result, any corrections to the thermodynamics must be proportional to the gradient of the dilaton. Since this is a constant for a linearly varying dilaton, the resulting thermodynamics behaves like that of a translationally invariant, but anisotropic system.

We find that when , with fixed , the resulting extremal black brane flows to an attractor geometry in the near horizon region. This shows that in the dual field theory the anisotropy is in fact a relevant perturbation and the theory flows to a new fixed point characterized by the symmetries of a dimensional CFT, unlike the theory in the UV which is a CFT in dimensions. Starting from this extremal geometry we then study the effects of turning on a small temperature, so that continues to be large but not infinite. The essential features of the thermodynamics can be understood analytically from the near-horizon region. We find that the system is thermodynamically stable, with a positive specific heat.

We then turn to analyzing the viscosity of the system in more detail. In a rotationally invariant theory there are two independent components of the viscosity, the shear and bulk viscosity. In the conformally invariant case the bulk viscosity vanishes. Once rotational invariance is broken the viscosity has many more independent components and should be thought of most generally as a fourth rank tensor, with appropriate symmetry properties. In our case, for a dilaton gradient along the direction, rotational invariance in the plane is preserved and can be used to classify the various components. We find that viscosity, , which corresponds to the spin two component of the metric perturbations continues to saturate the famous Kovtun-Son-Starinets (KSS) bound, see [14], [15], with

(1)

where is the entropy density. But the viscosity, , which corresponds to the spin one component can become much smaller, violating this bound. In fact, close to extremality it goes like,

(2)

so that the ratio vanishes as for fixed .

In section 5 we study the quasi normal modes of the system in considerable detail, see [16] for a general discussion. Our emphasis is on the regime close to extremality where, as mentioned above, components of the viscosity can become very small. In the rotationally invariant case, the KSS bound can be violated, see [17], [18], [19], [20], but attempts to make very small often lead to pathologies, like causality violation in the boundary theory, see [21], [22]. Here, in the anisotropic case, we find no signs of instability, close to extremality, for the quasi normal modes we study. All the frequencies of these modes we find lie safely in the lower complex plane. While we do not study all the quasi normal modes, we view the absence of any instability in the fairly large class we have studied as an encouraging sign of stability for the system.

In section 6 we turn to a more detailed analysis of the fluid mechanics. We first describe in fairly general terms how to set up the fluid mechanics, at least to first order in the derivative expansion for the stress tensor, for anisotropic phases. The breaking of rotational invariance is characterized by a four vector, , proportional to the dilaton gradient which enters in the fluid mechanics. As a result the number of allowed terms in the constitutive relation for the stress tensor proliferate. An added complication is that the system we are considering actually breaks translational invariance. While the equilibrium thermodynamics still continues to be translationally invariant, as was mentioned above, the breaking of translational invariance does need to be accounted for in out of equilibrium situations pertaining to fluid mechanics. We describe how to do so in a systematics manner in the derivative expansion as well. Towards the end of this section we consider a simple example of a flow between two plates with a relative velocity between them. This is a canonical situation where the shear viscosity results in a friction force being exerted by the fluid on the plates. Due to the breaking of rotational invariance we find that the friction force is different depending on how the plates are oriented. For a suitably chosen orientation it turns out to be proportion to , eq.(165), and can become very small close to extremality, as .

In section 7 we discuss string embeddings of our system. The simplest embedding is the celebrated example of IIB string theory on . However, we find that in this example there is an instability for the extremal and near-extremal brane which arises from a KK mode on . This mode saturates the BF bound in but lies below the BF bound for the geometry which arises in the near horizon limit of the near-extremal black brane.

Let us make some comments before concluding this introduction. First, it is worth pointing out that the equations which govern the fluid mechanics of an isotropic phase are always rotationally invariant since these equations inherit the symmetries of the underlying equilibrium configuration. The solutions of these equations in contrast of course need not be, and are often not, rotationally invariant. The rotational symmetry can be broken, for example, by initial conditions. This is what happens for example in relativistic heavy ion collision experiments. Rotational invariance is not broken in equilibrium by heating up QCD to temperatures attained at heavy ion colliders. However the initial conditions for the collisions are anisotropic resulting in anisotropic fluid flows. In contrast, the system we are studying has no rotational invariance in equilibrium, and the resulting equations of fluid mechanics themselves break rotational invariance regardless of initial or boundary conditions.

Second, some references which directly bear on the study being discussed here should be especially mentioned. The behavior of the gravity-dilaton system, subjected to a general slowly varying dilaton, was studied in [23]. Our results in this paper extend this analysis to the rapidly varying case, after restricting to the linearly varying profile. In fact the more exotic phenomenon arise in the rapidly varying limit as mentioned above. Anisotropic phases along with their viscosity was discussed in [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35]. It was found that the shear viscosity can acquire components in some cases which violate the KSS bound [30], [31], [32], [33], [35]. In particular, in [25] [30] and [31], a system quite similar to the one studied here was analyzed in some depth. These authors considered the gravity system coupled to an axion-dilaton which is subjected to a linearly varying axion instead of dilaton. The related paper by [32] found that some components of the viscosity tensor can become very small violating the KSS bound eq.(1). The main virtue of the dilaton system we have studied is that it is somewhat simpler and this allows the study of issues related to stability to be carried out in more detail. In the paper [33] the system, which is supersymmetric and thus stable, was studied and the ratio of viscosity to entropy density, for a component of viscosity, was found to go like , as in our case. Gravitational systems where the phase of a complex scalar is linearly varying have been studied in [36]. The near horizon extremal solution we obtain, with a linearly varying dilaton, was found earlier in section of [37], see also [38].

Let us end by mentioning that besides the contents of various sections summarized above, additional important details can also be found in appendices A - E. In particular appendix B contains details on how to set up the fluid mechanics in anisotropic situations, and appendix C and D discuss, from a fluid mechanics perspective, the response after coupling to a metric and the linearized fluctuations about equilibrium. These allow us to relate the transport coefficients which appear in fluid mechanics to results from the gravity side. Appendix A and E contain more details of the calculations of the stress energy tensor and quasi normal modes in the gravity theory and appendix F contains a discussion of instabilities in space. Appendix F analyses instabilities in and shows that they are relatively insensitive to the UV completion of the geometry and continue to persist independent of the details of this completion. This is due to the infinite throat of the ; the argument is also valid for other spaces and should be generalizable for other attractor geometries as well. Finally Appendix G contains an analysis of a similar theory in one lower dimension. In this case the spacetime is asymptotically , and the near horizon extremal geometry due to the linearly varying dilaton is . One finds that a component of the viscosity, compared to the entropy density, again becomes small. One advantage in this case is that some of the analysis of the quasi normal modes can be carried out analytically.

2 Anisotropic solution in dilaton gravity system

We consider a system consisting of gravity, a massless scalar field, , which we call the dilaton, and cosmological constant, , in space time dimensions with action,

(3)

Here is the gravitational coupling with being Newton’s Constant in 5-dimension. This system is well know to have an solution with metric

(4)

and with a constant dilaton. , the radius of space, is related to by

(5)

We will also use the constant defined by

(6)

In the discussion which follows we use units where

(7)

so that from eq.(5)

(8)

The system above is well known to arise as a consistent truncation of the bulk system in many cases, e.g., solution of IIB supergravity, and more generally several solutions where is an Einstein Manifold. In the dual field theory, the dilaton is dual to a dimension scalar operator, e.g., in the case dual to the SYM theory it is dual to the gauge coupling.

We will be interested in solutions which break the rotational invariance in the three spatial directions. This breaking of rotational invariance will be obtained by turning on the dilaton. We will show below that a set of black brane solutions can be obtained in which the dilaton varies linearly along one of the spatial directions, say , and takes the form:

(9)

Note that this will be the form of the dilaton in the bulk, in the solutions we consider 111In a string embedding, e.g., IIB theory on , as discussed towards the end of this paper, the gravity approximation can break down at large when the dilaton gets very large or small. We have in mind placing the system in a box of size along the direction to exclude this large region. Mostly our interest is in near-extremal situations and our conclusions will follow by taking .. In particular the dilaton will be independent of the radial direction . The parameter will be an additional scale which will characterizes the breaking of rotational invariance.

The resulting metric which arises after incorporating the back reaction due to the dilaton will preserve rotational invariance in the directions. In addition, since the dilaton stress energy only depends on gradients of the dilaton, it will be translationally invariant in . In this way by turning on a simple linear dilaton profile we will find a class of black brane solutions which correspond to homogeneous but anisotropic phases of the dual field theory.

The metric which arises after incorporating the dilaton back reaction can be written in the form,

(10)

The coefficient on the RHS above can be clearly set to unity by rescaling , but we will keep it explicitly in the metric, since this will prove convenient in some of the following analysis.

The trace reversed Einstein equations and the equation of motion for the dilaton are given by

(11)

In terms of the metric coefficients, the components of the trace reversed Einstein equations can be written explicitly as

(12)

It is easy to see that the linearly varying ansatz, eq.(9), for the dilaton satisfies its equation of motion. Moreover, while the dilaton does not become constant asymptotically, as , its back reaction on the metric can easily be seen to be sub dominant compared to in this region. Thus the metric becomes that of , eq.(4), as . In particular, we will assume that no non-normalizable deformation for the metric is turned on so that we are considering the dual field theory in flat space. We also see that the non-normalizable mode is turned on for the dilaton, this means that the dual field theory is subjected to a varying source term, e.g. for the case a linearly varying gauge coupling. The varying source term breaks rotational invariance and gives rise to the anisotropic phases which we study222It is worth noting that the source terms in the boundary by themselves do not preserve even translational invariance. However the response of the metric in the bulk only cares about gradients of the dilaton and thus is translationally invariant. In the boundary theory this implies that the resulting expectation values of the stress tensor or the operator dual to the dilaton are also translationally invariant. This happens from the field theory point of view because the expectation value of energy density, entropy etc in the CFT is independent of the value of the gauge coupling, , for any constant value of in the gravity approximation. Therefore any deviations from these results must be proportional to the gradient of the gauge coupling which preserves translational invariance..

We will choose so that the asymptotic form for the metric is , eq.(4), in effect setting the speed of light i.e. at the boundary333In general the asymptotic behavior of , and the additional constant would also then have to be taken into account in the determining the thermodynamics. In the analysis below we will see how this constant can be set equal to unity.. With these normalizations the thermodynamics can be easily computed. First, we note that the Euclidean continuation of the above metric in the plane takes the form

(13)

The requirement of regularity of the metric at the horizon then implies that the inverse of the period of the Euclidean time is identified with the temperature

(14)

The entropy can be calculated as follows. The area element on a is given by

(15)

The entropy is given by

(16)

so that the entropy density per unit volume in the -direction is

(17)

2.1 Gravity solutions

We will be interested in black brane solutions at finite temperature , with a dilaton of the form in 444The work in this section was jointly done with Prithvi Narayan. eq.(9). These solutions have two mass scales, and . High values of will be highly anisotropic, while low values of will be nearly isotropic solutions.

The solution in the regime where can be constructed by starting with the black brane in and incorporating the effects of the varying dilaton perturbatively in . Upto second order in one gets

(18)

where the coefficient functions are given by

(19)

This second order solution is a special case of the general discussion in [23]. We have also obtained the solution upto fourth order, the resulting expressions are quite complicated and can be found in [39].

The solution in the highly anisotropic region, where is more interesting. It is useful to first consider the extremal case where with . In this case the full black brane solution is hard to find analytically but a simple calculation shows that the near horizon region is of the form , with metric components,

(20)

Note that we have retained as in eq.(10), as a free parameter in the solution. In fact, as often happens with attractor geometries, eq.(20) is an exact solution of the equations of motion, with the dilaton given by eq.(9).

This attractor geometry will play an important role in the subsequent discussion. It is worth emphasizing that the linear variation of the dilaton gives rise to extra components in the stress tensor which prevent the direction from shrinking, resulting in the geometry. Also we note from eq.(20) that the geometry has a radius smaller than that of the asymptotic ,

(21)

Starting with the near horizon geometry in eq.(20) we can show that by adding a suitable perturbation which grows in the UV the solution matches asymptotically to the metric, eq.(4), for a suitable choice of . The form of this perturbation is

(22)

with

(23)

The analysis showing that after turning on this perturbation one matches with at large was carried out using mathematica. The resulting value of is

(24)

By adjusting the coefficient one can ensure that the asymptotic behavior of in the metric eq.(20), eq.(10) agrees with eq.(4).

Figure 1: Numerical interpolation of near horizon to asymptotic , for the metric coefficients with and .
Figure 2: Numerical interpolation of near horizon to asymptotic , for the metric coefficient with and .

A plot showing the resulting metric components as a function of is given in Fig. 1 and Fig. 2. These plots were obtained by numerical interpolation for the case and . Asymptotically, as , one finds that , while , with .

Now that we have understood the extremal solution, one can also turn on a small temperature, while still keeping . In the near-horizon region the metric components for the near extremal solution are given by

(25)

The horizon radius is related to the temperature by

(26)

Note that the effects of the temperature die out at large , thus the perturbation eq.(22), eq.(23), when added to this solution will flow at large to space, for small enough values of . Using the definition in eq.(17), the entropy density for the solution in eq.(25) can be obtained as

(27)

where we used the numerical value of the coefficient from eq.(24). It is fixed by requiring that the metric becomes of form eq.(4). To leading order in the resulting value of is the same as in the extremal case.

3 Thermodynamics

The aim of this section is to discuss the thermodynamics of the anisotropic solutions presented in the previous section from both field theory and gravity point of view. First we shall discuss basic facts in general about thermodynamics for the situation when we have anisotropy in one of the spatial directions from purely field theory point of view. Then we shall consider the gravity solutions explicitly and check that the thermodynamics obtained from the study of dual gravity background is consistent with that obtained from the field theory analysis.

3.1 Thermodynamics of anisotropic phase from boundary analysis

We consider a relativistically invariant quantum field theory at temperature . It’s partition function is given by

(28)

Here stand for the fields in the theory over which one must integrate. is the action which depends on the fields, , and also on the background metric and the dilaton .

The metric has Euclidean signature and is of the form

(29)

where , and where time has periodicity,

(30)

The resulting partition function is a functional of . With the system of interest here in mind we would like to consider cases where the dilaton is linearly varying with the profile eq.(9). The system is therefore characterized by two energy scales, . We will take the metric to be varying slowly compared to both .

For the systems we are studying this boundary partition function can be obtained by evaluating the dual path integral in the bulk. As was discussed above in the bulk it is clear that the dependence on can only arise through gradients of the dilaton. For the dilaton profile under consideration, eq.(9) this means the partition function would depend on

(31)

where

(32)

When the metric is varying slowly we can express the dependence of on the metric, , in terms of a derivative expansion. It will be enough for our case to set in the metric in eq.(29). This gives555Note that the dependence of partition function on and the subsequent discussions are similar in structure to the superfluid case studied in [40].,

(33)

where the terms in the ellipses stand for terms with one or more derivatives of the metric. It will be enough for our purposes to consider the leading term on the RHS above.

In the flat space limit, the stress tensor is defined in terms of the variation of with respect to the metric

(34)

where . With the metric in eq.(29) and we get

(35)

The component of the stress tensor can be obtained from eq.(35) as,

(36)

With the following definitions of the energy density

(37)

and the entropy density

(38)

one readily obtains the thermodynamic relation,

(39)

Next, we consider the component of the stress tensor

(40)

where we have defined

(41)

Using the definitions of entropy density in eq.(38) and of the function from eq.(41) one obtains the relation

(42)

From eq.(40) we obtain

(43)

The last equation in eq.(43) implies

(44)

Defining the extensive quantities i.e. total energy and total entropy in terms of the energy density, entropy density and the volume , one can further obtain a thermodynamic relation from eq.(39), and eq.(42),

(45)

This is the first law of thermodynamics but now also including a term which arises due to the presence of the dilaton gradient.

A few comments are now in order. First, we note from eq.(45) and eq.(43) that the pressure which is conjugate to the volume is or component of the stress tensor . Second, a useful analogy to keep in mind for comparison is that of a system in a magnetic field, . For such a system the first law takes the form,

(46)

with being the magnetization of the system. By comparing we see that , which is determined by the external dilaton’s gradient, plays the role of the external magnetic field and the role of .

The various components of the stress tensor in eq.(36), eq.(40) can be combined in the following simple form

(47)

where the velocity four-vector is given by

(48)

With this general analysis of the thermodynamics in hand we are now ready to turn to the system at hand. We will consider first the low anisotropy region and then the high anisotropy region below.

3.2 More on thermodynamics of the highly anisotropic regime:

We are now ready to consider some aspects of the highly anisotropic regime.

We begin with the extremal limit, where, . In this limit the thermodynamic identity eq.(39) becomes

(49)

where the subscript indicates the values. Notice that it follows from eq.(49) that if the energy density the pressure . We will comment on this feature further below. Finally, since the near horizon geometry is the entropy vanishes in the extremal case.

Close to extremality, for , the entropy density no longer vanishes. To leading order we see that

(50)

where . We can express the other thermodynamic variables as corrections about the extremal values,

(51)

Using the identity eq.(38) one obtains

(52)

Further using the relation eq.(39) along with eq.(49) one obtains,

(53)

Next, can also use the relation eq.(41) and eq.(52) to get

(54)

This in turn gives from eq.(43) that

(55)

Let us end this subsection with two comments. First, we will see from the gravity analysis that the trace anomaly in this system takes the form

(56)

In particular the RHS above is independent of . This means that the corrections to the energy and pressures must satisfy the relation

(57)

We see from the expressions obtained above that this relation is indeed met. We have also seen that vanishes at small temperature. This can be understood in terms of the solution which describes the near horizon geometry. The solution should be dual to a CFT which is Lorentz invariant in dim. Small temperature or energy excitations should be states in this CFT. The trace anomaly in this CFT takes the form

(58)

On the other hand the full asymptotically solution is dual to a Lorentz invariant dimensional theory (with a source turned on) with the trace anomaly eq.(56). This implies that eq.(57) must also be valid. Both these constraints then necessarily lead to vanishing of .

Second, we had mentioned above that the pressure at extremality can be negative. This might lead to the worry that the system is unstable. However, we see that the small temperature excitations above extremality have positive , and also is not negative. The stress energy of small excitations there does not show any instability. In addition we see from eq.(50) that the entropy density and therefore specific heat is positive, which shows that the system is also thermodynamically stable. An instability would be present if the compressibility, . But this is not true for our system. In fact a negative pressure at extremality is akin to what happens in a system with a positive cosmological constant and does not by itself signal an instability. Let us also mention that the parameter determines the magnitude of the external forcing function provided by the dilaton, it is the analogue of an external magnetic field . Thermodynamic stability does not require to necessarily have a definite sign, where denotes the Free energy. This is analogous to what happens in an external magnetic field. The susceptibility of a system coupled to a magnetic field can be positive or negative depending on whether the system is para or diamagnetic and stable systems of both kinds can arise.

3.3 Thermodynamics from gravity

To study the thermodynamics from gravity we need the near boundary behavior of the metric and the dilaton. From this behavior one can extract the stress energy tensor after suitably subtracting divergences using the procedure of holographic renormalization, see [41], [42]. For this purpose one needs to work with a total action of the form,

(59)

where is the bulk action given in eq.(3), is the Gibbons Hawking boundary term, and is the counter term action needed to subtract the divergences.

It is convenient to work in Fefferman-Graham coordinates in which the near boundary metric and dilaton take the form

(60)

Here is a coordinate which vanishes at the boundary and it is related to the coordinate by

(61)

where the ellipses stand for corrections which vanish near the boundary where .

We will follow the analysis in [43]. The counter term action is given in eq. B.12 of [43]. In our case the axion and the boundary metric is flat, . Since there is no normalizable mode for the dilaton turned on we have

(62)

where is the operator dual to the dilaton in the boundary. After subtracting the divergences, the stress tensor has an expectation value

(63)

From eq.(B.8), eq.(B.11), eq.(B.16) of [43] we have that666It is important to note that the dilaton kinetic term in the action, eq.(2.1) of [43] has a relative factor of with the dilaton kinetic term in our action eq.(3).

(64)

Note in particular that the RHS is independent of temperature and only dependent on .

Also, from eq.(63) we get that trace

(65)

where the last equality follows from eq.(56) and we have also used the relation between and given in eq.(6) with .

Note we see from eq.(B.21), eq.(B.22) of [43] that for the system at hand, the trace anomaly is given by

(66)

This agrees with eq.(65) after using eq.(6), eq.(9).

Let us end this subsection with one comment. The choice of counter-terms one makes does affect the the final result for the stress tensor. However important physical consequences are independent of this choice. For example, the choice of counter terms will affect the value of the energy density, , , etc, at extremality, as calculated below in eq.(78), but it will not change the additional finite temperature corrections, , etc which determine the physical consequences.

3.3.1 Low anisotropy regime:

We are now ready to study the low anisotropy region. The asymptotic form of the metric is given in eq.(176), eq.(178) appendix A.1. The resulting values for the stress energy tensor in terms of are given in eq.(179) of appendix A.1. The thermodynamic quantities can be obtained, as is explained in appendix A.1, as

(67)
(68)
(69)

The resulting entropy density becomes

(70)

Using eq.(67), eq.(68) and eq.(69) we can also obtain the value of the trace anomaly

(71)

which agrees with eq.(65).

It is also worth pointing out from eq.(178) and eq.(60) that

(72)

which agrees with eq.(64). In fact vanishes.

Let us mention that the low anisotropy regime was also studied in [23] in some generality.

3.3.2 High anisotropy regime:

In this case the full solution cannot be obtained analytically. Instead one can understand the near horizon region analytically and then numerically interpolate to go to the asymptotically region.

We start with the extremal case, with . The full solution preserves Lorentz invariance in the directions. As a result the metric components in the asymptotic, region take the form

(73)

where are unknown functions of . Here we are using the Fefferman -Graham coordinate defined in eq.(61).

Note that the coefficient of the term is fixed by the anomaly and coefficient of is fixed in terms of the dilaton in the FG expansion (see [43]).

Using eq.(73) in eq.(63) gives

(74)

The anomaly constraint eq.(65) allows us to solve for in terms of and yields,

(75)

From eq.(75), eq.(74) and eq.(44) we get a differential equation for which yields,

(76)

where is an integration constant. can be absorbed by introducing a suitable scale giving,

(77)

Finally, inserting eq.(77) back in eq.(74) gives

(78)

Note that at the leading order in eq.(74) and also in eq.(78), hence from eq.(39) the entropy is zero. It is worth noting that an additional scale has appeared in the expressions above. The value of can be determined from a numerical analysis where the full solution which interpolates between the near horizon region and asymptotic AdS space is constructed. Note also that eq.(77) determines all terms in the metric eq.(73) which go like . One can verify that these terms satisfy the relation eq.(64).

The behavior at small temperatures above extremality was discussed in subsection 3.2 above. Numerically we find that the coefficient in eq.(50) takes the value

(79)

The resulting values of thermodynamics quantities can then be obtained from the formula in subsection 3.2.

4 Computation of the viscosity from gravity

In isotropic situations it is well known that the shear viscosity for any system having a gravity dual is given by

(80)

where is the entropy density. This result holds as long as Einstein’s two derivative theory is a good approximation on the gravity side. The only other independent component is the bulk viscosity which vanishes for a conformally invariant theory.

More generally, for anisotropic situations, the viscosity should be thought of as a tensor where the indices, take values along the spatial directions. A Kubo formula can be written down relating the viscosity to the two point function of the stress tensor and takes the form:

(81)

where

(82)

is the retarded Green’s function, and denotes its imaginary part. Note that we are interested here in the viscosity at vanishing spatial momentum. It is clear from eq.(81) that the viscosity is a fourth rank tensor, , symmetric in and and also symmetric with respect to the exchange . This means it has independent components.

In the gravity theory, this two point function can be calculated by studying the behavior of metric perturbations in the corresponding black brane solution. The solution we are interested in preserves partial rotational invariance in the plane. We can use this unbroken subgroup, , to classify the perturbations. We write the metric as

(83)

Here , with stands for the background black brane metric while, , with correspond to general metric perturbations. We have made a gauge choice to set . Let us note before proceeding that for the brane solution we consider the black brane metric is given in eq.(10), in the following discussion we will take it to be more generally of the form

(84)

The two-point functions of the spatial components of the stress tensor, which appear in eq.(82) require us to study the behavior of the metric perturbations where . There are six independent components of this type. Two of these, carry spin with respect to . Two more, carry spin . The remaining two, and , are of spin . Correspondingly, we see that the viscosity tensor will have independent components 777There are three independent two point functions among the two spin perturbation.. Actually, the analysis of the spin sector is more complicated due to mixing with the dilaton perturbation. We will comment on this more below.

The spin and spin perturbations, as we will see below, satisfy an equation of the form,

(85)

where is a scalar field, whose precise relation to the metric perturbation will be discussed shortly, and the functions are determined in terms of background metric. The expectation value of the dual operator is determined, as per the standard AdS dictionary in terms of the canonical momentum

(86)

It then follows that the retarded Green’s function takes the form

(87)

And the response function which will enter in the definition of the viscosity eq.(81) is given by

(88)

An important fact, see [44], is that the RHS of eq.(88) can equally well be evaluated very close to the horizon, , instead of at the boundary of AdS space, . This follows from noting that to evaluate the RHS we are interested in the behavior of the ratio only upto correction, as . Now from the equation of motion, eq.(85), we see that unless we are very close to the horizon, where diverges, the second term can be neglected since it is proportional to . Thus