Joule-Thomson expansion in AdS black holes with momentum relaxation

The inner structure of realistic materials make them exhibit momentum relaxation. In this paper we study the holographic version of the Joule-Thomson effect on AdS black holes in which translational invariance is broken by two methods: First by considering planar black holes in general relativity supported by axion scalar fields with a linear dependence on the horizon coordinates and secondly by considering black holes in massive gravity models in which momentum relaxation is obtained by breaking the bulk diffeomorphism invariance of the theory. In contrast with black holes studied so far, for both theories it is possible to obtain inversion curves with two branches reproducing the behavior of Van der Wall fluids. Moreover in the specific case of the massive gravity model we show that black holes can heat up when crossing the inversion curve.


I. INTRODUCTION
From the very beginning black holes were described as extreme classical objects that absorb every kind of matter and energy without leaving anything out. Basically regarded as bald objects [1,2] they were supposed to be described just by few parameters; mass, angular momentum and electromagnetic charges. With the advent of quantum field theory, particularly in the context of curved spacetimes, it was demonstrated the fundamental relationship existing between the area of black holes and their entropy [3] and that they possess a temperature related with its surface gravity [4]. Moreover it was shown that black holes emit radiation resembling the spectrum of black bodies. All these considerations led to the development of black hole thermodynamics [5], constituting the first successful semiclassical description of gravitational phenomena, a deep insight into the understanding of a possible quantum description of the gravitational interaction.
When considering black holes in Anti-de Sitter spacetimes black hole thermodynamics becomes particularly interesting. The AdS/CFT correspondence defines a duality between gravitational theories on anti-de Sitter spacetimes in (D+1)-dimensions and conformal field theories in D-dimensions [6]. In this context black holes in the presence of a negative cosmological constant admits a dual description given by thermal states in a conformal field theory. Hawking and Page [7] demonstrated that AdS spacetimes suffer a phase transitions from the AdS background state to large Schwarzschild AdS black holes for a critical temperature.
Phase transition that was demonstrated to be dual to a confinement/deconfinement phase transition in the free energy of the dual field theory quark/gluon plasma [8,9]. Many applications of these ideas that combine black hole thermodynamics and AdS/CFT duality have been developed during the last decade providing a deeper understanding of the interplay between gravity and quantum physics in the context of condense matter physics, [10,11], the loss information paradox [12], quantum chromodynamics [13], to mention few examples.
When considering black hole thermodynamic for AdS black holes the cosmological constant parameter, Λ, is considered as a fixed parameter introduced in the action and does not appears in the first law of black hole thermodynamics. This ensures that we are comparing thermodynamical ensembles for solutions exhibiting the same asymptotic behavior, by fixing the AdS background.
Nevertheless, it is a well accepted idea that black hole thermodynamic is much richer when considering Λ as variable [14,15]. The cosmological constant, from a perfect fluid point of view incorpores the notion of Pressure, through the relation where d represents the dimension of the spacetime and l the AdS radius. This allows to obtain a more physical interpretation of what the volume of a black hole should be [16,17], This also allows to include the pressure-volume term of everyday thermodynamic P V into the first law of black hole thermodynamic [23]. In this case the mass M of the spacetime must be interpreted as the enthalpy of the thermodynamical system [14]. Novel new phenomenology 1 It has been stressed [16] that the thermodynamical volume seems to be equal or more than the corresponding Euclidean volume associated with the area/entropy. This implies that black holes are more efficient when storing information. This result is known as the reverse isoperimetric inequality (RII) and has been analyzed for several black hole solutions [18][19][20][21][22].
is obtained, new phase transitions like Van der Waals liquid-gas phase transitions [24,25], existence of triple points like the one encountered in the phase diagram of water [26], heat engines black hole analogous, just to mention few applications [27][28][29][30]. The subject has been dubbed generalized black hole thermodynamic or black hole chemistry [26].
An interesting classical thermodynamical effect is the so-called Joule-Thomson effect, also known as Joule-Thomson expansion [31]. This effect deals with the change of temperature of a gas or fluid when it is expanded adiabatically by using a valve. In fact this adiabatic expansion can be performed in several ways. The Joule-Thomson effect takes place when the thermodynamical process occurring during the expansion is irreversible and enthalpy remains constant. The change of temperature is measured by the Joule-Thomson coefficient µ JT which can be either positive or negative depending if the fluid is cooling or heating, respectively.
By working on the context of generalized black hole thermodynamic recently the Joule-Thomson effect have been studied for first time byÖkcü and Aydıner [32], in particular for the case of Reissner-Nordstrom black holes in anti-de Sitter spacetimes.
As we have stated previously, the Joule-Thomson expansion deals with the change of temperature of the fluid under expansion in an isenthalpic process. This change is quantitatively expressed by the sing of the Joule-Thomson coeficiente defined by [31] µ JT = ∂T ∂P H We observe that by computing this coefficient it is possible to determinate when heating or cooling is taking place. Even if pressure is always decreasing the change of temperature can be either positive or negative. When µ JT goes to zero it is possible to defined the inversion temperature T i , the particular point in the gradient of temperature of the black hole for which the system change form cooling to heating or vice versa. In the same manner it is defined the inversion pressure P i . Then (P i , T i ) gives inversion transition point. By making use of the generalized first law of black hole thermodynamic and taking into account the isenthalpic nature of the process, it is possible to define the Joule-Thomson coefficient in term of the volume and heat capacity at constant pressure [32], where C P = T ∂S ∂T P is the heat capacities at constant pressure. This definition has the advantage that allows to define easily the inversion temperature which will provide heating and cooling regions in the T − P plane.
The study of Joule-Thomson expansion has been generalized for several black hole solutions including arbitrary dimensional charge AdS black hole [33], Kerr-AdS black holes [34], Gauss-Bonnet AdS black holes [35], Lovelock gravity [36] and nonlinear electrodynamic gravity [37] to mention few examples.
It is known that real materials exhibit momentum dissipation, namely, that momentum is not continuously conserved. This implies that resistivity of materials has a non-vanishing value providing for finite electrical conductivities. When making use of the tools of the gauge/gravity duality to study condense matter systems in term of their gravitational duals it is not straightforward to include momentum dissipation. At this respect to well-known strategies to produce momentum dissipation are the inclusion of matter fields that breaks translation invariance in the dual field theory, as it is for example the case of axion scalar fields that depend linearly on the horizon coordinates [38], and the case of massive gravity theories which present a broken diffeomorphism invariance in the bulk [39].
In this paper we present the study of the Joule-Thomson effect for two models presenting momentum dissipation: (i) the Einstien-Maxwell-axions theory where the axions field act as spatial-dependent sources breaking the Ward identity so that the momentum is not conserved in the dual theory [38]; (ii) massive gravity theory where the momentum dissipation in the dual theory is implemented by breaking the diffeomorphism invariance in the bulk [39].
The paper is organized as follows. Sec. II is designated to analyze the Joule-Thomson expansion in the context of Einstien-Maxwell-axions theory. This is done for axion fields presenting a standard kinetic term but also for the case in which the kinetic term is modified by the so-called k-essence term [40,41]. Sec. III is devoted to the analysis of the Joule-Thomson expansion in the context of massive gravity theory. Finally we conclude in Sec.

A. Black holes in Einstien-Maxwell-axions theory
The Einstein-Maxwell-Axions gravity theory with scalar fields was first proposed in [38] by homogeneously distributing 2 massless scalar fields along the horizon coordinates. The action principle in four dimensions reads where the cosmological and the AdS radius are related by l 2 = − 3 Λ . By setting the scalar fields to depend on the 2 dimensional spatial coordinates and by considering a spacetime with a planar base manifold the Klein-Gordon equation for the scalars is easily integrated, yielding Subsequently one finds that the action admits the following charged black hole solution where the index a goes a = 1, 2, and the horizon r h satisfies f (r h ) = 0. It is worthwhile to point out that the scalar fields in the bulk source a spatially dependent field theory with momentum relaxation, which is dual to the homogeneous and isotropic black hole (8)  The mass and charge of the black hole are connected with the parameters m and q as 2 From a geometric point of view the axionic parameter β induces and effective negative curvature scale on the horizon, resembling the causal structure of hyperbolic black holes. This was first observed in [42]. 3 Recently axions fields of this type have been used to construct exact anti-de Sitter homogeneous black strings [43].
where V 2 is the volume of the 2 dimensional flat space and we will set it to be 1. On the other hand, by identifying the period of the Euclidean time in order to avoid conical singularities, the temperature of the black hole is given by and the entropy is obtained by the area law as

B. Joule-Thomson expansion
We shall apply the previous solution to the study of the Joule-Thomson expansion. To do so we consider the thermodynamical analysis of the Einstein-Maxwell-Axion theory provided in [29]. As it is known that the pressure is given by Making use of this result into the definition of mass, the mass of the black hole can be rewritten as which is taken as the enthalpy H of the system. On the other hand, let us use pressure (12) into the expression for the temperature (10), then In this manner we obtain our black hole equation of state. The thermodynamical volume is the conjugate variable of the pressure, then With these ingredients at hand we use the definition of the Joule-Thomson coefficient (4), As was stated before, the point µ JT defines the inversion temperature T i , which in our case with the corresponding pressure P i . As we see this expression depends apart from the inversion pressure on the horizon of the black hole. Nevertheless r h can be obtained from the temperature relation, relation that T i and P i must also satisfy, then the only positive root is Substituting (18) into (17), we obtain the analytical relation between the inversion temperature and pressure From this last equation we observe some analytical properties of the inversion temperature.
For the case without momentum relaxation, i.e, β = 0, we have T i = which is proportional to P 3 4 i with fixed q ≠ 0. This shows that for planar charged AdS black holes the minimum temperature goes to zero when the inversion pressure tends to zero, contrary to the case in which the horizon is spherical. Moreover there is no inversion temperature for uncharged solutions. On the other hand when q = 0, but we have momentum relaxation we 2π which is shifted from zero in contrast with the standard uncharged AdS black hole [32]. We show the explicit relation of inversion curve for different β in figure   1.
There is only one branch of inversion curves as that studied in [32][33][34][35][36][37], which differ from the one obtained for Van der Waals fluids. As momentum increases, the curve is higher. This effect is similar to the one produced by the electric charge. So the momentum relaxation enhances the inverse curve. We turn to study the isenthalpic curves with constant mass/enthalpy in the T − P plane.
The results for q = 5 and β = 2 is displayed in figure 2 where where X i = 1 2 ∇ µ ψ i ∇ µ ψ i with i = 1, 2. ψ i are massless scalar field. The above action goes back to that for the minimally coupled Einstein-Maxwell-axions gravity studied in [38] just by setting γ = 0. The exact black hole solution for this theory was found in [41] with unchanged matter fields respect to the previous analyzed solution The extended thermodynamics of the above solution was studied by us in [30]. The Hawking temperature and the entropy of this black hole are given by We see that the temperature is modified by γ while the entropy is the same as (11). The mass of the black hole and the charge are connected with m and q by means of where we have used the definition of pressure. The charge and the thermodynamical volume as the conjugation of the pressure are the same as (9) and (15), respectively.
In order to analyze quantitatively the effect of the k-essence contribution on the Joule-Thomson expansion we will consider the k = 2 case. By following the same strategy followed previously we obtain that for the inverse curve the horizon should satisfy the equation By allowing only positive values for γ, no phantom contributions, there is only one positive root for r h Then, our inversion temperature is related with the inverse pressure by The inverse curve for different values of the γ coupling are shown in figure 4. Similarly, the higher coupling also enhances the inverse curve.
Now we analyze the case in which γ can take negative values. When considering this phantom contribution equation (28) possesses two positive roots defined by It is straightforward to compute when − 4q 2 both r h− and r h+ , which implies that in this case, the Joule-Thomson expansion breaks down for negative γ. This is reasonable because negative γ involves instability.

A. Planar black holes in massive gravity
Now we turn to the study of the Joule-Thomson expansion in the context of a massive gravity theory. We shall focus on four dimensional black holes with planar horizon. The action of the four-dimensional massive gravity we are considering is given by [39] where m g is the parameter controlling the massive term. In the action, in contrast with Einstein gravity, the last terms represent massive potentials associated with the graviton mass which breaks the diffeomorphism invariance in the bulk producing momentum relaxation in the dual boundary theory. The couplings c i are series of constants while f and U i denote the reference metric and symmetric polynomials of the eigenvalue of the (4)×(4) matrix K µ ν ≡ √ g µα f αν , respectively. U i have the forms where [K ] = K µ µ and the square root in K can be interpreted as ( It is noticed that in AdS space, the stability of fluctuations of fields deserves analysis, here we do not take care of the sign of c i even though self-consistent massive gravity theory may require c i to be negative if m 2 g > 0 [44]. The static planar black hole solution of the above action yields [39,44] with U 1 = 2c 0 r, U 2 = 2c 2 0 r 2 , and U 3 = U 4 = 0 and where in the second line, we have used the definition of pressure (12) and set the volume of two dimensional space to be 1. The extended thermodynamics of massive gravity has been studied in [44][45][46]. The integral constant m and q are connected with the mass and charge of the black hole as M = m 8π and Q = q 8π, respectively. The mass of the black hole is The Hawking temperature T , the entropy S, the thermodynamic volume V , and the electric potential Φ were derived as We note that the formulas of the entropy and the thermodynamical volume are the same as those (11) and (15) in Einstein-Maxwell-axion theory we showed in previous sections. The first law and the Smarr relation of the black hole in the extended phase space have been generalized as We continue to study the Joule-Thomson expansion in massive gravity. Using (4), we get the Joule-Thomson expansion coefficient in massive gravity Using µ JT = 0 we find the inversion temperature Making use of the Hawking temperature, we obtain that for the inversion points, the horizon should satisfy the equation We can solve r h from the above equation and then substitute it into (45) to get the T i as a function of P i . From (45) and (46), we see that c 3 has no imprint on the inversion curve. In what follows, we shall fix c 0 = c 3 = 1 without loss of generality and mainly study the effects of c 1 and c 2 .
We firstly consider the case with q = 0. Consequently, beside r h = 0, there are two more solutions of (46) which are These observations are novel in the following aspects. Comparing with previous works [32][33][34][35][36][37], where only one branch was obtained, we first obtain two branches for the inversion curve, which is similar to the Van der Waals fluids case. However, the minimal inversion temperature is negative unlike the case of Van der Waals fluids. On the other hand, for the isenthalpic curves in figure 7, the points µ JT = (∂T ∂P ) M = 0 all fall in the inversion curve.
However, we see here that µ JT = 0 denotes the minimum but not maximal value which is a different behavior respect to the Van der Walls case and the other black hole cases that have been analyzed. This means that in the left side of the inversion curve, the isenthalpic process is a cooling process because of µ JT < 0 while it is a warming process with µ JT > 0 in the right side.  We then consider the case with q = 5. In this case, there are four solutions of equation (46) for r h , which we do not show due to esthetics purposes. We find with samples of c 1 and c 2 , that only one solution is real and positive, therefore getting one branch for the inversion curve. We show the inversion curves with c 1 = 1 for different c 2 in figure 8, which is similar to the ones previously obtained in the literature with only one branch. Then the isenthalpic curves and the related inversion curves for choices of c 2 are shown in figure 9.
Similarly, in each plot, the isenthalpic process in the left side of the inversion curve denotes warming process while those in the right side are for cooling process. Similar properties can be obtained for c 1 = −1.

IV. FINAL COMMENTS
By considering the cosmological constant as a thermodynamical quantity we have analyzed the Joule-Thomson expansion, this means, the expansion of gas from a higher pressure section to a lower one by maintaining the enthalpy of the process constant, this in the context of AdS planar black hole that exhibit momentum dissipation. Real materials relax momentum, behavior that from the point of view of gravitational dual theories, can be introduced by several methods that break translational invariance in the field theory side.
Two methods were investigated, first when linear axion fields, massless scalar fields that depend linearly on the horizon coordinates, are introduced, and secondly the case in which the Einstein-Hilbert action is supplemented with massive potentials that renders gravity massive and that break the bulk diffeomorphism invariance of the theory. By studying the Joule-Thomson coefficient, µ JT , which determinates the transition from warming/cooling phases, we have computed the inversion curves in the T i − P i plane as well as the corresponding isenthalpic curves.
We have observed that for the case of linear axions, when they possess standard kinetic term, the inversion curve possesses only one branch, similar to what was obtained in [32][33][34][35][36][37], behavior that differs from the case of Van der Wall fluids. The net effect of the momentum relaxation mechanism, which is controlled by our coupling β, is that the inversion curve is enlarged for higher values of β. This means that the temperature for which the heating/cooling transition takes place is greater when increasing β. In fact, the momentum relaxation parameter behave as an electric charge, not only enhancing the inversion curve, but also supporting the Joule-Thomson expansion in the absence of electric charge.
Next, we have modified the kinetic term for our axions by including a nonlinear kinetic term of the type (∂ µ ψ∂ µ ψ) k , contribution controlled by the parameter γ. We observe a similar behavior than in the previous case, this means, that considering greater values of γ we obtain enlarged inversion curves. Nevertheless we observe (for the k = 2 case) that allowing γ to be negative, this means, by considering possible phantom contributions the inversion curve presents two branches, similar to the case of Van der Wall fluids. However in this case, the range of possible values for the higher order coupling depends on other model parameters, restricting the range for which positive isenthalpic curves might be obtained.
We expect to recover similar isenthalpic curves as Van der Wall fluids.
For the case of the massive gravity theory we have found an interesting new behavior of the process, mostly related with the form of the isenthalpic curves. Respect to previous works, for the uncharged case we observe that for some values of the relevant parameters we obtained two branches, similar to what is obtained for Van der Wall fluids but with a minimum inversion temperature that takes a negative value. Moreover when constructing the isenthlapic curves we observe that the inversion point represent a minimum of the isenthalpic curve instead of a maximum as it has been typically found [32][33][34][35][36][37]. This implies that in the left side of the inversion curve, the isenthalpic process is a cooling process while it is a warming process in the right side. So far black holes where found to always cool when passing the inversion curve, nevertheless these solutions of massive gravity are able to heat when crossing it. The situation is restored to typical behaviors when including electric charge.