Ehrenfest Scheme of Higher Dimensional AdS Black holes in The Third Order Lovelock-Born-Infeld Gravity

# Ehrenfest Scheme of Higher Dimensional AdS Black holes in The Third Order Lovelock-Born-Infeld Gravity

A. Belhaj, M. Chabab, H. EL Moumni, K. Masmar, M. B. Sedra
Département de Physique, Faculté Polydisciplinaire, Université Sultan Moulay Slimane, Béni Mellal, Morocco
High Energy Physics and Astrophysics Laboratory, FSSM, Cadi Ayyad University, Marrakesh, Morocco
Département de Physique, LHESIR, Faculté des Sciences, Université Ibn Tofail, Kénitra, Morocco.
July 15, 2019
###### Abstract

Interpreting the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we reconsider the investigation of - critical behaviors of ()-dimensional AdS black holes in Lovelock-Born-Infeld gravity. In particular, we derive an explicit expression of the universal number in terms of the space dimension . Then, we examine the phase transitions at the critical points of such black holes for as required by the physical condition of the thermodynamical quantities including criticality behaviors. More precisely, the Ehrenfest equations have been checked and they reveal that the black hole system undergoes a second phase transition at the critical points.

Keywords: - criticality, AdS black holes, Lovelock-Born-Infeld gravity, Ehrenfest thermodynamical equations.

## 1 Introduction

Black holes in various dimensions have received an increasing attention in connection with higher dimensional supergravity models embedded either in superstrings or in M-theory moving on non trivial geometric backgrounds including Calabi-Yau manifolds. A particular interest has been devoted to the study of extremal black hole solutions using the attractor mechanism developed in [1, 2, 3, 4]. In this approach, the corresponding effective potentials and the entropy functions have been computed using the U-duality group theory applied to the black hole charge invariants. Models based on Calabi-Yau manifolds have been elaborated using complex and quaternionic geometries [4].

Recently, many efforts have been devoted to study thermodynamical properties of the black holes using techniques explored in statistical physics and fluids [5, 6, 7, 8, 9]. In particular, the critical behaviors have been obtained for several black holes in various dimensions using either numerical or analytic calculations [5, 10, 11, 12, 13, 14, 15, 16]. A special interest has been on Ads black holes in arbitrary dimensions [17]. More precisely, the state equations have been worked out by considering the cosmological constant as the thermodynamic pressure and its conjugate as the thermodynamic volume. This activity has revealed a nice interplay between the behavior of the RN-AdS black hole systems and the Van der Waals fluids which has been seriously investigated in many places. In fact, it has been shown that the corresponding - criticality can be related to the liquid-gas systems of statistical physics. Moreover, it has been seen that the criticality depends on the dimension of the spacetime in which the black holes live. This subject has been extensively investigated producing interesting results [16-23].

More recently, a special emphasis has been put on the thermodynamical properties of AdS black holes in Lovelock-Born-Infeld Gravity [26]. A critical behavior in seven dimensions has been obtained for uncharged and charged black holes. Particularly, the - diagram has been elaborated for such black holes with spherical geometries.

Motivated by black objects in string theory and the above mentioned studies, we consider in this work the criticality of ( dimensional AdS black holes in Lovelock-Born-Infeld gravity. Interpreting the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we investigate such behaviors in terms of the dimension of the spacetime and other parameters specified later on. Among others, we derive an explicit expression of the universal number , for as required by the reality of the thermodynamical quantities and criticality. Then, we discuss the phase transitions at the critical points of these black hole solutions. In particular, we show that the Ehrenfest thermodynamical equations are satisfied and find that the black hole system undergoes a second phase transition.

## 2 Thermodynamics of higher dimensional black holes in the third order Lovelock-Born-Infeld gravity

In this section, we focus on the study of thermodynamics of higher dimensional black holes in Lovelock-Born-Infeld gravity. In particular, we obtain an explicit expression for the corresponding state equation at critical points in dimensions. The analysis will be made in terms of three parameters: the space dimension , the Born-Infeld parameter and the curvature constant . The discussion of the critical behaviors will be given in next sections.

To start, we consider the physical action describing the third order Lovelock gravity in the presence of a nonlinear Born-Infeld electromagnetic gauge field as studied in [27, 28, 29]. This action, which has been investigated in different context, takes the following form

 IG=116π∫dn+1x√−g(−2Λ+L1+α2L2+α3L3+L(F)), (1)

where is the cosmological constant. , , and represent Einstein-Hilbert, Gauss-bonnet, the third order Lovelock and the Born-Infeld Lagrangians respectively. They are given by the following, expressions

 L1 = R, (2) L2 = RμνγδRμνγδ−4RμνRμν+R2, (3) L3 = 2RμνσκRσκρτRρτμν+8RμνσρRσκντRρτμκ+24RμνσκRσκνρRρμ + 3RRμνσκRσκμν+24RμνσκRσμRκν+16RμνRνσRσμ−12RRμνRμν+R3, L(F) = 4β2(1−√1+F22β). (5)

where the is the Born-Infeld parameter as proposed in [28, 29]. The constants read as follows

 α2=α(n−2)(n−3) (6)
 α3=α272(n−24), (7)

where is the Lovelock coupling constant. This choice has been made for simplicity reason. It’s used due to some properties which are absent in the Gauss- Bonnet gravity with the three fundamental constants. Other choices can be made as repported in [29].

The -dimensional static solution is given by

 ds2=−f(r)dt2+dr2f(r)+r2dΩ2n−1. (8)

In this solution, the black hole function takes the following form

 f(r)=1+r2α(1−g(r)1/3) (9)

where

 g(r)=1+3αmrn−12αβ2n(n−1)[1−√1+η−Λ2β2+(n−1)η(n−2)F(η)]. (10)

is the hypergeometric function

 F(η)=2F1([12,n−22n−2],[3n−42n−2],−η) (11)

with

 η=(n−1)(n−2)q22β2r2n−2 (12)

It is interesting to note that the dimensional line element depends on the geometry in question and it is given by

 dΩ2n−1=dθ21n−1∑i=2i−1∏j=1sin2θjdθ2i (13)

defining dimensional hypersurfaces with the constant curvature . For these configurations, the Hawking temperature is expressed as in [27]

 T=(n−1)[3(n−2)r4++3(n−4)αr2++(n−6)α2]+12r6+β2(1−√1+η)−6Λr6+12π(n−1)r+(r2++α)2 (14)

Similar calculations reveal that the entropy reads as

 S=∫r+01T(∂m∂r+)=Σk(n−1)rn−5+4(r4+n−1+2r2+αn−3+α2n−5). (15)

It follows that this physical solution requires that the integer is constrained by the condition: . It is also remarked that, for higher dimensional cases, the Lovelock gravity does not coincide with Einstein one. However, the usual four dimensional case can be recovered by deleting the Lovelock gravity terms [30].

It is recalled that that describes the volume of the dimensional hypersurface and the thermodynamic volume can be written as

 V=Σkrn+n. (16)

It is known that the Gibbs free energy is given by

 G = Σkrn−6+48πα(r2++α)2[(n−1)r6+(r2++α)2⎛⎜ ⎜⎝−1+(r2++α)3r6++12αβ2(1−Λ2β2−√η+1+(n−1)ηF(η)n−2)n(n−1)⎞⎟ ⎟⎠ (17) − α(r4+n−1+2r2+αn−3+α2n−5)((n−1)(3(n−2)r4++3(n−4)kαr2++(n−6)α2) − 6Λr6++12r6+β2(1−√η+1))]

We have considered only the internal energy to discuss criticality in the extended phase space. In fact, it has been realized that the internal energy has been needed to give a complete study on the corresponding criticality.

Since the thermodynamical pressure of the black hole is interpreted as the cosmological constant,

 P=−Λ8π, (18)

the first law of the black hole thermodynamics can be modified by the introduction of the variation of the cosmological contant (the pressure) in this law. Now, one can write the following mass equation

 dm=TdS+Φdq+PdV+Bdβ (19)

where is the electric potential and is a conjugate quantity to called Born-Infeld vacuum polarization [14, 16, 30]. This quantity is given by

 B = (∂m∂β)S,q,P,β=Σk8πnβr2+⎧⎪⎨⎪⎩2r2n+β2⎛⎜⎝2− ⎷4+2(n−1)(n−2)q2r2(1−n)+β2⎞⎟⎠ (20) + (n−2)(n−1)q2r2+2F1([12,n−22n−2],[n−22n−2],−(n−1)(n−2)q22β2r2(n−1))}.

In fact, we can obtain the pressure as a function of the temperature and the horizon radius. Indeed, after a lengthy but straightforward calculations, we show that the the pressure reads as

 P = Tv−((n−2)(n−1)v−32παT)π(n−1)2v3−16α((n−4)(n−1)v−16παT)π(n−1)4v5−256α2(n−6)3π(n−1)5v6 (21) + β2(√24n+1(n−2)(n−1)3q2v2((n−1)v)−2nβ2+64−8)32π.

By confronting this equation to the Van der Waals equation of state, we readily derive the specific volume as follows

 v=4r+n−1. (22)

Hereafter, we focus our analysis on critical behaviors and we show how to establish an explicit expression of the universal number in dimensions. Then, we study the phase diagram transitions using the classical thermodynamical physics.

## 3 Critical behavior description

As mentioned before, here we consider the study of the critical behaviors of the above black hole solutions. We first give a detail study on uncharged case. Then we shortly present the charged case.

### 3.1 Uncharged solutions

Roughly speaking, the computation leads to the following state equation

 P = (n−1)2v2(π(n−1)Tv−(n−2))−16α((n−4)−2π(n−1)Tv)π(n−1)3v4 (23) − 256α2(k(n−6)−3π(n−1)Tv)3π(n−1)5v6.

In fact, through numerical calculations, we plot the - diagrams in terms of the space dimension . The results are shown in figure 1.

From this figure, we observe that the - behavior is similar to the Van der Waals’one. This may allow to derive the critical point coordinates. To do this, we should first solve the following system of equations

 ∂P∂v=0,∂2P∂v2=0. (24)

Then, as a consequence, one can determine the explicit thermodynamical expressions for the critical values. They are given by,

 Tc = −((n−1)2(n(2n−29)+2)−√(11−n)(n−1)5(n+14))2π(n−2)(n−1)7((n+4)(n−1)2+2√(11−n)(n−1)5(n−2)(n−1)4)5/2√α vc =4  ⎷(n+4)(n−1)2+2√(11−n)(n−1)5(n−2)(n−1)4√α (25) Pc = (n−2)2(n−1)9C(n)48π((n+4)(n−1)2+2√(11−n)(n−1)5)5α.

where is a quantity depending on the space dimension which can be expressed as

 C(n) = (n(n(n(9n−536)+4772)+3328)−6048)(n−1)2 + 2√(11−n)(n−1)5(n((68−11n)n+1420)−432).

By combining the critical expressions shown in Eq. (3.1), we can deduce the explicit form of the universal number . Indeed, this number is given by

 χ=−(n−1)4C(n)6((n+4)(n−1)2+2√(11−n)(n−1)5)2D(n) (27)

 D(n)=(n−1)2(n(2n−29)+2)−√(11−n)(n−1)5(n+14). (28)

It is worth noting the system of equations (24) involves actually two real solutions. However, we exclude the critical one which yields a vanishing critical specific volume, hence , for producing a black hole without event horizon, known as nude singularity. Besides, unlike the excluded solution, the other solution reproduces exactly the critical coordinates derived in [27].

Here we stress that the generalized expression involves many interesting features. First, we recover the six dimension result given in [27]. Indeed, for , the critical coordinates are given by

 Tc=1π√5α,vc=4√α√5,Pc=17200πα,χ=PcvcTc=1750. (29)

Furthermore, the universal number behaves nicely in terms of the space dimension as illustrated in figure 2. Thus, we observe from figure 1 that critical behaviors with a clear inflexion point appear only when the space dimension lies in the range .

For , it follows that for a temperature less than the critical one, the behavior of the black hole does not show an inflexion point but a maximum. However, the latter can not be considered this point like a critical one. For this reason, we have considered only models associated with .

It is noted that do not exceed space dimension as required by the physical condition of the critical volume. This is not a surprising feature in high energy physics. In fact, a close inspection in higher dimensional theories shows that this critical dimension appears naturally in string theory and related topics. Indeed, corresponds to a non perturbative limit of eleven dimensional type IIB superstring. This limit is interpreted in terms of dimensional theory. As proposed by Vafa, this is known as F-theory which has been constructed using a geometric interpretation of the duality [31]. This observation motivates us to think about a string theory realization of these black holes in terms of the brane physics. We hope to come back to this issue in future works.

### 3.2 Charged solutions

Here, we shortly give the calculations for the charged black holes in the asymptotic limit of (). In this limit, the spherical topologies show critical behaviors for . The numerical evaluations are listed hereinafter which agree with [41]. In particular, we illustrate graphically the critical behaviors. We plot in figure 3 the - diagrams in various dimensions.

To make contact with charged case, we discuss the corresponding critical behavior. This has been presented in figure 4. In particular, we plot the equation of state with the Born-infeld parameter and make comparison between the Born-Infeld theory and the Maxwell one in the limit where .

It is observed form figure that the Born-Infeld parameter modify the critical points in the the plan. It has been shown that .

## 4 Ehrenfest scheme

Having discussed the - criticality of dimensional AdS black holes in Lovelock-Born-Infeld Gravity, we move now to the study of the corresponding phase transitions using classical thermodynamics principals. We note that the classification of such phases associated with the first order and higher orders can be done in terms of the Clausius-Clapeyron-Ehrenfest equations. Indeed, the first order transition is ensured when the Clausius-Clapeyron equation is satisfied at the critical points. However, the second order transition arises when the Ehrenfest thermodynamical equations are verified. In this section, we examine such equations using results obtained in classical thermodynamics [32, 33, 34]. In fact, the Ehrenfest equations read as

 (∂P∂T)S=CP2−CP1VT(Θ2−Θ1)=ΔCPVTΔΘ, (30)
 (∂P∂T)V=Θ2−Θ1κT2−κT1=ΔΘΔκT. (31)

In these equations, is the volume expansion and defines the isothermal compressibility coefficient.

In what follows, we compute the relevant thermodynamical quantities involved in the above equations for () dimensional AdS black holes in Lovelock-Born-Infeld gravity. Indeed, combining equations (14), (15) and (18), we get a general expression of the temperature

 (32)

where is the real positive root solving the entropy function equation (15). Performing similar calculations, we can also determine the specific heat at constant pressure and the volume expansion coefficient. They are respectively given by

 CP = ζ(S)(α+ζ(S)2)((n−1)((n−6)α2+3(n−4)αζ(S)2+3(n−2)ζ(S)4)+48πPζ(S)6)B(α,S,P)ζ′(S) Θ = 12π(n−1)nζ(S)(α+ζ(S)2)3B(α,S,P) (33)

 B(α,S,P) = −(n−6)(n−1)α3−2(n−9)(n−1)α2ζ(S)2+48πPζ(S)8 (34) +18(n−1)αζ(S)4−3ζ(S)6((n−3)n−80πPα+2)

Thanks to the famous thermodynamic relation

 (∂V∂P)T(∂P∂T)V(∂T∂V)P=−1, (35)

we obtain the expression of the isothermal compressibility coefficient

 KT=48πnζ(S)6(α+ζ(S)2)B(α,S,P). (36)

From these equations, we notice the existence of a special factor appearing in their dominators. This factor can be explored to stress critical behaviors for the above thermodynamic quantities. In fact, the constraint leading to a divergence of the heat capacity can easily be checked for the critical points.

To discuss the validity of Ehrenfest equations at the critical points, we should analyze the expression of the volume expansion coefficient . The latter is evaluated as

 VΘ=(∂V∂T)P=(∂V∂S)P(∂S∂T)P=(∂V∂S)P(CPT). (37)

Moreover, the right handed side of Eq. can be converted to

 ΔCPTVΔΘ=[(∂S∂V)P]c, (38)

where the index indicates the values of the thermodynamical variables at the critical points. Exploring Eqs. , and , we obtain

 ΔCPTVΔΘ=⎡⎢ ⎢⎣π12(−n−1)Γ(n+12)ζ(S)1−n2ζ′(S)⎤⎥ ⎥⎦c. (39)

Using Eq. , the left handed side of Eq. translates to

 [(∂P∂T)S]c=(n−1)(α+ζ(Sc)2)24ζ(Sc)5. (40)

Similar calculations can be done using Eqs.(15), (16) and (32). In this way, the left handed side of Eq. becomes

 [(∂P∂T)V]c=(n−1)(α+ζ(Sc)2)24ζ(Sc)5. (41)

A close inspection of the expressions of the isothermal compressibility coefficient and volume expansion coefficient shows that we have

 VKT=−(∂V∂P)T=(∂T∂P)V(∂V∂T)P=(∂T∂P)VVΘ. (42)

The right handed side of Eq. produces the following formula

 ΔΘΔKT=[(∂P∂T)V]c=(n−1)(α+ζ(Sc)2)24ζ(Sc)5 (43)

revealing the validity of the second Ehrenfest’s equation.

It is worth to note that the Prigogine-Defay (PD) ratio [35, 36] can also be computed. Indeed, the calculation shows the following expression,

 Π=ΔCPΔKTTV(ΔΘ)2=⎡⎢ ⎢⎣π12(−n−1)Γ(n−12)ζ(S)6−nζ′(S)(α+ζ(S)2)2⎤⎥ ⎥⎦c. (44)

To illustrate the above analysis, we consider the case of associated with an eight dimensional black solution. In this case, the expression of the is reduced to

 ζ(S)=√3√π4α3+12Sπ4/3−α. (45)

Moreover, Eqs. (39) and (40) become

 ΔCPTVΔΘ = [(∂S∂V)P]c=[(∂P∂T)S]c (46) = 3(π5α3+12πSc)2/32(3√π4α3+12Sc−π4/3α)5/2.

indicating clearly the validity of Ehrenfest first equation at the critical point. Furthermore Eqs. and give the relation

 ΔΘΔKT=[(∂P∂T)V]c=3(π5α3+12πSc)2/32(3√π4α3+12Sc−π4/3α)5/2. (47)

It follows that the Ehrenfest second equation is also valid at the critical point.

It is worth noting that the definition of PD ratio was proposed by Prigogine and Defay [35] and reviewed in many works including [36]. The second Ehrenfest equation is not always satisfied and the PD ratio can be used to measure the deviation from the second Ehrenfest equation [37]. Here, here it reduces to

 Π=1, (48)

showing a phase transition despite existence of the divergency near the critical point. Note that this matches perfectly with the second order equilibrium transition discussed in [38, 39]. For more detail on this case, we plot all quantities in figure 5 to illustrate that such quantities are divergent at the critical points.

## 5 Conclusion and open questions

In this paper, we have reconsidered criticality of dimensional AdS black holes in Lovelock-Born-Infeld gravity. Interpreting the cosmological constant as a thermodynamic pressure and its conjugate quantity as a thermodynamic volume, we have studied thermodynamical behaviours in terms of the space dimension . More precisely, we have derived an explicit expression of the universal number in terms of . Then, we have discussed the phase transitions at the critical points. In particular, the Ehrenfest thermodynamical equations have been verified showing that the black hole system undergoes a second phase transition.

This work comes up with many open questions. An interesting one concerns the space dimension . It has been realized that the integer is constrained to lie within the range

 6≤n<11

as required with the reality of the values of the physical quantities at the critical points. A fast inspection shows that dimensions between 6 and 11 appear naturally in the study of higher dimensional theories including superstrings and M-theory. This observation may provide a new challenge on such black holes and theirs connections with string theory compactification. We believe that the above range can be explored to investigate possible realizations in terms of brane physics. This issue deserves a more deeper study which could be addressed in coming works .

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