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

(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

(2)
(3)
(5)

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

(6)
(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

(8)

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

(9)

where

(10)

is the hypergeometric function

(11)

with

(12)

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

(13)

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

(14)

Similar calculations reveal that the entropy reads as

(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

(16)

It is known that the Gibbs free energy is given by

(17)

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,

(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

(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

(20)

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

(21)

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

(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

(23)

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

Figure 1: The diagrams for space dimension between and . is the critical temperature for .

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

(24)

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

(25)

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

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

(27)

where the quantity reads as

(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

(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 .

Figure 2: The universal number in term of space-time dimension.

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.

q
0.5 1 0.1855 1.3731 0.0452 0.3349
2 1 0.1839 1.4728 0.0438 0.3509
1 1 0.1851 1.4036 0.0448 0.3401
1 0.5 0.2537 1.1748 0.0807 0.3737
1 2 0.1313 1.9254 0.0226 0.3328
Table 1: Critical values for
q
0.5 1 0.2292 1.1308 0.0684 0.3374
2 1 0.2287 1.1561 0.0677 0.3426
1 1 0.2297 1.1071 0.0689 0.3323
1 0.5 0.3157 0.9437 0.1243 0.3716
1 2 0.1627 1.5188 0.0347 0.3245
Table 2: Critical values for
q
0.5 1 0.2747 0.9182 0.0987 0.3298
2 1 0.2742 0.9389 0.0979 0.3352
1 1 0.2751 0.8990 0.0993 0.3245
1 0.5 0.3786 0.7844 0.1786 0.3701
1 2 0.1949 1.2253 0.0500 0.3149
Table 3: Critical values for
q
0.5 1 0.3210 0.7631 0.1358 0.3229
2 1 0.3205 0.7816 0.1348 0.3287
1 1 0.3214 0.7453 0.1367 0.3170
1 0.5 0.4422 0.6688 0.2439 0.3689
1 2 0.2277 0.9944 0.0691 0.3018
Table 4: Critical values for
Figure 3: The diagrams for space dimension between and . is the critical temperature for , and .

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 .

Figure 4: The diagrams for space dimension between and . is the critical temperature for and .

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

(30)
(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

(33)

where the function reads as

(34)

Thanks to the famous thermodynamic relation

(35)

we obtain the expression of the isothermal compressibility coefficient

(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

(37)

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

(38)

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

(39)

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

(40)

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

(41)

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

(42)

The right handed side of Eq. produces the following formula

(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,

(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

(45)

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

(46)

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

(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

(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.

Figure 5: , and in terms of the entropy for .

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

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 .

References

  • [1] H. Ooguri, A. Strominger, C. Vafa, Black Hole Attractors and the Topological String, Phys. Rev. D 70, (2004) 106007, hep-th/0405146.
  • [2] S. Ferrara, R. Kallosh, Supersymmetry and Attractors, Phys. Rev. D54 (1996) 1514, hep-th/9602136.
  • [3] S. Ferrara, K. Hayakawa, A. Marrani, Erice Lectures on Black Holes and Attractors, arXiv:005.2498 [hep-th].
  • [4] A. Belhaj, L. B. Drissi, E. H. Saidi, A. Segui, N=2 Supersymmetric Black Attractors in Six and Seven Dimensions, Nucl. Phys. B796, (2008) 521-580, arXiv:0709.0398.
  • [5] S. Hawking, D. N. Page, Thermodynamics of Black Holes in Anti-de Sitter Space, Commun. Math. Phys. 83 (1987) 577.
  • [6] D. Kastor, S. Ray, J. Traschen, Enthalpy and the Mechanics of AdS Black Holes, Class. Quant. Grav. 261 (2009)95011, arXiv:0904.2765.
  • [7] O. Miskovic, R. Olea, Quantum Statistical Relation for black holes in nonlinear electrodynamics coupled to Einstein-Gauss-Bonnet AdS gravity, Phys. Rev. D83 (2011) 064017, arXiv:1012.4867.
  • [8] A. Belhaj, M. Chabab, H. El moumni, K. Masmar and M. B. Sedra, Maxwell‘s equal-area law for Gauss-Bonnet-Anti-de Sitter black holes, Eur. Phys. J. C 75, no. 2, 71 (2015) [arXiv:1412.2162 [hep-th]].
  • [9] A. Belhaj, M. Chabab, H. E. Moumni, K. Masmar, B. Sedra and A. Segui, On Heat Properties of AdS Black Holes in Higher Dimensions, JHEP 05 (2015) 149 arXiv:1503.07308 [hep-th].
  • [10] A. Chamblin, R. Emparan, C. Johnson, R. Myers, Charged AdS black holes and catastrophic holography, Phys. Rev. D 60, (1999) 064018.
  • [11] A. Chamblin, R. Emparan, C. Johnson, R. Myers, Holography, thermodynamics, and fluctuations of charged AdS black holes, Phys. Rev. D 60, (1999) 104026.
  • [12] M. Cvetic, G. W. Gibbons, D. Kubiznak, C. N. Pope, Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume, Phys. Rev. D 84 (2011) 024037, arXiv:1012.2888 [hep-th].
  • [13] B. P. Dolan, D. Kastor, D. Kubiznak, R. B. Mann, J. Traschen, Thermodynamic Volumes and Isoperimetric Inequalities for de Sitter Black Holes, arXiv:1301.5926 [hep- th].
  • [14] B. P. Dolan, Pressure and volume in the first law of black hole thermodynamics, Class. Quant. Grav. 28 (2011) 235017, arXiv:1106.6260 [gr-qc].
  • [15] J, Liang and B. Liu, Thermodynamics of noncommutative geometry inspired BTZ black hole based on Lorentzian smeared mass distribution, EPL. 100 (2012) 30001.
  • [16] D. Kubiznak and R. B. Mann, P-V criticality of charged AdS black holes, J. High Energy Phys. 1207 (2012) 033.
  • [17] A. Belhaj, M. Chabab, H. El Moumni, M. B. Sedra, On Thermodynamics of AdS Black Holes in Arbitrary Dimensions, Chin. Phys. Lett 29 (2012)100401.
  • [18] C. Song-Bai, L. Xiao-Fang, L. Chang-Qing, - Criticality of an AdS Black Hole in f(R) Gravity, Chin. Phys. Lett, 30 (2013) 060401.
  • [19] D. O’Connor, B. P. Dolan, M. Vachovski, Critical Behaviour of the Fuzzy Sphere,arXiv:1308.6512
  • [20] B.P. Dolan, The compressibility of rotating black holes in D-dimensions, arXiv:1308.5403
  • [21] S. Gunasekaran, D. Kubiznak, R. B. Mann, Extended phase space thermodynamics for charged and rotating black holes and Born-Infeld vacuum polarization, arXiv:hep-th/1208.6251v2.
  • [22] A. Belhaj, M. Chabab, H. El Moumni, L. Medari, M. B. Sedra, The Thermodynamical Behaviors of Kerr-Newman AdS Black Holes, Chin. Phys. Lett 30 (2013) 090402.
  • [23] A. Belhaj, M. Chabab, H. El Moumni, K. Masmar, M. B. Sedra, Critical Behaviors of 3D Black Holes with a Scalar Hair, International Journal of Geometric Methods in Modern Physics. 12, 2 (2015) 1550017, arXiv:hep-th/1306.2518
  • [24] De-Cheng Zou, Yunqi Liu, Bin Wang. Critical behavior of charged Gauss-Bonnet AdS black holes in the grand canonical ensemble, arXiv:hep-th/1404.5194
  • [25] R. Banerjee, S.K. Modak, D. Roychowdhury, A unified picture of phase transition: from liquid-vapour systems to AdS black holes, J. High Energy Phys. 1210 (2012) 125.
  • [26] Jie-Xiong Mo, Wen-Biao Liu. P-V Criticality of Topological Black Holes in Lovelock-Born-Infeld Gravity , Eur. Phys. J. C74 (2014) 2836, arXiv:gr/qc 1401.0785
  • [27] M. H. Dehghani, N. Alinejadi, S. H. Hendi, Topological Black Holes in Lovelock-Born-Infeld Gravity, Phys. Rev. D 77 (2008) 104025, [arXiv:0802.2637].
  • [28] S. H. Hendi and A. Dehghani, Thermodynamics of third-order Lovelock-AdS black holes in the presence of Born-Infeld type nonlinear electrodynamics, Phys. Rev. D 91 (2015) 6, 064045.
  • [29] M. H. Dehghani and M. Shamirzaie, Thermodynamics of asymptotic flat charged black holes in third order Lovelock gravity, Phys. Rev. D 72, 124015 (2005) [hep-th/0506227].
  • [30] D. -C. Zou, S. -J. Zhang and B. Wang, Critical behavior of Born-Infeld AdS black holes in the extended phase space thermodynamics, Phys. Rev. D 89, 044002 (2014), [arXiv:1311.7299 hep-th].
  • [31] C. Vafa, Evidence for F-theory, Nucl. Phys. B 469 (1996) 403415, hep-th/9602022.
  • [32] R. Banerjee, S.K. Modak, D. Roychowdhury, A unified picture of phase transition: from liquid-vapour systems to AdS black holes, J. High Energy Phys. 1210 (2012) 125.
  • [33] J. X. Mo, W. B. Liu, Ehrenfest scheme for P-V criticality in the extended phase space of black holes, Phys. Lett. B 727 (2013) 336-339.
  • [34] J. X. Mo, G. Q. Li, Wen-Biao Liu, Another novel Ehrenfest scheme for P-V criticality of RN-AdS black holes, Phys. Lett. B 730 (2014) 111-114.
  • [35] I. Prigogine, R. Defay, Chemical Thermodynamics, Longmans Green, New York, 1954.
  • [36] P.K. Gupta, C.T. Moynihan, Prigogine-Defay ratio for systems with more than one order parameter, J. Chem. Phys. 65 (1976) 4136.
  • [37] R. Banerjee, S. Ghosh, D. Roychowdhury, New type of phase transition in Reissner Nordstrom - AdS black hole and its thermodynamic geometry, Phys.Lett.B696:156-162 (2011), arXiv/gr-qc-1008.2644 .
  • [38] R. Banerjee, D. Roychowdhury, Thermodynamics of phase transition in higher dimensional AdS black holes, J. High Energy Phys. 1111 (2011) 004, [arXiv/gr-qc:1109.2433v2].
  • [39] T. M. Nieuwenhuizen, Phys. Rev. Lett. 79 (1997) 1317-1320.
  • [40] K. Samwer, R. Busch, W.L. Johnson, Phys. Rev. Lett. 82 (1999) 580-583 .
  • [41] H. Xu, W. Xu, L. Zhao, Extended phase space thermodynamics for third order Lovelock black holes in diverse dimensions, Eur.Phys.J. C74 (2014) 9, 3074, [arXiv:1405.4143].
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
""
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
   
Add comment
Cancel
Loading ...
114163
This is a comment super asjknd jkasnjk adsnkj
Upvote
Downvote
""
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters
Submit
Cancel

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test
Test description