Thermodynamics of Charged AdS Black Holes in Extended Phases Space via M2-branes Background

Motivated by a recent work on asymptotically Ad$S_4$ black holes in M-theory, we investigate both thermodynamics and thermodynamical geometry of Raissner-Nordstrom-AdS black holes from M2-branes. More precisely, we study AdS black holes in $AdS_{4}\times S^{7}$, with the number of M2-branes interpreted as a thermodynamical variable. In this context, we calculate various thermodynamical quantities including the chemical potential, and examine their phase transitions along with the corresponding stability behaviors. In addition, we also evaluate the thermodynamical curvatures of the Weinhold, Ruppeiner and Quevedo metrics for M2-branes geometry to study the stability of such black object. We show that the singularities of these scalar curvature's metrics reproduce similar stability results obtained by the phase transition program via the heat capacities in different ensembles either when the number of the M2 branes or the charge are held fixed. Also, we note that all results derived in [1] are recovered in the limit of the vanishing charge.


Introduction
A more increasing interest has been recently devoted to the black hole physics and the connection with both string theory and thermodynamical models.Particularly, many studies focus on the relationship between the gravity theories and the thermodynamical physics using Anti-De Sitter geometries.In this context, the laws of thermodynamics have been translated into laws of black holes [2][3][4][5][6].Hence, the phase transitions along with various critical phenomena for AdS black holes have been extensively analysed in the framework of different approaches [7][8][9][10][11].Also, the equations of state describing rotating black holes have been interpreted by confronting them to some known thermodynamical ones, as Van der Waals gas [12][13][14][15][16][17][18][19].Emphasis has also been put on the free energy ands its behavior in the fixed charge ensemble.These studies shed some light on the thermodynamical criticality, free energy, first order phase transition and on understanding of the behaviors near the critical points with respect to the liquid-gas systems.
In this context, very recently the thermodynamics and thermodynamical geometry for five dimensional AdS black hole in type IIB superstring background known by AdS 5 × S 5 [20][21][22] have been scrutinezed.this geometry has been studied in many places in connection with AdS/CFT correspondence provides a very useful framework to investigate such geometry via the equivalence between gravitational theories in d-dimensional AdS space and the conformal field theories (CFT) in a (d-1)-dimensional boundary of such AdS spaces [23][24][25][26].
The number of colors has been interpreted as a thermodynamical variable in these works.
In this respect, various thermodynamical quantities have been computed and the stability problem of AdS 5 × S 5 black holes analysed by identifying the cosmological constant in the bulk with the number of colors.
All these recent inspiring works on asymptotically AdS 4 black holes in M-theory [27][28][29][30] motivate us to study the thermodynamics and thermodynamical geometry of AdS 4 × S 7 , from the physics of M2-branes, where we interpret the number of M2 as a thermodynamical variable as in [1].We then discuss the stability of such solutions and examine the corresponding first phase transition by analysing various relevant quantities including the chemical potential, free energy and heat capacitites.Besides, we also evaluate the thermodynamical curvatures from the Weinhold, Ruppeiner and Quevedo metrics for M2-branes geometry and study the corresponding stability problems via their singularities.
The paper is arranged as follows: In section 2 we discuss thermodynamic properties and stability of the charged black holes in AdS 4 × S 7 , by assuming the number of M2-branes as a thermodynamical variable.Section 3 and 4 are devoted to show that similar results are recovered through thermodynamical curvature calculations associated with the Weinhold, Ruppeiner and Quevedo metrics.Our conclusion is drawn in section 5.

Thermodynamics of black holes in AdS 4 × S 7 space
In this section, we investigate the phase transition of the Reissner Nordstrom-AdS black holes in M-theory in the presence of solitonic objects.Here we recall that, at low energy, M-theory describes an eleven dimensional supergravity.This theory, as proposed by Witten , can produce some nonperturbative limits of superstring models after its compactification on particular geometries [31].
First, let us consider the case of M2-brane.The corresponding geometry is AdS 4 × S 7 .
In such a geometric background, the line element of the black M2-brane metric is given by [32,34] where dΩ 2  7 is the metric of seven-dimensional sphere with unit radius.In this solution, the metric function reads as follows where L is the AdS radius and m and q are integration constants.The cosmological constant is Form M-theory point of view, the eleven-dimensional spacetime in Eq.( 1) can be interpreted as the near horizon geometry of N coincident configurations of M2-branes.In this background, the AdS radius L is linked to the M2-brane number N via the relation [1,32,35] ( According to the proposition reported in [1,[20][21][22], we consider the cosmological constant as the number of M2-branes in the M theory background and its conjugate quantity as the associated chemical potential. The event horizon r h of the corresponding black hole is determined by solving the equation f = 0. From Eq.( 2), the mass of the black hole can be written as where the charge of the black hole Q is related to the constant q through the formula, (5) 1 where .
The Bekenstein-Hawking entropy formula of the black hole reads as, Here we recall that four-dimensional Newton gravitational constant is related to the elevendimensional one as For simplicity reason, we use G 11 = κ 11 = 1 in the remainder of the paper.In this way, the black hole mass can be expressed as a function of N and S, Using the standard thermodynamic relation dM = TdS + µ dN + Φ dQ, the corresponding temperature takes the following form This quantity can be identified with the Hawking temperature of the black hole.Using eq.
(8) the chemical potential µ conjugate to the number of M2-branes is given by It defines the measure of the energy cost to the system when one increases the variable while the electric potential reads as In terms of these quantities, the Helmholtz free energy is expressed by, Having calculated the relevant thermodynamical quantities, we turn now to the analysis of the corresponding phase transition.For this, we study the variation of the Hawking temperature as a function of the entropy.
This variation plotted in figure 1 shows that Hawking temperature is a monotonic func- , it presents a critical point to be determined by solving the The solution of this equation is easily derived, In figure 2, we illustrate the Helmholtz free energy as function of the Hawking temperature T for some fixed values of N.
It can be seen that the "swallow tail", a type signal for the first phase transition, between small black hole and large one.
To study the phase transition, we vary the chemical potential in terms of the entropy, and plot in figure 3 such a variation for a fixed value of N.
Figure 3: The chemical potential µ as function of the entropy for N = 3.
From the figure we can see that he chemical potential becomes positive when the entropy lies within the interval S − ≤ S ≤ S + with Furthermore, we also plot in figure 4 the behavior of the chemical potential as a function of temperature T for a fixed N .From figure 4 we can see that there exists a multivalued region, which just corresponds to the unstable region of the black hole with a negative heat capacity (red line in figure 1).
To illustrate the effect of the number of the M2-branes, we discuss the behavior of the chemical potential µ in terms of such a variable as shown in figure 12.
Figure 5: The chemical potential µ as function of N, we have set S 4 = 4.
We clearly see that the chemical potential µ presents a maximum at We remark that this is quite different from the classical gas having a negative chemical potential.In the case where the chemical potential approaches to zero and becomes positive, quantum effects should be considered and become relevant in the discussion [22].
In the subsequent sections, we consider thermodynamical geometry of the M2-branes black holes in the extended phase space and study the stability problem when either N or the charge is held fixed.

Geothermodynamics and phase transition of charged AdS black holes with fixed N case
Here we discuss the geothermodynamics of the charged AdS black holes in AdS 4 × S 7 : Our analysis will focus on the singular limits of certain thermodynamical quantities, including the heat capacities and scalar curvatures, which are relevant in the study of the stability of such black hole solution.
To do this, the number of branes N should be held fixed to consider the thermodynamics in the canonical ensemble.For a fixed N, the heat capacities for M2-branes AdS black hole are given respectively by, In the canonical ensemble with fixed N, a critical point exists and is given by the Eq.( 14).
The behavior of the C Φ,N as function of the entropy is plotted in the figure 6.
Figure 6: The heat capacity in the case with a fixed N = 3 as a function of entropy S for From the figure we can see that the C Φ,N presents two singularities at The heat capacity C Q,N is plotted in figure 7 Figure 7: The heat capacity in the case with a fixed N = 3 as a function of entropy S.
We see that it is consistent with the temperature shown in the figure 1 (red line).For small and large black hole the heat capacity is always positive, while for the intermediate black holes it is negative when Q is less than the critical value, whereas it is always positive in the case when the charge is larger than the critical point.
The heat capacity C Q,N under the critical case has two singularities at these two singularities coincide for Q = Q c , and (21) becomes We turn now our attention to the thermodynamical geometry of the black hole to see whether the thermodynamical curvature can reveal the singularities of these two specific heats.The Weinhold metric [36] is defined as the second derivative of the internal energy with respect to the entropy and other extensive quantities in the energy representation, while the Ruppeiner metric [37] is related the Weinhold metric by a conformal factor of the temperature [38].
Notice that the Weinhold and Ruppeiner metrics, which depend on the choice of thermodynamic potentials, are not Legendre invariant.The Quevedo metric defined as [39][40][41][42] is a Legendre invariant.φ denotes the thermodynamic potential, E a and I a represent respectively the set of extensive variables and the set of the intensive variable, while a = In this context we can evaluate the thermodynamical curvature of the black hole.For the Weinhold metric, where M ij stands for ∂ 2 M/∂x i ∂x j , and x 1 = S, x 2 = Q, we can see that its scalar curvature is derived via a direct calculation, simply by substituting Eq. ( 8) into Eq.( 25), In the next figure, we plot R Q 1 in terms of the entropy for a fixed N (here N = 3).In this section we study the thermodynamics geometry of the M2-branes black holes in the canonical ensemble (fixed charge).That means that the charge of the black hole is not treated as thermodynamical variable but as a fixed external parameter.The critical number of the M2-brane reads as, The heat capacity C µ,Q with a fixed chemical potential is given by The full expression of the C µ,Q , quite lengthy, is not given here.Instead we plot, in figure 10, C µ,Q in terms of the entropy in the critical sector.In the fixed charge case the Weinhold metric can be expressed as From a treatment similar to the calculation performed in the previous section, one can derive the full expression of the scalar curvatures of both the Weinhold and Ruppeiner metrics respectively.For the former one finds, with, while the result of the latter metric is, with Next we revisit the Quevedo' metric in the fixed charge case, and compute its corresponding scalar curvature given by, branes as a thermodynamical variable, we have considered AdS black holes in AdS 4 × S 7 .
Then, we have discussed the corresponding phase transition by computing the relevant quantities.In particular, we have computed the chemical potential and discussed the corresponding stabilities, the critical coordinates and the Helmoltez free energy.In addition, we have also studied the thermodynamical geometry associated with such AdS black holes.
More precisely, we have derived the scalar curvatures from the Weinhold, Ruppeiner and Quevedo metrics and demonstrated that these thermodynamical properties are similar to those which show up in the phase transition program.In the limit of the of the vanishing charge we recover all the results of [1].We aim to extend this work to other geometries and black hole configurations.

Aknowledgements
This work is supported in part by the GDRI project entitled: "Physique de l'infiniment petit et de l'infiniment grand" -P2IM (France -Maroc).

Figure 1 :
Figure 1: The temperature as function of the entropy S, with N = 3

Figure 4 :
Figure 4: The chemical potential µ as function of the temperature T, with N = 3.

Figure 9 :
Figure 9: The scalar curvature vs entropy for the Quevedo metric case with N = 3 .Right side: Q= Q c , Left side: Q < Q c .

Figure 10 :
Figure 10: The specific heat C µ,Q vs entropy with Q = .55,for N > N c and N = N c .

Figure 11 :
Figure 11: The scalar curvature of the Weinhold and Ruppeiner metrics vs entropy for N > N c , with Q = 0.55.