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

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

M. Chabab, H. EL Moumni, K. Masmar
High Energy Physics and Astrophysics Laboratory, FSSM, Cadi Ayyad University, Marrakesh, Morocco.
Département de Physique, Faculté des Sciences, Université Ibn Zohr, Agadir, Morocco.
July 12, 2019
###### Abstract

Motivated by a recent work on asymptotically Ad 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 , 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.

## 1 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 [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 black holes analysed by identifying the cosmological constant in the bulk with the number of colors.

All these recent inspiring works on asymptotically Ad black holes in M-theory [27, 28, 29, 30] motivate us to study the thermodynamics and thermodynamical geometry of , 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 , 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.

## 2 Thermodynamics of black holes in AdS4×S7 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 . In such a geometric background, the line element of the black M2-brane metric is given by [34, 32]

 ds2=r4L4(−fdt2+2∑i=1dx2i)+L2r2f−1dr2+L2dΩ27, (1)

where is the metric of seven-dimensional sphere with unit radius. In this solution, the metric function reads as follows

 f=1−mr+q2r2+r2L2, (2)

where is the AdS radius and and 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 coincident configurations of M2-branes. In this background, the AdS radius is linked to the M2-brane number via the relation [32, 35, 1]

 L9=N3/2κ211√2π5. (3)

According to the proposition reported in [20, 21, 22, 1], 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 of the corresponding black hole is determined by solving the equation . From Eq.(2), the mass of the black hole can be written as

 M=mω28πG4=rω2(L2+r2)8πG4L2+2πG4Q2rω2.\lx@notefootnotewhere$ωd=2πd+12Γ(d+12)$. (4)

where the charge of the black hole is related to the constant through the formula,

 Q=ω24πG4q. (5)

The Bekenstein-Hawking entropy formula of the black hole reads as,

 S=A4G4=ω2r24G4. (6)

Here we recall that four-dimensional Newton gravitational constant is related to the eleven-dimensional one as

 G4=3G112πω2L4. (7)

For simplicity reason, we use in the remainder of the paper. In this way, the black hole mass can be expressed as a function of and ,

 M(S,N)=39√2π11/93√NQ2+33√πS2+8√[3]2NS4 213/18√3π11/18N2/3√S (8)

Using the standard thermodynamic relation , the corresponding temperature takes the following form

 T=∂M(S,N)∂S∣∣∣N=−39√2π11/93√NQ2+93√πS2+83√2NS8 213/18√3π11/18N2/3S3/2. (9)

This quantity can be identified with the Hawking temperature of the black hole. Using eq. (8) the chemical potential conjugate to the number of M2-branes is given by

 μ=∂M4(S,N)∂N∣∣∣S=−39√2π11/93√NQ2−63√πS2+83√2NS12 213/18√3π11/18N5/3√S. (10)

It defines the measure of the energy cost to the system when one increases the variable .

while the electric potential reads as

 Φ=∂M(S,N)∂Q∣∣∣S=√3π11/18Q2 211/183√N√S. (11)

In terms of these quantities, the Helmholtz free energy is expressed by,

 F(T,N)=M−T S=99√2π11/93√NQ2−33√πS2+83√2NS8 213/18√3π11/18N2/3√S. (12)

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 function if , but when , it presents a critical point to be determined by solving the system,

 (∂T∂S)Qc=(∂2T∂S2)Qc=0 (13)

The solution of this equation is easily derived,

 Qc=4 25/18N5/69π7/9,Sc=493√2πN. (14)

In figure 2, we illustrate the Helmholtz free energy as function of the Hawking temperature for some fixed values of .

The sign change of the free energy indicates Hawking-Page phase transition, which occurs at

 SHP=18√2(4 25/18N+√16 25/9N2+27π14/93√NQ2)33√π. (15)

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 .

From the figure we can see that he chemical potential becomes positive when the entropy lies within the interval with

 S±=43√2N±25/9√8 25/9N2−9π14/93√NQ263√π (16)

Furthermore, we also plot in figure 4 the behavior of the chemical potential as a function of temperature for a fixed .

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 in terms of such a variable as shown in figure 12.

We clearly see that the chemical potential presents a maximum at

 Nmax=33√π(2π2Q63√15Q6S5/2+√225Q12S5−4π3Q18+3√2π3√15Q6S5/2+√225Q12S5−4π3Q18+5 22/3S5/2)16S3/2 (17)

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 or the charge is held fixed.

## 3 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 : 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 should be held fixed to consider the thermodynamics in the canonical ensemble. For a fixed , the heat capacities for M2-branes AdS black hole are given respectively by,

 CQ,N=T(∂S∂T)Q,N=T4(83√2N+183√πS−39√2π11/93√NQ2+93√πS2+83√2NS−32S)−1 (18)
 CΦ,N=T(∂S∂T)Φ,N=⎛⎜ ⎜⎝9S−39√2π8/93√NQ2+9S2+83√2πNS−12S⎞⎟ ⎟⎠−1 (19)

In the canonical ensemble with fixed , a critical point exists and is given by the Eq.(14). The behavior of the as function of the entropy is plotted in the figure 6.

From the figure we can see that the presents two singularities at

 SΦ,±=19⎛⎝43√2πN±√16(2π)2/3N2−279√2π8/93√NQ2⎞⎠ (20)

The heat capacity is plotted in figure 7

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 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 under the critical case has two singularities at

 SQ.±=83√2N±2√16 22/3N2−819√2π14/93√NQ2183√π (21)

these two singularities coincide for , and (21) becomes

 SQ=493√2π(√N2−3 32/33√N+N) (22)

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].

 ds2R=1Tds2W (23)

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]

 g=(Ec∂ϕ∂Ec)(ηabδbc∂2ϕ∂Ec∂Ed),ηcd=diag(−1,1,⋯,1) (24)

is a Legendre invariant. denotes the thermodynamic potential, and represent respectively the set of extensive variables and the set of the intensive variable, while .

In this context we can evaluate the thermodynamical curvature of the black hole. For the Weinhold metric,

 gW=(MSSMSQMQSMQQ), (25)

where stands for , and , , we can see that its scalar curvature is derived via a direct calculation, simply by substituting Eq. (8) into Eq. (25),

 RW1=−6418√2√3π11/18N5/3S3/2(3√[9]2π11/93√NQ2+93√πS2−83√2NS)2 (26)

While the Ruppeiner metric, deduced from Eq.(23), is given by

 gR=1T(MSSMSQMQSMQQ), (27)

with the following curvature,

 RR1=A1B2 (28)

where,

 A1 = 162π2/3S[−486 24/9π7N2Q12+6480 22/3π52/9N8/3Q10S+1296 22/9π2/3N4/3S9 (29) × (896 25/9N5/3−351π14/9Q2)−4323√2π41/9N5/3Q8S2(80 25/9N5/3−9π14/9Q2) + 1449√2π10/3N7/3Q6S3(1280N5/3−261 24/9π14/9Q2)+183√πN2/3S8 × (901129√2N10/3+5103π28/9Q4−62208 25/9π14/9N5/3Q2)−963√2N5/3S7 × (−40969√2N10/3−3645π28/9Q4+8640 25/9π14/9N5/3Q2)+489√2π11/9NQ2S6 × (−40969√2N10/3−729π28/9Q4+5184 25/9π14/9N5/3Q2)−32π8/9N2Q2S5 × (−4096 25/9N10/3+243 24/9π28/9Q4+8064π14/9N5/3Q2)+6 22/9π19/9N4/3Q4S4 × (−409609√2N10/3−729π28/9Q4+20736 25/9π14/9N5/3Q2)−59049 27/9π5/3S12 − 1049769√2π4/3NS11+559872 24/9πN2S10] B1 = (39√2π11/93√NQ2−93√πS2−83√2NS)3 × (−189√2π22/9N2/3Q4+963√2π11/9N4/3Q2S+81 28/9π2/3S4−128 25/9N2S2)2

In the figure 8 we plot the scalar curvatures of the Weinhold and Ruppeiner metrics where the charge is less than critical one.

From Fig.8 we see that scalar curvature of Weinhold and Ruppeiner metrics reveal both the same singularities of the heat capacity . The Ruppeiner metric presents a further singularity in where the black hole is extremal . Hence both Weinhold and Ruppeiner metric are able to show phase transition of the black hole in the fixed ensemble.

The Quevedo metric is defined by,

 gQ=(ST+QΦ)(−MSS00MQQ), (31)

Using Eqs.(8,9) and Eq.(31) we show that the scalar curvature reads as,

 RQ1=A2B2 (32)

with,

 A2 = −7689√2π11/9N4/3S[43749√2π44/9N4/3Q8+131220π4NQ6S2+194403√2π11/3N2Q6S (33) + 104976 28/9π28/9N2/3Q4S4+85536 22/9π25/9N5/3Q4S3+17280 25/9π22/9N8/3Q4S2 + 21870 27/9π20/93√NQ2S6+194409√2π17/9N4/3Q2S5−34560 24/9π14/9N7/3Q2S4 + 21504 27/9π11/9N10/3Q2S3−19683 22/3π4/3S8−46656πNS7−103683√2π2/3N2S6 + 18432 22/33√πN3S5+32768N4S4] B2 = (9 27/9π11/93√NQ2+9 22/33√πS2+16NS)2(99√2π11/93√NQ2+93√πS2−83√2NS)2 (34) × (99√2π11/93√NQ2+93√πS2+83√2NS)2

In the next figure, we plot in terms of the entropy for a fixed (here ).

Under the critical scheme the scalar curvature of the Quevedo metric presents two singularities at which are the same as those of the heat capacity shown in figure 7 (red line). When , the two singularities coincide to one (represented by dashed black line). That mean that the Quevedo metric can reveal the phase transition in the fixed charge ensemble.

## 4 Geothermodynamics and phase transition of charged AdS black holes with fixed charge Q

In this section we study the thermodynamics geometry of the -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,

 Nc=9 32/5π14/15Q6/54 211/15 (35)

The heat capacity with a fixed chemical potential is given by

 Cμ,Q=T(∂T∂S)−1μ,Q (36)

The full expression of the , quite lengthy, is not given here. Instead we plot, in figure 10, in terms of the entropy in the critical sector.

From the left panel, we see that the heat capacity presents two divergencies up to the critical regime, given numerically by and for . When , these two singularities coincide, as shown in the right panel, to only one singularity.

In the fixed charge case the Weinhold metric can be expressed as

 gW=(MSSMSNMNSMNN), (37)

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,

 RW2=A3B3 (38)

with,

 A3 = 120 25/6√3π11/6NQ2S3/2[−408 24/9π11/9N4/3Q2+8379√2π14/93√NQ2S+1404π2/3S3 (39) + 643√2NS(73√2N−273√πS)] B3 = (−99 22/9π22/9N2/3Q4−4869√2π14/93√NQ2S2+288 24/9π11/9N4/3Q2S+54π2/3S4 (40) + 963√2πNS3−64 22/3N2S2)2

while the result of the latter metric is,

 RR2=A4B4 (41)

with

 A4 = 153√π3√N[7683√πN8/3S(27π3Q6−16S5)+991443√2π4N2/3Q6S3−432(2π)2/3N5/3S2 (42) × (261π3Q6+8S5)+111537 22/9π28/93√NQ4S5−126360 25/9π25/9N4/3Q4S4 + 123264 28/9π22/9N7/3Q4S3−55296 22/9π19/9N10/3Q4S2+991449√2π20/9Q2S7 − 1536009√2π11/9N3Q2S4+49152 24/9π8/9N4Q2S3−432 27/9π14/9N2Q2(3π3Q6+100S5) + 81 24/9π17/9NQ2S(357π3Q6+640S5)+81923√2N11/3S5] B4 = (−39√2π11/93√NQ2+93√πS2+83√2NS)(−99 22/9π22/9N2/3Q4−4869√2π14/93√NQ2S2 (43) + 288 24/9π11/9N4/3Q2S+54π2/3S4+963√2πNS3−64 22/3N2S2)2

These two scalar curvatures are plotted as functions of the entropy in figure 11 which shows that the two metrics reproduce the results obtained in the previous section regarding the singularities of the heat capacity . Furthermore, the figure also shows the divergency of the scalar curvature of the Weinhold (left) and Ruppeiner (right) metrics when the entropy tends to the value for which the black hole becomes extremal .

Next we revisit the Quevedo’ metric in the fixed charge case,

 gQ=(ST+Nμ)(−MSS00MNN), (44)

and compute its corresponding scalar curvature given by,

 RQ2=A5B5 (45)

The expression of and are found to be,

 A5 = 864π14/9N5/3S2[768 24/93√πN8/3S(123π3Q6−88S5)−243 27/9πN2/3S3(953π3Q6−48S5) (46) − 1449√2π2/3N5/3S2(5817π3Q6+2176S5)−522450 22/3π28/93√NQ4S5+490320π25/9N4/3Q4S4 + 21162243√2π22/9N7/3Q4S3−190464 22/3π19/9N10/3Q4S2−24786 25/9π20/9Q2S7 − 1588224 25/9π11/9N3Q2S4+114688(2π)8/9N4Q2S3−864 22/9π14/9N2Q2(17π3Q6−792S5) + 81 28/9π17/9NQ2S(459π3Q6+12520S5)+131072 27/9N11/3S5] B5 = 5(39√2π11/93√NQ2−33√πS2−83√2NS)3(69√2π11/93√NQ2+153√πS2−83√2NS)2 (47) × (99√2π11/93√NQ2+93√πS2−83√2NS)2 (48)

Illustration of behaviour as a function of the entropy is seen in the next figure when .

From the figure 12 we can see that the Quevedo metric presents similar singularity’s features, here denoted by , as in the previous analysis of the for fixed shown in Eq.18. In addition we note that at , becomes also singular, while at an additional singularity shows up signaling the extremal case.

## 5 Conclusion

In this paper, we have explored the thermodynamics and thermodynamical geometry of charged AdS black holes from M2-branes. More concretely, by assuming the number of M2-branes as a thermodynamical variable, we have considered AdS black holes in . 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).

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