#  Black holes thermodynamics with CFT re-scaling

Yahya Ladghami<EMAIL_ADDRESS>Laboratory of Physics of Matter and
Radiation, Mohammed I University, BP 717, Oujda, Morocco  Astrophysical and
Cosmological Center, Faculty of Sciences, BP 717, Oujda, Morocco  Taoufik
Ouali<EMAIL_ADDRESS>Laboratory of Physics of Matter and Radiation,
Mohammed I University, BP 717, Oujda, Morocco  Astrophysical and Cosmological
Center, Faculty of Sciences, BP 717, Oujda, Morocco

###### Abstract

In this paper, we study the thermodynamic behavior of charged AdS black holes
in a conformal holographic extended thermodynamic. Our setup is constructed
using a new dictionary that relates AdS black hole quantities to the
corresponding dual conformal fields theory (CFT) one, with the conformal
factor being treated as a variable thermodynamic. In this thermodynamic study,
we investigate the critical phenomena of charged AdS black holes and their
relationship to the central charge value, C. Additionally, we examine the
phase transitions and black holes stability using the free energy and the heat
capacity, respectively. Furthermore, by examining the chemical potential, we
establish a criteria that differentiate between quantum and classical black
hole’s behaviors. Our setup highlights one of the key findings, namely
traditional black hole thermodynamics acts as a boundary between quantum and
classical regimes.

##  1 Introduction

Black hole thermodynamics is a fundamental field of study in theoretical
physics that has garnered significant attention. This interest arises from its
ability to elucidate the connection between gravity and quantum physics [1].
The exploration of black hole thermodynamics commenced with Hawking and a
group of physicists introducing a new perspective on our comprehension of
black holes. They uncovered that black holes emit radiation, a phenomenon
referred to as black holes evaporation in the literature, exhibit temperature
and entropy [2, 3, 4]. Additionally, they formulated four laws analogous to
those of ordinary thermodynamics [5]. But the process of a black hole
evaporation results in the loss of information, a puzzling phenomenon that
presents a profound challenge to the fundamental properties of quantum theory
[6, 7, 8]. This conundrum raises questions about the conservation of
information in the context of black holes. Another important phenomenon
related to black holes in anti-de Sitter (AdS) spacetime is the Hawking-Page
phase transition [9], which describes a transition between a black hole state
and a pure thermal AdS. This transition can be interpreted as a
confinement/deconfinement phase transition in the dual quark-gluon plasma
[10]. Additionally, thermodynamic of black holes with negative entropy [11],
quantum corrections to thermodynamics of black holes [12, 13], anti-
evaporation of black holes in $F(R)$ gravity [14, 15], shadow of the black
holes [16] were also studied. In the literature, this kind of black holes
thermodynamics is referred to as “the traditional black hole thermodynamics
(TBHT)”.

In the last decade, thermodynamics of black holes has witnessed significant
development, due to the inclusion of the cosmological constant as a
thermodynamic variable, interpreted as thermodynamic pressure and its
conjugate quantity as a thermodynamic volume. This approach is called
“extended phase space thermodynamics (EPST)” or “black holes chemistry” [17,
18, 19, 20, 21, 22]. Within this framework, many phenomena and behaviors of
black holes have been discovered, including critical phenomena, phase
transitions between small and large black holes, and their analogy with Van
der Waals fluids [23, 24, 25, 26, 27, 28, 29]. Additionally, polymer-type
phase transitions [30], stretched quintessence phase [31] multi-critical
points [32, 33], snapping transitions [34], superfluid [35], Joule-Thomson
expansion of black holes [36, 37, 38, 39], EGB black holes [40, 41, 42, 43,
44, 45, 46, 47] were also analyzed. Despite all this results, Gao and Zhao
[48, 49] recently pointed out some problems that the EPST formalism suffers
from, such as the problem of the “ensemble of theories”, the non-homogeneity
of Smarr relation, and the interpretation of black holes mass as enthalpy.
They then proposed another formalism competing with the EPST one, known as
“restricted phase space thermodynamics (RPST).

Building upon Maldacena duality or Anti-de Sitter/Conformal Field Theory
(AdS/CFT) correspondence [50], Visser introduced a new thermodynamic framework
that incorporates central charge, $C$, and chemical potential, $\mu$, as novel
thermodynamic variables [51]. However, it should be noted that Visser was not
the first to include $C$ and $\mu$, as they were introduced previously in [52,
53, 54, 55]. Visser’s contribution lies in his different interpretations of
the pressure and the volume compared to earlier works. The RPST formalism is a
version of the Visser thermodynamics in which the cosmological constant is
fixed, while the Newton constant is variable. The RSPT formalism excludes the
pressure-volume $(P,V)$ pair and includes $(C,\mu)$ one. In this context, the
Smarr relation is a first-order homogeneous for the extensive variables, with
mass interpreted as internal energy. This formalism has been subject of many
interesting studies, including black holes in higher dimensions [56], Taub–NUT
AdS black holes [57], black holes in a weak gravity conjecture [58], Einstein-
Power-Yang–Mills AdS black holes [59] and 4D-EGB black holes [60]. While the
RPST formalism addresses the issues raised in EPST formalism, it does not have
any connection to the original black hole chemistry. This is because it
assumes the AdS radius to be constant, implying that the size of CFT remains
constant. Consequently, the variation of the central charge stems from the
variation of Newton’s constant originating from the bulk [61].

Based on the conformal symmetry of the dual conformal field theory, the
authors of [61] introduced a new holographic approach. A pivotal aspect of
this approach involves a reevaluation of the Anti-de Sitter boundary conformal
factor, $\omega$, considering it as a thermodynamic variable and does not
depend on the AdS radius, $L$. This consideration contrasts with the approach
taken in Visser’s thermodynamics and RPST [51, 48], where they employed
$\omega=1$. In another study [62], a different choice was made by utilizing a
factor $R/L$ instead of the conformal factor $\omega$, with $R$ is the
curvature radius at the boundary. Further details on these distinctions will
be explored in Section 2. Through this setup, they established a new
dictionary linking quantities of AdS black holes with those of the dual
conformal field theory. Additionally, they successfully formulated a
holographic first law that corresponds to the first law of extended black hole
thermodynamics. Notably, this approach circumvents challenges encountered in
the EPST formalism, all the while avoiding the need to treat Newton’s constant
as a variable. In this research, we call this formalism “conformal holographic
extended thermodynamic (CHET)”. This new formalism represents an extension of
black hole chemistry. More importantly, it offers a comprehensive holographic
perspective that can enhance our understanding of black hole phase transitions
and their stability. It may also provide insights into the relationship
between gravity and quantum physics. Furthermore, it holds the potential for
applications in various fields, including cosmology.

In this work, we examine the CHET formalism applied to charged black holes in
Anti-de Sitter spacetime to gain insights into their thermodynamic behavior
and phase transitions. We have chosen these black holes due to their
simplicity and significance in the context of black hole thermodynamics. This
work, at our knowledge, is the first application of the CHET formalism to
black holes.

This paper is organized as follows: In Section 2, we present the new
dictionary and the construction of conformal holographic extended
thermodynamics. In Section 3, we define thermodynamic quantities of charged
AdS black holes in the CHET formalism. In Section 4, we study the behavior of
thermodynamics and the critical phenomena of charged AdS black holes in CHET.
We discuss the results and conclude in Section 5.

In this paper, we adopt the units $\hbar=c=k_{B}=1$.

##  2 Conformal Holographic Extended Thermodynamics

In this section, we review the construction of conformal holographic extended
thermodynamics, including the central charge, the new expression for the CFT
volume, as well as the first law and Smarr relation. This construction is
carried out within the framework of AdS/CFT correspondence. The cornerstone of
this approach is to consider the conformal factor of the Anti-de Sitter
boundary as thermodynamic variable.

###  2.1 Conformal Factor

In the majority of instances, as in the standard case, the AdS radius, $L$, is
typically selected to match the boundary curvature radius of the bulk AdS
spacetime, $R$. Under this condition, the CFT metric can be described as
follows [51]

$ds^{2}=-dt^{2}+L^{2}d\Omega_{d-2}^{2},$ (1)

where $d\Omega_{d-2}^{2}$ and $d$ represent the metric on the round unit
$(d-2)$-sphere and the dimension number of the bulk, respectively.
Additionally, the CFT volume, $\mathcal{V}$, and the central charge, $C$, are
related as follows

$\mathcal{V}\propto L^{d-2}\quad\text{and}\quad C\propto\frac{L^{d-2}}{G},$
(2)

where $G$ is the Newton constant. The definition of the boundary curvature
radius, $R=L$, as adopted in the Visser thermodynamics and RPST formalism [48,
51], introduces a key issue. This issue stems from the fact that the central
charge and the volume are directly proportional to the AdS radius.
Consequently, any alterations made to the radius $L$ will result in
corresponding changes to both the central charge $C$ and volume $\mathcal{V}$,
rendering them interdependent. An alternative, broader definition for the
curvature radius at the boundary, i.e. at $R\neq L$, were employed in [62], in
which the conformal field theory’s metric undergoes a re-scaling, as depicted
below

$ds^{2}=\frac{R^{2}}{L^{2}}\left(-dt^{2}+L^{2}d\Omega_{d-2}\right).$ (3)

This formulation is constructed in order to distinguish the central charge and
the volume of the CFT. Hence, the volume of CFT is proportional to $R^{d-2}$
rather than to the AdS radius. This means that the variation of the central
charge and the CFT volume are independent. Belhadj et al. [61], replaced the
$R/L$ factor with an arbitrary dimensionless factor, $\omega$, independent of
the AdS radius

$ds^{2}=\omega^{2}\left(-dt^{2}+L^{2}d\Omega_{d-2}^{2}\right).$ (4)

This factor, called the conformal factor, reflects the conformal symmetry of
CFT and is regarded as a thermodynamic variable. In this approach, the volume
is redefined by

$\mathcal{V}\propto(\omega L)^{d-2}.$ (5)

This modification exemplifies its innovative approach to addressing the
problems aforementioned above.

###  2.2 AdS/CFT dictionary

Based upon the idea of [61], the CFT time used in Eq. (1) differs from the one
in the re-scaling case, Eq. (4), by the conformal factor. Indeed, expressing
the CFT time by $\tilde{t}$ one has

$\tilde{t}=\omega t.$ (6)

The CFT re-scaled energy, $\tilde{E}$, is given by the relation

$\tilde{E}=E\frac{dt}{d\tilde{t}}=\frac{M}{\omega},$ (7)

where $M$ denotes the black hole’s energy. We can derive the rest of the CFT
quantities from Eq. (7) as

$\tilde{T}=\dfrac{\partial\tilde{E}}{\partial
S},\quad\tilde{\Phi}=\dfrac{\partial\tilde{E}}{\partial\tilde{Q}}\quad\text{and}\quad\tilde{\Omega}=\dfrac{\partial\tilde{E}}{\partial
J}.$ (8)

Here, $\tilde{Q}=QL/\sqrt{G}$ is the re-scaled electric charge. The quantities
of AdS black holes are found through the following equations

$T=\dfrac{\partial M}{\partial S},\quad\Phi=\dfrac{\partial M}{\partial
Q}\quad\text{and}\quad\Omega=\dfrac{\partial M}{\partial J}.$ (9)

From Eqs. (7)-(9), the holographic dictionary relating the quantities of AdS
black holes (without tildes) to the quantities of CFT (with tildes) is as
follows [61]

$\displaystyle\tilde{S}=S=\frac{A}{4G},\quad\tilde{E}=\frac{M}{\omega},\quad\tilde{T}=\frac{T}{\omega},\
$
$\displaystyle\tilde{\Omega}=\frac{\Omega}{\omega},\quad\tilde{J}=J,\quad\tilde{\Phi}=\frac{\Phi\sqrt{G}}{\omega
L},\quad\tilde{Q}=\frac{QL}{\sqrt{G}}.$ (10)

where $S$, $T$, $M$, $\Omega$, $J$, $\Phi$, $Q$, $\tilde{E}$ are the
Bekenstein-Hawking entropy, Hawking temperature, black hole mass, the angular
velocity, the angular velocity, the electrical potential, the electric charge,
the CFT energy, respectively. In this approach, the new dictionary is the most
general, and the mass of black holes is proportional to internal energy by the
conformal factor, in contrast to the old interpretation in EPST , where it is
interpreted as enthalpy [17, 64], or in the RPST formalism, where the mass of
black holes equals the internal energy [48]. However, the entropy is invariant
under this re-scaling, which means that the re-scaling does not affect the
partition function of the boundary theory. When the CFT metric is re-scaling,
it leads to a re-scaling of CFT time, $t\to\omega t$, and of the most CFT
quantities, including internal energy and temperature.

###  2.3 First Law and Smarr Relation

We can get the first law of conformal holography extended thermodynamics, by
re-scaling with the conformal factor and treating it as a thermodynamic
variable, where the derivation of internal energy is expressed by

$d\left(\frac{M}{\omega}\right)=\frac{dM}{\omega}-\frac{M}{\omega^{2}}\,d\,\omega$
(11)

where $dM$ is given by [51]

$\displaystyle dM=$ $\displaystyle Td\left(\frac{A}{4G}\right)+\Omega
dJ+\frac{\Phi}{L}d(QL)-\frac{M}{d-2}\frac{dL^{d-2}}{L^{d-2}}$ (12)
$\displaystyle+\left(M-TS-\Omega
J-\frac{\Phi}{L}QL\right)\frac{d\left(L^{d-2}/G\right)}{L^{d-2}/G}.$

By using Eqs. (11) and (12), we can derive the first law as follows [61]

$\displaystyle d\left(\frac{M}{\omega}\right)=$
$\displaystyle\frac{T}{\omega}d\left(\frac{A}{4G}\right)+\frac{\Omega}{\omega}dJ+\frac{\Phi\sqrt{G}}{\omega
L}d\left(\frac{QL}{\sqrt{G}}\right)$ (13)
$\displaystyle+\left(\frac{M}{\omega}-\frac{TS}{\omega}-\frac{\Omega
J}{\omega}-\frac{\Phi
Q}{\omega}\right)\frac{d\left(L^{d-2}/G\right)}{L^{d-2}/G}-\frac{M}{\omega(d-2)}\frac{d(\omega
L)^{d-2}}{(\omega L)^{d-2}}.$

Using Eqs. (5)-(10), we simplify the holography first law as

$d\tilde{E}=\tilde{T}dS+\tilde{\Omega}dJ+\tilde{\Phi}d\tilde{Q}+\mu dC-
pd\mathcal{V}$ (14)

where the chemical potential $\mu$ is

$\mu=\frac{1}{C}(\tilde{E}-\tilde{T}S-\tilde{\Omega}J-\tilde{\Phi}\tilde{Q})$
(15)

and

$(d-2)p\mathcal{V}=\tilde{E}.$ (16)

The last equation represents an equation of state (EoS). To eliminate the term
$-pd\mathcal{V}$ from the first law, Eq. (14), we re-scale certain CFT
quantities by a factor of $\omega L$ using the following formulas [61]

$\hat{E}=\omega L\tilde{E},\quad\hat{T}=\omega
L\tilde{T},\quad\hat{\Omega}=\omega L\tilde{\Omega},\quad\hat{\Phi}=\omega
L\tilde{\Phi},\quad\hat{\mu}=\omega L\mu.$ (17)

We obtain a first law that does not contain pressure and volume [61]

$d\hat{E}=\hat{T}dS+\hat{\Omega}dJ+\hat{\Phi}d\tilde{Q}+\hat{\mu}dC.$ (18)

By using Eqs. (15) and (17), we derive the Smarr relation [61]

$\hat{E}=\hat{T}S+\hat{\Omega}J+\hat{\Phi}\tilde{Q}+\hat{\mu}C.$ (19)

Conformal holographic extended thermodynamics is based on three key ideas. The
first idea involves a re-scaling of the CFT metric using the conformal factor.
The second idea is to treat the AdS radius as a thermodynamic variable while
keeping Newton constant fixed. The final idea is to regard the conformal
factor as a thermodynamic parameter independent of the AdS radius. This
framework yields a fresh formulation for the CFT volume and prevents the
degeneration of the first law, allowing the central charge and the CFT volume
to vary independently, without adopting the approach proposed in the RPST
formalism. In Smarr relation, the internal energy is a first-order homogeneous
function as RPST formalism.

##  3 Charged AdS Black hole

In this section, we use the results of the conformal holographic extended
thermodynamic to define thermodynamic quantities of a Reissner-Nordström black
holes in AdS spacetime (RN-AdS black holes). The metric of this black hole is

$\mathrm{d}s^{2}=-f(r)\mathrm{d}t^{2}+f(r)^{-1}\mathrm{d}r^{2}+r^{2}\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\phi^{2}\right),$
(20)

where the metric function $f(r)$ is

$f(r)=1-\frac{2MG}{r}+\frac{Q^{2}G}{r^{2}}+\frac{r^{2}}{L^{2}},$ (21)

where M and Q are the mass and the electrical charge of black holes,
respectively. By solving the equation $f(r)=0$, two solutions are found,
$r_{\pm}$, where $r_{+}$ corresponds to the event horizon of the black hole
and $r_{-}$ is the Cauchy horizon. For $r=r_{+}$, the expression of black
holes mass write

$M=\frac{r_{+}}{2G}\left(1+\frac{GQ^{2}}{r_{+}^{2}}+\frac{r_{+}^{2}}{L^{2}}\right).$
(22)

To construct holographic extended thermodynamics of RN-AdS black holes, we
need to determine the thermodynamic quantities of this black hole. To this
aim, the RN-AdS black holes entropy, the electrical potential and the Hawking
temperature at the event horizon are given by

$S=\frac{A}{4G}=\frac{\pi r_{+}^{2}}{G},$ (23)

$\Phi=\frac{Q}{r_{+}},$ (24)

and

$T=\frac{f^{\prime}\left(r_{+}\right)}{4\pi}=\frac{r_{+}^{2}+3r_{+}^{4}L^{-2}-GQ^{2}}{4\pi
r_{+}^{3}},$ (25)

respectively. By using the new dictionary between quantities of AdS black
holes and CFT, Eqs.(10) and (17), we find the CFT quantities that correspond
to those of the RN-AdS black hole as follow

$\hat{E}\equiv\hat{M}=M\,L,\quad\hat{T}=T\,L,\quad\tilde{S}=S,\quad\tilde{Q}=\frac{QL}{\sqrt{G}},\quad\hat{\Phi}=\Phi\sqrt{G},\quad
C=\frac{L^{2}}{G},\quad\hat{\mu}=\omega L\mu.$ (26)

From Eqs. (18) and (19), the first law and Smarr relation in CHET of the
charged AdS black holes write as

$d\hat{M}=\hat{T}dS+\hat{\Phi}d\tilde{Q}+\hat{\mu}dC,$ (27)

and

$\hat{M}=\hat{T}S+\hat{\Phi}\tilde{Q}+\hat{\mu}C,$ (28)

respectively. Finally, we get thermodynamic variables from the first law, Eq.
(28), and the mass of black holes, Eq. (22), as

$\hat{T}=\left(\frac{\partial\hat{M}}{\partial
S}\right)_{\tilde{Q},C}=\frac{\pi
CS-\pi^{2}\tilde{Q}^{2}+3S^{2}}{4\pi^{3/2}S\sqrt{CS}},$ (29)
$\hat{\Phi}=\left(\frac{\partial\hat{M}}{\partial\tilde{Q}}\right)_{S,C}=\frac{\sqrt{\pi}\tilde{Q}}{\sqrt{CS}},$
(30) $\hat{\mu}=\left(\frac{\partial\hat{M}}{\partial
C}\right)_{S,\tilde{Q}}=\frac{\pi
CS-\pi^{2}\tilde{Q}^{2}-S^{2}}{4\pi^{3/2}C\sqrt{CS}}.$ (31)

##  4 Phase structure

###  4.1 Critical phenomena

In this section, we study the phase transition and the critical central charge
for the charged AdS black holes in the conformal extended holographic
thermodynamic. To find the coordinate of the critical point i.e. the event
horizon and the critical central charge, we solve the following system

$\left(\frac{\partial\hat{T}}{\partial
S}\right)_{\tilde{Q},C}=0\quad\text{and}\quad\left(\frac{\partial^{2}\hat{T}}{\partial
S^{2}}\right)_{\tilde{Q},C}=0,$ (32)

one gets

$S_{c}=\pi\,\tilde{Q},\qquad
C_{c}=6\,\tilde{Q},\qquad\tilde{T}_{c}=\frac{1}{\pi}\sqrt{\frac{2}{3}}.$ (33)

Using the following normalization

$t=\frac{\tilde{T}}{\tilde{T}_{c}},\qquad s=\frac{S}{S_{c}},\qquad
c=\frac{C}{C_{c}},$ (34)

we can rewrite the equation of state, Eq. (29), in the following form

$t=\frac{3s^{2}+6s\,c-1}{8s\,\sqrt{s\,c}}.$ (35)

Figure 1: t - s curves in the iso-central-charge processes

Fig. 1 shows the evolution of the Hawking temperature in terms of the entropy
or the event horizon of the black hole, for different values of the central
charge. For $C=C_{c}$, a phase transition of the second order occurs between a
small and a large black hole. For $C>C_{c}$, we notice a non-monotonous
curves, where a first order phase transition occurs, between a large and a
small black hole, passing through a medium black hole. Finally, no phase
transition occurs for $C<C_{c}$. This thermodynamic behavior is similar to
that of the Van der Waals fluids, as well as to that of the thermodynamic
behavior for RN-AdS black holes in the extended phase space thermodynamics
[65] and the restricted phase space thermodynamics [48]. The main discrepancy
resides in the effect of the central charge on the thermodynamic behavior
rather than the effect of the electrical charge.

###  4.2 Free Energy

In the context of the phase structure of RN-AdS black holes in the CHET
formalism, we can also study the thermodynamic behavior in terms of the
evolution of the free energy and its relation to the dual CFT. We express the
Helmholtz free energy $\hat{F}$ in conformal holography extended
thermodynamics by

$\hat{F}=\hat{M}-\hat{T}S$ (36)

By normalizing the free energy as $f=\hat{F}/\hat{F}_{c}$, where
$\hat{F}_{c}=\sqrt{\frac{2}{3}}Q$, Eq. (36) becomes

$f=\frac{s\left(-4\sqrt{cs}\,t+6c+s\right)+1}{4\sqrt{cs}}.$ (37)

Figure 2: f - t curves in the iso-central-charge processes

Fig. 2 represents the evolution of the normalized Helmholtz free energy in
terms of the normalized temperature for different values of the central
charge. In the free energy evolution curve, a swallow’s tail appears for a
central charge greater than the critical one corresponding to a first-order
phase transition. This swallow’s tail disappears for a central charge equals
or less than the critical central charge.

###  4.3 Heat Capacity

The heat capacity gives us an idea of the stability/instability of black
holes. It is known that the system is stable (unstable) for a positive
(negative) heat capacity. The heat capacity is expressed by

$C_{\tilde{Q}}=\hat{T}\left(\frac{\partial{S}}{\partial{\hat{T}}}\right)_{\tilde{Q}}=\frac{2S\left(\pi
CS-\pi^{2}\tilde{Q}^{2}+3S^{2}\right)}{3S^{2}-\pi CS+3\pi^{2}\tilde{Q}^{2}}.$
(38)

We normalize the heat capacity by

$\eta=\frac{C_{\tilde{Q}}}{2\pi\tilde{Q}}=\frac{3s\left(s+2sc\right)-s}{3s(s-2c)+3}.$
(39)

Figure 3: Heat capacity in term of the entropy for different values of C.

Fig. 3 represents the normalized heat capacity in terms of the normalized
entropy for different values of the central charge. From Figs. 1 and 3, we
conclude that small and large black holes are stable, while medium black holes
are unstable. The thermodynamic behavior of black holes depends on the dual
CFT i.e. on the central charge. Indeed, for a large central charge
($C>6\hat{Q}$), a phase transition between small and large black holes is
observed. While for a small central charge no phase transition occurs.

###  4.4 Chemical Potential

In statistical physics of ordinary matter, the sign of chemical potential
plays a role in determining the type of regime we are studying, i.e. classical
or quantum regime. By analogy between statistical physics of ordinary matter
and conformal holography extended thermodynamics of AdS black holes, we define
conditions on the central charge C to distinguish quantum regime from
classical one. If the chemical potential is positive, then the regime
corresponds to a quantum regime and if it is negative, then it corresponds to
a classical regime [66]. Using Eqs. (17), (31) and (34), we find

$\hat{\mu}=\frac{6\,c\,s-s^{2}-1}{24\,c\,\sqrt{6\,c\,s}},$ (40)

Thus, the system corresponds to a quantum regime for

$C>\tilde{Q}\left(s+\frac{1}{s}\right),$ (41)

and to a classical one for the case of $C<\hat{Q}\left(s+s^{-1}\right)$. The
chemical potential is zero for $C=\hat{Q}\left(s+s^{-1}\right)$. This
condition may be interpreted as a non holographic point as the element
$\hat{\mu}dC$ and $\hat{\mu}C$ disappear from the first law and Smarr
relation, respectively. In other words, we recover the traditional black hole
thermodynamic (TBHT) formalism.

Figure 4: Chemical potential in term of entropy for different values of C.

By comparing Figs. 1, 3 and 4, we conclude that the phases transition region
are located in the quantum regime i.e. for $\hat{\mu}>0$ and $0.5<s<2$, while
for the classical regime, i.e. $\hat{\mu}<0$ and large entropy, there is no
phases transition. For the Schwarzschild-AdS black hole ($\tilde{Q}=0$) we
find that the quantum regime , from Eq. (31), is characterized by

$S<\pi C\quad$ (42)

and the classical regime by

$S>\pi C\quad$ (43)

##  5 Discussion and Conclusion

####

In this work, we have studied a charged AdS black holes in the conformal
holography extended thermodynamic by using the new dictionary between
quantities of the extended black hole thermodynamic and the quantities of the
dual CFT in the context of the AdS/CFT correspondence. Through this
dictionary, we get the first law and Smarr relation of the conformal
holography extended thermodynamic formalism. In this formalism, the
interpretation of black hole mass differs from the previous formalisms i.e.
TBHT, EPST, and RPST. In CHET, the mass is a homogeneous function of the
first-order without resorting to making Newton constant $G$ a variable as it
will be in the RPST formalism. By re-scaling some quantities of CFT, the
pressure-volume disappear from the first law. Furthermore, in this formalism,
the volume is not the conjugate quantity of the thermodynamic pressure of the
black hole as in the EPST formalism, but is the CFT volume from which a new
pressure is defined as its conjugate quantity.

####

By using the results of the CHET formalism, we have studied the thermodynamic
behavior and the property of a charged AdS black hole and its relationship to
the states of the dual CFT for a fixed electrical charge. We found expressions
of the thermodynamic quantities and the equation of state in this approach. We
have also studied the effect of the CFT’s states for the thermal evolution of
black holes. For a critical central charge, $C=C_{c}$, an emergency second-
order phase transition appears between a small and a large black hole. For the
large central charge, i.e. $C>C_{c}$, we find a first-order phase transition
between small and large black holes, passing through an unstable medium black
hole with a negative heat capacity, in contrast to large and small black
holes, which are stable system, as shown in Figs. 1 and 3. Finally, no phase
transitions occurs for a small central charge ($C<C_{c}$). This thermodynamic
behavior is similar to that of Van der Waals fluids. After normalizing the
equation of state, the electric charge no longer exists. Indeed, the value of
the electric charge has no effect on the thermodynamic behavior of charged AdS
black holes, but this does not mean that the critical phenomena are achieved
in an electrically neutral black hole. Whereas, the state of CFT depends on
the electric charge as $C_{c}=6\tilde{Q}$ and discussed above.

####

Finally, we have studied the regime of the system in terms of the chemical
potential. We find that the case of $C>\hat{Q}\left(s+s^{-1}\right)$
corresponds to a quantum regime while the case of
$C<\hat{Q}\left(s+s^{-1}\right)$ corresponds to the classical one. We notice
that the non holographic point, $C=\hat{Q}\left(s+s^{-1}\right)$, corresponds
to the TBHT formalism.

## References

  * [1] R. Wald, Living Reviews In Relativity. 4 pp. 1-44 (2001).
  * [2] J.D.Bekenstein, Lettere al Nuovo Cimento. 4, 737-740 (1972).
  * [3] J.D.Bekenstein,  Physical Review D, 7, 2333-2346 (1973).
  * [4] S.W.Hawking,  Communications in Mathematical Physics. 43, 199-220 (1975).
  * [5] J.M.Bardeen, B.Carter and S.W.Hawking,  Communications in Mathematical Physics. 31, 161-170 (1973).
  * [6] S. Hawking, Physical Review D. 14, 2460 (1976).
  * [7] S. Mathur, Classical And Quantum Gravity. 26, 224001 (2009).
  * [8] P. Chen, Y. Ong, & D. Yeom, Physics Reports. 603 pp. 1-45 (2015).
  * [9] S. Hawking, & D. Page, Communications In Mathematical Physics. 87 pp. 577-588 (1983).
  * [10] E. Witten, Advances In Theoretical And Mathematical Physics. 2, 505-532 (1998).
  * [11] M. Cvetič, S. Nojiri & S. Odintsov, Nuclear Physics B. 628, 295-330 (2002).
  * [12] S. Nojiri & S. Odintsov, International Journal Of Modern Physics A. 15, 989-1010 (2000).
  * [13] S. Nojiri & S. Odintsov, Physical Review D. 59, 044026 (1999).
  * [14] S. Nojiri & S. Odintsov, Classical And Quantum Gravity. 30, 125003 (2013).
  * [15] S. Nojiri & S. Odintsov, Physics Letters B. 735, pp. 376-382 (2014).
  * [16] S. Vagnozzi, R. Roy, Y.D. Tsai, L. Visinelli, M. Afrin, et al. Classical And Quantum Gravity. 40, 165007 (2022).
  * [17] D.Kastor, S.Ray and J.Traschen, Classical And Quantum Gravity 26 (2009), 195011.
  * [18] B. Dolan, Classical And Quantum Gravity. 28, 125020 (2010).
  * [19] B. Dolan, Classical And Quantum Gravity. 28, 235017 (2011).
  * [20] B. Dolan, Physical Review D. 84 (2011).
  * [21] D. Kubizňák, & R. Mann, Canadian Journal Of Physics. 93, 999-1002 (2015).
  * [22] D. Kubizňák, R. Mann, & M. Teo, Classical And Quantum Gravity. 34, 063001 (2017)
  * [23] D. Kubizňák, & R. Mann, Journal Of High Energy Physics. 2012, 1-25 (2012).
  * [24] R. Cai, L. Cao, L. Li, & R. Yang, Journal Of High Energy Physics. 2013, 1-22 (2013).
  * [25] S. Wei, & Y. Liu, Physical Review D. 101, 104018 (2020).
  * [26] S. Hendi, & M. Vahidinia, Physical Review D. 88, 084045 (2013).
  * [27] B. Majhi, & S. Samanta, Physics Letters B. 773 pp. 203-207 (2017).
  * [28] M. Dehghani, S. Kamrani & A. Sheykhi, Physical Review D. 90, 104020 (2014).
  * [29] H Liu, & X. Meng, Modern Physics Letters A. 31, 1650199 (2016).
  * [30] B. Dolan, A. Kostouki, D. Kubizňák & R. Mann. Classical And Quantum Gravity. 31, 242001 (2014).
  * [31] Y. Ladghami, B. Asfour, A. Bargach, A Errahmani, & T. Ouali, Annals Of Physics. 460 pp. 169575 (2024).
  * [32] M. Tavakoli, J. Wu, & R. Mann, Journal Of High Energy Physics. 2022, 1-13 (2022).
  * [33] J. Wu & R Mann, Classical And Quantum Gravity. 40, 06LT01 (2023).
  * [34] N. Abbasvandi, W. Cong, D. Kubizňák & R. Mann, Classical And Quantum Gravity. 36, 104001 (2019).
  * [35] R. Hennigar, R. Mann & E. Tjoa, Physical Review Letters. 118, 021301 (2017).
  * [36] Ö. Ökcü, The European Physical Journal C. 77 pp. 1-7 (2017).
  * [37] Ö. Ökcü, & E. Aydıner, The European Physical Journal C. 78 pp. 1-6 (2018).
  * [38] J. Liang, B. Mu & P. Wang, Physical Review D. 104, 124003 (2021).
  * [39] J. Mo, G. Li, S. Lan & X. Xu, Physical Review D. 98, 124032 (2018).
  * [40] M. Bousder & M. Bennai. Physics Letters B. 817 pp. 136343 (2021).
  * [41] M. Bousder, K. El Bourakadi & M. Bennai. Physics Of The Dark Universe. 32 pp. 100839 (2021).
  * [42] M. Bousder & M. Bennai. Nuclear Physics B. 969 pp. 115476 (2021).
  * [43] Y. Sekhmani, H. Lekbich, A. El Boukili & M. Sedra, The European Physical Journal C. 82, 1087 (2022).
  * [44] H. Lekbich, A. El Boukili, N. Mansour, & M. Sedra. Annals Of Physics. 458 pp. 169451 (2023).
  * [45] H. Lekbich, A. El Boukili, N. Mansour & M. Sedra. The European Physical Journal Plus. 137, 1-11 (2022).
  * [46] Y. Sekhmani & D. Gogoi. International Journal Of Geometric Methods In Modern Physics. pp. 2350160 (2023).
  * [47] J. Sadeghi, M. Salahshour, S. Noori, & M. Alipour. Physica Scripta. (2023).
  * [48] Z. Gao, & L. Zhao, Classical And Quantum Gravity. 39, 075019 (2022).
  * [49] Z. Gao, X. Kong & L. Zhao, The European Physical Journal C. 82, 112 (2022).
  * [50] J.M.Maldacena, Advances in Theoretical and Mathematical Physics. 2, 231-252 (1998).
  * [51] M. Visser,  Physical Review D. 105, 106014 (2022).
  * [52] D. Kastor, S. Ray, & J. Traschen, Journal Of High Energy Physics. 2014, 1-17 (2014).
  * [53] J. Zhang, R. Cai & H. Yu, Physical Review D. 91, 044028 (2015).
  * [54] A. Karch, & B. Robinson, Journal Of High Energy Physics. 2015, 1-15 (2015).
  * [55] R. Maity, P. Roy & T. Sarkar, Physics Letters B. 765 pp. 386-394 (2017).
  * [56] X. Kong, T. Wang, Z. Gao & L. Zhao, Entropy. 24, 1131 (2022).
  * [57] J. Sadeghi, M. Shokri, S. Gashti & M. Alipour, General Relativity And Gravitation. 54, 129 (2022).
  * [58] M. Alipour, J. Sadeghi, & M. Shokri, Nuclear Physics B. 990 pp. 116184 (2023).
  * [59] Y. Du, H. Li, Y. Zhang, X. Zhou & J. Zhao, Entropy. 25, 687 (2023).
  * [60] Y. Ladghami, B. Asfour, A. Bouali, A. Errahmani & T. Ouali, Physics Of The Dark Universe. 101261 (2023).
  * [61] M. Belhaj, W. Cong, D. Kubizňák, R. Mann, & M. Visser, Physical Review Letters. 130, 181401 (2023).
  * [62] W.Cong, D.Kubiznak, R.B.Mann and M.R.Visser, Journal Of High Energy Physics. 08, 174 (2022).
  * [63] M. Belhaj, W. Cong, D. Kubizňák, R. Mann, & M. Visser, Journal Of High Energy Physics. 2023, 1-32 (2023).
  * [64] D.Kubiznak, R.B.Mann and M.Teo, Classical And Quantum Gravity. 34, no.6, 063001 (2017).
  * [65] W.Xu and L.Zhao, Physics Letters B 736, 214-220 (2014).
  * [66] W. Cong, D. Kubiznak & R. Mann, Physical Review Letters. 127, 091301 (2021).