#### Validity of Maxwell equal area law for black holes conformally coupled to scalar fields in \(\text {AdS}_5\) spacetime

Eur. Phys. J. C
Validity of Maxwell equal area law for black holes conformally coupled to scalar fields in AdS5 spacetime
Yan-Gang Miao 0
Zhen-Ming Xu 0
0 School of Physics, Nankai University , Tianjin 300071 , China
We investigate the P−V criticality and the Maxwell equal area law for a five-dimensional spherically symmetric AdS black hole with a scalar hair in the absence of and in the presence of a Maxwell field, respectively. Especially in the charged case, we give the exact P−V critical values. More importantly, we analyze the validity and invalidity of the Maxwell equal area law for the AdS hairy black hole in the scenarios without and with charges, respectively. Within the scope of validity of the Maxwell equal area law, we point out that there exists a representative van der Waals-type oscillation in the P−V diagram. This oscillating part, which indicates the phase transition from a small black hole to a large one, can be replaced by an isobar. The small and large black holes have the same Gibbs free energy. We also give the distribution of the critical points in the parameter space both without and with charges, and we obtain for the uncharged case the fitting formula of the co-existence curve. Meanwhile, the latent heat is calculated, which gives the energy released or absorbed between the small and large black hole phases in the isothermal-isobaric procedure.
1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2 Analytic solution in D = 5 dimensions . . . . . . . .
3 Maxwell equal area law . . . . . . . . . . . . . . . .
3.1 Uncharged case: e = 0 . . . . . . . . . . . . . .
3.1.1 Critical values . . . . . . . . . . . . . . . .
3.1.2 Maxwell equal area law . . . . . . . . . .
3.2 Charged case: e = 0 . . . . . . . . . . . . . . . .
3.2.1 Critical values . . . . . . . . . . . . . . . .
3.2.2 Maxwell equal area law . . . . . . . . . .
3.2.3 The case of q < 0 . . . . . . . . . . . . . .
3.2.4 The case of q > 0 . . . . . . . . . . . . . .
a e-mail:
b e-mail:
4 Conclusion . . . . . . . . . . . . . . . . . . . . . . .
Appendix A: Derivation of Eqs. (3.10) and
(3.21) . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix B: Root of Eq. (3.16) . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
Since the seminal work by Hawking and Bekenstein on
the radiation of black holes, the exploration of
thermodynamic properties of black holes has received a wide range
of attention [1–3] and also acquired great progress [4–7].
Of more particular interest is the thermodynamics of
antide Sitter (AdS) black holes [8,9] where the AdS/CFT
duality plays a pivotal role in recent developments of
theoretical physics [10,11]. In the context of AdS/CFT
correspondence [12–14], the Hawking–Page phase transition [15] of
five-dimensional AdS black holes can be explained as the
phenomenon of the confinement/deconfinement transition in
the four-dimensional Yang–Mills gauge field theory [16].
Another archetypal example of the AdS/CFT correspondence
is the holographic superconductor, which can be regarded
as the scalar field condensation around a four-dimensional
charged AdS black hole [17].
With the cosmological constant being treated as a
thermodynamic pressure variable [18–23] and its conjugate
variable being considered as the thermodynamic volume, the
thermodynamics in the extended phase space has been
getting more and more attentions. In this paradigm, the mass of
black holes is identified as the enthalpy rather than the
internal energy. This idea has also been applied to other known
parameters, such as the Born–Infeld parameter [24,25], the
Gauss–Bonnet coupling constant [26], the noncommutative
parameter [27], and the Horndeski non-minimal kinetic
coupling strength [28], etc. All the parameters just mentioned
can be regarded as a kind of thermodynamic pressure.
Furthermore, there exists a similar situation in the exploration of
charged AdS hairy black holes [29] of Einstein–Maxwell
theory conformally coupled to a scalar field in five dimensions.
The model’s action has been given by [30–33]
1
I = κ
d5x √−g
R − 2
− 41 F 2 + κ Lm (φ, ∇φ) ,
where κ = 16π , R is the scalar curvature, F the
electromagnetic field strength, gμν the metric with mostly plus
signatures, and g = det(gμν ). In addition, the Lagrangian matter
Lm (φ, ∇φ) takes the following form in five dimensions:
Lm (φ, ∇φ) = b0φ15 + b1φ7 Sμν μν + b2φ−1(Sμλμλ Sνδνδ
− 4Sμλνλ Sνδμδ + Sμν λδ Sνμλδ),
Sμν λδ = φ2 Rμν λδ − 12δ[[μλδνδ]]∇ρ φ∇ρ φ
− 48φδ[[μ∇ν]∇δ]φ + 18δ[[μλ∇ν]φ∇δ]φ
λ
has been shown [31–34] to transform covariantly, Sμν λδ →
−8/3 Sμν λδ, under the Weyl transformation, gμν → 2gμν
and φ → −1/3φ.
In fact, one can see that the above model is the most general
scalar field/gravity coupling formulation whose field
equations are of second order for both gravity and matter. Hence,
we can say that this formulation is a generalization of the
Horndeski theory [35] whose action contains a non-minimal
kinetic coupling of a massless real scalar field and the
Einstein tensor. Even more importantly, being a simple and
tractable model, it provides a significant advantage for
studying the phase transition of hairy black holes on the AdS
spacetime where the back-reaction of the scalar field on the metric
can be solved analytically in five dimensions. In the paradigm
where there exists a complete physical analogy between the
four-dimensional Reissner–Nordström AdS black hole and
the real van der Waals fluid [20] in the phase transition, the
recent research [29] shows that this charged AdS hairy black
hole also exhibits the van der Waals-type thermodynamic
behavior; moreover, such a black hole undergoes a reentrant
phase transition which usually occurs in higher curvature
gravity theory. We note that all the interesting results just
mentioned are available by making the coupling parameter
dynamical, i.e. treating the coupling parameter as a certain
thermodynamic variable.
Based on the results pointed out above, we take advantage
of the well-established Maxwell equal area law [36–41] to
make a further investigation of the van der Waals-type phase
transition, of the co-existence curve, and of the P−V critical
phenomenon for this charged AdS hairy black hole. We give
analytically the critical values of the charged hairy black hole
and show in detail the behavior of the phase transition from a
small black hole to a large one. We also give the distribution
of the critical points in the parameter space of q (the
coupling constant of the scalar field) and e (the electric charge
of the black hole), and we obtain for the uncharged case the
fitting formula of the co-existence curve. More importantly,
we analyze the validity and invalidity of the Maxwell equal
area law for the five-dimensional charged AdS hairy black
hole and determine the conditions for the law to hold.
Meanwhile, the latent heat is calculated, which gives the energy
released or absorbed between the small and large black hole
phases in the isothermal–isobaric procedure.
The paper is organized as follows. In Sect. 2, we review
the analytic solution of the charged AdS hairy black hole
in D = 5 dimensions and some relevant thermodynamic
quantities. In Sect. 3, we calculate the P−V critical
values and investigate the Maxwell equal area law for this
fivedimensional charged AdS hairy black hole. This section
contains two subsections, which correspond to the scenarios
without and with charges, respectively. Finally, Sect. 4 is
devoted to drawing our conclusion.
2 Analytic solution in D = 5 dimensions
The model described by Eqs. (1.1) and (1.2) admits [29,32]
an exactly electrically charged solution in five dimensions,
ds2 = − f dt 2 +
where the function f takes the form
m q e2 r 2
f (r ) = k − r 2 − r 3 + r 4 + l2 ,
d 23(k) is the metric of the three-dimensional surface with a
constant curvature, the curvature is positive for k = 1, zero
for k = 0, and negative for k = −1, and m and e are two
integration constants corresponding to the mass and the electric
charge of the black hole, respectively. Here the parameter l
represents the curvature radius of the AdS spacetime, which
is associated with the cosmological constant , whose role
is analogous to the thermodynamic pressure,
Moreover, the parameter q is characterized as the coupling
constant of the scalar field,
q =
where ε = −1, 0, 1 and there exists an additional constraint:
10b0b2 = 9b12, to ensure the existence of this black hole
solution. These conditions imply that q only takes values
0, ±|q|. Meanwhile, the scalar field configuration takes the
form
and the Maxwell gauge potential reads
A =
√3e
∂μ Aν − ∂ν Aμ.
The location of this hairy black hole horizon is denoted
by rh , which is taken to be the largest real positive root of
f (r ) = 0. That is, the horizon radius rh satisfies the following
polynomial equation [32]:
rh + kl2rh4 − ml2rh2 − ql2rh + e2l2 = 0.
6
For the horizon thermodynamic properties of this hairy
black hole, some thermodynamic quantities have been
calculated in Refs. [29,32] and are listed below for later use.
The thermodynamic enthalpy M , temperature T , entropy S,
and charge Q take the following forms:
S =
Q = −
where ω3(k) denotes the area of the compact
three-dimensional manifold with the metric d 32(k). In order to develop
the first law and the Smarr relation, the coupling parameter
is dealt with [29] as a dynamical variable, which means that
q is extended to be a continuous and real parameter. Thus, q
should appear in the Smarr relation and its variation should
be included in the first law of thermodynamics to make the
first law of black hole thermodynamics be consistent with
the Smarr relation.
Overall, the extended first law of thermodynamics can be
written in terms of the thermodynamic quantities mentioned
above as follows:
dM = T dS + V d P +
where the thermodynamic volume V , the electric potential
, and the extensive variable K conjugate to the coupling
parameter q have the forms
V ≡
K ≡
2M = 3T S − 2 P V + 2 Q + 3q K .
Meanwhile, the Gibbs free energy and the equation of state
for this hairy black hole can be written in terms of Eqs. (2.8)–
(2.10) as follows:
G ≡ M − T S
5k − 4 q
3T 3k 3q 3e2
P(rh , T ) = 4rh − 8πrh2 − 16πrh5 + 8πrh6
Next, we shall discuss the Maxwell equal area law for
this hairy black hole in order to investigate its critical
behavior and the co-existence curve of two phases resorting to
Eqs. (2.13) and (2.18) in the ( P, V ) plane. Note that, in the
planar case, i.e. k = 0, according to Eq. (2.4), we see q = 0.
It implies that there are no hairs. In addition, for the
hyperbolic case, i.e. k = −1, there are no physically critical values
as pointed out in Ref. [29]. Hence, we shall focus only on the
spherical case, i.e. k = 1, in the following context.
3 Maxwell equal area law
The thermodynamic behavior of black holes in the AdS
background is analogous to that of the real van der Waals fluid.
As was known, the critical behavior of the van der Waals
fluid occurs at the critical isotherm T = Tc when the P−V
diagram has an inflection point,
∂ P
∂rh
= 0,
When T < Tc, there is an oscillating part in the P−V
diagram. We have to replace this oscillating part by an isobar
in order to describe it in such a way that the areas above and
below the isobar are equal to each other. This treatment is
based on the Maxwell equal area law. Thus, by making an
analogy between the black hole in the AdS background and
the real van der Waals fluid, we find that there also exists an
oscillating part below the critical temperature Tc in the P−V
diagram, which indicates that the first order phase transition
occurs from a small black hole to a large one. This isobar,
which satisfies the Maxwell equal area law, represents the
co-existence curve of small and large black holes [41].
Normally, the Maxwell equal area law is constructed in
the ( P, V ) plane for a constant temperature, and it can also
be made in the (T , S) or ( , Q) plane. These constructions
are equivalent. Theoretically, the law can be established from
the variation of the Gibbs free energy defined by Eq. (2.17),
dG = dM − T dS − SdT .
On resorting to the first law of thermodynamics (2.12) and
keeping in mind that the co-existing phases have the same
Gibbs free energy, one thus arrives at the Maxwell equal area
law in the ( P, V ) plane by integrating Eq. (3.2) at constant
T , Q, and q,
P(r1, T ) = P(r2, T ) = P∗,
r2
P∗ · (V2 − V1) =
P(rh , T ) dV ,
where P∗ stands for an isobar, and V1 and V2 denote the
thermodynamic volume defined by Eq. (2.13) for the small
and large black holes with the horizon radii r1 and r2,
respectively. Thanks to the Maxwell equal area law Eq. (3.3), we
also obtain the latent heat which represents the amount of
energy released or absorbed from one phase to the other in
the isothermal–isobaric condition,
L = T [S2 − S1],
where S1 and S2 denote the entropy defined by Eq. (2.10) for
the small and large black holes with the horizon radii r1 and
r2, respectively. It is worth mentioning that the co-existence
curve of the two phases, i.e. the small and large black holes
that are described by an isobar in the Maxwell equal area law,
is governed by the Clausius–Clapeyron equation,
In the following we specialize to the Maxwell equal area law,
the critical values, and the co-existing phases for this AdS
hairy black hole.
3.1 Uncharged case: e = 0
For this hairy black hole with the spherical symmetry in the
absence of a Maxwell field, according to Eqs. (2.18) and
(2.17), we see that the equation of state reduces to
and that the Gibbs free energy becomes1
3.1.1 Critical values The critical values defined by Eq. (3.1) have been obtained in Ref. [29],
rc = (−5q)1/3,
(−5q)2/3
− q
which exist only for q < 0, and the relevant ratio reads
We observe that this ratio is a constant that does not depend
on the parameter q, and that it is different from that of the
real van der Waals fluid or the five-dimensional Reissner–
Nordström AdS black hole [20].
3.1.2 Maxwell equal area law
Inserting Eqs. (3.6) and (2.13) into Eq. (3.3), we give2 the
Maxwell equal area law,
2r13r23(r1 + r2) + q[(r1 + r2)4 + 4r12r22] = 0,
4π T r13r23(r12 + r1r2 + r2 ) + 3q[(r1 + r2)4 − r1 r2 ] = 0.
2 2 2
When taking the critical limit r1 = r2 = rc, we see that
Eq. (3.10) turns back to Eq. (3.8). Moreover, with the help of
Eqs. (2.10) and (3.10) we obtain the latent heat L between
the small and large black hole phases,
L =
(r23 − r13).
2
In order to highlight the outstanding thermodynamic
properties of the uncharged case, the numerical calculations
as regards the Maxwell equal area law, Eq. (3.10), in the
( P, V ) plane, are displayed in Table 1, meanwhile the P−V
critical behavior and the Gibbs free energy described by
Eqs. (3.6), (3.7), and (2.13) are portrayed in Fig. 1 for the
specific value of q = −2. From Table 1, we see clearly that
the horizon radii of the small and large black holes, r1 and r2,
shrink into the critical horizon radius rc = 2.15443 when the
1 In the spherical case, i.e. k = 1, the area of a compact three
dimensional manifold ω3(k) equals 2π2.
2 For the details of the derivation, see Appendix A.
Table 1 The numerical results of the Maxwell equal area law Eqs. (3.3)
and (3.10), the Gibbs free energy Eq. (3.7), and the latent heat Eq. (3.11)
for the hairy black hole with the spherical symmetry in the absence of
a Maxwell field
P∗
isotherm condition T = Tc = 0.11081 is taken. It implies
that no phase transitions occur and correspondingly the latent
heat L is of course equal to zero. At this moment, the isobar
P∗ becomes the critical pressure Pc = 0.01543. With
gradually decreasing of the temperature T from the critical value
Tc = 0.11081, it is clear that r1 decreases while r2 increases,
and the Maxwell equal area law is always valid. Furthermore,
the small black hole with the radius r1 and the large one with
the radius r2 have the same Gibbs free energy, where the
values of the Gibbs free energy are listed in the fifth column
of Table 1. Quantitatively, the latent heat L between the two
phases takes a sharp increase with the temperature
decreasing. Qualitatively, our analysis is given below. As we have
known, r1 decreases while r2 increases when T decreases.
Combining Eq. (3.10) with Eq. (3.11), we get L ∼ 34π r22
when T is decreasing. Due to r2 increasing, the latent heat L
increases in the quadratic form of r2. As a result, the
qualitative analysis coincides with the quantitative one shown in
Table 1.
Next, we take a close look at Fig. 1 which portrays the
critical behavior of the uncharged case. The left diagram
demonstrates the representative P−V critical curves. When
the temperature T exceeds the critical temperature Tc, see
the black curve, there is no criticality at all. The middle
diagram illustrates the typical equal area law, where we indeed
observe a van der Waals-type oscillation shown in the red
curve. By analogy with the real van der Waals fluid, this
oscillating part must be replaced by an isobar in the P−V
diagram. Adopting the Maxwell equal area law, i.e., the areas
of the regions surrounded by an oscillation (red dashed curve)
and an isobar (black solid line) are equal to each other, we
find this isobar P∗ and correspondingly the thermodynamic
volumes V1 and V2. In other words, the phase transition from
the small black hole to the large one occurs in this
situation, and these two types of black holes have the same Gibbs
free energy, displayed by the crossing point A in the right
diagram.
With the aid of the Clausius–Clapeyron equation Eq. (3.5)
and the Maxwell equal area law Eq. (3.3), we obtain the
co-existence curve of two phases. Considering that the
coexistence curve for the real van der Waals fluid has a positive
slope everywhere and terminates at the critical point, we try
to fit it using a polynomial. Thanks to the simple forms of
the critical point Eq. (3.8), such a treatment can be realized.
By introducing the reduced parameters,
we eventually obtain the parametrization form of the
coexistence curve,
p = 0.828355t 2 + 0.096202t 3 − 0.558203t 4
+ 2.43626t 5 − 4.74242t 6 + 5.05509t 7 − 2.53495t 8
+ 0.241581t 9 + 0.178083t 10, (3.13)
where t ∈ (0, 1). Then we plot the numerical values
governed by the Clausius–Clapeyron equation Eq. (3.5) and the
Maxwell equal area law Eq. (3.3), and also plot the fitting
formula Eq. (3.13) of the co-existence curve of the small
and large black holes in Fig. 2. We can see clearly that the
numerical values and the fitting formula match well with
Fig. 1 P−V diagrams described by Eqs. (3.6) and (2.13) and P−G
diagram described by Eq. (3.7) at q = −2. Left The temperature
takes the values of T = 0.12000 (black), T = Tc = 0.11081
(blue dotted), T = 0.09500 (orange), T = 0.07520 (purple), and
T = 0.05500 (green), respectively. Middle The temperature takes the
value of T = 0.11000, and for the isobar P∗ = 0.01515, V1 and V2
correspond to r1 = 1.94812 and r2 = 2.40186, respectively.
Corresponding to the middle diagram, the Gibbs free energy is depicted in
the right diagram, showing the characteristic swallowtail behavior
Fig. 2 Left The co-existence curve for the five-dimensional AdS hairy
black hole without charges, the blue points for the numerical values, and
the red solid curve for the fitting formula. Right The co-existence curve
in the parameter q space. Each blue dotted curve indicates a co-existence
curve at a fixed q and the red curve at the boundary corresponds to the
critical points
each other. Both of them terminate at the critical point, i.e.
the point C (1, 1). With the increasing of q, the critical
values, see Eq. (3.8), take an increasing trend shown in the right
diagram of Fig. 2.
At last, we make a brief summary of this subsection. For
the AdS hairy black hole with the spherical symmetry in
the absence of a Maxwell field, the P −V critical behavior is
available only under the condition of q < 0, and the Maxwell
equal area law always holds under this condition. Meanwhile,
we take the specific value of q = −2 as an example to
highlight the salient features of this black hole in Table 1 and
Fig. 1. Furthermore, because the critical point Eq. (3.8) has
a simple form, we obtain the fitting formula (3.13) of the
co-existence curve; see Fig. 2 for the illustration. As to the
charged case, it shows complications, as exhibited in the
following subsection.
3.2 Charged case: e = 0
For the spherically symmetric AdS hairy black hole with the
Maxwell field, the equation of state takes the form
and the Gibbs free energy reads
3.2.1 Critical values
Substituting Eq. (3.14) into Eq. (3.1), we obtain the critical
radius, which satisfies the following equation:
rc4 + 5qrc − 15e2 = 0,
and derive the critical temperature,
Tc =
Pc =
and the critical pressure,
Then, using Eqs. (3.16)–(3.18), we give the ratio of three
critical values,
which leads us back to Eq. (3.9), i.e. the uncharged case, when
e = 0. Moreover, if q = 0, this ratio equals 5/16, which
coincides with the ratio of the five-dimensional Reissner–
Nordström AdS black hole [20]. Nonetheless, this ratio
depends in general on the values of the parameters e and q.
4
Let us try to solve Eq. (3.16). At first, we set w(rc) ≡ rc +
5qrc − 15e2 and take a look at its asymptotic behavior. When
rc → 0, we get a negative value, w(rc) → −15e2. When
rc → +∞, we see w(rc) → +∞. In addition, we notice
that for q > 0 the function w(rc) is monotone increasing at
the interval [0, +∞), and for q < 0 it is monotone increasing
at the interval [(−5q/4)1/3, +∞) but monotone decreasing
at the interval [0, (−5q/4)1/3]. Hence, we conclude that the
equation w(rc) = 0 must have one and only one positive
root,3
− tc
√tc ,
where the plus sign corresponds to the case of q < 0 and the
minus sign to the case of q > 0, and the newly introduced
parameter tc is defined as
tc ≡ 4
√5e sinh
sinh−1
√5q2
3 For the details of the derivation; see Appendix B.
3.2.2 Maxwell equal area law
Substituting Eqs. (3.14) and (2.13) into Eq. (3.3), we give4
the Maxwell equal area law,
2r14r24(r1 + r2) + qr1r2[(r1 + r2)4 + 4r12r22]
− 2e2[(r1 + r2)5 − r1r2(r13 + r2 )] = 0,
3
4π T r14r24(r12 + r1r2 + r2 ) + 3qr1r2[(r1 + r2)4 − r1 r2 ]
2 2 2
− 6e2(r1 + r2)3(r12 + r1r2 + r2 ) = 0.
2
If e = 0, i.e. for the uncharged case, the above set of equations
turns back to Eq. (3.10). When taking the critical limit, r1 =
r2 = rc, we see that Eq. (3.21) reduces to Eqs. (3.16) and
(3.17), that is, it coincides with the critical behavior. It is
remarkable for the charged case that the entropy defined by
Eq. (2.10) probably becomes negative in the case of q > 0,
which brings about an extra constraint on the P−V criticality
and the equal area law.
3.2.3 The case of q < 0
In this situation, the entropy of the hairy black hole with
the Maxwell field, Eq. (2.10), is always positive. There
are no additional restrictive conditions for investigating the
Maxwell equal area law and the phase transition. The
treatment is the same as that of the scenario without charges. We
can directly write down the condition under which the van
der Waals-type phase transition exists and the Maxwell equal
area law holds, i.e., the temperature takes the values T < Tc,
where Tc is given by Eq. (3.17) together with the constraint
q < 0.
Table 2 displays the numerical results of the Maxwell
equal area law (Eq. (3.21)), and Fig. 3 depicts the critical
behavior of the equation of state (Eqs. (3.14) and (2.13))
and the Gibbs free energy (Eq. (3.15)) for an example of the
charged hairy black hole at q = −2 and e = 2. We can see a
similar critical behavior to that of the uncharged case. When
T > Tc, there is no criticality; when T < Tc, there exists
a representative van der Waals-type oscillation in the P−V
diagram. By using the Maxwell equal area law and
replacing the oscillating part by an isobar, we observe the phase
transition from the small black hole to the large one in the
middle diagram of Fig. 3, and find that the two phases have
the same Gibbs free energy displayed by the crossing point
A in the right diagram. Moreover, when T is decreasing, the
horizon radius of the small black hole, r1, decreases while
that of the large black hole, r2, increases, and the latent heat L
between the two phases presents a sharp increasing tendency.
This observation means that the phase transition needs more
energy at the lower temperature of the isothermal–isobaric
procedure.
4 For the details of the calculations, see Appendix A.
Table 2 The numerical results of the Maxwell equal area law Eqs. (3.3)
and (3.21), the Gibbs free energy Eq. (3.15), and the latent heat Eq. (3.4)
for the spherically symmetric AdS hairy black hole with the Maxwell
field
P∗
3.2.4 The case of q > 0
In the situation of a positive parameter q, the analysis
becomes complicated. Due to the entropy described by
Eq. (2.10) being probably negative, we have to impose an
extra constraint to the horizon radius,
rh ≥ rs ≡
The purpose is to avoid the negative entropy that appears
if rh < rs , where the negative entropy is regarded as an
unphysical variable at present.
The Maxwell equal area law holds under the condition
T < Tc together with the constraint Eq. (3.22), where Tc is
given by Eq. (3.17) under the condition q > 0.
Now we consider an extreme situation, i.e., rc = rs , which
leads to the critical value of the electric charge,
|es | =
This critical value (rs or es ) gives the boundary of violating
or maintaining the P−V criticality and the Maxwell equal
area law. That is, if |e| < |es |, we have rc < rs ,
resulting in the violation of the P−V criticality (Eqs. (3.16)–
(3.18)) and of the Maxwell equal area law (Eq. (3.21)); if
|e| > |es |, we have rc > rs , maintaining the validity of
the P−V critical values, but we have to add the condition
r1 > rs in order to establish the Maxwell equal area law.
A sample of the latter situation is depicted in Fig. 4 for the
specific fixing of q = 2 and e = 1.5. In this sample, we
have e > es = 1.30766 and figure out rs = 1.70998 and
rc = 1.94471. Hence, the validity of the Maxwell equal area
law depends on the temperature with the range of T < Tc,
where Tc = 0.14207. The left two diagrams of Fig. 4
correspond to T = 0.14000 < Tc = 0.14207, which leads to
r1 > rs and illustrates the equal area law and the
characterisFig. 3 P−V diagrams described by Eqs. (3.14) and (2.13) at q = −2
and e = 2. Left The temperature takes the values of T = 0.08800
(black), T = Tc = 0.08072 (blue dotted), T = 0.06650 (orange),
T = 0.05364 (purple dashed), and T = 0.04800 (green), respectively.
Middle The temperature takes the value of T = 0.07520, and for the
isobar P∗ = 0.00681, V1 and V2 correspond to r1 = 2.38204 and
r2 = 4.32768, respectively. Corresponding to the middle diagram, the
Gibbs free energy described by Eq. (3.15) is depicted in the right
diagram, showing the characteristic swallowtail behavior
Fig. 4 P−V diagrams (Eqs. (3.14) and (2.13)) and the corresponding
Gibbs free energy (Eq. (3.15)) at q = 2 and e = 1.5. Left The
temperature takes the value of T = 0.14000. For the isobar P∗ = 0.02311,
Vs , V1, and V2 correspond to rs = 1.70998, r1 = 1.73921, and
r2 = 2.20835, respectively. Note that r1 > rs . Right The temperature
takes the value of T = 0.13800. For the isobar P∗ = 0.02237, Vs , V1,
and V2 correspond to rs = 1.70998, r1 = 1.66931, and r2 = 2.33574,
respectively. Note that r1 < rs
tic swallowtail behavior. The red dashed curve corresponds to
a negative entropy, which is unphysical, but it does not affect
the application for the equal area law and the characteristic
swallowtail structure owing to r1 > rs . On the contrary, if
r1 < rs , see the right two diagrams of Fig. 4, the red dashed
curve corresponding to the negative entropy indeed violates
the equal area law and the characteristic swallowtail
structure. Let us take a close look at this kind of violation. We can
still determine the isobar P ∗ because the initial point at r1
(or V1) and the terminal one at r2 (or V2) are independent of
Eq. (3.22). However, when T = 0.13800, that brings about
r1 < rs , i.e. the breaking of Eq. (3.22), and we find that the
branch of negative entropy terminates at rs (or Vs ), which
is larger than r1, which evidently leads to a violation of the
Maxwell equal area law and the characteristic swallowtail
structure.
In addition, we take a look at the co-existence curve
governed by the Clausius–Clapeyron equation, Eq. (3.5), and the
Maxwell equal area law, Eq. (3.21), for the five-dimensional
AdS hairy black hole with charges. As the formulas of the
critical values, Eqs. (3.17) and (3.18), are much more
complicated than Eq. (3.8), it is hard to give a perfect fitting
formula of the co-existence curve of the small and large black
holes. Nonetheless, we know through the above analysis that
the thermodynamic behavior for the case with charges is the
same as that of the case without charges within the scope
of validity of the Maxwell equal area law, and that the
coexistence curve always has a positive slope and terminates at
Fig. 5 The distribution of the critical points described by Eqs. (3.17)
and (3.18) for different values of e and q. The left diagram corresponds
to e = 1.90, 2.05, 2.20, and 2.35, respectively, from top to bottom, the
green dotted curve for q < 0, the purple dotted curve for q > 0, and the
red dots for q = 0. The middle diagram depicts the case of q ≥ 0 with
discrete values of q = 0 (red), 0.3 (green), 0.6 (blue), and 1.2 (purple),
and the right diagram depicts the case of q ≤ 0 with discrete values of
q = 0 (red), −0.2 (green), −0.3 (blue), and −0.5 (purple)
Fig. 6 The Gibbs free energy (Eq. (3.15)) at q = 2, e = 1.5, and
T = 0.13960. P∗ corresponds to the pressure of the intersection (point
d), Pt corresponds to the pressure of the cusp (point a), and Ps
corresponds to the pressure of point c with zero entropy. The red dashed
curve corresponds to the negative entropy. A black arrow indicates an
increasing trend of the horizon radius rh
the critical point. Figure 5 shows the critical points for
different values of e and q, from which we see that the critical
values Tc and Pc show an increasing trend with an increasing
q but a fixed e, while they show a decreasing trend with an
increasing e but a fixed q.
Before the end of this section, let us take a look at the
physical meaning of the appearance of a negative entropy in
the charged case with positive q. We illustrate this situation
by setting q = 2 and e = 1.5, where the behavior of the
Gibbs free energy (Eq. (3.15)) can be classified into three
types, which are shown in Fig. 6 and the lower-left and
lowerright diagrams of Fig. 4. We know that the stable black hole is
thermodynamically favorable to the lower Gibbs free energy.
• If P ∗ < Ps < Pt , see Fig. 6, one can see that the global
minimum of the Gibbs free energy emerges as a
discontinuous characteristic due to the negative entropy (see the
red dashed curve). More precisely, when Ps < P < P ,
t
the global minimum is located on the curve a → b. When
P ∗ < P < Ps , the global minimum is located on the
curve c → d. When P < P ∗, the global minimum is
located on the curve d → e. At P = Ps , the Gibbs
free energy has a discontinuous global minimum, which
means the occurrence of the zeroth order phase transition
as reported in Ref. [29]. At P = P ∗, the global minimum
of the Gibbs free energy has an inflection point, which
implies a standard first order small/large black hole phase
transition, i.e. a van der Waals-type phase transition, and
the Maxwell equal area law holds.
• If Ps > Pt , see the lower-left diagram of Fig. 4, the
global minimum of the Gibbs free energy is continuous
and also has an inflection point (corresponding to the
pressure P ∗), indicating the existence of a van der
Waalstype phase transition.
• If Ps < P ∗, see the lower-right diagram of Fig. 4, the
global minimum of the Gibbs free energy is continuous
but has no inflection points, which indicates
nonexistence of phase transitions and results in invalidity of the
Maxwell equal area law.
All in all, it is the positivity/negativity of the entropy that
affects the behavior of the global minimum of the Gibbs free
energy. The discontinuity of the Gibbs free energy causes a
zeroth order phase transition at the pressure Ps ∈ [ P ∗, Pt ]
under a certain temperature. Moreover, if Ps < P ∗, the
disappearance of inflection points leads to the failure of the
Maxwell equal area law, so that no real physical isobar exists
for phase transitions, as shown in the upper-right diagram of
Fig. 4.
4 Conclusion
In this paper, we investigate the P −V criticality, the Maxwell
equal area law, and the co-existence curve for the spherically
symmetric AdS black hole with a scalar hair [29, 32] both in
the absence of and in the presence of a Maxwell field,
respectively. Especially in the charged case, we give the exactly
analytical P −V critical values; see Eqs. (3.16)–(3.18), and
(3.20). Meanwhile, we provide the conditions of validity of
the Maxwell equal area law for the hairy black hole without
Acknowledgements This work was supported in part by the National
Natural Science Foundation of China under Grant No. 11675081.
Finally, the authors would like to thank the anonymous referee for the
helpful comment that indeed greatly improved this work.
Open Access This article is distributed under the terms of the Creative
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ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Derivation of Eqs. (3.10) and (3.21)
Inserting Eqs. (2.18) and (2.13) into Eq. (3.3), we obtain the
following three equations:
and with charges, respectively. Our results can be
summarized as follows:
– Scenario without charges: the Maxwell equal area
law holds under the conditions T < Tc and Eq. (3.8).
– Scenario with charges: the Maxwell equal area law
holds under the conditions T < Tc and Eq. (3.17).
– Scenario without charges: the Maxwell equal area
law is violated.
– Scenario with charges: besides the conditions T <
Tc and Eq. (3.17) together with the extra constraint
Eq. (3.22), whether the Maxwell equal area law holds
or not depends on the charge and the relation between
the temperature and the horizon radius, which can be
classified into the following two situations:
– ∗ when |e| < |es |, the law does not hold.
– ∗ when |e| > |es |, the law holds if r1 > rs , but fails
if r1 < rs , where r1 depends on the temperature T
shown in Fig. 4.
Within the scope in which the Maxwell equal area law
holds, we point out that there exists a representative van der
Waals-type oscillation in the P −V diagram. This oscillating
part that indicates the phase transition from a small black
hole to a large one can be replaced by an isobar and the
small and large black holes have the same Gibbs free energy.
These salient features have been illustrated in Tables 1 and 2,
and Figs. 1 and 3, from which we conclude that when the
temperature T is decreasing, the horizon radius of the small
black hole r1 decreases, while that of the large black hole
r2 increases; moreover, the latent heat L between the two
phases presents a sharp increasing tendency. This
observation means that the phase transition needs more energy
at the lower temperature of the isothermal–isobaric
procedure. Furthermore, for the uncharged case we obtain the
fitting formula (3.13) of the co-existence curve depicted in
Fig. 2 due to the simple form of the critical point
equation (3.8). For the charged case, we give the distribution
of the critical points described by Eqs. (3.17) and (3.18)
in the parameter space of q and e in Fig. 5. Finally, we
point out that the positivity/negativity of the entropy has
effect on the global minimum of the Gibbs free energy.
The inflection point of the Gibbs free energy leads to the
van der Waals-type phase transition, but the Maxwell equal
area law does not hold and the real physical isobar does
not exist for phase transitions if no inflection points exist
as shown in Fig. 4. The discontinuity of the Gibbs free
energy causes a zeroth order phase transition as shown in
Fig. 6.
Combining Eq. (A1) with Eq. (A2), we have
4π T r15r25(r2 − r1) + 2e2(r26 − r1 ) − qr1r2(r25 − r1 )
6 5
− 2kr14r24(r22 − r1 ) = 0,
2
and inserting Eqs. (A1) and (A2) into Eq. (A3) yields
4π T r12r22(r23 − r1 ) + 18e2(r22 − r1 ) − 15qr1r2(r2 − r1)
3 2
− 6kr12r22(r22 − r1 ) = 0.
2
Comparing Eq. (A4) with Eq. (A5) and eliminating the
parameter T , we obtain
2kr14r24(r1 + r2) + qr1r2[(r1 + r2)4 + 4r12r22]
− 2e2[(r1 + r2)5 − r1r2(r13 + r23)] = 0.
Substituting Eq. (A6) into Eq. (A5) and eliminating the term
containing the parameter k, we get
4π T r14r24(r12 + r1r2 + r22) + 3qr1r2[(r1 + r2)4 − r12r22]
− 6e2(r1 + r2)3(r12 + r1r2 + r2 ) = 0.
2
For the case of k = 1, Eqs. (A6) and (A7) turn out to be
Eq. (3.21), and Eq. (3.21) reduces to Eq. (3.10) when e = 0
is taken.
Appendix B: Root of Eq. (3.16)
x 4 + bx − c
2 = (x 2 + γ x + α)(x 2 − γ x + β),
where α, β, and γ are parameters to be determined. At first,
we write the four roots of the above quartic equation by
solving the two quadratic equations separated from the quartic
one,
x1,2 =
x3,4 =
Next, expanding the right hand side of Eq. (B1) and
comparing the terms with the same power to x , we have
Solving Eqs. (B3) and (B4), we first determine two of the
three parameters, α and β,
1
β = 2
In order to fix the parameter γ , we then insert Eq. (B6) into
Eq. (B5) and introduce the new parameter y ≡ γ 2, which
can change the sixth-order equation with respect to γ into a
cubic equation with respect to the new parameter y,
y3 + 4c2 y − b2
= 0.
For this kind of cubic equations, there is only one real root
that is given by a hyperbolic form of Viète’s solution,
y =
sinh−1
3√3b2
which actually determines the last parameter γ . As a result,
Eqs. (B2), (B6), and (B8) give the exactly analytical roots of
the original quartic equation.
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