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 . . . . . . . . . . . . .
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Contents
4 Conclusion . . . . . . . . . . . . . . . . . . . . . . 9
Appendix A: Derivation of Eqs. (3.10) and
(3.21) . . . . . . . . . . . . . . . . . . . . . . . . . 10
Appendix B: Root of Eq. (3.16) . . . . . . . . . . . . . 11
References . . . . . . . . . . . . . . . . . . . . . . . . 11
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) ,
(1.1)
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),
(1.2)
where b0, b1, and b2 are coupling constants and the four-rank
tensor
Sμν λδ = φ2 Rμν λδ − 12δ[[μλδνδ]]∇ρ φ∇ρ φ
− 48φδ[[μ∇ν]∇δ]φ + 18δ[[μλ∇ν]φ∇δ]φ
λ
(1.3)
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 c (...truncated)