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

The European Physical Journal C, Jun 2017

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.

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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 . . . . . . . . . . . . . - 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)


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Yan-Gang Miao, Zhen-Ming Xu. Validity of Maxwell equal area law for black holes conformally coupled to scalar fields in \(\text {AdS}_5\) spacetime, The European Physical Journal C, 2017, pp. 403, Volume 77, Issue 6, DOI: 10.1140/epjc/s10052-017-4978-3