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Reentrant phase transitions and van der Waals behaviour for hairy black holes

Robie A. Hennigar, Robert B. Mann

arXiv:1509.06798v1hep-thgr-qc

TL;DR

The paper examines the previously unexplored extended phase-space thermodynamics of five-dimensional hairy AdS black holes. It finds van der Waals behavior in spherical cases with and without charge, and charged reentrant transitions, the first such example in five-dimensional gravity without higher-curvature corrections.

  • Problem

    Extended phase-space thermodynamics had not yet been explored for these five-dimensional hairy AdS black holes.

    Source: §I · p. 1

  • Method

    The paper studies extended phase-space thermodynamics for five-dimensional Einstein-Maxwell-Λ black holes conformally coupled to a scalar field, considering spherical, planar, and hyperbolic horizons.

    Source: §I · p. 1, §IV · p. 8

  • Results

    Spherical solutions show van der Waals behavior in charged and uncharged cases, while charged cases include reentrant transitions.

    Source: §2 · p. 5, §IV · p. 8

  • Takeaways & Limitations

    This is the first example of a reentrant phase transition in a five-dimensional gravitational system without higher-curvature corrections.

    Source: §IV · p. 8

Abstract

from arXiv · show

We study the extended phase space thermodynamics for hairy AdS black hole solutions to Einstein-Maxwell-$\Lambda$ theory conformally coupled to a scalar field in five dimensions. We find these solutions to exhibit van der Waals behaviour in both the charged/uncharged cases, and reentrant phase transitions in the charged case. This is the first example of reentrant phase transitions in a five dimensional gravitational system which does not include higher curvature corrections.

I. INTRODUCTION

This paper begins the study of extended phase-space thermodynamics for five-dimensional hairy AdS black holes with a conformally coupled scalar. It finds van der Waals behavior and charged reentrant transitions in the spherical case, extending phase-structure results beyond previously studied settings.

I. INTRODUCTION

  • Extended phase space treats the cosmological constant as pressure and identifies black-hole mass with enthalpy, linking black-hole thermodynamics to ordinary thermodynamic systems.Earlier work found a complete analogy between four-dimensional Reissner–Nordström–AdS critical behavior and the van der Waals system.Source: §I · p. 1
  • The broader literature had found reentrant transitions for six-dimensional doubly rotating Kerr-AdS black holes and more exotic behavior in higher-curvature gravity.Other applications include thermodynamically inspired black holes, entropy inequalities, and holographic heat engines.Source: §I · p. 1
  • The paper first explores extended phase-space thermodynamics for five-dimensional hairy AdS black holes in Einstein-Maxwell-Λ theory with a conformally coupled scalar.The model includes the scalar field’s back-reaction on the metric.Source: §I · p. 1
  • Spherical solutions exhibit van der Waals behavior and can undergo reentrant phase transitions; these are the first five-dimensional examples outside higher-curvature gravity.The flat and hyperbolic cases yield no interesting results.Source: §I · p. 1

II. SOLUTION & THERMODYNAMICS

The paper studies five-dimensional hairy black-hole solutions and their thermodynamics across horizon curvatures. It defines the thermodynamic variables, imposes physicality conditions, and uses a specific-volume choice to analyze phase behavior.

II. SOLUTION & THERMODYNAMICS

  • The paper presents the theory’s action and five-dimensional solution, with the Maxwell field strength defined in the standard way and the scalar configuration specified for k = ±1.The scalar couplings constrain the solution parameters, while q can take discrete values.Source: §II · p. 2
  • The planar solutions have no hair, so the analysis considers only spherical and hyperbolic horizons.The hair parameter q takes discrete values determined by the scalar couplings, and the planar case has q = 0.Source: §II · p. 2
  • The solution’s hair parameter q is not a conserved charge and is treated as continuous when the couplings are dynamical.The extended first law includes a Kdq term, and the corresponding Smarr formula is satisfied.Source: §II · p. 2
  • The thermodynamic quantities are extended to the hyperbolic case, and negative-entropy black holes are treated as unphysical.The entropy for k = −1 is checked using Wald entropy; for positive q, entropy can be negative.Source: §II · p. 2
  • The specific volume is defined as v = 4r+/3 because the usual area-based degrees-of-freedom formula is inappropriate when entropy is not one quarter of horizon area.The paper says its qualitative results do not differ from those obtained using thermodynamic volume, though some quantities can depend meaningfully on volume choice.Source: §II · p. 3

III. CRITICAL BEHAVIOUR

The criticality analysis focuses on nonzero hair and compares spherical and hyperbolic black holes. It also considers the zero-entropy limit at the critical point.

III. CRITICAL BEHAVIOUR

  • The analysis restricts attention to nonzero q, since q = 0 gives the five-dimensional Reissner–Nordström–AdS solution studied previously.The stated aim is to test whether hair changes thermodynamic behavior.Source: §III · p. 3
  • The paper treats spherical black holes with k = 1 and hyperbolic black holes with k = −1 in separate subsections.Source: §III · p. 3
  • The analysis then specializes to cases where entropy vanishes at the critical point.Source: §III · p. 3

1. Uncharged case

For uncharged spherical black holes, negative hair yields a critical point with standard mean-field exponents and van der Waals behavior. Positive hair has no interesting uncharged phase-transition behavior under the entropy-positivity condition.

1. Uncharged case

  • For q < 0, the uncharged equation of state has a single critical point, characterized by standard mean-field theory exponents.The ratio of critical values differs from both the five-dimensional Reissner–Nordström–AdS solution and the van der Waals fluid.Source: §1 · p. 3
  • For q < 0, the Gibbs free energy has a swallowtail below critical pressure, while the P–v curves show van der Waals oscillation below critical temperature.The coexistence line separates small and large black-hole phases and terminates at the critical point.Source: §1 · pp. 3–4, Figure 1 · p. 4
  • For q > 0, the Gibbs free energy has a cusp, but entropy positivity truncates its upper branch and produces no interesting uncharged phase-transition behavior.Below the cusp, no black-hole solutions exist.Source: §1 · p. 4, Figure 2 · p. 4

2. Charged case

For spherical black holes, the charged case retains standard van der Waals behavior for q < 0 and can exhibit reentrant transitions when a positive-q critical point is physical. Positive-entropy constraints determine which critical points and Gibbs-free-energy branches are physically admissible.

2. Charged case

  • For q < 0, every charge value gives one critical point, standard van der Waals behavior, a first-order transition, and mean-field critical exponents.Entropy positivity is automatically satisfied in this case.Source: §2 · p. 5
  • For q > 0, the equation of state has one inflection point, but it is physical only when the entropy is positive; no phase transitions were found where it is unphysical.For fixed electric charge, sufficiently large q makes the would-be critical point a negative-entropy solution.Source: §2 · p. 5
  • At q = e = 1, a narrow pressure window produces a reentrant transition: increasing temperature takes the preferred state from large to small and back to large black holes.The sequence is a zeroth-order transition at t0 followed by a first-order transition at t1.Source: §2 · p. 5
  • The reentrant window ends above a pressure threshold, where the entropy-forbidden branch terminates below the Gibbs-energy cusp and only a first-order transition remains below critical pressure.As the zero-entropy boundary is approached, the pressure range for transitions narrows; at q = 1.33 and e = 1 its width is about 2×10−5.Source: §2 · p. 5

B. Hyperbolic

Hyperbolic black holes have no physical critical points in the uncharged case, and neither sign of q yields phase transitions in the cases discussed. For charged q > 0, entropy positivity also excludes the candidate critical points.

B. Hyperbolic

  • In the uncharged hyperbolic case, neither sign of q gives a physical critical point with positive critical volume, temperature, and pressure.The analysis therefore proceeds to the charged case.Source: B. Hyperbolic · p. 5
  • For q < 0, the hyperbolic equation of state has no inflection point, and no phase transitions occur.Its Gibbs free energy has only one physical branch, with entropy positivity satisfied.Source: B. Hyperbolic · pp. 5–6
  • For charged hyperbolic black holes with q > 0, entropy positivity excludes the critical points, and neither zeroth- nor first-order transitions are observed.The candidate points are unphysical under the entropy condition.Source: B. Hyperbolic · p. 7

C. Black holes with zero entropy

At zero entropy at the critical point, only one Gibbs-free-energy branch remains physical, so the system has no phase transitions. The argument uses the requirement for at least two physical branches and the entropy signs of the near-critical branches.

C. Black holes with zero entropy

IV. CONCLUSIONS

The phase structure depends on horizon topology: the spherical case has van der Waals behaviour, while the planar and hyperbolic cases yield no interesting new phase structure. In the charged spherical case, positive-entropy solutions include reentrant transitions, the first such example in five-dimensional gravity without higher-curvature corrections.

IV. CONCLUSIONS

  • The planar case has no hair, while hyperbolic entropy and critical-value constraints eliminate the physical parameter space and its phase structure.Source: §IV · p. 8
  • Spherical black holes show van der Waals behaviour in both charged and uncharged cases, with reentrant phase transitions in the charged case when entropy is positive.Source: §IV · p. 8
  • The charged spherical case provides the first example of reentrant phase transitions in five-dimensional gravity without higher-curvature corrections.Source: §IV · p. 8
  • At zero entropy, all critical behaviour ceases; this limit occurs at nonzero r+ for positive q.Source: §IV · p. 8

Appendix A: Black holes with zero or negative mass

The appendix examines zero- and negative-mass solutions and their horizon structure across horizon topologies. It identifies extremal and multiple-horizon limits, while noting that a near-horizon curvature singularity in one expansion is outside that expansion’s domain of validity.

Appendix A: Black holes with zero or negative mass

  • For spherical horizons with q > 0, massless or negative-mass black holes are possible; the negative-mass solutions have negative entropy and are excluded by the study’s positivity requirement.Source: Appendix A · p. 8
  • For spherical negative-mass solutions with nonzero electric charge and m0 ≤ 0, a minimum q0 is required for horizons to exist.Source: Appendix A · p. 8
  • For the charged solution, q > q0 gives inner and outer horizons, q = q0 is extremal, and q < q0 gives a naked singularity.Source: Appendix A · p. 8
  • For 0 > q > q0, the solution has inner and outer horizons; at q = q0 they coincide, and for q < q0 the solution has a naked singularity.Source: Appendix A · p. 8
  • For q > 0, negative-mass solutions can have up to three horizons when 0 ≥ m ≥ −9l2/20 and q− ≤ q ≤ q+; outside this range there is one horizon.Source: Appendix A · p. 9
  • A triple coincidence of horizons is treated as a triple root of the metric function, and its Kretschmann scalar is regular at the horizon.Source: Appendix A · p. 9
  • The near-horizon expansion’s curvature singularity at R = 0 is not a physical pathology because that limit lies outside the expansion’s near-horizon assumption.Source: Appendix A · p. 9
  • With nonzero electric charge, up to four horizons are possible; the appendix focuses on their coincidence, where the Kretschmann scalar is K = 300/l4.Source: Appendix A · pp. 9–10
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