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arxiv: 2412.08337 · v3 · pith:7WBTSSYAnew · submitted 2024-12-11 · ✦ hep-th

Thermodynamic Behavior of a 4D Nonminimal Maxwell-AdS Black Hole

Mehdi Sadeghi , Faramarz Rahmani This is my paper

Pith reviewed 2026-05-25 08:13 UTC · model grok-4.3

classification ✦ hep-th
keywords black hole thermodynamicsnonminimal couplingvan der Waals phase transitionHawking-Page transitionAdS black holesextended phase spacecanonical ensemblegrand canonical ensemble
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The pith

A perturbative nonminimal Ricci-Maxwell coupling produces black hole solutions whose thermodynamics follows a van der Waals equation of state in both canonical and grand canonical ensembles.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors construct a 4D black hole metric in Einstein-Maxwell theory with nonminimal coupling between the Ricci tensor and the Maxwell field strength by expanding perturbatively around an AdS background. They then examine the thermodynamics in the extended phase space, treating the cosmological constant as pressure. The resulting pressure-volume isotherms display the characteristic van der Waals loop and critical point in both the fixed-charge and fixed-potential ensembles. When the charge or Maxwell potential is small enough, the dominant transition instead becomes the Hawking-Page transition between thermal AdS space and the black hole.

Core claim

Using a perturbative expansion, the authors obtain a consistent 4D black hole metric satisfying the modified Einstein-Maxwell equations with nonminimal coupling. In the extended phase space, the thermodynamic quantities satisfy an equation of state that exhibits van der Waals-like behavior, including a first-order phase transition between small and large black holes, in both the canonical ensemble (fixed charge) and the grand canonical ensemble (fixed potential). When the charge or potential is sufficiently small, the system undergoes a Hawking-Page phase transition between thermal AdS and the black hole phase.

What carries the argument

The perturbative black hole metric obtained from the nonminimal Ricci-Maxwell coupling term, which alters the field equations and produces the observed thermodynamic equation of state in extended phase space.

If this is right

  • The black hole equation of state mimics a van der Waals fluid, so the system possesses critical points and regions of phase coexistence.
  • The same qualitative phase structure appears in both ensembles, indicating the behavior is not an artifact of ensemble choice.
  • For sufficiently small charge or potential the Hawking-Page transition replaces the van der Waals transition as the dominant feature.
  • Treating the cosmological constant as thermodynamic pressure reveals the extended phase space structure of the nonminimal theory.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the perturbative metric approximates the exact solution, nonminimal couplings of this type may produce fluid-like thermodynamics in higher dimensions or with additional matter fields.
  • The result suggests that the dual holographic fluid in the AdS/CFT correspondence would also exhibit van der Waals behavior when the bulk coupling is nonminimal.
  • Checking the thermodynamic stability and quasinormal modes of the perturbative solution beyond the working order would delimit the regime where the reported transitions remain valid.

Load-bearing premise

The perturbative expansion around the background solution yields a consistent black hole metric that satisfies the modified field equations to the working order.

What would settle it

Deriving an exact non-perturbative solution and verifying whether its pressure-volume relation still contains a van der Waals loop and critical point would directly test the claim.

Figures

Figures reproduced from arXiv: 2412.08337 by Faramarz Rahmani, Mehdi Sadeghi.

Figure 1
Figure 1. Figure 1: Profile of the metric function f(r) for different Maxwell charges Q, showing the emergence of multiple horizons. As the charge increases, the number of roots of f(r) also increases. The analysis reveals that up to three distinct roots may emerge for sufficiently large values of Q. In the absence of electromagnetic charge (Q = 0), the function f(r) admits a single real root, corresponding to a unique event … view at source ↗
Figure 2
Figure 2. Figure 2: P-rh diagrams of the system: (a) Temperature-dependent behavior with fixed charge Q = 0.5; (b) Charge-dependent behavior (Q) at fixed temperature. For both cases, λ = 0.001. The black hole exhibits Van der Waals-like phase transition in general, but for Q = 0.1, the pressure profile closely resembles that of Hawking-Page phase transitions. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: T − rh diagrams of the system: (a) when the system pressure changes, and (b) when the Maxwell charge varies. For both panels, λ = 0.001. In the right panel, for Q = 0.1, the temperature profile closely resembles that of a Hawking-Page phase transition. Phase transitions in canonical ensemble can be classified using the Helmholtz free energy by examining how the free energy and its derivatives change during… view at source ↗
Figure 4
Figure 4. Figure 4: F − T diagrams of the system: (a) with a pressure of P = 0.01 (below the critical pressure), (b) at the critical pressure Pc = 0.026, and (c) with a pressure of P = 0.06 (above the critical pressure). For all diagrams, λ = 0.001. The swallowtail shape in the left panel is characteristic of Van der Waals fluids. The middle panel shows the Helmholtz function corresponding to the critical pressure. A kink is … view at source ↗
Figure 5
Figure 5. Figure 5: S−T diagrams of the system: (a) with a pressure of P = 0.01, which is less than the critical pressure, (b) at the critical pressure of P = Pc = 0.026, and (c) with a pressure of P = 0.06, which is higher than the critical pressure. For all diagrams, λ = 0.001. A jump is seen in the left panel which is characteristic of a first order phase transions. meet each other, that corresponds to the critical point o… view at source ↗
Figure 6
Figure 6. Figure 6: F-T diagrams of the black hole: (a) Different regions of the Helmholtz free energy in a Hawking￾Page-like phase transition, (b) Changes in the pressure of the black hole with Q = 0.1, (c) Variations in Q with a fixed pressure of P = 0.08. For all panels, λ = 0.001 is used as a small perturbation coefficient. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: C − rh diagrams of the system: (a) at the critical pressure, P = Pc = 0.026, (b) at pressures less than the critical pressure, and (c) at pressures higher than the critical pressure. For all diagrams, λ = 0.001 and Q = 0.5. At the critical pressure, the intermediate region disappears. (a) a (b) b [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: C-rh diagrams of the black hole for fixed charge Q = 0.1: (a) Dependence of heat capacity (C) on the horizon radius (rh) under varying pressure at Q = 0.1, (b) Behavior at fixed pressure P = 0.026 with varying charge Q. For comparison, curves for both Q = 0.5 and Q = 0.1 are plotted together. The perturbation coefficient is fixed at λ = 0.001 in both cases. 21 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: T-rh diagrams of the black hole in the grand canonical ensemble: (a) for varying black hole pressure, and (b) for fixed pressure at the critical value corresponding to the potential Φ = 0.5, while varying Φ. At Φ = 0.1, the temperature behavior approaches that of a Hawking-Page phase transition, exhibiting a critical minimum. The study of Gibbs free energy versus temperature reveals that the black hole exh… view at source ↗
Figure 10
Figure 10. Figure 10: Gibbs free energy (G) versus temperature (T): (a) Swallow-tail structure, indicative of van der Waals-like phase transitions. (b) Transition between thermal AdS space and black hole states, exhibiting Hawking-Page-like behavior at Φ = 0.05. For comparison, the curve corresponding to Φ = 0.5 is also plotted. (a) a (b) b [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: C-rh diagrams of the black hole: (a) For Φ = 0.5 and various values of pressure. (b) For fixed pressure and various values of Φ. At the small value Φ = 0.1 (compared to Φ = 0.5), only two unstable and stable regions exist, exhibiting Hawking-Page-like behavior. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_11.png] view at source ↗
read the original abstract

In this paper, we derive a black hole solution within the Einstein Maxwell framework incorporating a nonminimal coupling between the Ricci tensor and the Maxwell field strength tensor, using a perturbative approach. We subsequently explore the thermodynamic phase transitions of the black hole in an extended phase space, analyzing both canonical and grand canonical ensembles. Our findings reveal that the system exhibits Van der Waals like behavior in both ensembles. Moreover, for sufficiently small values of electric charge and Maxwell potential, the thermodynamics is dominated by a Hawking Page phase transition.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The paper derives a perturbative black hole solution in 4D Einstein-Maxwell theory with nonminimal coupling between the Ricci tensor and Maxwell field strength tensor. It then studies the thermodynamics in extended phase space, reporting Van der Waals-like behavior in both the canonical and grand canonical ensembles, and dominance of the Hawking-Page phase transition for sufficiently small electric charge and Maxwell potential.

Significance. If the perturbative metric is shown to satisfy the modified field equations, the work would extend known thermodynamic phase structures to nonminimal couplings and demonstrate ensemble-independent Van der Waals behavior plus parameter-controlled Hawking-Page dominance, adding to the literature on black hole thermodynamics in modified gravity.

major comments (1)
  1. [metric derivation section] The section deriving the perturbative metric: no explicit verification is supplied that the obtained metric satisfies the nonminimal Einstein-Maxwell field equations to the working perturbative order (e.g., via residual computation of the modified Einstein tensor or on-shell action evaluation). This is load-bearing, as all thermodynamic potentials, temperatures, entropies, and the subsequent phase diagrams are constructed from this metric.
minor comments (1)
  1. [Abstract] Abstract: the claim of a 'perturbative approach' would benefit from specifying the expansion order and any truncation error estimate to allow readers to assess the regime of validity for the reported phase behaviors.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of our manuscript and for identifying an important point regarding the verification of the perturbative solution. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [metric derivation section] The section deriving the perturbative metric: no explicit verification is supplied that the obtained metric satisfies the nonminimal Einstein-Maxwell field equations to the working perturbative order (e.g., via residual computation of the modified Einstein tensor or on-shell action evaluation). This is load-bearing, as all thermodynamic potentials, temperatures, entropies, and the subsequent phase diagrams are constructed from this metric.

    Authors: We agree that an explicit verification of the field equations strengthens the manuscript. The perturbative metric was derived by substituting the ansatz into the modified Einstein-Maxwell equations and solving order by order in the nonminimal coupling parameter, but the original text omitted a direct residual computation. In the revised version we will add this check (either in the metric section or an appendix) by explicitly evaluating the modified Einstein tensor components and confirming that all residuals vanish to the working perturbative order. This will directly support the subsequent thermodynamic quantities and phase diagrams. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain.

full rationale

The paper derives a perturbative black hole metric from the nonminimal Einstein-Maxwell-AdS field equations and then computes thermodynamic potentials and phase structure directly from that metric in both ensembles. No quoted step shows a self-definitional loop, a fitted parameter renamed as a prediction, or a load-bearing result justified only by self-citation. The Van der Waals-like behavior and Hawking-Page dominance are computed outputs, not inputs imposed by construction, so the chain remains independent of its target results.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the Einstein-Maxwell action with an added nonminimal term and on the perturbative method being sufficient to obtain a consistent solution; no new entities are introduced.

free parameters (1)
  • nonminimal coupling constant
    The coefficient controlling the strength of the Ricci-Maxwell interaction term is a free parameter of the model whose value is not fixed by the abstract.
axioms (1)
  • domain assumption Einstein-Maxwell theory extended by a nonminimal Ricci-Maxwell coupling term
    The starting action is assumed without derivation; the perturbative solution is constructed within this framework.

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