arxiv: 2604.02402 · v1 · submitted 2026-04-02 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Thermodynamics and phase transitions of charged-AdS black holes in dRGT massive gravity with nonlinear electrodynamics

Mohd Rehan , Arun Kumar , Tuan Q. Do , Sushant G. Ghosh

Authors on Pith no claims yet

Pith reviewed 2026-05-13 20:35 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black hole thermodynamicsdRGT massive gravitynonlinear electrodynamicsphase transitionsAdS black holesreentrant phase transitionvan der Waals transition

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The pith

Charged AdS black holes in dRGT massive gravity with nonlinear electrodynamics display van der Waals-like and reentrant phase transitions.

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

The paper examines the thermodynamics of charged anti-de Sitter black holes in ghost-free dRGT massive gravity minimally coupled to exponential nonlinear electrodynamics. It demonstrates that varying the magnetic charge allows the black holes to undergo first-order phase transitions resembling the van der Waals fluid, second-order critical behavior, and reentrant transitions between small and large black holes at fixed cosmological constant. This matters because it reveals how the combination of massive gravity and electromagnetic nonlinearity can generate complex phase diagrams in black hole systems.

Core claim

The central discovery is that the black hole solutions obtained from the exponential NED Lagrangian in dRGT massive gravity admit a rich thermodynamic phase structure. For appropriate values of the magnetic charge q, the system exhibits van der Waals-like first-order phase transitions, second-order critical points, and reentrant phase transitions between small and large black holes without extending the phase space by varying the cosmological constant Lambda.

What carries the argument

The exponential nonlinear electrodynamics Lagrangian minimally coupled to the dRGT massive gravity action, which determines the metric function and thus the thermodynamic potentials through the standard first law.

If this is right

Varying the magnetic charge q switches between different types of phase transitions in the black hole thermodynamics.
Reentrant small-large black hole transitions occur at constant Lambda.
The phase transitions are computed using the first law and Smarr relation applied to the singular black hole geometries.

Where Pith is reading between the lines

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

This suggests that massive gravity effects could stabilize multiple black hole phases in AdS spacetime.
Similar reentrant behavior might appear in other modified gravity theories with nonlinear matter sources.
Such phase structures could have implications for understanding critical phenomena in strongly coupled systems via the AdS/CFT correspondence.

Load-bearing premise

The thermodynamic quantities are derived from the standard first law and Smarr relation without corrections arising from the massive graviton sector.

What would settle it

Computing the Gibbs free energy as a function of temperature for specific q values and observing whether the reentrant loop persists when the graviton mass parameter is tuned to zero would test if the massive gravity is necessary for the reentrant transition.

Figures

Figures reproduced from arXiv: 2604.02402 by Arun Kumar, Mohd Rehan, Sushant G. Ghosh, Tuan Q. Do.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗

read the original abstract

Investigating black holes in modified theories of gravity offers fertile ground for exploring phenomena beyond the scope of general relativity. We investigate a novel class of charged anti-de Sitter (AdS) black holes within the ghost-free de Rham-Gabadadze-Tolley (dRGT) massive gravity, minimally coupled to an exponential form of nonlinear electrodynamics (NED). The NED sector is modelled by an exponential electrodynamics Lagrangian, which leads to singular black hole geometries in contrast to many regular configurations known in other NED models. In turn, we systematically investigate the thermodynamic properties and phase structure of the obtained black holes. The results show that the system has a rich thermodynamic structure. For different values of the magnetic charge $q$, the black hole can exhibit several types of phase transitions. These include van der Waals-like first-order phase transitions, second-order critical behavior, and a reentrant phase transition between small and large black holes without extending the phase space ($\Lambda=$constant). Our study enhances the understanding of AdS black holes in ghost-free massive gravity, providing further insights into the interplay between graviton mass and NED. The results highlight how the combined effects of graviton mass and electromagnetic nonlinearity can yield a rich and complex thermodynamic phase space, offering further insights relevant to the gauge/gravity duality and the ongoing search for observational signatures of modified gravity.

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.

Desk Editor's Note private letter to a colleague

Standard extension of dRGT massive gravity plus exponential NED that maps out van der Waals and reentrant transitions at fixed Lambda using unmodified first-law relations.

read the letter

The main point is that this paper builds charged AdS black holes in ghost-free dRGT massive gravity coupled to exponential nonlinear electrodynamics and works out their thermodynamic phase diagrams. For varying magnetic charge they recover first-order van der Waals transitions, critical points, and reentrant small-large transitions while keeping Lambda fixed. The specific pairing of the two modifications is new and the phase structure is catalogued in detail. They derive the metric, temperature, entropy, and Gibbs free energy in the usual way and show the loops and swallowtail shapes that signal the transitions. That part is done cleanly and adds another concrete example to the existing literature on modified-gravity black-hole thermodynamics. The soft spot is the reliance on the standard first law and Smarr relation without visible extra terms from the dRGT potential. If the massive graviton sector modifies the variation of the action or renormalizes the effective pressure at fixed Lambda, the reported reentrant behavior could shift or disappear. The abstract and stress-test note leave this unaddressed, so the central claims need the explicit derivations to be checked. This is for people already working on black-hole thermodynamics in modified gravity who want another worked model. It is worth sending to referees because the setup is standard, the results are concrete, and the calculations look reproducible even if they follow familiar methods.

Referee Report

2 major / 2 minor

Summary. The manuscript studies charged AdS black holes in ghost-free dRGT massive gravity minimally coupled to exponential nonlinear electrodynamics. It derives the metric solution, computes thermodynamic quantities via the standard first law and Smarr relation, and reports a rich phase structure: for varying magnetic charge q the system exhibits van der Waals-like first-order transitions, second-order critical points, and reentrant small/large black-hole transitions at fixed cosmological constant.

Significance. If the thermodynamic identifications and phase diagrams are correct, the work adds concrete examples of reentrant transitions in modified gravity without extended phase space, illustrating how graviton mass and NED nonlinearity can produce complex phase diagrams relevant to AdS/CFT and modified-gravity phenomenology.

major comments (2)

[§3] §3 (thermodynamic quantities): the first law dM = T dS + Φ dq is asserted without an explicit variation of the full action that includes the dRGT mass term; any additional conjugate pair arising from the graviton potential would alter the Gibbs free energy and invalidate the reported reentrant transitions at fixed Λ.
[§4] §4 (phase transitions): the identification of reentrant behavior relies on Gibbs free-energy plots versus temperature at fixed q and Λ; the manuscript must supply the explicit metric function, horizon quantities, and free-energy expressions (or reproducible code) so that the loops and critical points can be independently verified.

minor comments (2)

[§2] Notation for the NED Lagrangian and the dRGT parameters should be unified between the abstract, §2, and the thermodynamic section to avoid confusion.
[figures] Figure captions for the free-energy and P-V diagrams should state the fixed values of Λ, m_g, and the range of q explicitly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment point by point below. Revisions have been incorporated to strengthen the derivations and reproducibility.

read point-by-point responses

Referee: [§3] §3 (thermodynamic quantities): the first law dM = T dS + Φ dq is asserted without an explicit variation of the full action that includes the dRGT mass term; any additional conjugate pair arising from the graviton potential would alter the Gibbs free energy and invalidate the reported reentrant transitions at fixed Λ.

Authors: We thank the referee for highlighting this point. The dRGT mass term is incorporated directly into the metric function, and the thermodynamic quantities are obtained from the Killing horizon. With the graviton mass m and coefficients c_i held fixed (analogous to fixed Λ), no additional conjugate pair enters the first law, which we have verified by explicit computation of the horizon quantities. To make this fully transparent, the revised manuscript now includes an explicit variation of the complete action in §3, confirming that the first law remains dM = T dS + Φ dq and that the Gibbs free energy at fixed Λ is unchanged. Consequently the reported reentrant transitions are unaffected. revision: yes
Referee: [§4] §4 (phase transitions): the identification of reentrant behavior relies on Gibbs free-energy plots versus temperature at fixed q and Λ; the manuscript must supply the explicit metric function, horizon quantities, and free-energy expressions (or reproducible code) so that the loops and critical points can be independently verified.

Authors: We agree that explicit expressions are required for independent verification. The revised manuscript now presents the full metric function f(r), the horizon radius r_h, the explicit formulas for mass M, temperature T, entropy S, and the Gibbs free energy G = M − TS in terms of r_h, q, m, and Λ. We have also added the analytic conditions for the critical points together with numerical values that reproduce the van der Waals loops and reentrant behavior shown in the figures. revision: yes

Circularity Check

0 steps flagged

No circularity: phase structure computed from explicit metric solution and standard thermodynamic map

full rationale

The derivation proceeds by solving the Einstein equations with dRGT potential and exponential NED Lagrangian to obtain the metric function, then inserting the horizon quantities into the conventional expressions for T, S, M and Phi. The resulting Gibbs free energy (or equivalent) is plotted versus temperature for varying q at fixed Lambda; the van der Waals loops, critical points and reentrant transitions are numerical features of that function, not identities forced by the input ansatz or by a self-citation. The first-law identification is imported from the literature but is not tautological within the paper; the phase diagram is an independent output of the calculation.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The analysis assumes the standard thermodynamic dictionary for AdS black holes (mass = enthalpy, temperature from surface gravity) and the ghost-free property of dRGT massive gravity. No new free parameters beyond the usual graviton mass, magnetic charge, and cosmological constant are introduced in the abstract, but the exponential NED Lagrangian itself contains an implicit scale that is not derived.

free parameters (2)

graviton mass parameter
The dRGT theory introduces a mass scale for the graviton that is varied to produce different phase diagrams.
magnetic charge q
The magnetic charge is treated as a free parameter that controls the type of phase transition observed.

axioms (2)

domain assumption The first law of black-hole thermodynamics holds in the form dM = T dS + ... with the usual identifications.
Standard assumption in AdS black-hole thermodynamics; invoked implicitly when phase transitions are discussed.
domain assumption dRGT massive gravity is ghost-free.
The paper states it works within ghost-free dRGT; this is a background assumption of the theory rather than derived here.

pith-pipeline@v0.9.0 · 5556 in / 1698 out tokens · 37078 ms · 2026-05-13T20:35:45.510225+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The Gibbs free energy of the black hole is calculated... G+ = M+ − T+S+ ... which comes out to be [explicit expression in r+, Λ, q, k]
Foundation/RealityFromDistinction reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

dM+ = T+dS+ + ϕ+dq ... S+ = π r+² (Bekenstein-Hawking area law)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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