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arxiv: 2606.26210 · v1 · pith:BOC5QKP2new · submitted 2026-06-24 · 🌀 gr-qc · hep-th

Extended Thermodynamics and Throttling Process of Charged AdS Black Holes in ModMax-dRGT Massive Gravity with Sharma-Mittal Entropy

Naba Jyoti Gogoi , Hassan Hassanabadi , Himasri Pinapothu , Dhruba Jyoti Gogoi This is my paper

Pith reviewed 2026-06-26 01:27 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords AdS black holesextended thermodynamicsJoule-Thomson expansionModMax electrodynamicsmassive gravitySharma-Mittal entropyphase transitions
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The pith

ModMax nonlinearities expand the cooling domain for charged AdS black holes in massive gravity by shifting inversion to smaller radii.

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

This paper studies the extended thermodynamics of charged AdS black holes combining ModMax nonlinear electrodynamics and dRGT massive gravity, using Sharma-Mittal entropy to capture non-extensive effects. It shows that the ModMax parameter enlarges the region of cooling during the throttling process by moving the inversion transition to smaller horizon radii. Sharma-Mittal parameters control local stability while the massive gravity background fixes the global phase boundary. The Gibbs free energy displays a van der Waals-like first-order phase transition with a swallow-tail structure. The work separates the influences of each modification on thermodynamic observables.

Core claim

The conformal nonlinearities of the ModMax field expand the physically accessible cooling domain by shifting the inversion transition to smaller horizon radii. While the Sharma-Mittal parameters critically govern local thermodynamic stability and the inversion radius, the global inversion phase boundary remains fundamentally dictated by the massive graviton background. An analysis of the Gibbs free energy uncovers a van der Waals-like first-order phase transition characterized by a distinct swallow-tail structure.

What carries the argument

The inversion curve of the Joule-Thomson expansion in extended phase space, modified by the ModMax nonlinearity parameter γ and Sharma-Mittal entropy parameters.

If this is right

  • The physically accessible cooling domain enlarges with increasing ModMax nonlinearity.
  • Local thermodynamic stability and inversion radius are controlled by Sharma-Mittal parameters δ and R.
  • The global inversion phase boundary is set by the massive graviton background.
  • A first-order phase transition appears with swallow-tail structure in the Gibbs free energy.

Where Pith is reading between the lines

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

  • Thermodynamic observables can act as diagnostics to distinguish the separate contributions of nonlinear electrodynamics, non-extensive entropy, and massive gravity.
  • The observed decoupling of effects suggests that similar combined models may allow independent tuning of different thermodynamic features.

Load-bearing premise

The generalised Sharma-Mittal entropy correctly accounts for non-extensive statistical correlations in this black hole system.

What would settle it

A direct computation of the inversion radius as a function of the ModMax parameter γ to check whether it decreases and enlarges the cooling domain.

Figures

Figures reproduced from arXiv: 2606.26210 by Dhruba Jyoti Gogoi, Hassan Hassanabadi, Himasri Pinapothu, Naba Jyoti Gogoi.

Figure 1
Figure 1. Figure 1: FIG. 1: Specific heat [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Joule-Thomson coefficient [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Isenthalpic curves in the [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Inversion curves in the [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Combined isenthalpic and inversion curves in the [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Representative Gibbs free energy [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Subcritical Gibbs free energy showing the swallow-tail structure. The small and large black-hole branches correspond [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
read the original abstract

We investigate the extended thermodynamics, including the Joule-Thomson expansion and $P-V$ criticality, of a four-dimensional charged anti-de Sitter (AdS) black hole within the combined framework of ModMax nonlinear electrodynamics and dRGT-like massive gravity. Operating in the extended phase space and employing the generalised Sharma-Mittal entropy to account for non-extensive statistical correlations, we derive exact analytical expressions for the modified Hawking temperature, specific heat, Joule-Thomson coefficient, and the equation of state. Our analysis of the throttling process reveals that the conformal nonlinearities of the ModMax field ($\gamma$) expand the physically accessible cooling domain by shifting the inversion transition to smaller horizon radii. While the Sharma-Mittal parameters ($\delta$, $R$) critically govern local thermodynamic stability and the inversion radius, the global inversion phase boundary remains fundamentally dictated by the massive graviton background. Furthermore, an analysis of the Gibbs free energy uncovers a van der Waals-like first-order phase transition characterized by a distinct swallow-tail structure. We observe a clear physical decoupling in the critical regime: ModMax nonlinearities modify the critical phase boundary by suppressing electromagnetic interactions, Sharma-Mittal parameters dictate the relative thermal stability of competing phases, and massive gravity governs the overarching macroscopic phase landscape. These results highlight the sensitivity of thermodynamic phase phenomena as robust diagnostic tools for distinguishing nonlinear and non-extensive modifications to black hole physics.

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

2 major / 1 minor

Summary. The paper claims to derive exact analytical expressions for the modified Hawking temperature, specific heat, Joule-Thomson coefficient, and equation of state for four-dimensional charged AdS black holes in the combined ModMax nonlinear electrodynamics and dRGT massive gravity framework, using the generalised Sharma-Mittal entropy in extended phase space. It reports that ModMax nonlinearity γ expands the cooling domain by shifting the inversion transition to smaller radii, that δ and R govern local stability and inversion radius while massive gravity sets the global boundary, and that the Gibbs free energy exhibits a van der Waals-like first-order phase transition with swallow-tail structure, with a claimed physical decoupling of the three modifications.

Significance. If the entropy substitution and derivations are shown to be consistent, the results would illustrate how nonlinear electrodynamics, massive gravity, and non-extensive entropy separately affect inversion curves and phase transitions, offering potential diagnostics for black hole modifications. The work does not provide machine-checked proofs, reproducible code, or parameter-free derivations.

major comments (2)
  1. [Abstract and entropy introduction] Abstract (paragraph on entropy choice) and the section introducing the thermodynamic quantities: the central claims rest on substituting the area law with the generalised Sharma-Mittal entropy S_δ,R while retaining the standard extended first law dM = T dS + V dP + Φ dQ + … without any derivation or consistency check against the ModMax stress-energy tensor, the dRGT graviton mass terms, or the resulting Smarr relation; this substitution is load-bearing for all reported analytical expressions, stability conclusions, and the inversion/phase-transition results.
  2. [Abstract and throttling-process analysis] Abstract and throttling-process analysis: the statements that γ 'expands the physically accessible cooling domain' and that 'clear physical decoupling' occurs in the critical regime are presented as outcomes of the calculation, yet no explicit verification is supplied that δ and R are not adjusted post-hoc to produce the reported inversion radius and stability behaviors, raising the possibility that the results reduce to input choices by construction.
minor comments (1)
  1. [Introduction or entropy section] Notation for the Sharma-Mittal parameters δ and R should be defined explicitly at first use with their physical interpretation, and any relation to the standard Bekenstein-Hawking entropy should be stated.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to the major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract and entropy introduction] Abstract (paragraph on entropy choice) and the section introducing the thermodynamic quantities: the central claims rest on substituting the area law with the generalised Sharma-Mittal entropy S_δ,R while retaining the standard extended first law dM = T dS + V dP + Φ dQ + … without any derivation or consistency check against the ModMax stress-energy tensor, the dRGT graviton mass terms, or the resulting Smarr relation; this substitution is load-bearing for all reported analytical expressions, stability conclusions, and the inversion/phase-transition results.

    Authors: The extended first law is postulated in the extended phase space as the fundamental relation, with temperature defined by T = (∂M/∂S) holding the other extensive variables fixed. Substituting the Sharma-Mittal form for S modifies the resulting T and equation of state in a manner consistent with this definition. The Smarr relation then follows directly from Euler homogeneity applied to the scaling properties of the extended thermodynamic variables, independent of the explicit functional form of S. The metric and mass function are determined by the field equations involving the ModMax stress-energy and dRGT terms, after which the thermodynamic quantities are obtained from the first law; no re-derivation of the stress-energy tensor is needed for the entropy substitution, which is a standard phenomenological step in the literature on generalized entropies. We maintain that the procedure is internally consistent and do not plan to alter the derivations. revision: no

  2. Referee: [Abstract and throttling-process analysis] Abstract and throttling-process analysis: the statements that γ 'expands the physically accessible cooling domain' and that 'clear physical decoupling' occurs in the critical regime are presented as outcomes of the calculation, yet no explicit verification is supplied that δ and R are not adjusted post-hoc to produce the reported inversion radius and stability behaviors, raising the possibility that the results reduce to input choices by construction.

    Authors: The inversion temperature, Joule-Thomson coefficient, and critical points are obtained from the closed-form analytical expressions in which γ, δ, R, and the massive gravity parameters appear as independent variables. The reported effects are demonstrated by explicit differentiation and by comparing families of curves in which one class of parameters is varied while the others are held fixed (including the limiting cases that recover the standard entropy). The functional dependence itself produces the observed expansion of the cooling domain under γ and the separation of influences in the critical regime; no auxiliary tuning of δ or R is performed to force these outcomes. revision: no

Circularity Check

0 steps flagged

No significant circularity; derivations are model-driven but independent

full rationale

The paper derives exact analytical expressions for temperature, specific heat, Joule-Thomson coefficient and equation of state after substituting the Sharma-Mittal entropy into the extended first law. Parameters δ and R enter as explicit inputs that modify the resulting formulas; the reported effects on inversion radius, stability and swallow-tail structure are direct algebraic consequences of those formulas rather than fits performed on the output quantities. No self-citation chain, uniqueness theorem, or ansatz smuggling is invoked to justify the entropy choice or the central claims. The derivation chain therefore remains self-contained once the entropy model is accepted as given.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the Sharma-Mittal entropy form, the ModMax Lagrangian, the dRGT massive gravity action, and the extended phase space identification of cosmological constant with pressure; none of these are derived in the paper.

free parameters (2)
  • γ (ModMax nonlinearity)
    Controls conformal nonlinearities and shifts inversion radius; introduced as a free parameter in the model.
  • δ, R (Sharma-Mittal)
    Govern local stability and inversion radius; fitted or chosen to produce the reported thermodynamic behavior.
axioms (2)
  • domain assumption Extended phase space formalism remains valid when combining ModMax, dRGT and Sharma-Mittal entropy
    Invoked throughout the abstract for P-V criticality and Joule-Thomson analysis.
  • ad hoc to paper Sharma-Mittal entropy correctly captures non-extensive correlations for these black holes
    Stated as the reason for employing the generalised entropy.

pith-pipeline@v0.9.1-grok · 5811 in / 1589 out tokens · 21259 ms · 2026-06-26T01:27:58.388164+00:00 · methodology

discussion (0)

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Reference graph

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