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arxiv: 2606.28468 · v1 · pith:L3MJI2PZnew · submitted 2026-06-26 · 🌀 gr-qc

Thermodynamic Geometry, Heat Engines, and Topology of Sharma--Mittal ModMax-dRGT Black Holes

Hassan Hassanabadi , Abdullah Guvendi , Dhruba Jyoti Gogoi , Semra Gurtas Dogan , Farokhnaz Hosseinifar This is my paper

Pith reviewed 2026-06-30 01:28 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole thermodynamicsModMax electrodynamicsdRGT massive gravitySharma-Mittal entropyheat enginesthermodynamic geometrytopological chargeAdS black holes
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0 comments X

The pith

Sharma-Mittal entropy corrections and ModMax-dRGT parameters alter black hole heat engine efficiency and thermodynamic topology.

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

The paper studies charged AdS black holes in ModMax nonlinear electrodynamics combined with dRGT massive gravity under Sharma-Mittal entropy corrections. Thermodynamic geometry is examined via the Weinhold metric in horizon radius and charge space, where the Ricci scalar indicates microscopic interactions and singularities at extremal limits. A rectangular heat engine is built in extended phase space to obtain an exact efficiency formula showing direct effects from ModMax and massive gravity terms on enthalpy conversion, while Sharma-Mittal adjustments change the black hole temperature and thus the Carnot efficiency bound. Topological study of the corrected temperature and generalized free energy identifies conventional and new critical points with their conserved topological charges. Readers care because the work connects geometric descriptions of interactions, practical work extraction, and global topological invariants in one framework.

Core claim

In the ModMax-dRGT black hole model with Sharma-Mittal entropy, the Weinhold thermodynamic geometry produces a Ricci scalar whose singularities mark extremal boundaries, the rectangular heat engine efficiency receives explicit modifications from the ModMax parameter and dRGT couplings via the enthalpy, and the Sharma-Mittal parameters correct the temperature to change the Carnot bound, while topological analysis of the generalized free energy reveals both standard and novel critical points carrying conserved topological charge.

What carries the argument

Weinhold metric on the space of horizon radius and electric charge, together with the generalized free energy used for topological classification of critical points.

If this is right

  • The heat engine efficiency receives direct contributions from the ModMax parameter and dRGT couplings through the enthalpy.
  • Sharma-Mittal parameters modify the black hole temperature and thereby shift the Carnot efficiency bound.
  • The thermodynamic Ricci scalar exhibits singularities at extremal boundaries and metric degeneracies.
  • Topological analysis identifies both conventional and novel critical points, each carrying a conserved topological charge.

Where Pith is reading between the lines

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

  • If the corrected efficiency can be compared with future observations, the ModMax and dRGT parameters could be bounded by thermodynamic data.
  • The novel critical points may signal phase-transition behaviors absent from standard entropy models.
  • Similar topological methods applied to other nonlinear electrodynamics models could test whether charge conservation holds across different corrections.

Load-bearing premise

The Sharma-Mittal entropy correction and the ModMax-dRGT black hole solution apply directly to charged AdS black holes without further consistency requirements.

What would settle it

A computation showing that the derived efficiency formula recovers the standard Carnot value and that the novel critical points disappear when the Sharma-Mittal parameters are set to zero.

read the original abstract

We investigate the thermodynamic structure of charged AdS black holes in ModMax nonlinear electrodynamics coupled to dRGT-like massive gravity, incorporating Sharma--Mittal entropy corrections. The thermodynamic geometry is analyzed using the Weinhold metric in the parameter space spanned by the horizon radius and electric charge. The resulting thermodynamic Ricci scalar characterizes effective microscopic interactions, with curvature singularities signaling extremal boundaries and degeneracies of the thermodynamic metric. We further construct a rectangular black hole heat engine in the extended phase space and derive an exact expression for its efficiency, demonstrating how the ModMax parameter and massive-gravity couplings influence the enthalpy-based conversion of heat into work, while the Sharma--Mittal parameters modify the Carnot bound through corrections to the black-hole temperature. Finally, a topological analysis of the corrected temperature and generalized free energy reveals both conventional and novel critical points, and the associated conserved topological charge is investigated.

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 examines charged AdS black holes in ModMax nonlinear electrodynamics coupled to dRGT massive gravity, incorporating Sharma-Mittal entropy corrections. It analyzes the Weinhold thermodynamic metric in the space of horizon radius and charge, constructs a rectangular heat engine in extended phase space to derive an efficiency expression, and performs a topological analysis of the corrected temperature and generalized free energy to identify conventional and novel critical points along with their conserved topological charges.

Significance. If the first-law consistency after entropy replacement is verified, the results would quantify how the ModMax parameter and massive-gravity couplings affect enthalpy-to-work conversion in heat engines while Sharma-Mittal parameters alter the temperature and thus the Carnot bound, with topology revealing additional critical points. No machine-checked proofs, reproducible code, or parameter-free derivations are reported.

major comments (2)
  1. [Abstract and thermodynamic analysis sections] The central claims on efficiency, temperature corrections to the Carnot bound, and topological charge rest on replacing the Bekenstein-Hawking entropy with the two-parameter Sharma-Mittal form and recomputing T = (∂M/∂S). No derivation from the ModMax-dRGT action or explicit verification that dM = T dS + Φ dQ + V dP continues to hold is provided; this assumption is load-bearing for all reported expressions.
  2. [Heat engine construction and topological analysis sections] The efficiency expression and topological charge calculations inherit the unexamined replacement of entropy; without a consistency check, the reported influences of the ModMax parameter, massive-gravity couplings, and Sharma-Mittal parameters on heat-engine performance and critical points cannot be taken as established.
minor comments (1)
  1. Notation for the Weinhold metric components and the explicit form of the Sharma-Mittal entropy should be stated at first use for clarity.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments on thermodynamic consistency. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract and thermodynamic analysis sections] The central claims on efficiency, temperature corrections to the Carnot bound, and topological charge rest on replacing the Bekenstein-Hawking entropy with the two-parameter Sharma-Mittal form and recomputing T = (∂M/∂S). No derivation from the ModMax-dRGT action or explicit verification that dM = T dS + Φ dQ + V dP continues to hold is provided; this assumption is load-bearing for all reported expressions.

    Authors: The Sharma-Mittal entropy is introduced as a phenomenological correction, following the standard practice in the literature on quantum-corrected black-hole thermodynamics. Temperature is obtained from the thermodynamic identity T = (∂M/∂S) with the corrected entropy, and the first law is retained in its conventional form. No derivation from the underlying action is possible because the entropy correction itself is not obtained by varying the ModMax-dRGT action. We will add an explicit numerical verification that dM = T dS + Φ dQ + V dP holds to within numerical precision for the parameter ranges used in the efficiency and topology calculations. revision: yes

  2. Referee: [Heat engine construction and topological analysis sections] The efficiency expression and topological charge calculations inherit the unexamined replacement of entropy; without a consistency check, the reported influences of the ModMax parameter, massive-gravity couplings, and Sharma-Mittal parameters on heat-engine performance and critical points cannot be taken as established.

    Authors: The reported parameter dependencies are derived within the corrected thermodynamic framework. Once the first-law consistency check is included (as noted in the response to the first comment), the efficiency formula and the topological charges follow directly from the corrected temperature and free energy. We will therefore incorporate the verification step into the revised manuscript so that the claimed influences rest on an explicitly checked foundation. revision: yes

standing simulated objections not resolved
  • Derivation of the first law directly from the ModMax-dRGT action when the Sharma-Mittal entropy is used, because the entropy correction is phenomenological and not generated by the action.

Circularity Check

0 steps flagged

No significant circularity; derivation applies external entropy ansatz as input

full rationale

The paper takes the Sharma-Mittal entropy form as an explicit modeling choice (an input ansatz) and recomputes thermodynamic quantities such as temperature from the first law applied to the modified entropy. No quoted step shows a self-definitional loop, a fitted parameter renamed as a prediction, or a load-bearing result that reduces by construction to prior self-citations. The ModMax and dRGT parameters are likewise treated as free inputs whose effects are then computed; the resulting efficiency and topological-charge expressions are direct consequences of those inputs rather than tautological restatements. The derivation chain therefore remains self-contained once the entropy correction is granted, with no reduction of outputs to the inputs by algebraic identity.

Axiom & Free-Parameter Ledger

3 free parameters · 3 axioms · 0 invented entities

Abstract-only; the central claims rest on the applicability of the Weinhold metric, the validity of Sharma-Mittal entropy corrections, and the physical relevance of the ModMax-dRGT model, all treated as given without independent derivation in the provided text. Multiple model parameters are introduced without stated values or fitting procedures.

free parameters (3)
  • ModMax parameter
    Modifies nonlinear electrodynamics; appears as a free input controlling efficiency and geometry.
  • massive-gravity couplings
    dRGT parameters controlling massive gravity effects on enthalpy conversion.
  • Sharma-Mittal parameters
    Entropy correction parameters that modify temperature and Carnot bound.
axioms (3)
  • domain assumption Weinhold metric is the appropriate thermodynamic geometry for this system
    Invoked when analyzing curvature singularities and microscopic interactions.
  • ad hoc to paper Sharma-Mittal entropy corrections apply directly to the ModMax-dRGT black hole
    Used to modify temperature and free energy without further justification in abstract.
  • domain assumption Extended phase space thermodynamics with enthalpy applies
    Basis for rectangular heat engine construction.

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discussion (0)

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Reference graph

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