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REVIEW 3 major objections 8 minor 95 references

Entropy deformation reshapes black hole stability and cooling

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2026-07-09 16:29 UTC pith:GKFIXLB6

load-bearing objection Internally consistent algebraic exercise, but the identical equation of state for both entropy frameworks undermines the central comparative claim. the 3 major comments →

arxiv 2607.07234 v1 pith:GKFIXLB6 submitted 2026-07-08 hep-th gr-qc

Thermodynamics of Deformed AdS-Schwarzschild Black Holes Beyond the Bekenstein Paradigm

Jafar Sadeghi , Fatemeh Khosravani , Alireza Amani This is my paper
classification hep-th gr-qc
keywords blackentropythermodynamicads-schwarzschildbehaviorbeyondclassicalcorrections
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks what happens to the thermodynamics of a black hole when you replace the standard Bekenstein-Hawking entropy formula with two generalized alternatives: Tsallis entropy, which accounts for long-range gravitational interactions, and Barrow entropy, which accounts for quantum-scale fractal structure of the event horizon. The authors apply both to a deformed AdS-Schwarzschild black hole—a spacetime metric modified by an extra gravitational source—and compute the full thermodynamic profile: mass, temperature, heat capacity, Gibbs free energy, enthalpy, internal energy, and the Joule-Thomson coefficient that governs cooling during adiabatic expansion. The central finding is that both entropy deformations shift phase transition points, expand the thermodynamic stability region as their deformation parameters increase, and keep the black hole in a cooling regime during Joule-Thomson expansion. In the limit where the deformation parameters vanish, both frameworks recover standard Bekenstein-Hawking thermodynamics, confirming consistency. The two frameworks operate through different mechanisms—Tsallis through statistical non-extensivity, Barrow through geometric fractality of the horizon—producing quantitatively different but qualitatively parallel modifications to the black hole's phase structure.

Core claim

The paper demonstrates that substituting Tsallis or Barrow non-extensive entropy for standard Bekenstein-Hawking entropy in a deformed AdS-Schwarzschild black hole shifts second-order phase transition points, expands the region of thermal stability, and maintains a positive Joule-Thomson coefficient (cooling regime) across a wide entropy range, with both frameworks reducing to classical behavior as their respective deformation parameters approach zero. The equation of state takes identical functional form in both frameworks; the difference enters through how thermodynamic volume depends on the entropy parameter, meaning the statistical or geometric deformation acts implicitly through the体积-熵

What carries the argument

The deformed AdS-Schwarzschild metric (Eq. 2) with deformation parameter α and energy-density parameter β; Tsallis entropy S_T = γ(πr_h²)^δ with non-extensivity parameter δ; Barrow entropy S_B = (A/A_P)^{(1+Δ)/2} with fractal deformation parameter Δ; the Joule-Thomson coefficient μ = (∂T/∂P)_M computed via Maxwell relations and heat capacity; mechanical stability condition (∂P/∂V)_T < 0 yielding critical temperatures T_c^T and T_c^B for Tsallis and Barrow respectively.

Load-bearing premise

The paper treats the deformation parameters (α for the metric, δ for Tsallis, Δ for Barrow) as independent free knobs, inserting non-extensive entropy formulas into the first law without deriving them from a specific underlying quantum-gravity or statistical-mechanics model. If the entropy deformation is not physically grounded, the shifted phase transitions could be artifacts of the parameterization rather than genuine physics.

What would settle it

A derivation showing that the Tsallis parameter δ and the Barrow parameter Δ are not independent but are both constrained to specific functions of horizon radius and Planck length, which would collapse the apparent freedom in the phase diagrams and eliminate the claimed distinction between statistical and geometric deformation mechanisms.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If non-extensive entropy corrections are physically real, black holes in strong gravitational fields should exhibit shifted phase transition temperatures and modified Hawking-Page transition points compared to standard predictions.
  • The Joule-Thomson cooling rate of a black hole could serve as a diagnostic probe distinguishing between statistical (Tsallis-type) and geometric (Barrow-type) quantum corrections to the horizon.
  • The framework extends naturally to charged and rotating black holes, where the interplay between electromagnetic or rotational parameters and entropy deformation may produce richer phase diagrams.
  • Critical exponents near the shifted phase transition points may differ from the standard van der Waals universality class, potentially offering observational signatures.

Where Pith is reading between the lines

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

  • If the deformation parameters δ and Δ are not independent but are both functions of a common quantum-gravity scale (such as the Planck length relative to horizon radius), the two entropy frameworks may be limiting cases of a single unified correction, and their quantitative differences could be used to identify which regime a given black hole occupies.
  • The fact that increasing deformation expands the stability region suggests that quantum-gravity corrections may act as a stabilizing mechanism for small black holes, potentially suppressing the runaway evaporation that leads to information-loss paradoxes in the classical theory.
  • The identical functional form of the equation of state across both entropy frameworks hints that the thermodynamic volume, rather than the entropy formula itself, is the carrier of physical deformation effects—a separation that could be tested by computing geometric thermodynamic curvature (Ruppeiner geometry) to see whether the two frameworks produce genuinely different thermodynamic geometries o

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 8 minor

Summary. This paper studies the thermodynamics of a deformed AdS-Schwarzschild black hole (metric from Ref. [94]) under two non-extensive entropy frameworks: Tsallis and Barrow. The authors derive mass, temperature, heat capacity, Gibbs free energy, enthalpy, internal energy, and the Joule-Thomson (JT) coefficient as functions of the respective entropy variable, and analyze thermal stability, mechanical stability, and JT cooling/heating behavior. The central claim is that both generalized entropy frameworks produce richer thermodynamic structure than standard Bekenstein-Hawking entropy, with the deformation parameters (δ for Tsallis, Δ for Barrow) controlling stability regions, phase transition locations, and JT cooling intensity, while recovering standard results in the limit of vanishing deformation.

Significance. The paper provides a systematic, algebraically complete derivation of thermodynamic quantities for the deformed AdS-Schwarzschild black hole under two well-known non-extensive entropy frameworks. The recovery of standard BH thermodynamics in the limit δ → 1 and Δ → 0 serves as a consistency check. The stability analysis combining heat capacity, isothermal compressibility, and the JT coefficient into a three-tier framework is a reasonable organizational structure. However, the physical significance is limited by the fact that the entropy deformation parameters are treated as free knobs without derivation from an underlying microscopic or quantum-gravity model, and the central comparative claim between Tsallis and Barrow frameworks faces a conceptual challenge (see Major Comment 2).

major comments (3)
  1. First-law consistency (load-bearing). The paper substitutes the generalized entropy expressions (Eqs. 6, 19) into the mass formula (Eq. 3) and temperature (Eq. 4) derived from the standard BH first law dM = T dS_BH + V dP. However, if the entropy is deformed (S_T or S_B), the first law should be modified accordingly: dM = T dS_deformed + V dP, which generically changes the relation between T and r_h. The paper appears to use the standard BH temperature (Eq. 4) and then re-expresses it in terms of S_T or S_B via a change of variables, without re-deriving T from the modified first law. The authors should clarify whether the temperature in Eqs. (8) and (21) is obtained from T = (∂M/∂S)|_P using the deformed entropy, or simply by substituting r_h(S) into the standard expression. If the latter, the heat capacity, stability, and JT results need justification.
  2. Identical equation of state undermines the comparative claim (load-bearing). Section III, Eq. (27) and the surrounding text explicitly state that the equation of state has 'exactly the same form for both the Tsallis and Barrow entropies,' with the difference appearing only in how the thermodynamic volume is defined in terms of entropy. Since all thermodynamic quantities (M, T, V, C_P, G, H, U, μ) are ultimately functions of r_h and the metric parameters (α, β, Λ), and since both S_T(r_h) and S_B(r_h) are monotonic reparameterizations of the horizon radius, the differences in the plotted behavior of C_P and μ between the two frameworks may be artifacts of plotting against different horizontal axes (S_T vs. S_B) rather than genuinely distinct physical predictions. The authors should demonstrate that the two frameworks produce observably different predictions when compared at fixed r_h (or,
  3. Physical motivation for parameter independence (load-bearing for interpretation). The deformation parameter α (geometric) and the entropy parameters (δ, Δ) are treated as independent free parameters. The paper does not discuss whether there is a physical relationship between the spacetime deformation (controlled by α, β) and the entropy deformation (controlled by δ, Δ). If the geometric deformation of the horizon is what gives rise to quantum-gravitational corrections to the entropy, one might expect δ or Δ to be functions of α. The authors should either motivate the independence of these parameters or discuss the physical regime in which treating them as independent is justified.
minor comments (8)
  1. Eq. (2): The metric function expression is difficult to parse due to ambiguous parenthesization in the deformation term. The term α(β² + 3r² + 3βr) / [3r(β+r)³] should be typeset more clearly.
  2. The notation 'a' in Eqs. (7)–(14) and 'b' in Eqs. (20)–(26) as composite variables is compact but not self-explanatory; a brief reminder of their definitions near the JT expressions would help readability.
  3. Fig. 1 caption: 'for several deformation parameters δ' — the caption should specify that δ is the Tsallis non-extensivity parameter, not the geometric deformation parameter α.
  4. Fig. 3 caption: mentions 'deformation parameters δ' but the figure and surrounding text refer to the Barrow parameter Δ. This appears to be a copy-paste error.
  5. The paper uses both 'non-expansive' and 'non-extensive' to describe the entropy frameworks. These terms have distinct meanings in the literature; the usage should be made consistent.
  6. Some references (e.g., [25]–[28], [73]–[78]) are cited in the introduction but their relevance to the specific thermodynamic analysis is not clear. Consider trimming or connecting them more explicitly.
  7. Section III, Eq. (27): the statement that the equation of state is identical for both entropies is important and should be highlighted earlier (e.g., in the abstract or introduction) to set expectations for the comparison.
  8. The conclusion (Section IV) is lengthy and somewhat repetitive. It could be tightened to highlight the key quantitative results more sharply.

Circularity Check

0 steps flagged

No significant circularity; the derivation is self-contained with standard entropy definitions from external sources

full rationale

The paper's derivation chain is straightforward and non-circular: (1) The deformed AdS-Schwarzschild metric (Eq. 2) is taken from external reference [94] (Khosravipoor & Farhoudi, different authors). (2) Mass, temperature, and volume (Eqs. 3-5) are derived from the metric by standard methods (setting f(r_h)=0, using the first law). (3) Tsallis entropy (Eq. 6) and Barrow entropy (Eq. 19) are standard definitions cited to external sources [95]. (4) All subsequent thermodynamic quantities (heat capacity, Gibbs free energy, JT coefficient, etc.) are computed by substituting these entropy definitions into standard thermodynamic relations. No step reduces to its own inputs by construction. The paper does not fit parameters to data and then 'predict' the same data. No load-bearing self-citation chain exists: references [87, 88] by overlapping authors concern generalized entropy formalisms generally and are not invoked as uniqueness theorems or ansatz justifications for the specific calculations here. The skeptic's observation that the equation of state (Eq. 27) is identical for both entropy frameworks is a valid correctness/depth concern — the 'distinct behavior' seen in plots of C_P vs S_T versus C_P vs S_B is partly a reparameterization effect since all quantities are ultimately functions of r_h — but the paper explicitly acknowledges this fact ('the difference between the two frameworks appears not in the form of the equation of state, but in the way the thermodynamic volume is defined'), so it is not hiding a circular reduction. The functional forms expressed in terms of S_T and S_B are genuinely different mathematical objects (different exponents, different variable dependencies), even if they encode the same underlying r_h physics. This is a matter of physical interpretation depth, not circularity.

Axiom & Free-Parameter Ledger

5 free parameters · 3 axioms · 0 invented entities

The paper introduces no new physical entities. It uses existing entropy frameworks (Tsallis, Barrow) and an existing deformed metric. The free parameters (α, β, δ, Δ, γ) are the main handles controlling the thermodynamic behavior, and their values are chosen by hand for parametric exploration rather than fitted to observational data.

free parameters (5)
  • α (deformation parameter) = 0.45 (in figures)
    Controls the strength of geometric deformation of the background spacetime; chosen ad hoc for plots.
  • β (energy density parameter) = 0.7 (in figures)
    Controls the behavior of energy density at zero radius; chosen ad hoc for plots.
  • δ (Tsallis entropy parameter) = 0.3, 0.5, 0.7, 1.2 (in figures)
    Quantifies deviation from BH entropy in the Tsallis framework; varied parametrically.
  • Δ (Barrow entropy parameter) = 0, 0.3, 0.6, 0.9 (in figures)
    Quantifies fractal deformation of the event horizon in the Barrow framework; varied parametrically.
  • γ (Tsallis normalization)
    Normalization coefficient for Tsallis entropy; introduced but not assigned a numerical value in the plots.
axioms (3)
  • domain assumption The first law of black hole thermodynamics dM = TdS + VdP holds in the presence of non-extensive entropy deformations.
    Used throughout to derive thermodynamic quantities; standard in extended phase space thermodynamics but unproven for the specific deformed metric.
  • domain assumption The deformed AdS-Schwarzschild metric (Eq. 2) represents a physically valid black hole solution with well-defined thermodynamic properties.
    The metric is taken from [94]; the paper notes potential singularity issues but proceeds with thermodynamic analysis.
  • ad hoc to paper Non-extensive entropy formulas (Tsallis, Barrow) can be consistently substituted into black hole thermodynamic relations derived under Boltzmann-Gibbs statistics.
    The substitution is performed without deriving the entropy from a microscopic statistical model of the deformed black hole.

pith-pipeline@v1.1.0-glm · 20911 in / 2505 out tokens · 453925 ms · 2026-07-09T16:29:13.855795+00:00 · methodology

0 comments
read the original abstract

This work investigates the thermodynamic behavior of deformed AdS-Schwarzschild black holes by incorporating higher-order corrections within non-interacting spacetime models and extended entropy frameworks. To address the inadequacies of classical statistical mechanics in describing gravitational systems with non-local and long-range interactions, we employ non-extensive entropy formalisms, specifically Tsallis and Barrow entropies, which capture quantum-scale deviations and extended correlations. The resulting thermodynamic analysis reveals significant departures from conventional black hole behavior under strong entropy deformations. Notably, as the degree of non-extensivity decreases, the system asymptotically recovers classical features, indicating an emergent universality across statistical regimes. Furthermore, the Joule-Thomson (JT) expansion is examined to analyze the temperature-pressure response during adiabatic processes. Key thermodynamic quantities, including mass, temperature, heat capacity, Gibbs free energy, enthalpy, internal energy, and the JT coefficient, are computed under the influence of non-extensive entropy corrections. These results provide deeper insight into black hole thermodynamics in quantum-corrected spacetimes and offer new avenues for exploring gravitational systems beyond the traditional Bekenstein-Hawking (BH) framework.

Figures

Figures reproduced from arXiv: 2607.07234 by Alireza Amani, Fatemeh Khosravani, Jafar Sadeghi.

Figure 1
Figure 1. Figure 1: FIG. 1. Heat capacity [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The JT expansion coefficient [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Heat capacity [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Analyzing the JT expansion coefficient of deformed AdS-Schwarzschild black holes within the framework of Barrow [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗

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Reference graph

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