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REVIEW 3 major objections 5 minor 114 references

Black Hole Thermodynamics via Tsallis Statistical Mechanics

T0 review · 3 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Tsallis statistics predicts a thermodynamically stable Schwarzschild black hole when the non-extensive parameter is small and negative.

desk verdict A genuinely new Tsallis-statistics derivation of black hole entropy that fails at a key algebraic step: the Γ_q(3) cancellation does not happen, so the main result and stability window are not established. read the letter →

arxiv 2502.02522 v2 pith:6FPMFJ7J submitted 2025-02-04 gr-qc

Phuwadon Chunaksorn , Ratchaphat Nakarachinda , Pitayuth Wongjun This is my paper · ORCID
classification gr-qc MSC 83C5782B3080A10 PACS 04.70.Dy
keywords Tsallisentropyblackholethermodynamicsnon-extensivestatisticalmechanicsSchwarzschildthermodynamicstabilityBekenstein-HawkingSmarrformulaHawking-Pagetransition
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper derives a generalized Bekenstein–Hawking entropy by applying Tsallis (non-extensive) statistical mechanics to a thin gas shell just outside a Schwarzschild horizon, instead of the usual Gibbs–Boltzmann counting. The resulting entropy, Eq. (67), reduces to the area law in the limit $\eta\to0$ but otherwise carries an exponential dependence on $M^2$. Feeding this entropy into black hole thermodynamics, the paper claims that for $-\frac{1}{36}<\eta<0$ a large Schwarzschild black hole is both locally stable (positive heat capacity) and globally stable (negative free energy). If true, the standard result that a Schwarzschild black hole cannot be in stable equilibrium would hold only at the Gibbs–Boltzmann limit, with non-extensivity supplying the missing stabilizing interaction.

What carries the argument

The load-bearing object is the Tsallis $q$-entropy of the near-horizon gas, computed through the $q$-Laplace transform whose algebraic factor is the $q$-gamma function $\Gamma_q(3)=2/[(2-q)(3-2q)(4-3q)]$. The paper's key move is to choose the internal-energy relation $U_{q,\mathrm{loc}}=u_1\Gamma_q(3)^{u_2}U_{\mathrm{loc}}$ with $u_1=2^{1/3}$, $u_2=-1/3$; this choice cancels $\Gamma_q(3)$ from the gas entropy and makes the resulting black hole entropy a homogeneous function of $M^2$ and $\eta^{-1}$. Homogeneity lets the authors apply Euler's theorem to obtain the generalized Smarr formula $M=2S_{\eta,\mathrm{BH}}T_{\eta,H}-2\eta\Phi_\eta$ and the first law $dM=T_{\eta,H}dS_{\eta,\mathrm{BH}}+\Phi_\eta d\eta$, with $\eta$ acting as a state variable. All subsequent stability conclusions follow from the sign of the heat capacity and free energy computed from this thermodynamic phase space.

What would settle it

Recompute the black hole $q$-entropy and stability window using an alternative legitimate choice of the internal-energy relation, for example $U_{q,\mathrm{loc}}=U_{\mathrm{loc}}$; if the resulting entropy is not homogeneous or admits no negative-$\eta$ window for $S_{\eta,\mathrm{BH}}>0$, then the claimed stabilization is an artifact of the chosen map between Tsallis and ordinary energies rather than a consequence of non-extensive statistics.

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Extended reading notes

Core claim

In the paper's own terms, the discovery is that the non-extensivity of Tsallis statistics changes black hole entropy from the area-proportional Bekenstein–Hawking form to the closed expression $S_{\eta,\mathrm{BH}} = \frac{1}{\eta}\left[(1+48\pi M^2\eta)\exp\!\left(-\frac{44\pi M^2\eta}{1+48\pi M^2\eta}\right)-1\right]$, with $\eta=1-q$. This entropy is homogeneous when the Tsallis internal energy is linked to the ordinary local internal energy by the specific relation $U_{q,\mathrm{loc}}=2^{1/3}\Gamma_q(3)^{-1/3}U_{\mathrm{loc}}$, which removes the $q$-generalized gamma function from the entropy. Because the entropy is homogeneous, Euler's theorem gives a generalized Smarr formula and first law in which $\eta$ itself is a thermodynamic variable. The stability analysis then yields a window $-\frac{1}{36}<\eta<0$ in which the entropy has a local maximum at $A_{\max}=-1/(36\eta)$, the heat capacity is positive between $A_C^\eta$ and $A_{\max}$, and the Gibbs free energy is negative above $A_G\approx0.695 A_{\max}$, so a stable large-black-hole phase exists and a Hawking–Page transition appears at $T_{\mathrm{HP}}\approx5.935\sqrt{-\eta}$.

Load-bearing premise

The load-bearing premise is the chosen relation between the Tsallis gas energy and the ordinary gas energy, $U_{q,\mathrm{loc}}=u_1\Gamma_q(3)^{u_2}U_{\mathrm{loc}}$ with $u_1=2^{1/3}$, $u_2=-1/3$; the paper itself says $J(q)$ is not unique, and this specific choice is what removes $\Gamma_q(3)$ and makes the entropy a homogeneous function, so if another allowed choice is used the stability window changes or disappears.

Editorial extensions

If this is right

  • A Schwarzschild black hole, usually thought to be thermodynamically unstable because its heat capacity is negative, has a large-size phase with positive heat capacity whenever $-\frac{1}{36}<\eta<0$.
  • The generalized entropy is not the power-law form $S\propto A^{\delta}$ often assumed for Tsallis black hole entropy; it contains an exponential factor, so earlier power-law treatments of Tsallis black hole thermodynamics would need to be reconsidered.
  • A Hawking–Page type transition from thermal radiation to a stable large black hole occurs at temperature $T_{\mathrm{HP}}\approx5.935\sqrt{-\eta}$, driven by non-extensivity alone in an asymptotically flat spacetime.
  • In the limit $\eta\to0$ the standard Bekenstein–Hawking entropy, negative heat capacity, and positive free energy are all recovered, so the construction is a genuine one-parameter deformation of Schwarzschild thermodynamics.
  • Along fixed $\Phi_\eta$ processes, the black hole cannot be simultaneously locally and globally stable for any allowed $\eta$, so the stabilization is specific to the fixed-$\eta$ ensemble.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the stability window is real, it suggests that the microscopic degrees of freedom counted by black hole entropy may be effectively non-extensive with a small negative $\eta$, and that the long-range correlations producing stability could be observable in the thermodynamics of astrophysical black holes.
  • The paper leaves open which physical mechanism fixes $\eta$; a testable extension would be to compare the predicted Hawking–Page temperature and the stable-phase heat capacity with the thermodynamics extracted from gravitational-wave ringdown or Lyapunov-exponent analyses of black hole spacetimes.
  • Because the paper notes that $J(q)$ and the internal-energy relation are not unique, the most conservative reading is that the stability window is a feature of this particular consistent closure of Tsallis thermodynamics, not a universal prediction of non-extensive statistics; checking other legitimate closures is the natural next step.
  • The same construction could be applied to rotating or charged black holes; if the $-\frac{1}{36}<\eta<0$ window survives there, it would give a general mechanism for stabilizing black holes without a cosmological constant.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper revisits the derivation of black hole entropy from the statistical mechanics of a thin gas shell near a Schwarzschild horizon. It first reproduces the standard Bekenstein-Hawking result in the Gibbs-Boltzmann limit, then extends the derivation to Tsallis statistics and obtains a modified black hole entropy S_{eta,BH}(Sch) = (1/eta)[(1 + 48 pi M^2 eta) exp(-44 pi M^2 eta/(1 + 48 pi M^2 eta)) - 1] (Eq. 67). Using Euler's theorem on this homogeneity-adapted entropy, the authors construct a generalized Smarr formula and first law, and analyze local and global stability. They conclude that a Schwarzschild black hole becomes locally and globally stable for -1/36 < eta < 0, with a Hawking-Page-like phase transition.

Significance. If correct, this would be an interesting statistical-mechanics route to non-extensive black hole thermodynamics, with a concrete falsifiable prediction: Schwarzschild black holes are thermodynamically stable for a narrow interval of negative non-extensive parameter. The paper is self-contained and careful in several respects: the phase-space calculation of Sec. II is standard and explicit, the q-Laplace transform and Gamma_q(3) are presented in closed form, and the homogeneous-function/Smarr construction is internally consistent once Eq. (54) is taken as given. However, the central new formula currently rests on an algebraic error and on an unconstrained internal-energy ansatz, so the significance is conditional.

major comments (3)
  1. [III, Eq. (54)] Equation (54) does not follow from Eqs. (47), (49) and (50). Setting M = N(1-q), r = 1 + 3M and X = 4 pi e V_loc/(h^3 c^3 beta_loc^3 N), Eq. (50) gives J^r = (Gamma_q(3)^{1/3}/u1) (X/Gamma_q(3))^M. Substituting this into Eq. (47) yields [X/(J^3 Gamma_q(3))]^M = (2X)^{M/r} Gamma_q(3)^{-2M/r} for u1 = 2^{1/3}. The factor Gamma_q(3)^{-2M/r} is missing from Eq. (54). Since Gamma_q(3) depends on q, equivalently on eta, the resulting black hole entropy is not homogeneous of degree 1 in (M^2, eta^{-1}); consequently Eqs. (68)-(72), the temperature (74), and the stability window (79) do not follow from the derivation as written. The cancellation the authors intend would require u2 = +1/3 in Eq. (49) rather than the value u2 = -1/3 stated in the paper.
  2. [III, Eq. (49)] Even after the sign of u2 is corrected, the derivation depends on the ansatz U_{q,loc} = u1 Gamma_q(3)^{u2} U_loc with free constants u1 and u2. The paper itself states that J(q) is not unique and that u2 is chosen so that Gamma_q(3) cancels from the q-entropy. This makes Eq. (67) and the stability window (79) calibration-dependent rather than robust outputs of Tsallis statistics. The authors should either derive the internal-energy relation from a physical or thermodynamic principle, or demonstrate that the final claims are invariant under alternative choices of J(q).
  3. [III, after Eq. (54)] The identification of the gas q-entropy with black hole entropy uses the Gibbs-Boltzmann inverse temperature beta_loc = 4 pi^2 l_loc/(h c) after Eq. (54), while the gas is characterized by beta_q = J(q) beta_loc. For q != 1 these temperatures differ. The paper should justify that a gas in q-equilibrium with this beta_q is in the appropriate equilibrium with a black hole whose Hawking temperature is set by beta_loc, or explain how the q-temperature is related to the physical temperature (for example via Ref. [41]). As written, the identification of S^{(0)}_{q,loc}(bh) with the black hole entropy assumes an equilibrium condition that has not been established for q != 1.
minor comments (5)
  1. [IV, Fig. 2 description] The sentence 'it becomes at a certain horizon radius' is incomplete; presumably 'becomes zero' or 'becomes negative' was intended.
  2. [IV, Eq. (76)] The expression '112/3 (3 sqrt(7869) + 236)^{1/3}' is typeset ambiguously; please use explicit brackets or a cleared decimal form.
  3. [IV, Figs. 4 and 5] Axis and legend labels such as 'eta C eta(Sch)' and 'eta C eta,BH (Sch)' contain typesetting artifacts; please ensure subscripts and superscripts render correctly.
  4. [References] Reference [24] is missing its volume and page fields and should be completed.
  5. [III and IV notation] The paper switches from (1-q) in Sec. III to eta = 1-q in Sec. IV; introducing eta explicitly before Eq. (55) would improve readability.

Circularity Check

2 steps flagged · score 6.0 of 10

The homogeneity behind the claimed generalized Bekenstein-Hawking entropy is imposed by the free choice u2 = −1/3 in the internal-energy relation, and the asserted cancellation of Γq(3) in Eq. (54) is algebraically incorrect; the stability window −1/36 < η < 0 is therefore a property of an implicit ansatz rather than a derived consequence of Tsallis statistics.

  1. self definitional [Sec. III, after Eq. (49), and Sec. IV, after Eq. (67)]
    "As will be discussed in Sec. IV, by choosing u2 = −1/3, Γq(3) is eliminated in the q-entropy and then the q-entropy can be treated as a homogeneous function."

    Eq. (49) introduces arbitrary constants u1 and u2 in Uq,loc = u1 Γq(3)^{u2} Uloc, and Eq. (50) fixes J(q) through that relation. The text then declares u2 = −1/3 to be the choice that eliminates Γq(3) from Eq. (47), making the black hole entropy homogeneous. The Smarr formula, the first law, the temperature, and the stability window (Eqs. 67–81) all follow from that imposed homogeneity. Thus the advertised result—a locally and globally stable Schwarzschild phase for −1/36 < η < 0—is a property of the entropy form that the free exponent was selected to produce, not an independent output of Tsallis statistics. The paper is explicit about the choice, so this is partial circularity by construction.

  2. other [Sec. III, Eq. (54) (claimed substitution of Eqs. (47) and (50))]
    "As expected, this resulting entropy is independent of Γq(3)."

    Direct substitution of Eq. (50) into Eq. (47) leaves a Γq(3) factor. With M = N(1−q) and X = 4πe Vloc/(h^3 c^3 β_loc^3 N), Eq. (50) gives J^{1+3M} = (Γq(3)^{1/3}/u1)(X/Γq(3))^M, so (X/(J^3 Γq(3)))^M = 2^{M/(1+3M)} X^{M/(1+3M)} Γq(3)^{−4M/(1+3M)} for u1 = 2^{1/3}. Eq. (54) omits Γq(3)^{−4M/(1+3M)}. Consequently Eq. (55) and Eq. (67), along with the Smarr formula, first law, temperature, and stability bounds, are not consequences of the stated Tsallis calculation; the homogeneous exponential entropy is effectively an implicit ansatz. This is a load-bearing algebraic gap rather than a circular equivalence, but it reinforces that the final prediction is not forced by the stated inputs.

full rationale

The main circularity concern is real but limited. The paper is transparent that the internal-energy relation (49) is a free choice and that u2 = −1/3 is selected to eliminate Γq(3) in order to make the entropy homogeneous. All subsequent thermodynamic structure is built on that homogeneity, so the stability result is a consequence of the chosen construction rather than a unique prediction of Tsallis statistics. Additionally, the text's claim that the elimination works is algebraically incorrect: the Γq(3) factor survives in Eq. (54), so the derivation chain from Tsallis statistics to Eq. (67) has a gap. The self-citations in the paper (Refs. [42] and [89]) are not load-bearing for these claims; the Euler-homogeneity method and the hydrostatic-equilibrium remark are standard or contextual. The paper does not claim machine-checked or externally falsified support for the central entropy, so the score is set at 6 rather than 0–2. The result is partially circular and partly unsupported by the written derivation, but because the free choice is openly identified and the stability analysis itself is carried out consistently from the assumed entropy, this is not a fully tautological score of 8 or 10.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central q-entropy is obtained only after fixing an arbitrary internal-energy relation (u1, u2), identifying gas particles with horizon Planck areas, and adopting a specific Tsallis expectation-value scheme. These are inputs, not outputs of the derivation; consequently the stability bound is conditional on them.

free parameters (4)
  • u1 = 2^(1/3)
    Scale factor in the assumed internal-energy relation U_q,loc = u1 Gamma_q(3)^(u2) U_loc, fixed by requiring J(q) to approach 1 at q = 1.
  • u2 = -1/3
    Chosen to eliminate Gamma_q(3) from the q-entropy so that S_eta can be a homogeneous function; this choice drives the final entropy formula and stability window.
  • lloc = e^(15/8) l_P / (4 pi^(5/2)) approximately 0.1 l_P
    Proper distance of the gas shell from the horizon, fixed in the Gibbs-Boltzmann derivation so the gas entropy equals the Bekenstein-Hawking entropy; carried over to the Tsallis calculation.
  • q (non-extensive parameter) = not fitted; stability restricts -1/36 < eta = 1-q < 0
    Model parameter of Tsallis statistics; the paper determines allowed ranges from positivity and stability conditions rather than from data.
assumptions (6)
  • domain assumption Gas shell near the horizon is in thermal equilibrium with the black hole and its particles can be identified with horizon degrees of freedom via N = Ah/lP^2.
    Established in Sec. II following Refs. [9,12]; it is the bridge between matter entropy and black hole entropy and is assumed, not derived.
  • domain assumption The Tsallis total energy of N particles takes the non-additive factorized form Eq. (35) so the q-partition function factorizes as Eq. (33).
    Invoked in Sec. III when writing Z_q as (Z_q^(1))^N / N!; this is a choice from the Tsallis literature [85].
  • domain assumption Expectation values are defined with normalized f and the f^q-weighted energy: integral dV f = 1, integral dV f^q E = U_q (Eq. 25).
    One of several possible Tsallis expectation-value prescriptions; the paper follows Ref. [65], and the final thermodynamic quantities depend on this choice.
  • ad hoc to paper Internal-energy relation U_q,loc = u1 Gamma_q(3)^(u2) U_loc with u1 = 2^(1/3), u2 = -1/3 (Eq. 49).
    Postulated to make Gamma_q(3) cancel from the entropy; without it the homogeneous-function/Smarr construction in Sec. IV fails.
  • standard math q-Laplace transform and q-gamma function identities Eqs. (38)-(39) with the stated convergence ranges.
    Used to evaluate the one-particle q-partition function; standard results from Ref. [87].
  • standard math Euler's theorem for homogeneous functions is applicable to S_eta,BH viewed as a function of M^2 and eta^(-1).
    Used in Sec. IV to derive the Smarr formula and first law.

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Pith. "Pith review of Black Hole Thermodynamics via Tsallis Statistical Mechanics." pith.science (2026). https://pith.science/paper/6FPMFJ7J

@misc{pith2026250202522,
  author       = {Pith},
  title        = {Pith review of: Black Hole Thermodynamics via Tsallis Statistical Mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6FPMFJ7J}},
  note         = {Machine review of arXiv:2502.02522}
}
read the original abstract

An investigation of black hole thermodynamics based on Tsallis statistical mechanics is explored through the study of the thermodynamics of a gas system located near the horizon of a black hole. In spite of the difficulty in exploring black hole thermodynamics through statistical mechanics, the entropy of the nearby gas system is found to be proportional to the black hole's horizon area using Gibbs-Boltzmann statistical mechanics. This allows us to study black hole thermodynamics by using statistical mechanics through the thermodynamic behaviors of the gas system. Since the entropy of the black hole is proportional to the horizon area, it is more suitable to use non-extensive statistical mechanics instead of the usual Gibbs-Boltzmann ones. In this work, the black hole entropy is derived based on Tsallis statistical mechanics, one of well-known non-extensive statistical mechanics. It is found that the black hole entropy gets a modification due to non-extensivity. By using such an entropy, the black hole can be stabilized due to the non-extensivity, and the bound on the non-extensive parameter is also determined.

Figures

Figures reproduced from arXiv: 2502.02522 by the authors.

Figure 1
Figure 1. FIG. 1: The profiles of the gas entropy versus [PITH_FULL_IMAGE:figures/full_fig_p017_1.png] view at source ↗
Figure 2
Figure 2. It is very important to point out that the entropy with negative [PITH_FULL_IMAGE:figures/full_fig_p021_2.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The black hole entropy versus [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figures from the paper (5 more)
Figure 3
Figure 3. Figure 3: FIG. 3: Profiles of the temperature of the black hole with negative [PITH_FULL_IMAGE:figures/full_fig_p024_3.png]
Figure 4
Figure 4. Figure 4: FIG. 4: Profiles of heat capacity for negative [PITH_FULL_IMAGE:figures/full_fig_p025_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The left panel shows the profiles of the heat capacity and Gibbs free energy for [PITH_FULL_IMAGE:figures/full_fig_p026_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The profiles of Φ [PITH_FULL_IMAGE:figures/full_fig_p027_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: The top panels show the profiles of the black hole thermodynamic quantities under [PITH_FULL_IMAGE:figures/full_fig_p029_7.png]

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