From modified Tsallis-Renyi entropy to a MOND-like force law, Bekenstein bound, and Landauer principle for black holes

Everton M. C. Abreu; Jorge Ananias Neto

REVIEW 3 major objections 4 minor 77 references

From modified Tsallis-Renyi entropy to a MOND-like force law, Bekenstein bound, and Landauer principle for black holes

T0 review · 3 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read This paper claims that substituting a modified Rényi entropy for Bekenstein–Hawking entropy in the entropic-force law yields a MOND-like gravitational force, a Bekenstein bound that is always respected, and a Landauer-principle mass loss…

desk verdict The abstract promises MOND, the body delivers a 1/R^4 force law and says so; the Bekenstein check is a tautology. read the letter →

arxiv 2505.03061 v2 pith:7ZHBNPVR submitted 2025-05-05 gr-qc

Everton M. C. Abreu , Jorge Ananias Neto This is my paper · ORCID

classification gr-qc PACS 04.70.Dy89.70.Cf

keywords TsallisentropyRényiMONDBekensteinboundLandauerprincipleblackholethermodynamicsnonextensivestatisticsentropicgravity

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 tries to show that replacing the Bekenstein–Hawking entropy with a modified Rényi entropy produces several black-hole results from one statistical starting point. The derivation rests on the conjecture that the original Tsallis entropy equals the Bekenstein–Hawking entropy, which fixes the microstate count and defines the modified Rényi entropy. Within that framework, the entropic-force formula gives an effective force whose interpolating function is exactly the simple MOND function, the Bekenstein bound holds for typical deformation parameters, and the Landauer principle yields a new mass-loss expression. A sympathetic reader should care because the paper offers a single nonextensive-statistics origin for these separate results, while stating openly that the force law it derives decays as $1/R^4$ and therefore does not actually reproduce MOND's characteristic $1/R$ galactic behavior.

What carries the argument

The carrying object is the modified Rényi entropy $S_R=(k_B/\lambda)\ln(1+\lambda S_{BH}/k_B)$, an entropic deformation with parameter $\lambda=1-q$, together with the entropic-force derivative $dS/dA$. The entropy provides the input, and the derivative $dS_R/dA=(dS_{BH}/dA)/(1+\lambda S_{BH}/k_B)$ converts area dependence into a modified gravitational force; the same entropy feeds the Bekenstein-bound inequality and the Landauer mass-loss calculation. In the limit $\lambda\to 0$, the modified Rényi entropy reduces to $S_{BH}$, and the force law, temperature, Bekenstein bound, and Landauer mass loss all reduce to their standard black-hole forms.

What would settle it

For a Schwarzschild black hole of mass $M$, compute $W=(1+\lambda S_{BH}/k_B)^{1/\lambda}$ with $S_{BH}=4\pi k_B G M^2/(\hbar c)$; the conjecture (3) requires $W$ to be an integer microstate count for any $\lambda\neq0$, and a non-integer result would falsify it.

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Extended reading notes

Core claim

The central claim is that the modified Rényi entropy $S_R=(k_B/\lambda)\ln(1+\lambda S_{BH}/k_B)$, obtained by conjecturing Tsallis entropy equals the Bekenstein–Hawking entropy, plugged into the entropic-force formula $F=(GMm/R^2)(4l_p^2/k_B)(dS/dA)$, yields the effective force $F_{\rm eff}=GMm/(R^2(1+\lambda\pi R^2/l_p^2))$, which can be written as $ma_N/(1+a_0/a_N)$ with interpolating function $\mu(a_N/a_0)=a_N/(a_0+a_N)$. The same entropy satisfies the Bekenstein bound in the form $S_R\le(e^{\lambda S_R}-1)/\lambda$, and the Landauer principle gives a black-hole mass loss $\Delta M=(\ln 2/8\pi)(1+4\pi\lambda M^2)/M$. The paper also emphasizes that the derived force decays as $1/R^4$ at large $R$, so the model is MOND-like only through the interpolating function and does not reproduce MOND's deep-infrared $1/R$ acceleration.

Load-bearing premise

The load-bearing premise is that the original Tsallis entropy, $k_B(W^{1-q}-1)/(1-q)$, equals the Bekenstein–Hawking entropy $S_{BH}$ for a black hole; if that equality fails, the microstate count, the modified Rényi entropy, and all three derived results lose their foundation.

Editorial extensions

If this is right

For $\lambda\to0$, the modified Rényi entropy reduces to $S_{BH}$, and the force law, temperature, Bekenstein bound, and Landauer mass loss all reduce to their standard black-hole forms.
The interpolating function $\mu(x)=x/(1+x)$ is exactly the simple MOND interpolating function, even though the underlying force does not share MOND's $1/R$ large-distance decay.
The Bekenstein bound holds under modified Rényi entropy for typical $\lambda\ge0$, with the ratio $R_R=(e^{\lambda S_R}-1)/(\lambda S_R)\ge1$.
Landauer's principle applied to this entropy predicts a black-hole mass loss that, for $\lambda>0$, has a minimum and then grows with mass, differing qualitatively from the standard $1/M$ Hawking behavior.
The framework connects nonextensive statistics to gravitational dynamics and information theory, giving a common origin for a MOND-like interpolation, an entropy bound, and an information-erasure cost.

Reading between the lines

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

If the $1/R^4$ large-distance decay is taken literally, the model would predict galaxy rotation curves that fall off rather than flatten; existing low-acceleration rotation-curve data could test this directly, and the paper's own admission indicates the test would fail.
The microstate-count conjecture $W=(1+\lambda S_{BH}/k_B)^{1/\lambda}$ could be checked for consistency: for a given physical black hole, $W$ must be an integer for every allowed $\lambda$, a constraint the paper does not examine.
The modified Rényi temperature has a minimum at finite mass, suggesting a phase transition and a heat-capacity sign change; this stability feature could be probed in analogue-gravity or condensed-matter realizations of nonextensive entropy, though the paper does not propose such tests.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a clean and transparent substitution exercise, but the headline result is not what it seems. The authors take the modified Renyi entropy of Ref. [13], plug it into their own entropic-force template from Ref. [42], and find F = (GMm/R^2)/(1 + lambda pi R^2/l_p^2). For large R that is 1/R^4, not MOND's 1/R. The paper says this explicitly in Section 3: the model 'does not reproduce MOND either.' So the abstract's claim that a MOND-like force law emerges naturally is contradicted by the body. The interpolating function x/(1+x) satisfies the formal limits (11)-(12), but the actual force law does not have MOND's deep-MOND scaling. That is a load-bearing inconsistency.

What is good: the algebra is correct, the derivations are easy to follow, and the authors are honest about the antecedents—the modified entropy is not theirs, the entropic-force formula is from their earlier work, and they cite the Kaniadakis paper [18] that already did the same program. The Landauer part is a routine derivative but correctly done. The Bekenstein bound check, however, is a tautology: since SR = (1/lambda) ln(1 + lambda S_BH) and ln(1+x) < x, SR < S_BH automatically, so verifying SR <= pi R^2 is just plotting e^x >= 1+x. Nothing new there.

Novelty is low: this is a new entropy inserted into an old template, and the MOND connection is formally similar to Ref. [18] but actually comes out worse (1/R^4 vs the Kaniadakis interpolating function). The paper would need a major revision to rescope the claims and delete the vacuous bound check. As it stands, I would not accept it for peer review; the abstract misrepresents the result and the remaining content is minor. If the authors want to salvage it, they should reframe as a modified Newtonian force with a MOND-like interpolation function in a formal sense, and remove the claim that it reproduces MOND.

Referee Report

3 major / 4 minor

Summary. The paper proposes a 'modified Rényi entropy' S_R = (k_B/λ) ln(1 + λ S_BH/k_B), obtained from the conjecture that the original Tsallis entropy equals the Bekenstein–Hawking entropy (Eq. 3). Using the entropic-force formula (Eq. 13), the authors derive an effective force law (Eq. 15) and claim that its interpolating function (Eq. 19) makes the model MOND-like. They further claim to verify the Bekenstein bound within this framework (Eq. 26, Fig. 1) and use the Landauer principle to obtain a black-hole mass-loss formula (Eq. 36). The central advertised result is that a MOND-like force law emerges naturally from entropic considerations, together with a verification of the Bekenstein bound and a Landauer-based mass-loss prediction.

Significance. If correct, the paper would connect nonextensive thermodynamics to MOND phenomenology and to information-theoretic bounds for black holes. The algebraic manipulations leading to Eqs. (9), (15), and (16) are internally consistent, and the paper is commendably transparent in Section 3 when it acknowledges that its model does not reproduce MOND. However, the main claims are not supported: the MOND-like force law is contradicted by the model's own large-distance scaling, and the Bekenstein-bound 'verification' is a mathematical identity following from the definition of the modified Rényi entropy. The Landauer result is a straightforward application of ΔS = (dS/dM)ΔM with ΔS = k_B ln 2, rather than a new physical principle. The paper's significance therefore reduces to a formal exercise rather than a substantive contribution.

major comments (3)

[Abstract and Section 3, Eqs. (15)–(19)] The advertised MOND-like force law is contradicted by the paper's own algebra. Equation (15) gives F_eff = (GMm/R^2)/(1 + λπR^2/l_p^2), which decays as 1/R^4 for large R, yielding an acceleration a ≈ a_N^2/a_0 ~ 1/R^4 in the deep-MOND regime. MOND requires a ~ sqrt(a_0 a_N) ~ 1/R. The paper itself states in Section 3 that the model 'does not reproduce MOND either.' Moreover, Eq. (19) defines μ(a_N/a_0), while the MOND force law requires μ(a/a_0) with the actual acceleration a satisfying μ(a/a_0) a = a_N. Thus Eq. (19) is not a MOND interpolating function in the dynamical sense. This contradiction undermines the central claim of the abstract and cannot be resolved by rewording alone.
[Section 4, Eqs. (26)–(27) and Fig. 1] The claimed verification of the Bekenstein bound is an identity, not a physical test. From Eq. (5) in natural units, S_R = (1/λ) ln(1 + λ S_BH), so S_BH = (e^{λ S_R} - 1)/λ. Inequality (26), S_R ≤ (e^{λ S_R} - 1)/λ, is exactly the elementary inequality ln(1+x) ≤ x (equivalently e^u ≥ 1+u), and Fig. 1 merely plots (e^{λ S_R} - 1)/(λ S_R) ≥ 1. Consequently, the Bekenstein bound is satisfied by construction for all λ > 0 and carries no independent physical content.
[Section 2, Eq. (3)] The load-bearing premise is the conjecture that the Tsallis entropy in Eq. (2) exactly equals the Bekenstein–Hawking entropy, as stated in Eq. (3). No independent evidence or physical motivation is provided for this equality, and all subsequent results—the force law, the Bekenstein-bound check, and the Landauer mass loss—follow from it. Since the MOND-like claim fails on its own algebra and the Bekenstein check is tautological, the remaining content reduces to algebraic consequences of this unproven conjecture.

minor comments (4)

[Section 2, after Eq. (1)] The text says the Tsallis entropy 'recovers the Boltzmann-Gibbs entropy when q approaches 0'; the correct limit is q → 1, as the following paragraph correctly states.
[Section 2, Eq. (1)] Equation (1) is poorly typeset in the manuscript; the denominator should be presented unambiguously as (q − 1) with the numerator k_B(1 − Σ p_i^q), matching the standard Tsallis definition.
[Section 4, Eq. (26)] The sentence following Eq. (26) is grammatically incomplete; it should be completed or merged with the definition of the ratio R_R in Eq. (27).
[Section 6] The phrase 'the use of the modified Rényi entropy could be explored in the context of BH area quantization' is vague; the future-work paragraph would benefit from a concrete proposal, such as which area spectrum is expected.

Circularity Check

2 steps flagged · score 6.0 of 10

Bekenstein-bound 'verification' is the defining inequality ln(1+x)<x, and the 'MOND-like' force law is Eq. (15) relabeled via a0=λπGM/l_p^2; the paper itself admits it does not reproduce MOND.

self definitional [Section 4, Eqs. (25)-(27) and Fig. 1]
"Hence, considering that the modified Rényi entropy governs BH thermodynamics, Eq. (5) with SBH = πR^2 allows us to recast the Bekenstein bound in Eq. (25) as SR ≤ (e^{λSR}−1)/λ ... To verify the validity of inequality (26), we have plotted in Figure 1 the ratio RR ... This result indicates that the Bekenstein bound conjecture holds when the modified Rényi entropy is applied to BH thermodynamics."

By the defining relation SR = (1/λ)ln(1+λSBH), we have e^{λSR}−1 = λSBH. Thus inequality (26), SR ≤ (e^{λSR}−1)/λ, is exactly SR ≤ SBH, which with SBH = πR^2 is the very Bekenstein bound being 'tested.' Since ln(1+x)<x for x>0, the inequality is true for every λ>0 by construction; Figure 1's plot of RR = (e^{λSR}−1)/(λSR) = SBH/SR > 1 simply records that definitional identity. No independent input from a confined quantum system, its radius, or its energy is used, so the verification reduces to the definition of the modified Rényi entropy.
renaming known result [Section 3, Eqs. (15)-(19) and the final paragraph of Section 3]
"After some algebra, we obtain Feffective = maN/(1 + a0/aN), where aN ≡ GM/R^2, and a0 ≡ λπGM/l_p^2. Thus, from the effective gravitational force, Eq. (16), we can identify μ(aN/a0) = 1/(1+a0/aN) = (aN/a0)(1+aN/a0)^{-1}. ... our model, based on the modified Rényi entropy, leads to a 1/R^4 decay at large distances, as previously discussed, and thus does not reproduce MOND either."

Equation (16) is just Eq. (15), the force obtained from the entropy-derivative formula, rewritten after defining a0 ≡ λπGM/l_p^2. The interpolating function (19) is therefore a relabeling of the same expression, not an independent MOND prediction. In MOND, a0 is a universal acceleration constant and the acceleration satisfies the implicit equation a = a_N μ(a/a0); here a0 is mass-dependent and the paper instead sets a = a_N/(1+a0/a_N), whose deep-MOND limit is a ∝ 1/R^4, not MOND's 1/R behavior. The paper explicitly concedes this, so the advertised 'MOND-like force law emerges' is a renamed, self-admittedly non-MOND result.

full rationale

Two of the three advertised results reduce by construction. First, the Bekenstein-bound check is tautological: starting from SR=(1/λ)ln(1+λSBH), inequality (26) is literally SR≤SBH, i.e., the bound being tested, and it holds for all λ>0 solely because ln(1+x)<x. Figure 1 plots that definitional identity. Second, the MOND-like section does legitimate algebra from Eq. (13) and Eq. (5) to obtain Eq. (15), but the 'MOND-like' interpretation is a renaming: Eq. (16) is Eq. (15) with a0≡λπGM/l_p^2, and Eq. (19) is that rewritten expression. Because a0 is mass-dependent and the resulting deep-MOND acceleration is a_N^2/(a_N+a0)∝1/R^4, the force law does not have MOND's 1/R behavior; the paper itself states 'our model ... does not reproduce MOND either.' The Landauer-principle calculation, Eqs. (35)-(36), is an independent consequence of differentiating SR and is not circular. The conjecture in Eq. (3) is an input assumption rather than a circular step. Self-citations (e.g., [56], [18]) are used for context and comparison, not as load-bearing uniqueness theorems. Overall, the Bekenstein-bound 'verification' is forced by definition and the MOND claim is a self-admittedly non-MOND relabeling, giving partial circularity.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or dimensions are introduced. The central machinery relies entirely on the conjectured Tsallis equals Bekenstein-Hawking equality, the entropic-force formula, and the standard information bounds.

free parameters (1)

lambda (deformation parameter) = unspecified; lambda = 0.2 used in Fig. 2, lambda >= 0 in Fig. 1
lambda = 1-q is the non-extensivity parameter of Tsallis entropy; it is free, with no independent constraint. The MOND acceleration scale a0 = lambda*pi*G*M/l_p^2 absorbs lambda, so matching observed galaxy data would require tuning lambda to an extremely small value.

assumptions (5)

ad hoc to paper Tsallis entropy exactly equals Bekenstein-Hawking entropy (Eq. 3): kB(W^(1-q)-1)/(1-q) = S_BH
Foundation of the modified Renyi entropy; the paper states 'we propose' this equality, i.e., a conjecture with no independent evidence.
domain assumption Entropic force formula F = (GMm/R^2)(4l_p^2/kB)(dS/dA) (Eq. 13)
Taken from Refs. [42,43]; assumed to hold for any entropy, not derived here.
domain assumption Bekenstein bound S <= 2*pi*R*E/(hbar*c) (Eq. 20)
Invoked as a conjecture to test; for Schwarzschild with natural units it reduces to S <= pi*R^2.
domain assumption Identification E = M and R = 2M for Schwarzschild black holes
Used to convert the bound to S <= pi*R^2; the paper acknowledges Ref. [57] offers an alternative identification that could change conclusions.
domain assumption Landauer erasure cost Delta E >= k_B*T*ln 2 and Delta S = k_B*ln 2 per bit
Used in Section 5 to extract mass loss; standard information-theoretic assumption extended to black holes.

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Pith. "Pith review of From modified Tsallis-Renyi entropy to a MOND-like force law, Bekenstein bound, and Landauer principle for black holes." pith.science (2026). https://pith.science/paper/7ZHBNPVR

@misc{pith2026250503061,
  author       = {Pith},
  title        = {Pith review of: From modified Tsallis-Renyi entropy to a MOND-like force law, Bekenstein bound, and Landauer principle for black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZHBNPVR}},
  note         = {Machine review of arXiv:2505.03061}
}

read the original abstract

We examine black hole thermodynamics within the framework of modified Renyi entropy and explore its implications in Modified Newtonian Dynamics (MOND), an extension of Newton second law proposed to explain galaxy rotation curves without invoking dark matter. We conjecture that Tsallis entropy provides an exact description of Bekenstein Hawking entropy, from which the modified Renyi entropy is derived. Using this formulation, we show that a MOND like force law emerges naturally from entropic considerations. We also analyze the Bekenstein bound conjecture, which imposes an upper limit on the entropy of confined quantum systems, and verify its validity under the Renyi modified framework for typical values of the deformation parameter. Furthermore, by invoking the Landauer principle, we obtain an expression for the mass loss due to black hole evaporation. These results suggest that modified Renyi statistics, originating from Tsallis entropy, provides a coherent and promising approach to gravitational dynamics and information theoretic aspects of black hole physics.

Figures

Figures reproduced from arXiv: 2505.03061 by the authors.

**Figure 2.** FIG. 2: Values of the normalized black hole mass loss [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

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BEKENSTEIN BOUND CONJECTURE The Bekenstein-Hawking entropy formula states that the entropy of a BH is directly proportional to the area of its event horizon. This principle plays a crucial role in BH thermodynamics, defining the connection between entropy and the surface area of the horizon. First proposed by Jacob Bekenstein and later refined by Stephen ...

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LANDAUER PRINCIPLE The Landauer principle 1, proposed by Rolf Landauer in 1961 [12], is a fundamental concept linking information theory and thermodynamics. It asserts that the erasure of a single bit of information entails a minimum energy cost, given by the equation ∆E≥kBT ln 2, (28) where T signifies the absolute temperature. The ln 2 term in Eq. (28) ...

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