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REVIEW 3 major objections 4 minor 66 references

In modified gravity, apparent-horizon entropy gains an integral correction that dictates whether the generalized second law holds.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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An integral-corrected apparent-horizon entropy is derived from modified Friedmann equations and used to re-evaluate the generalized second law in f(T) and f(R) gravity.

T0 review reviewed 2026-08-03 challenge →

load-bearing objection The central entropy formula and GSL condition are clean and mostly correct, but dimensional mistakes and a missing numerical procedure make the model-specific claims hard to trust as printed. the 3 major comments →

arxiv 2512.17861 v2 pith:4EDMYBS2 submitted 2025-12-19 gr-qc astro-ph.CO

Revisited apparent horizon entropy and GSL in modified gravity

Soma Heydari , Parastoo Askari , Kayoomars Karami This is my paper
classification gr-qc astro-ph.CO
keywords apparent horizon entropygeneralized second lawmodified gravityf(T) gravityf(R) gravityClausius relationFriedmann equationsdark energy
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 reading

This paper tries to establish a universal formula for apparent-horizon entropy in modified gravity: whenever the modified Friedmann equations can be rewritten in standard form with an effective density and pressure, the entropy is S_A = A/4G − 8π²∫ H r̃_A^4(ρ_e+p_e)dt, with the integral term generated by the modification. The formula follows by forcing the Clausius relation δQ = T_A dS_A to reproduce the field equations, and it reduces to the Bekenstein-Hawking area law in general relativity. From this entropy the paper derives a compact generalized-second-law condition, T_A Ṡ_tot = 8π²GH r̃_A^5(ρ_m+p_m)(ρ_t+p_t), and applies it to viable f(T) and f(R) models. A sympathetic reader would care because the result turns horizon thermodynamics into a concrete, model-by-model test of whether a modified-gravity dark-energy scenario is thermodynamically consistent.

Core claim

The paper's central claim is Eq. (15): for any modified gravity whose Friedmann equations can be recast in standard form with an effective density ρ_e and pressure p_e, the apparent-horizon entropy consistent with the Clausius relation is S_A = A/4G − 8π²∫ H r̃_A^4(ρ_e+p_e)dt. The horizon is treated as momentarily static when computing the Hawking temperature, and only matter energy is counted as crossing the horizon. In general relativity the integral vanishes and the Bekenstein-Hawking area law is recovered. Combining this horizon entropy with the matter entropy through the Gibbs equation yields the compact GSL criterion Eq. (21), which the paper then evaluates numerically for two f(T) and

What carries the argument

The load-bearing object is the revisited entropy formula Eq. (15), derived by integrating the Clausius relation against the modified Friedmann equations. Its new piece is the integral over the effective fluid combination ρ_e+p_e; specialized to f(T) gravity it becomes Eq. (35), S_A = Af_T/4G + α∫ f_TT d ln T, and to f(R) gravity Eq. (56), S_A = Af_R/4G − β∫ f̈_R r̃_A^3 dt. The companion criterion Eq. (21), T_A Ṡ_tot = (1/2)A(ρ_m+p_m)ṙ_A, is the universal GSL test: non-negative in GR, sign-dependent in modified gravity.

Load-bearing premise

The argument stands or falls on treating the apparent horizon as momentarily static when assigning its Hawking temperature and counting only matter energy as crossing the horizon; if the true dynamical temperature or the effective fluid's energy must enter the Clausius heat flux, the derived entropy and GSL criterion are not the physical ones.

What would settle it

Recompute δQ = T_A dS_A using the full dynamical Hawking temperature T_A = (1/2πr̃_A)(1 − ṙ_A/2Hr̃_A) instead of the static-horizon value and see whether the same entropy formula still satisfies the Clausius relation; alternatively, compare Eq. (56) with the Noether-charge entropy for the same f(R) models — a mismatch would show the integral-corrected entropy is not the physical horizon entropy.

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

If this is right

  • In general relativity the revisited entropy and GSL reduce to the standard Bekenstein-Hawking area law and an always-satisfied second law, T_A Ṡ_tot = 8π²GH r̃_A^5(ρ_m+p_m)^2 ≥ 0.
  • The GSL condition Eq. (21) is a single model-independent inequality that any modified gravity with reformulated Friedmann equations must pass at every redshift.
  • For f(T) gravity Model 2, the integral term removes the late-time (z = −1) GSL violation that appears when the entropy is approximated as Af_T/4G; Model 1 already satisfies the GSL with either prescription.
  • For the five f(R) models, the correction leaves AB, Starobinsky, and Hu-Sawicki unchanged, partially narrows the violation window in the Exponential model, and removes the late-time violation in the Tsujikawa model.
  • Because Eq. (15) is derived directly from the field equations, the same construction applies to any modified gravity with an effective fluid description, including models with extra components or interactions.

Where Pith is reading between the lines

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

  • The derivation's reliance on a momentarily static horizon means the entropy formula may be an instantaneous equilibrium entropy; if the full dynamical Hawking temperature were used, the correction term could acquire different coefficients or additional terms, changing the GSL verdicts for borderline models.
  • The sign of the integral correction is controlled by ρ_e+p_e, so models whose effective fluid crosses the null-energy condition will naturally produce late-time entropy corrections; this may explain why the correction matters most near de Sitter asymptotics.
  • One could test the construction by comparing Eq. (56) with the Noether-charge entropy for the same f(R) models, or with holographic entanglement entropy on the same horizon; agreement would strengthen the claim that this is the physical horizon entropy rather than a bookkeeping device.
  • A practical extension would be to feed the same formula into perturbed cosmology: since entropy perturbations couple to horizon dynamics, the correction term could leave signatures in gravitational-wave or CMB observables if modified gravity is realized in nature.
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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 / 4 minor

Summary. The paper proposes a universal expression for the apparent-horizon entropy in modified gravity, S_A = A/4G - 8π² ∫ H r̃_A^4(ρ_e + p_e) dt (Eq. 15), obtained by combining the Clausius relation with the reformulated Friedmann equations. It then derives a compact Generalized Second Law (GSL) condition, Eq. (21), and applies the formalism to two f(T) models and five f(R) models, comparing the standard entropy prescriptions (α=0, β=0) with the revisited ones (α=18π/G, β=π/G). The authors report that the integral correction improves late-time GSL validity for some models (f(T) Model 2, exponential and Tsujikawa f(R)) while leaving others unchanged.

Significance. If the central construction is accepted, Eq. (15) provides a compact, model-independent entropy formula for the apparent horizon, and Eq. (21) gives a simple GSL criterion applicable to any modified gravity that admits a reformulated Friedmann description. The algebra is mostly self-consistent and correctly reduces to the Bekenstein-Hawking entropy in GR and to earlier α=0 and β=0 results in the f(T) and f(R) literature. The numerical application to viable models is useful and falsifiable. However, the entropy is constructed so that the Clausius relation reproduces the modified Friedmann equations, so the GSL check is partly a consistency condition rather than an independent test. The paper's contribution is therefore a useful organizing framework and a set of concrete model comparisons, not a derivation from a more fundamental entropy functional.

major comments (3)
  1. [§III, Eqs. (16)–(17) and (21)] The entropy S_A in Eq. (15) is derived from the Clausius relation using the momentarily static temperature T_A = 1/(2π r̃_A), Eq. (8), with r̃̇_A=0. However, Eq. (16) computes T_A Ṡ_A with the full dynamical temperature of Eq. (6). Substituting Eq. (14) then gives T_A Ṡ_A = 4π H r̃_A^3(ρ_m+p_m)[1 − 2πG r̃_A^2(ρ_t+p_t)], which differs from the Clausius heat flux 4π H r̃_A^3(ρ_m+p_m) used to define S_A. This is an internal inconsistency in the derivation of Eq. (21). The final sign condition may survive if one instead works directly with Ṡ_tot and assumes T_A>0, but the manuscript does not say this; it simply switches temperatures without comment. Please either derive the GSL from Ṡ_tot using a single temperature convention, or explicitly state that the GSL is formulated as T_A Ṡ_tot ≥ 0 and justify why this is equivalent to the second law (including verifying T_A>0 for the models and reds
  2. [§II, Eqs. (8)–(15)] The derivation of the energy flux, Eq. (10), assumes r̃̇_A=0. But the apparent horizon is not static in the cosmological solutions considered; indeed Eq. (14) gives a nonzero r̃̇_A for any non-de Sitter evolution. The paper should clarify the logical status of Eq. (15): is it an exact dynamical entropy, or is it an entropy defined by an instantaneous static-horizon matching that is then integrated along the evolution? The standard first-law derivation for apparent horizons uses the energy flux through the moving horizon and does not require r̃̇_A=0; the manuscript should either provide that derivation or state clearly that Eq. (15) is a construction valid only under an additional approximation. This is central because Eq. (15) is the main result on which the GSL analysis is built.
  3. [§III, Eqs. (21)–(22) and Tables I–II] The manuscript presents Eq. (21) as a universal GSL criterion, but for pressureless matter it reduces to T_A Ṡ_tot ∝ ρ_m (ρ_t+p_t) r̃̇_A, i.e. the sign condition is essentially the null energy condition of the total fluid (assuming T_A>0). This observation should be stated explicitly, and the paper should temper the claim that the GSL is an independent test of modified gravity; it is a consistency condition between the chosen entropy construction and the field equations. The tables would be more informative if the authors also reported the sign of T_A over the relevant redshift intervals, since T_A Ṡ_tot ≥ 0 is equivalent to the GSL only when T_A > 0.
minor comments (4)
  1. [§IVC, Eqs. (47)–(48)] The expressions for G S_A are not derived in the text and contain notation that is ambiguous or dimensionally inconsistent as printed: “A1.96” should presumably read A^{1.96}, and the exponential e^{-7.41A} requires that A be dimensionless. Please specify the unit system (e.g. Planck area, G=1) and explain how these numerical fits were obtained from Eq. (35).
  2. [§II, Eq. (15)] Eq. (15) contains an indefinite integral, so S_A is defined only up to an integration constant. The constant does not affect Ṡ_A but does affect entropy positivity and the entropy-area fits in Eqs. (47)–(48). Please specify the lower limit or the normalization condition used.
  3. [§IVB, §VB] The reductions to α=0 and β=0 are said to match Refs. [23] and [26], but the f(R) numerical scheme is imported from Ref. [26] without specifying the background equations or initial conditions. For reproducibility, please include at least a concise description of the numerical procedure used to compute H(z) for the f(R) models.
  4. [Abstract and conclusions] There are typographical errors: “living others unchanged” should be “leaving others unchanged,” and “give rises” should be “give rise.” Please also correct the notation in Eq. (47) and the figure caption where μ1 is written with a comma instead of a decimal separator.

Circularity Check

2 steps flagged

GSL 'test' is partly an algebraic restatement of the input Friedmann equations, and the full-temperature evaluation in Section III is inconsistent with the static-temperature construction of S_A.

specific steps
  1. self definitional [Abstract; Section II Eqs. (9)-(15); Section III Eq. (21)]
    "This revisited horizon entropy is constructed directly from the modified Friedmann equations ... SA = A/4G − 8π² ∫ H r̃_A^4 (ρ_e + p_e) dt. ... TA ˙Stot = 8π²GH r̃_A^5 (ρ_m + p_m)(ρ_t + p_t) = 1/2 A(ρ_m + p_m) ˙r̃_A."

    S_A is obtained by substituting Eq. (11), which is the second modified Friedmann equation, into the matter heat flux Eq. (10) and integrating. Thus the Clausius relation is satisfied by construction. The GSL expression Eq. (21) is then just 1/2 A(ρ_m+p_m) ˙r_A with ˙r_A given by Eq. (14), i.e. the same Friedmann equation. Consequently the condition T_A ˙S_tot ≥ 0 is an algebraic identity following from the input equations; using it to 'test' a model does not provide an independent thermodynamic check beyond the background dynamics already assumed.

  2. other [Section II Eq. (8) vs Section III Eqs. (16)-(21)]
    "TA|˙r̃_A=0 = 1/(2π r̃_A). ... TA ˙SA = (1 − ˙r̃_A/(2H r̃_A))[ ˙r̃_A/G − 4πH r̃_A^3 (ρ_e + p_e)]."

    The entropy S_A was fixed using the momentarily static horizon temperature (8), but the GSL computation multiplies the same S_A by the full dynamical T_A of Eq. (6). With the full temperature, T_A dS_A no longer equals the heat flux −dE from Eq. (10); the discrepancy is the factor [1 − 2πG r̃_A^2 (ρ_t+p_t)] appearing in Eq. (17). The GSL condition (21), and so the claimed late-time improvements in Tables I and II, are therefore not consequences of the Clausius-consistent entropy but artifacts of an unmotivated replacement of the temperature.

full rationale

The paper is explicit that the entropy is constructed from the modified Friedmann equations, so the Clausius relation is guaranteed by design rather than by a hidden assumption. The GR limits and the α=0 / β=0 reductions to Refs. [23] and [26] are algebraic consistency checks, not load-bearing self-citations. The central formula (15) therefore has independent content as a proposed entropy assignment. However, the GSL analysis is partially circular: once S_A is defined via the Friedmann equations, the GSL condition (21) is an algebraic restatement of those equations, so the model-by-model 'tests' in Section IV-V do not probe thermodynamics independently of the input dynamics. In addition, the derivation of S_A uses the static-horizon temperature (8), while the GSL is evaluated with the full dynamical T_A (6); with the full temperature the Clausius relation used to define S_A no longer holds. This makes the claimed GSL improvements dependent on an inconsistent temperature substitution rather than on the thermodynamics of the derived entropy. These issues make the central GSL claims partially circular, though the entropy formula itself is not merely a renamed known result.

Axiom & Free-Parameter Ledger

5 free parameters · 7 axioms · 0 invented entities

The paper introduces no new particles or forces; its free parameters are standard model/cosmological parameters from prior literature. The key axiomatic input is the Clausius relation imposed to define S_A, plus the momentary-static-horizon assumption.

free parameters (5)
  • f(T) Model 1 exponent n = 0.04
    Taken from Linder [11,42]; the GSL outcome for Model 1 in Table I depends on this value but it is not fitted to the GSL.
  • f(T) Model 2 parameter β = -0.02
    Taken from Wu & Yu [42]; the improvement of GSL for Model 2 (Table I) is assessed at this value.
  • cosmological parameters H0 and Ωm0 = H0=68.22 km/s/Mpc, Ωm0=0.3032
    ACT [43] values used in all numerical GSL plots and tables.
  • f(T) model amplitudes μ1, μ2 = μ1=-14043, μ2=-35.9
    Derived from present-day Friedmann matching; enter entropy expressions Eqs(47)-(48) and the GSL numerics.
  • f(R) model parameters (b, n, λ, γ, c1, c2) = b=1.4, n=2, λ=1, γ=1.8, c1=1.25e-3, c2=6.56e-5
    Chosen from prior literature for AB, Starobinsky, Exponential, Hu-Sawicki, and Tsujikawa models; Table II outcomes depend on them.
axioms (7)
  • domain assumption Modified Friedmann equations with effective energy density and pressure (Eqs1-2)
    Background for all modified gravity models; assumes a perfect-fluid effective description.
  • domain assumption Clausius relation δQ = -dE = T_A dS_A for the apparent horizon
    Used in Eq(7) to define horizon entropy; this is the thermodynamic law being imposed.
  • ad hoc to paper Apparent horizon is momentarily static (ṙ_A = 0) so T_A ≈ 1/(2π r̃_A)
    Stated in Section II before Eq(8) as a simplifying assumption for the temperature; it makes the integration for S_A possible.
  • domain assumption Matter and effective fluids conserve separately (Eqs3-4)
    Assumed no interaction between matter and modified-gravity effective fluid.
  • domain assumption Gibbs equation T_A dS_m = dE + p_m dV for matter entropy
    Used in Eq(18) to compute the matter entropy contribution.
  • standard math f(T) effective density and pressure formulae (Eqs24-25) and f(R) ones (Eqs50-51)
    Standard effective-fluid reformulations from cited literature.
  • domain assumption Selected f(T)/f(R) models are viable and parameter choices come from literature
    The numerical GSL conclusions in Tables I-II assume these models and parameters.

reviewed 2026-08-03 · how reviews work

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Cite this review

Pith. "Pith review of Revisited apparent horizon entropy and GSL in modified gravity." pith.science (2026). https://pith.science/paper/4EDMYBS2

@misc{pith2026251217861,
  author       = {Pith},
  title        = {Pith review of: Revisited apparent horizon entropy and GSL in modified gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4EDMYBS2}},
  note         = {Machine review of arXiv:2512.17861}
}
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read the original abstract

This work presents a universal and revisited formalism for the entropy of the apparent horizon in modified gravity to investigate the validity of the Generalized Second Law (GSL) of thermodynamics. This revisited horizon entropy is constructed directly from the modified Friedmann equations in a Friedmann-Robertson-Walker (FRW) universe. The resulting entropy relation contains, beside the standard Bekenstein-Hawking term, an additional integral contribution that encodes the effective energy density and pressure generated by deviations from general relativity. Using this universal entropy formula, a compact expression for the GSL is derived. This formalism is then applied to some viable $f(T)$ and $f(R)$ gravity models, in order to re-evaluate the validity of the GSL as a function of redshift. The analysis demonstrates that including the integral term in the revisited entropy can relatively improve the late-time validity of the GSL for some of these models while living others unchanged, thereby reinforcing the profound connection between thermodynamics and gravity.

Figures

Figures reproduced from arXiv: 2512.17861 by Kayoomars Karami, Parastoo Askari, Soma Heydari.

Figure 1
Figure 1. Figure 1: FIG. 1: Evolutions of [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Evolutions of [PITH_FULL_IMAGE:figures/full_fig_p018_2.png] view at source ↗

discussion (0)

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Reference graph

Works this paper leans on

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This paper was first reviewed by deepseek-v4-flash on August 3, 2026.