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Static black holes in a class of f(R,K) modified gravity theories carry an Iyer–Wald entropy that departs from the area law by a power-law correction controlled by a single parameter n.

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2026-08-01 06:43 UTC pith:ONGTWELN

load-bearing objection A clean perturbative entropy result for a specific f(R,K) theory; the stress-test's horizon-location objection does not survive contact with the paper's own equations. the 1 major comments →

arxiv 2607.21786 v3 pith:ONGTWELN submitted 2026-07-23 gr-qc

Static Black Holes and Iyer-Wald Entropy in f(mathcal{R},mathcal{K}) Gravity

I. D\'iaz-Salda\~na , J. L\'opez-Dom\'inguez , Wilfredo Yupanqui , Javier Chagoya , M. Sabido This is my paper
classification gr-qc PACS 04.70.-s04.50.Kd
keywords modified gravityf(R,K) gravityKretschmann scalarhigher-curvature correctionsSchwarzschild black holeIyer–Wald entropyentropy-area relationBekenstein–Hawking entropy
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.

This paper claims that in a class of modified gravity theories whose action adds a function of the Ricci and Kretschmann scalars to the Einstein–Hilbert term, the entropy of a static, spherically symmetric black hole gains a correction to the Bekenstein–Hawking area law that is itself a power of the area, with the exponent set by a free parameter n of the theory. The authors work perturbatively: the higher-curvature terms are treated as a small deformation of the Schwarzschild metric, and the first-order corrections to the geometry are solved in closed form. The Iyer–Wald entropy is then computed to the same order, yielding a compact formula that reduces to constant, logarithmic, and area-proportional corrections for particular n. This matters because the same functional form was previously found for the apparent-horizon entropy of cosmological solutions in this theory, suggesting that a single entropy-area relation may govern both cosmological and black hole sectors.

Core claim

The paper establishes that for the action S = (1/16πG)∫d⁴x√−g (R + λ/ℓ^{2n} f(R,K)) with f(R,K) = −[R + √(6K − R²)]^{1−n}, the Iyer–Wald entropy of the perturbative static black hole takes the form S_W = A_H/(4G) + (λ/ℓ^{2n}) (n−1)(1+√2) Gⁿ / ((6√2π)ⁿ) (A_H/(4G))^{n+1} to first order in λ. The correction is a pure power law in the Bekenstein–Hawking entropy, with exponent n+1; it vanishes only for n=1, which corresponds to the pure cosmological-constant term, and for n=−1 it becomes a constant independent of the horizon area. The authors emphasize that, because the Noether charge is evaluated locally on the bifurcation surface, the entropy result does not depend on the (possibly non-asymptot

What carries the argument

The central object is the Lagrangian f(R,K) = −X^{1−n} with X = R + √(6K − R²), where R is the Ricci scalar and K = R_{αβγδ}R^{αβγδ} is the Kretschmann scalar, added to the Einstein–Hilbert action with coupling λ/ℓ^{2n}. The argument uses a perturbative expansion of the metric around Schwarzschild in powers of λ, reducing the modified field equations to a pair of first-order ODEs for the metric functions; the entropy is then obtained from the Iyer–Wald Noether charge, whose integrand involves the functional derivative of the Lagrangian with respect to the Riemann tensor contracted with the horizon binormal.

Load-bearing premise

The entire calculation rests on the coupling combination λ(M/ℓ)^{2n} being much smaller than one; if it is of order one, the first-order metric is not a controlled deformation of Schwarzschild and the Wald entropy evaluated on the unperturbed horizon would miss the leading correction.

What would settle it

For n=1 the solution is exactly Schwarzschild–(A)dS; compute the Wald entropy of that exact metric to first order in the cosmological constant and verify it equals A_H/4G. For n=−1, evaluate the exact four-dimensional Gauss–Bonnet entropy on the Schwarzschild background and compare its coefficient with the constant shift in Eq. (29) — any mismatch indicates that the perturbative evaluation is dropping terms of the same order.

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

If this is right

  • If correct, the same power-law entropy formula applies to the black hole sector and the cosmological apparent-horizon sector of this theory, supporting a unified thermodynamic picture across regimes.
  • The n=1 case reduces to Einstein gravity with a cosmological constant (Schwarzschild–(A)dS) and the correction vanishes, providing a cross-check of the perturbative calculation.
  • For n=−1 the correction is a constant, matching the perturbative entropy of Einstein–Gauss–Bonnet gravity even though the field equations are not dynamically equivalent; entropy measurements alone would not distinguish the two theories.
  • Near n=−1, the formula yields the standard logarithmic correction as a limiting case, so a single expression covers three well-known entropy modifications without extra assumptions.

Where Pith is reading between the lines

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

  • The n=1/2 branch, which yields an A^{3/2} correction, was previously linked (in the cosmological context) to a self-accelerating universe and the DGP model; if the black hole entropy formula holds, the same branch predicts a specific area scaling that could be probed in holographic or semiclassical settings.
  • Because the first-order Wald entropy is insensitive to the metric perturbation's asymptotic behavior and even to the horizon shift, the entropy-area relation may be fixed almost entirely by the Lagrangian structure; extending the calculation to rotating or charged black holes would test that universality.
  • The discrepancy between the coefficient found here and the one from the apparent-horizon derivation suggests the thermodynamic construction at the apparent horizon (temperature choice or first law) may be ambiguous; a careful comparison could single out which entropy definition is physically preferred.

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

1 major / 4 minor

Summary. The paper studies a class of f(R,K) gravity, with f(R,K) = -X^{1-n} and X = R + sqrt(6K - R^2), as proposed in a recent cosmological work. It constructs static, spherically symmetric black hole solutions perturbatively around Schwarzschild, treating the higher-curvature term as a first-order correction controlled by a small parameter. The authors solve the linearized field equations for the metric perturbations and then compute the Iyer-Wald entropy of the resulting black hole. Their central result is a power-law correction to the Bekenstein–Hawking entropy: S_W = A_H/(4G) + (λ/ℓ^{2n})(n-1)(1+√2)G^n/((6√2π)^n)(A_H/(4G))^{n+1} (Eq. 29), which vanishes only for n=1 and reproduces constant, logarithmic, and area-proportional corrections in different limits. They compare this with entropy-area relations previously obtained from apparent-horizon thermodynamics in the same theory.

Significance. If correct, the paper would provide an explicit, closed-form Iyer–Wald entropy for a non-trivial higher-curvature theory, linking black hole entropy to cosmological entropy-area relations in a unified power-law framework. The perturbative construction of the metric and the entropy computation are carried out analytically, which is valuable and goes beyond a formal statement. However, the central result is undermined by a first-order error in the choice of horizon surface: the Wald entropy is evaluated on the trapping horizon (g_rr=0) rather than the Killing horizon (g_tt=0), and the two surfaces differ at O(λ) for the generic solution. Since the Einstein–Hilbert term contributes A/(4G), this misidentification introduces an O(λ) error in the entropy, of the same order as the claimed higher-curvature correction. The paper's own emphasis on the Killing horizon in Section IV makes this an internal inconsistency. The approach is salvageable, but the main formula needs to be recomputed on the correct surface.

major comments (1)
  1. [Section IV, Eqs. (16), (17), (29)] The Iyer–Wald entropy is defined on the bifurcation surface of the Killing horizon, where g_tt=0. The paper instead defines the horizon by g_rr=0 (Eq. 16) and uses the corresponding area A_H in the final formula (29). For the generic solution with c1=0, the two surfaces differ at first order: r_K = 2M - 2MλA(2M) + O(λ^2) and r_T = 2M - 2MλB(2M) + O(λ^2), with A(2M) ≠ B(2M) for n≠1 (and also for n=0 if one uses the explicit expressions, unless c1 is tuned as in Eq. 22). Consequently A_K - A_T = O(λ), and the Bekenstein–Hawking term changes by (A_K - A_T)/(4G) = O(λ). The paper's statement in Section IV that 'the correction induced by the horizon displacement contributes only at order O(λ^2)' applies only to the higher-curvature term S_HC, whose integrand is O(λ); it does not apply to the Einstein–Hilbert area term. Thus Eq. (29) is not the correct first-order Wald entropy unless A_H is re
minor comments (4)
  1. [Section III, after Eq. (15)] The perturbative expansion is controlled by the combination λ(M/ℓ)^{2n}, but the paper does not explicitly state the range of M/ℓ for which the solution is valid for each n. For n>0, one needs M/ℓ << 1; for n<0, M/ℓ >> 1. The paper should specify these regimes to avoid overclaiming the validity of the solution and the entropy result for all n.
  2. [Section III, Eq. (22)] The paper notes that a specific choice of c1 makes the trapping and Killing horizons coincide, but this choice is not adopted in the entropy calculation. The entropy result (29) implicitly uses c1=0. The authors should clarify this and discuss whether the final coefficient depends on the horizon-surface choice.
  3. [Section IV, paragraph after Eq. (27)] The sentence 'replacing the classical background area A_H^(0) with the full perturbed horizon area A_H only introduces negligible terms of order O(λ^2)' is misleading because it is true only for S_HC, not for the total entropy. Rephrase to avoid ambiguity.
  4. [General] The notation ∂L/∂R_{μναβ} in Eq. (24) is standard for the Wald formula, but the distinction between the scalar Lagrangian and the Lagrangian 4-form is not spelled out. A brief comment would improve readability.

Circularity Check

0 steps flagged

No significant circularity: the Iyer–Wald entropy is computed from the stated action, field equations, and Wald formula, without fitting the quoted entropy-area relation or importing it as an input.

full rationale

The derivation chain is self-contained. The action (1) and the f(R,K) choice (2) are inputs taken from ref. [16] (which is not by the present authors). The field equations (3) are obtained by variation, the perturbative metric (10)-(11) solves the resulting equations (12), and the Wald entropy (23) is evaluated using the background derivatives (7), yielding Eq. (29). The comparison with the entropy-area relation of [16] occurs only after the derivation: the paper states the functional dependence 'coincides' but explicitly notes that the coefficient and the n dependence differ from [16] (the correction vanishes at n=1 here, and at n=1/2 there), so Eq. (29) is not a renaming or a fitted reproduction of the cited result. The self-citations [13,31,39,45] are used as contextual examples (DGP equivalence, MOND limit, logarithmic corrections) and do not support the central entropy computation. The paper's own scoping note (footnote 4) limits the analysis to the Noether-charge entropy rather than the first law, and its use of the unperturbed horizon for the higher-curvature term is a consistency/correctness question about O(λ) horizon-displacement effects, not a circular reduction: the correction coefficient is computed from the Lagrangian and background geometry, not assumed from the target entropy relation.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 0 invented entities

The paper introduces no new dynamical entities. It relies on standard GR background, the Iyer-Wald formalism, and the stated perturbative smallness of λ. The parameters λ, n, ℓ are theory parameters, not fitted to data.

free parameters (3)
  • λ = not specified (assumed |λ|≪1)
    Dimensionless coupling controlling the higher-curvature term; perturbative expansion parameter chosen by hand to be small.
  • n = free parameter
    Dimensionless exponent in f=-X^{1-n}; controls the entropy power (n+1) and the asymptotic behavior of the metric corrections.
  • ℓ = not specified
    Length scale suppressing the higher-curvature terms; enters the effective coupling λ(M/ℓ)^{2n}.
axioms (4)
  • domain assumption Schwarzschild spacetime is a valid background solution of the zeroth-order theory
    The perturbative expansion in Section III takes g^(0) as Schwarzschild with R^(0)_μν=0, which is the vacuum solution of the leading Einstein-Hilbert term.
  • standard math The Iyer-Wald Noether charge formula applies to this higher-derivative Lagrangian
    Section IV uses Eq. (23) to compute entropy from ∂L/∂R_{μνρσ}, standard for diffeomorphism-invariant higher-curvature theories.
  • standard math On the Schwarzschild background, ∇_α∇_β(f_K R_{μαβν}) reduces to (∇_α∇_β f_K) R_{μαβν} via Bianchi identities and Ricci-flatness
    The paper states this in Section III to simplify Eq. (4); it depends on R^(0)_μν=0 and the vanishing of the divergence of the Riemann tensor.
  • domain assumption The perturbative expansion in λ commutes with taking the horizon limit; the O(λ) horizon shift only affects entropy at O(λ^2)
    Section IV argues that because the higher-curvature entropy term is already O(λ), evaluating on the unperturbed horizon is sufficient to first order.

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0 comments
read the original abstract

We investigate static, spherically symmetric black hole solutions in a class of higher-curvature gravitational theories described by a Lagrangian that depends on the Ricci and Kretschmann scalars. The modified Einstein equations contain higher-order curvature terms. We develop a perturbative scheme to look for a black hole solution that deviates parametrically from the Schwarzschild geometry. We obtain first-order corrections to the metric by solving the resulting coupled differential equations. The corresponding Iyer--Wald entropy is then evaluated consistently to first order in the perturbative coupling. We show that the higher-curvature contribution gives rise to a power-law correction to the Bekenstein--Hawking entropy.

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