Is The Internal Entropy of F(R)-Gravity Really An Entropy?

Bohuslav Matou\v{s}

REVIEW 4 major objections 6 minor 19 references

Is The Internal Entropy of F(R)-Gravity Really An Entropy?

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper argues that the internal entropy of F(R)-gravity, when derived from a combined entropy-functional construction, depends on the second derivative of F and so disagrees with the earlier formula; it concludes the term is a…

desk verdict A genuinely new recombination of two entropy-functional methods, but the central incompatibility claim rests on an erroneous integral and a comparison of unlike quantities. read the letter →

arxiv 2411.09276 v1 pith:76OSCZ5M submitted 2024-11-14 gr-qc

Bohuslav Matouš This is my paper · ORCID

classification gr-qc

keywords thermodynamicsofgravityF(R)-gravitymodifiedtheoriestheoryelasticityinternalentropyfunctionalhorizon

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

The paper tries to settle whether the extra term in F(R)-gravity thermodynamics that is usually called internal entropy deserves that name. It combines two constructions of the entropy functional, one based on spacetime elasticity and one based on the entropy tensor of emergent gravity, into a single functional. When the internal part of that functional is integrated on a static spherically symmetric horizon, it depends on the second derivative of F, whereas the previously derived internal-entropy formula depends only on F and its first derivative. Because the two expressions generally disagree, the paper concludes that the earlier "internal entropy" is not an entropy and that the same effects are better described as a pressure generated by higher-curvature terms. If correct, this changes which terms belong in the horizon first law for modified gravity.

What carries the argument

The central object is the combined entropy functional of Eq. (16), obtained by inserting the F(R)-gravity entropy tensor $P^{cd}_{ab}=\frac{F'}{32\pi}(\delta^c_a\delta^d_b-\delta^d_a\delta^c_b)$ into the elasticity-based entropy functional. After rewriting the first two terms as a divergence, the functional separates into the classical external entropy (19) and the internal entropy (20), whose integrand is a divergence of the vector $U^i=\xi_k\xi^k\nabla^iF'-\xi_k\xi^i\nabla^kF'$. The derivation uses the assumption that H vanishes in Eq. (13) and the displacement vector (27), $\xi^0=A/a$, $\xi^1=C/r^2$, to reduce the internal integral to Eq. (33), where the factor $\partial_r(r^2\partial_rF')$ appears. That factor is what disagrees with the expected form (28), which contains only F and F'.

What would settle it

For $F=R+\alpha R^2$, evaluating the horizon variation of Eq. (33) on the static spherically symmetric metric (23) and comparing it with Eq. (28) would settle the claim; agreement for nonzero $\alpha$ would refute the incompatibility.

Watch

Extended reading notes

Core claim

The central claim is that the internal entropy functional for F(R)-gravity, obtained by following the elasticity-based derivation after substituting the F(R)-gravity entropy tensor, is incompatible with the internal-entropy formula derived in earlier work: the integrated expression contains the second derivative of F, while the expected formula contains only F and its first derivative. The paper therefore concludes that the quantity previously identified as internal entropy is not an entropy at all, and that the interpretation of these effects as a pressure caused by higher-curvature terms is more suitable. This conclusion is reached by splitting the combined functional into a divergence that looks like the classical entropy with an F' factor and a second divergence that carries the internal part, then integrating the second divergence with a rotation-free, static spherically symmetric displacement ansatz.

Load-bearing premise

The argument depends on the assumption that energy conservation forces the coefficient H in Eq. (13) to vanish; if a non-zero H could instead balance the other terms, the derived entropy functional and the mismatch would change.

Editorial extensions

If this is right

Any horizon thermodynamic calculation that uses the old internal-entropy formula will generally disagree with the combined entropy-functional result for nonlinear F(R).
The mismatch means the higher-curvature terms in F(R)-gravity are more naturally assigned to an effective pressure term rather than to an entropy term in the horizon first law.
The classical part of the entropy functional remains proportional to F'(R), so the standard area-law horizon entropy is recovered with the familiar modification factor in the limit where F becomes R.
The combined construction has enough information to fix all coefficients without introducing an auxiliary scalar field, removing an ambiguity in the elasticity-based approach.

Reading between the lines

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

Extension: The divergence form of the internal part suggests the extra term may be a boundary contribution; one could check whether Eq. (33) changes when the integration surface is moved, which would distinguish a true bulk entropy from a pressure term.
Extension: The same construction could be applied to other modified-gravity entropy functionals; if the mismatch persists generally, it would strengthen the case that "internal entropy" is a bookkeeping term for higher-curvature pressure in a wider class of theories.
Extension: For a concrete F(R), such as F=R+alpha R^2, one could write the horizon first law with the pressure interpretation and compute the coefficient that replaces the entropy correction, giving a concrete prediction for the next-order correction to the area law.

Share X Bluesky LinkedIn Reddit HN

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

Colleague,

You should know this paper before citing it: it combines Padmanabhan's entropy tensor with Hammad's entropy functional in a new way, and claims the internal entropy term in F(R)-gravity from this combined approach is incompatible with the earlier form, so it should be viewed as pressure. The combination is genuinely new and does something useful: it fixes the coefficients in Hammad's functional without needing an extra scalar field, and the resulting external entropy has the expected F' factor. The paper is also honest, explicitly saying the comparison method has limits.

The problem is the central incompatibility is not demonstrated. Three soft spots, in increasing severity. First, the derivation imposes H=0 by requiring energy conservation, but that is a modeling choice; a nonzero H could be balanced against the gradient term. Second, Eq. (33) does not follow from Eq. (32). The metric (23) has √-g = r^2 sinϑ, not r^2 sin^2ϑ, so the angular integration gives a factor π, not 1/2, and the radial integral of ∂r(r^2∂rF') is a boundary term, so leaving it under a t-integral is incorrect. Third, and fatally, Eq. (33) is the integrated entropy S(i), while Eq. (28) is δS_i/δr_h, a variational derivative with respect to horizon radius. The paper never computes δS_i/δr_h from its own expression; it just notes second derivatives appear and declares a contradiction. That comparison is like comparing an integral to a derivative.

The earlier sections up to Eq. (16) are plausible and worth reading, but the punchline does not survive contact with the integration. A substantial revision could fix this: redo the integral, compute the variational derivative, and then see if the incompatibility actually holds.

Who is this for? Researchers in thermodynamics of gravity, particularly those working on F(R) and entropy functionals. It is a useful cautionary example of how easy it is to compare unlike objects. I would send it to peer review—the idea is fresh and the errors are repairable—but I would not cite it as evidence for the pressure interpretation until the derivation is fixed.

Referee Report

4 major / 6 minor

Summary. The paper combines Padmanabhan's emergent-gravity entropy functional with Hammad's elasticity-based functional to construct an entropy functional for F(R)-gravity. The functional is split into an 'external' part, written as a divergence, and an 'internal' remainder. Using the static spherically symmetric metric (23), the author evaluates the internal entropy and obtains Eq. (33), which appears to depend on second derivatives of F. This is contrasted with Eq. (28), the author's earlier result from a Clausius-type approach, which involves only F and F'. The paper concludes that the internal entropy found in previous works is not an entropy and that a pressure interpretation is more suitable. The referee finds that the central comparison is invalid: the integration leading to Eq. (33) is incorrect, and Eq. (33) is compared with an expression of a different type.

Significance. The question addressed—whether the higher-curvature contribution in F(R) thermodynamics is an entropy or a pressure—is interesting, and the unification of two entropy-functional frameworks is a worthwhile goal. The separation of the entropy functional into a divergence (external) part and a remainder (internal) part is a clean structural observation. However, the main conclusion is not supported by the derivation as presented. The paper contains no machine-checked proofs, no reproducibility artifacts, and no falsifiable predictions beyond the reinterpretation. If the calculation were corrected, the claimed incompatibility might disappear; as it stands, the paper's contribution is a suggestive derivation rather than an established result.

major comments (4)

[Section 4, Eqs. (32)-(33)] The volume element used in Eq. (32) is incorrect: for the metric (23) one has sqrt(-g) = r^2 sinϑ, not r^2 sin^2ϑ. Independently of this prefactor, the radial integral in Eq. (32) is a total derivative, ∫ dr ∂_r(r^2 ∂_r F') = [r^2 ∂_r F'] at the boundaries, so it cannot be left as an r-dependent factor inside the t-integral as done in Eq. (33). With the correct boundary evaluation, S(i) is proportional to the boundary term r_h^2 ∂_rF'(r_h) (assuming the contribution at infinity vanishes), which contains only the first derivative of F. The claimed dependence on second derivatives is therefore an artifact of stopping the integration before the radial part is evaluated.
[Section 4, comparison of Eq. (33) with Eq. (28)] The two expressions being compared are not of the same kind. Eq. (28) is the variational derivative δS_i/δr_h, whereas Eq. (33) is the integrated internal entropy S(i). The paper never computes δS(i)/δr_h from its own expression (33), or from the correctly integrated boundary form. Without that step, the assertion that the two 'generally do not match' is not justified; it is a comparison of an integrated quantity with a derivative with respect to the horizon radius.
[Section 3, after Eq. (13)] The condition H=0 is imposed rather than derived. Equation (13) is a sum of a gradient term and 2R_{ik}∇^iH, and there is no argument that each term must vanish separately; the two terms could in principle cancel. The entropy functional (16), the internal entropy (20), and all subsequent conclusions depend on this choice. This assumption should either be derived from a physical principle or explicitly labelled as a simplifying assumption, and its effect on the conclusion should be assessed.
[Section 3, Eq. (9)] The variation δS is stated without derivation. This variation is the basis for the conditions (10)-(15) that fix λ, E, and H, and therefore for the entire functional (16). No intermediate steps or references are provided for a nontrivial calculation involving integration by parts and the commutation of covariant derivatives. The paper is not self-contained at this load-bearing point; an appendix with the complete derivation is needed.

minor comments (6)

[Throughout] There are numerous typos and inconsistent spellings: 'Padmanbhan'/'Padamanabhan', 'disipative' for 'dissipative', 'Lammé' for 'Lamé', and 'unspeciﬁed' for 'unspecified'.
[Eq. (28)] The formula is ambiguous: the placement of parentheses around A/4 and the divisor 2κ should be clarified.
[Eqs. (7)-(8)] The notation D,i, E,k, H,l is not defined; it should be stated explicitly whether these denote ordinary partial derivatives or covariant derivatives.
[Introduction/Section 4] The comparison benchmark Eq. (28) comes from the author's previous paper [9]; the manuscript should briefly summarize the assumptions and definitions leading to (28) so the comparison is self-contained.
[Section 1] The phrase 'East Coast convention' for the metric signature (−,+,+,+) is nonstandard; consider using 'mostly-plus signature' to avoid confusion.
[Eq. (16)] The physical role of the matter stress-energy tensor T^{ik} in the entropy functional should be discussed; the functional is supposed to describe spacetime entropy, and inserting the matter content through the field equations may deserve a comment.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new entropy-functional derivation is self-contained, and the self-cited benchmark is used as an external comparison rather than as an input that forces the conclusion.

full rationale

The paper derives its entropy functional from Padmanabhan's entropy tensor and Hammad's elasticity-based functional, then extracts an internal-entropy piece and integrates it. The final comparison with Eq. (28) uses a previously published result of the same author as a benchmark, but this is not circular: Eq. (28) is an independent external target, and the derivation of Eq. (33) does not presuppose it. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and the central incompatibility claim rests on a calculation rather than on an identity constructed from the expected answer. The paper does contain serious technical weaknesses, including the questionable step from Eq. (32) to Eq. (33), the comparison of an integrated S(i) with a variational derivative delta S_i/delta r_h, and the imposed condition H=0 in Eq. (13). These are correctness concerns, not circularity: they do not make the conclusion equivalent to its inputs by definition.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a chain of modeling choices: the H=0 restriction, the Schwarzschild-like metric ansatz, the displacement solution from the elasticity conditions, and the benchmark (28) from the author's own previous work. No new entities are postulated.

free parameters (5)

Lambda (cosmological constant) = 0
Set to zero after Eq. (15); affects the stress-energy tensor substitution and would contribute to the internal entropy if retained.
beta = 0
Integration constant for D; set to zero because only derivatives of D appear in the functional.
gamma = 0
Integration constant for E; set to zero because only derivatives of E appear in the functional.
A (displacement amplitude) = arbitrary constant
Integration constant in the solution for ξ0; cancels or factors out in the entropy expressions.
C (displacement amplitude) = arbitrary constant
Integration constant in the solution for ξ1; cancels or factors out in the entropy expressions.

assumptions (6)

domain assumption Extremization of the entropy functional yields the gravitational field equations
Inherited from Padmanabhan's emergent gravity paradigm; used to derive the consistency conditions and Eq. (15).
domain assumption Vanishing local rotation tensor, ∇iξj - ∇jξi = 0
Inherited from Hammad's elasticity approach; used to solve for the displacement vector in Section 4.
domain assumption Generalized Hooke's law (22) with zero external forces, with independent vanishing of the µ- and ν-parts
Used to determine the displacement vector (27); the ν-part is allowed to be nonzero in F(R)-gravity.
domain assumption Static spherically symmetric metric ansatz ds^2 = -a dt^2 + 1/a dr^2 + r^2 dΩ^2
Restricts the horizon geometry; needed to integrate the internal entropy.
ad hoc to paper Energy conservation in Eq. (13) forces the gradient term and 2R_ik∇^iH to vanish separately, implying H=0
This is the paper's key modeling choice; it is asserted, not derived, and it is load-bearing for the entropy functional (16).
domain assumption The expected internal entropy is given by Eq. (28) from prior literature
Used as the benchmark in the comparison; comes from Ref. [9], which is authored by the present paper's author.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Is The Internal Entropy of F(R)-Gravity Really An Entropy?." pith.science (2026). https://pith.science/paper/76OSCZ5M

@misc{pith2026241109276,
  author       = {Pith},
  title        = {Pith review of: Is The Internal Entropy of F(R)-Gravity Really An Entropy?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/76OSCZ5M}},
  note         = {Machine review of arXiv:2411.09276}
}

read the original abstract

This paper connects two methods for finding the functional of entropy in F(R)-Gravity: Padmanabhan's and Hammad's. The resulting approach is simple to follow and yields entropy functional, which can be separated into two parts. The part unknown in General Relativity is often called in the literature as an internal entropy and this paper points on incompatibility between the internal entropy found from the entropy functional and the one found using conventional approach.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 6 canonical work pages

[1]

f(R) theories

A. De Felice and S. Tsujikawa, “f(R) theories” , Living Rev. Rel. 13 (2010), 3 doi:10.12942/lrr-2010-3 [arXiv:1002.4928 [gr- qc]]

arXiv 2010
[2]

Uniﬁed cosmic history in modiﬁed gravity: from F(R) theory to Lorentz non-invariant models

S. Nojiri and S. D. Odintsov, “Uniﬁed cosmic history in modiﬁed gravity: from F(R) theory to Lorentz non-invariant models” , Phys. Rept. 505, 59 (2011) doi:10.1016/j.physrep.2011.04.001 [arXiv:1011.0544 [g r-qc]]

arXiv 2011
[3]

Modiﬁed Gravity Theories on a Nutshell: Inﬂation, Bounce and Late-time Evolu- tion

S. Nojiri, S. D. Odintsov and V. K. Oikonomou, “Modiﬁed Gravity Theories on a Nutshell: Inﬂation, Bounce and Late-time Evolu- tion”, Phys. Rept. 692 (2017) 1 doi:10.1016/j.physrep.2017.06.001 [arXiv:1705.11098 [ gr-qc]]

arXiv 2017
[4]

A Panorama of Viable $F(R)$ Gravity Dark Energy Models

V. K. Oikonomou and I. Giannakoudi, “A panorama of viable F(R) gravity dark energy models” , Int. J. Mod. Phys. D 31 (2022) no.09, 2250075 doi:10.1142/S0218271822500754 [arXiv:2205.08599 [gr-q c]]. 7

work page Pith review arXiv 2022
[5]

Eling, R

C. Eling, R. Guedens and T. Jacobson, Non-equilibrium thermodynamics of spacetime, Phys. Rev. Lett. 96 121301 (2006) doi:10.1103/PhysRevLett.96.121301 [arXiv:gr-qc/0602001 [gr-qc]]

arXiv 2006
[6]

Chirco and S

G. Chirco and S. Liberati, Non-equilibrium Thermodynamics of Spacetime: The Role of Gravitational Dissipation , Phys. Rev. D 81 024016 (2010) doi:10.1103/PhysRevD.81.024016 [arXiv:0909.4194 [gr-q c]]

arXiv 2010
[7]

Bamba, C

K. Bamba, C. Q. Geng, S. Nojiri and S. D. Odintsov, Equivalence of modiﬁed grav- ity equation to the Clausius relation , EPL 89 no.5, 50003 (2010) doi:10.1209/0295- 5075/89/50003 [arXiv:0909.4397 [hep-th]]

arXiv 2010
[8]

Akbar and R

M. Akbar and R. G. Cai, Thermodynamic Behavior of Field Equations for f(R) Gravity , Phys. Lett. B 648 243-248 (2007) doi:10.1016/j.physletb.2007.03.005 [arXiv:gr-qc/0612089 [gr-qc]]

arXiv 2007

Show all 19 references

[9]

Matouš, Clausius equation for horizons in F(R)-gravity , Indian J

B. Matouš, Clausius equation for horizons in F(R)-gravity , Indian J. Phys. 98 (2024) no.6, 2249-2255 doi:10.1007/s12648-023-02993-5 [arXiv: 2111.14753 [gr-qc]]

2024 arXiv
[10]

General Relativity from a Thermodynamic Perspective

T. Padmanabhan, “General Relativity from a Thermodynamic Perspective” , Gen. Rel. Grav. 46 (2014), 1673 doi:10.1007/s10714-014-1673-7 [arXiv:1312.3253 [gr-qc ]]

2014 arXiv
[11]

Emergent Gravity Paradigm: Recent Progress

T. Padmanabhan, “Emergent Gravity Paradigm: Recent Progress” , Mod. Phys. Lett. A 30 (2015) no.03n04, 1540007 doi:10.1142/S0217732315400076 [arXiv:1410.6285 [gr-qc ]]

2015 arXiv
[12]

Distribution function of the Atoms of Spacetime and the Natu re of Gravity

T. Padmanabhan, “Distribution function of the Atoms of Spacetime and the Natu re of Gravity” , Entropy 17 (2015), 7420-7452 doi:10.3390/e17117420 [arXiv:1508.06 286 [gr-qc]]

2015 doi
[13]

Gravity and/is Thermodynamics

T. Padmanabhan, “Gravity and/is Thermodynamics” , Curr. Sci. 109 (2015), 2236-2242 doi:10.18520/v109/i12/2236-2242 [arXiv:1512.06546 [gr -qc]]

2015 arXiv
[14]

Exploring the Nature of Gravity

T. Padmanabhan, “Exploring the Nature of Gravity” , [arXiv:1602.01474 [gr-qc]]

arXiv
[15]

F. Hammad, Thermodynamics of Black Holes from an Entropy Functional: An Other Approach Using Generalized Elasticity , Int J Theor Phys 49, 1055–1064 (2010) doi:10.1007/s10773-010-0284-7

2010 doi
[16]

Hammad, Modiﬁed gravity from an entropy functional , Int

F. Hammad, Modiﬁed gravity from an entropy functional , Int. J. Mod. Phys. D 23 (2014) no.09, 1450073 doi:10.1142/S0218271814500734 [ar Xiv:1408.6577 [gr-qc]]

2014 arXiv
[17]

Entropy of null surfaces and dynamics of spacetime

T. Padmanabhan and A. Paranjape, “Entropy of null surfaces and dynamics of spacetime” , Phys. Rev. D 75 (2007), 064004 doi:10.1103/PhysRevD.75.064004 [arXiv:gr-qc/0701003 [ gr-qc]]

2007 arXiv
[18]

Thermodynamical interpretation of the ge- ometrical variables associated with null surfaces

S. Chakraborty and T. Padmanabhan, “Thermodynamical interpretation of the ge- ometrical variables associated with null surfaces” , Phys. Rev. D 92 (2015) no.10, 104011 doi:10.1103/PhysRevD.92.104011 [arXiv:1508.040 60 [gr-qc]]

2015 doi
[19]

Lanczos-Lovelock models of gravity,

T. Padmanabhan and D. Kothawala, “Lanczos-Lovelock models of gravity,” , Phys. Rept. 531 (2013), 115-171 doi:10.1016/j.physrep.2013.05.007 [arXiv:1302.2151 [g r-qc]]. 8

2013 arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

f(R) theories

A. De Felice and S. Tsujikawa, “f(R) theories” , Living Rev. Rel. 13 (2010), 3 doi:10.12942/lrr-2010-3 [arXiv:1002.4928 [gr- qc]]

arXiv 2010

[2] [2]

Uniﬁed cosmic history in modiﬁed gravity: from F(R) theory to Lorentz non-invariant models

S. Nojiri and S. D. Odintsov, “Uniﬁed cosmic history in modiﬁed gravity: from F(R) theory to Lorentz non-invariant models” , Phys. Rept. 505, 59 (2011) doi:10.1016/j.physrep.2011.04.001 [arXiv:1011.0544 [g r-qc]]

arXiv 2011

[3] [3]

Modiﬁed Gravity Theories on a Nutshell: Inﬂation, Bounce and Late-time Evolu- tion

S. Nojiri, S. D. Odintsov and V. K. Oikonomou, “Modiﬁed Gravity Theories on a Nutshell: Inﬂation, Bounce and Late-time Evolu- tion”, Phys. Rept. 692 (2017) 1 doi:10.1016/j.physrep.2017.06.001 [arXiv:1705.11098 [ gr-qc]]

arXiv 2017

[4] [4]

A Panorama of Viable $F(R)$ Gravity Dark Energy Models

V. K. Oikonomou and I. Giannakoudi, “A panorama of viable F(R) gravity dark energy models” , Int. J. Mod. Phys. D 31 (2022) no.09, 2250075 doi:10.1142/S0218271822500754 [arXiv:2205.08599 [gr-q c]]. 7

work page Pith review arXiv 2022

[5] [5]

Eling, R

C. Eling, R. Guedens and T. Jacobson, Non-equilibrium thermodynamics of spacetime, Phys. Rev. Lett. 96 121301 (2006) doi:10.1103/PhysRevLett.96.121301 [arXiv:gr-qc/0602001 [gr-qc]]

arXiv 2006

[6] [6]

Chirco and S

G. Chirco and S. Liberati, Non-equilibrium Thermodynamics of Spacetime: The Role of Gravitational Dissipation , Phys. Rev. D 81 024016 (2010) doi:10.1103/PhysRevD.81.024016 [arXiv:0909.4194 [gr-q c]]

arXiv 2010

[7] [7]

Bamba, C

K. Bamba, C. Q. Geng, S. Nojiri and S. D. Odintsov, Equivalence of modiﬁed grav- ity equation to the Clausius relation , EPL 89 no.5, 50003 (2010) doi:10.1209/0295- 5075/89/50003 [arXiv:0909.4397 [hep-th]]

arXiv 2010

[8] [8]

Akbar and R

M. Akbar and R. G. Cai, Thermodynamic Behavior of Field Equations for f(R) Gravity , Phys. Lett. B 648 243-248 (2007) doi:10.1016/j.physletb.2007.03.005 [arXiv:gr-qc/0612089 [gr-qc]]

arXiv 2007