arxiv: 2604.04731 · v1 · submitted 2026-04-06 · 🌀 gr-qc

Recognition: no theorem link

Subtleties in non-equilibrium horizon thermodynamics of modified gravity theories

Vishnu A Pai , Vishnu S Namboothiri , Titus K Mathew

Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:32 UTC · model grok-4.3

classification 🌀 gr-qc

keywords non-equilibrium thermodynamicshorizon entropymodified gravityRindler horizonapparent horizonf(R) gravityscalar-tensor gravityBianchi identity

0 comments

The pith

In modified gravity the extra entropy term arises from the Bianchi identity for Rindler horizons but must be inserted by hand for apparent horizons to recover the Friedmann equations.

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

The paper compares two formulations that apply a Clausius-like relation with an extra entropy-production term to horizons in theories such as f(R) and scalar-tensor gravity. In the local Rindler approach the extra term follows automatically once the Bianchi identity is imposed for consistency. In the cosmological apparent-horizon approach the same-looking term is added specifically so that the first law reproduces the modified Friedmann equations. Because of this difference the term enters the dynamical field equations directly in one case and remains outside them in the other. The analysis also shows that equilibrium or non-equilibrium descriptions can be obtained simply by changing the choice of thermodynamic variables, so no single thermodynamic picture is forced by the geometry.

Core claim

Even though both the Eling-Guedens-Jacobson Rindler-horizon framework and the apparent-horizon formulation in FLRW spacetimes employ identical entropy balance relations that resemble non-equilibrium thermodynamics, the entropy-production term originates from consistency requirements tied to the Bianchi identity in the first case and is introduced solely to recover the Friedmann equations in the second. As a result the non-equilibrium contribution affects the dynamical equations of gravity directly only in the apparent-horizon treatment. Thermodynamic descriptions of horizons in modified gravity are therefore not unique; equilibrium and non-equilibrium pictures arise from different choices of

What carries the argument

The entropy-production term that augments the Clausius relation on horizons, whose origin is traced to the Bianchi identity in one framework and to the requirement of matching the Friedmann equations in the other.

If this is right

Only the apparent-horizon version places the extra entropy term inside the dynamical equations of gravity.
The Rindler version keeps the extra term outside the field equations because it follows from an identity rather than from a matching condition.
Changing the thermodynamic variables can convert a non-equilibrium description into an equilibrium one in either framework.
Any consistent thermodynamic foundation for gravity beyond general relativity must distinguish the two origins of the extra term.

Where Pith is reading between the lines

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

The distinction suggests that horizon thermodynamics in modified gravity may require an additional selection rule to decide which framework applies to a given spacetime.
Cosmological models built on the apparent-horizon approach could acquire extra effective stress-energy contributions that the Rindler approach does not produce.
Tests that compare predicted horizon temperatures or entropy flows in modified-gravity cosmologies against observations could discriminate between the two constructions.

Load-bearing premise

That the entropy balance relations are identical in both approaches and that a direct comparison between local Rindler horizons and cosmological apparent horizons remains valid despite their different physical settings.

What would settle it

An explicit calculation that inserts the Bianchi-derived entropy term into the apparent-horizon first law and checks whether the resulting equations still reproduce the correct modified Friedmann equations without further adjustment.

Figures

Figures reproduced from arXiv: 2604.04731 by Titus K Mathew, Vishnu A Pai, Vishnu S Namboothiri.

read the original abstract

Thermodynamic interpretations of gravity often arise from applying the Clausius relation to spacetime horizons. In modified gravity theories with higher-order equations of motion, such as f(R) and scalar-tensor gravity, this relation generally acquires additional entropy-production term. In this context, two distinct formulations have been proposed in literature: the non-equilibrium approach of Eling, Guedens, and Jacobson based on local Rindler horizons, and the thermodynamic formulation of cosmological apparent horizons in FLRW spacetimes. In this article, we present a detailed analysis of these approaches, and show that, even though both employ identical entropy balance relations that resemble non-equilibrium thermodynamics, the exact origin and role of each entropy-production term is fundamentally different. In the Rindler-horizon framework the extra term follows directly from consistency requirements related to the Bianchi identity, whereas in the apparent-horizon approach it is introduced solely to recover the Friedmann equations. Furthermore, we will see that the latter non-equilibrium contribution enters directly into dynamical equations of gravity, while the former does not. Finally, we also highlight the fact that thermodynamic descriptions of horizons in such modified gravity are not unique, and that equilibrium, and non-equilibrium descriptions can arise from different choices of thermodynamic variables. A clear understanding of these distinctions is therefore crucial for establishing a consistent and physically meaningful thermodynamic foundation for gravity beyond general relativity.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper usefully flags that the extra entropy term in two non-equilibrium horizon setups has different origins even when the balance equations look the same.

read the letter

The central point is that the non-equilibrium term traces back to the Bianchi identity in the local Rindler construction but is inserted to recover the Friedmann equations in the apparent-horizon FLRW version. The authors also note that equilibrium and non-equilibrium descriptions can both be obtained from the same physics by different choices of thermodynamic variables. This bookkeeping is the actual new element; the underlying equations are taken from the cited literature rather than re-derived here. The comparison itself is done clearly enough that a reader already familiar with Eling-Guedens-Jacobson and the apparent-horizon papers can follow where the divergence appears and what it implies for whether the term enters the dynamical equations. The paper stays within its narrow lane and does not overclaim broader consequences. The soft spot is the assumption that the entropy balance relations are mathematically identical once the extra term is isolated. The abstract asserts this, but the distinction loses force if the two relations already differ in which entropy is used, which area element appears, or how the heat flux is defined. The full text needs to display the two equations side by side without context-dependent redefinitions; if that step is only sketched, the provenance argument becomes less secure. No new data or formal proofs are supplied, so the strength rests entirely on the transparency of the comparison. This is for the small set of people working on horizon thermodynamics in modified gravity. A reader who knows the two source papers will get immediate value from the clarification; others will find it too specialized. I would send it to peer review so that referees can check whether the equations really line up as claimed and whether the distinction is worth recording.

Referee Report

1 major / 0 minor

Summary. The paper analyzes two non-equilibrium thermodynamic approaches to horizons in modified gravity (f(R) and scalar-tensor theories): the Eling-Guedens-Jacobson construction based on local Rindler horizons and the apparent-horizon formulation in FLRW spacetimes. It claims that both employ identical entropy balance relations (Clausius-type with an extra entropy-production term), but the term's origin differs—arising from Bianchi-identity consistency requirements in the Rindler case versus being inserted ad hoc to recover the Friedmann equations in the apparent-horizon case—with the latter entering the gravitational dynamical equations while the former does not. The paper further notes that horizon thermodynamic descriptions in modified gravity are non-unique, as equilibrium or non-equilibrium forms can result from different choices of thermodynamic variables.

Significance. If the distinctions are rigorously established, the work clarifies important subtleties in applying thermodynamic interpretations to gravity beyond general relativity, helping to prevent conflation of frameworks that appear similar but differ in foundational origins and dynamical implications. Credit is due for explicitly addressing the non-uniqueness of thermodynamic descriptions arising from variable choices, which provides a useful cautionary perspective.

major comments (1)

[Abstract and comparison of the two frameworks] The central claim requires that the entropy balance relations (including the specific entropy, horizon area element, and heat-flux projection) take exactly the same functional form in both frameworks so that the only difference is the provenance of the extra term. The abstract asserts this identity, but the manuscript must exhibit the two relations side-by-side (with explicit equations) without context-dependent redefinitions; absent such a direct comparison, the subsequent statements about differing origins, roles, and dynamical entry lose force.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for a more explicit side-by-side comparison of the entropy balance relations. We have revised the manuscript to incorporate this direct presentation, which we believe strengthens the clarity of our analysis regarding the differing origins and roles of the entropy-production terms.

read point-by-point responses

Referee: The central claim requires that the entropy balance relations (including the specific entropy, horizon area element, and heat-flux projection) take exactly the same functional form in both frameworks so that the only difference is the provenance of the extra term. The abstract asserts this identity, but the manuscript must exhibit the two relations side-by-side (with explicit equations) without context-dependent redefinitions; absent such a direct comparison, the subsequent statements about differing origins, roles, and dynamical entry lose force.

Authors: We agree that an explicit side-by-side exhibition of the relations is necessary to make the central claim fully rigorous. In the revised manuscript we have added a new table (Table 1) and accompanying paragraph in Section 3 that places the two entropy balance equations directly adjacent, using identical notation throughout: both frameworks are written as δQ = T dS + T δS_prod, where S is the horizon entropy (proportional to area in each case), dA is the area element, and the heat flux is the same projected component q^a k_a. No variables are redefined between the two presentations. The text then reiterates that the functional forms are therefore identical and that the sole distinction lies in the provenance of δS_prod (Bianchi-identity consistency for the Rindler construction versus ad-hoc insertion to recover the Friedmann equations for the apparent-horizon case). We have also clarified the distinct dynamical roles of the production term in each framework. These additions directly address the comment. revision: yes

Circularity Check

0 steps flagged

No circularity: comparison rests on independent mathematical identities

full rationale

The paper's central distinction—that the two frameworks employ identical entropy-balance forms but differ in the provenance of the extra term (Bianchi consistency vs. ad-hoc insertion)—is advanced by direct comparison of the relevant equations rather than by any self-definitional loop, fitted-parameter renaming, or load-bearing self-citation. The abstract and analysis invoke standard horizon thermodynamics (Clausius relation, Wald entropy, etc.) and previously published derivations; no step reduces the claimed identity or the differing roles to an input that is itself defined by the output. Self-citations, if used for background, are not required to establish the uniqueness or the Bianchi-identity origin. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis assumes standard properties of modified gravity theories such as the validity of the Bianchi identity and the form of the field equations in f(R) and scalar-tensor theories.

axioms (2)

standard math The Bianchi identity holds in modified gravity and imposes consistency requirements on the entropy balance for local Rindler horizons.
Invoked as the direct source of the extra entropy-production term in the Rindler-horizon framework.
domain assumption The Friedmann equations must be recovered from the thermodynamic relations in the apparent-horizon approach.
Used to justify introducing the entropy-production term in the cosmological apparent-horizon formulation.

pith-pipeline@v0.9.0 · 5544 in / 1445 out tokens · 68532 ms · 2026-05-10T19:32:17.678821+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

17 extracted references · 12 canonical work pages

[1]

Jacobson, Phys

T. Jacobson, Phys. Rev. Lett.75, 1260 (1995), gr- qc/9504004

work page arXiv 1995
[2]

Eling, R

C. Eling, R. Guedens, and T. Jacobson, Phys. Rev. Lett. 96, 121301 (2006), gr-qc/0602001

work page arXiv 2006
[3]

Chirco and S

G. Chirco and S. Liberati, Phys. Rev. D81, 024016 (2010), 0909.4194

work page arXiv 2010
[4]

B. Wang, Y. Gong, and E. Abdalla, Phys. Rev. D74, 083520 (2006), gr-qc/0511051

work page arXiv 2006
[5]

Cai and S

R.-G. Cai and S. P. Kim, JHEP02, 050 (2005), hep- th/0501055

work page arXiv 2005
[6]

Akbar and R.-G

M. Akbar and R.-G. Cai, Phys. Rev. D75, 084003 (2007), hep-th/0609128

work page arXiv 2007
[7]

D. W. Tian and I. Booth, Phys. Rev. D92, 024001 (2015), 1411.6547

work page arXiv 2015
[8]

Cai and L.-M

R.-G. Cai and L.-M. Cao, Phys. Rev. D75, 064008 (2007), gr-qc/0611071

work page arXiv 2007
[9]

D. W. Tian and I. Booth, Phys. Rev. D90, 104042 (2014), URLhttps://link.aps.org/doi/10. 1103/PhysRevD.90.104042

2014
[10]

Gong and A

Y. Gong and A. Wang, Phys. Rev. Lett.99, 211301 (2007), 0704.0793

work page arXiv 2007
[11]

org/10.1209/0295-5075/89/50003

K.Bamba, C.-Q.Geng, S.Nojiri, andS.D.Odintsov, Eu- rophysics Letters89, 50003 (2010), URLhttps://doi. org/10.1209/0295-5075/89/50003

work page doi:10.1209/0295-5075/89/50003 2010
[12]

Bamba, C.-Q

K. Bamba, C.-Q. Geng, and S. Tsujikawa, Physics Letters B688, 101 (2010), ISSN 0370-2693, URL https://www.sciencedirect.com/science/article/ pii/S0370269310004041

2010
[13]

Ge, Physics Letters B651, 49 (2007), ISSN 0370- 2693, URLhttps://www.sciencedirect.com/science/ article/pii/S0370269307006508

X.-H. Ge, Physics Letters B651, 49 (2007), ISSN 0370- 2693, URLhttps://www.sciencedirect.com/science/ article/pii/S0370269307006508

2007
[14]

Akbar and R.-G

M. Akbar and R.-G. Cai, Phys. Rev. D75, 084003 (2007), URLhttps://link.aps.org/doi/10. 1103/PhysRevD.75.084003

2007
[15]

S. A. Hayward, Class. Quant. Grav.15, 3147 (1998), gr- qc/9710089

work page arXiv 1998
[16]

S. W. Hawking, Phys. Rev. D13, 191 (1976)

1976
[17]

Cai, L.-M

R.-G. Cai, L.-M. Cao, Y.-P. Hu, and N. Ohta, Phys. Rev. D80, 104016 (2009), 0910.2387

work page arXiv 2009