Friedmann Equations of FRW Universe in Scalar-tensor Gravity, f(R) Gravity and First Law of Thermodynamics

M. Akbar; Rong-Gen Cai

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Friedmann Equations of FRW Universe in Scalar-tensor Gravity, f(R) Gravity and First Law of Thermodynamics

M. Akbar, Rong-Gen Cai

High Energy Physics - Theory · Submitted 16 Feb 2006 · 16 references

Pith's verdict: A short follow-up to Cai-Kim whose positive half is a repackaging and whose negative half is asserted rather than shown, and is not a fair test as constructed.

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arXiv IDhep-th/0602156 (version 2)

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CitationM. Akbar, Rong-Gen Cai. "Friedmann Equations of FRW Universe in Scalar-tensor Gravity, f(R) Gravity and First Law of Thermodynamics." arXiv:hep-th/0602156 (2006). https://pith.science/paper/HAAMXDN3

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Short version: this is a follow-up to Cai-Kim asking whether the first-law derivation of the Friedmann equations survives in scalar-tensor and f(R) gravity. The answer they report is no for the modified-gravity entropy formulas, yes if you keep S = A/4G and fold the scalar and curvature terms into an effective stress tensor. The yes half is a repackaging of Einstein gravity with a source; the no half is the part that would matter, and it isn't shown. What's actually new: Cai-Kim got success in Gauss-Bonnet and Lovelock using those theories' entropy formulas. Here the same move fails. That is a real data point about the scope of the method, and the authors are honest that it is a scope limitation rather than a breakdown. Where it's soft: the failure is asserted, not demonstrated. The text says 'we are no able to derive' and 'we fail to re-produce' and never displays the algebra. More seriously, the comparison isn't apples-to-apples. The flux in Eq. (18) is built from the matter perfect fluid alone — Eq. (17) has only rho and p. The entropy in Eq. (19) carries f(phi), so dS picks up an f-dot term with no counterpart in the flux. A mismatch is then guaranteed by construction. The consistent calculation — full flux including T^phi paired with S = A f/4G — is never done, and that is the only version that would test non-universality. Same for f(R) with T^curv and S = A f'(R)/4G. So the headline negative result doesn't land as stated. The positive half is close to tautological: Eq. (10) defines T-tilde so that G = 8 pi G T-tilde holds identically, and then Cai-Kim applies verbatim. That is a clean way to see the structure, not a new derivation. Also, the step from (21) to (25) uses the effective continuity equation (22), which is asserted rather than derived from the scalar equation and Bianchi identity, and the integration constant is set to zero by hand. Minor, but it should be flagged. Who it's for: people in the thermodynamics-of-gravity and emergent-gravity corner who want to know where the Cai-Kim method stops working. It is a short note with modest scope. Recommendation: worth a referee, but I would want the failed algebra shown and the consistent flux-entropy pairing done before I would believe the negative claim.

Referee report

Major objections (5)

[p. 5, after Eq. (20)] The negative result is asserted, not shown. The text states 'we are no able to derive the Friedmann equations' and later 'we fail to re-produce corresponding Friedmann equations' without displaying the algebra that fails. Since this is the load-bearing claim of the paper, the authors should show explicitly the residual terms that prevent the first law from closing when S = A f(φ)/4G is used, and similarly for S = A f'(R)/4G. Without this, the reader cannot distinguish a genuine non-universality from an algebraic or bookkeeping error.
[Eqs. (17)-(19)] The comparison is not apples-to-apples. The energy flux in Eq. (18) is built from the matter perfect fluid alone (Eq. 17 contains only ρ and p), while the entropy in Eq. (19) carries f(φ), so dS = (f/4G) dA + (A/4G) f' dφ includes a scalar contribution with no counterpart in the flux. A mismatch is then guaranteed by construction. The consistent calculation — full flux including T^φ_μν paired with S = A f(φ)/4G — is never performed. The same gap applies to f(R) with T^curv_μν and S = A f'(R)/4G. The authors should either perform this calculation or soften the claim to the specific inconsistent pairing they actually test.
[Eqs. (10) and (27)] The positive result is close to construction. Eq. (10) defines T̃_μν = (T^m_μν + T^φ_μν)/f(φ) so that G_μν = 8πG T̃_μν holds identically, and Eq. (27) does the same for f(R). The Cai-Kim derivation with S = A/4G then applies verbatim with T̃ in place of T. The recovered equations (35) and (36) are the same field equations rewritten. The paper should state explicitly that this is a rewriting rather than an independent derivation, and should not present the success as evidence for the area law over the black-hole entropy formula.
[Eq. (22)] The continuity equation for the effective fluid is asserted rather than derived. Eq. (22) is what allows Eq. (21) to be integrated into Eq. (25), and the paper also sets the resulting integration constant to zero by hand. Without a derivation of Eq. (22) from the scalar field equation (8) and the Bianchi identity ∇^μ G_μν = 0, the recovery of the Friedmann equations is not self-contained. The authors should supply this derivation or cite it.
[Conclusion, p. 8] The conclusion states that 'the simplicity of the ansatz for the relation between the entropy and horizon area is not enough to reproduce corresponding Friedmann equations in those theories.' This overreaches the evidence. The failure observed could be due to the inconsistent pairing of a matter-only flux with a scalar-inclusive entropy rather than to the entropy formula itself. The conclusion should be revised to reflect what is actually shown, or the consistent calculation should be performed.

Minor objections (7)

[p. 5, p. 3, p. 8] Typos: 'we are no able' (p. 5) should be 'we are not able'; 'one the other hand' (p. 3) should be 'on the other hand'; 'every things' (p. 8) should be 'everything'; 'FR W' spacing appears throughout and should be 'FRW'.
[Eq. (21)] Eq. (21) is identical to Eq. (12). The paper should state explicitly that the first law reproduces the second Friedmann equation, and that the first Friedmann equation follows by integration with the continuity equation.
[Eq. (20)] The temperature ansatz T = 1/2πr_A is assumed without discussion of whether the surface gravity at the apparent horizon in scalar-tensor or f(R) gravity is modified. The authors should comment on this.
[Ref. [14]] Reference [14] for the entropy formula should be checked for the (n+1)-dimensional case and the specific f(φ) coupling used here.
[Abstract] The abstract's 'discuss the implication of results' is vague; the abstract should state the main conclusion.
[Eq. (7)] The paper should discuss the sign conventions and the definition of ∇² used in Eq. (7).
[References] The paper should cite other work on thermodynamics of apparent horizons in modified gravity, e.g., Eling-Guedens-Jacobson on non-equilibrium thermodynamics, if relevant.

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The paper

The paper claims

Applying the first law of thermodynamics at the apparent horizon of an expanding universe reproduces the Friedmann equations in scalar-tensor and $f(R)$ gravity only if one keeps the quarter-area entropy law and treats the new scalar and曲率—

The reading

The paper asks whether a trick that works in Einstein gravity — deriving the Friedmann equations of an expanding universe purely from the first law of thermodynamics applied to the apparent horizon — survives in two modifications of gravity that add new degrees of freedom: scalar-tensor gravity and $f(R)$ gravity. In the earlier work it extends, the quarter-area entropy law plus the temperature $T = 1/(2π r_A)$ was enough to reproduce the Friedmann equations in Einstein, Gauss-Bonnet and Lovelock gravity. Here the authors find that if you instead plug in the entropy formulas those two theories give for static black holes, $S = A f(φ)/4G$ and $S = A f'(R)/4G$, the derivation fails: the thermodynamic identity does not return the correct equations of motion. It works again if you keep the plain quarter-area law and move the extra scalar and curvature terms into an effective energy-momentum tensor, so that the new physics enters as matter rather than as entropy. The upshot is a limit on the thermodynamic reading of gravity: which entropy you assign to a horizon, not just the first law, decides whether the equations of motion come out.

What carries the argument

Two ingredients carry the argument. The first is the apparent horizon of an FRW universe, the surface $r_A = 1/√(H² + k/a²)$ defined by $h^{ab} ∂_a r ∂_b r = 0$, which the authors treat as the causal horizon that actually admits a thermodynamic description in a dynamical spacetime, unlike the event or particle horizon. The second is the Clausius relation $-dE = T dS$ applied across that horizon during an infinitesimal time $dt$, where the energy flux is built from Hayward's work density $W = -½ T^{ab} h_{ab}$ and energy-supply vector $ψ_a = T^a{}_b ∂_a r + W ∂_a r$, giving $-dE = A(ρ+p) H r_A dt$ for a perfect fluid. The test is then whether inserting a given entropy formula into $T dS$, with $T = 1/(2π r_A)$, reproduces the Friedmann equations that follow from varying the action.

Extended reading notes

Core claim

The central claim is a negative result with a constructive repair. Working in an $(n+1)$-dimensional FRW universe with arbitrary spatial curvature $k$, the authors apply $-dE = T dS$ at the apparent horizon $r_A = 1/√(H² + k/a²)$, taking the temperature to be the standard $T = 1/(2π r_A)$ and the heat flow to be the energy crossing the horizon, computed from Hayward's work density and energy-supply vector. When they use the black-hole entropy each theory prescribes — $S = A f(φ)/4G$ for scalar-tensor gravity and $S = A f'(R)/4G$ for $f(R)$ gravity — the resulting equation does not match the Friedmann equation obtained by solving the field equations. When they instead keep $S = A/4G$ and fold the scalar-field terms (or the extra curvature terms) into an effective perfect fluid with its own energy density and pressure, the first law reproduces both Friedmann equations, including the $k/a²$ curvature term, provided the effective fluid obeys a continuity equation and the integration constant is set to zero. The authors read this as evidence that the entropy-area relation alone does not carry the dynamical information of the new degrees of freedom.

Load-bearing premise

The repaired derivation needs the effective fluid to satisfy a continuity equation, $ρ̇ + nH(ρ+p) = 0$, which the paper states rather than derives from the scalar field equation and the Bianchi identity, and it also sets the resulting integration constant to zero by hand.

Editorial extensions

If this is right

If the result holds, the thermodynamic derivation of gravitational field equations is not universal: a modified theory's black-hole entropy formula cannot simply be transplanted onto a cosmological apparent horizon.
The derivation reproduces the Friedmann equations for any spatial curvature $k = -1, 0, 1$ in both theories, so the method covers closed and open FRW universes, not only flat ones.
Because the repair works by relabelling new degrees of freedom as effective matter, any modified gravity whose extra terms can be written as a perfect fluid with a continuity equation should admit the same treatment.
The authors single out the generalized second law of thermodynamics in these two theories, using the entropy formulas $S = A f(φ)/4G$ and $S = A f'(R)/4G$, as the natural next test.

Reading between the lines

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

The failure pattern suggests the dividing line is not 'modified gravity versus Einstein gravity' but whether the extra degrees of freedom can be encoded in a local effective stress tensor; theories where they cannot be so encoded would be the sharper test of the claim.
That the quarter-area law keeps working once the new fields are moved into matter hints that the area law is tied to the Einstein-Hilbert form of the gravitational action rather than to the horizon itself, a statement that could be checked in theories with extra propagating tensor modes.
The un-derived continuity equation is the natural place for an independent check: deriving it from the scalar field equation and the Bianchi identity would turn the recovery from a consistency observation into a proof.
Because the temperature ansatz $T = 1/(2π r_A)$ is held fixed throughout, the paper cannot separate 'wrong entropy' from 'wrong temperature' as the cause of the failure; a variant calculation with a modified horizon temperature would distinguish the two.

Original abstract

In the paper, hep-th/0501055 (R.G. Cai and S.P. Kim, JHEP {\bf 0502}, 050 (2005)), it is shown that by applying the first law of thermodynamics to the apparent horizon of an FRW universe and assuming the geometric entropy given by a quarter of the apparent horizon area, one can derive the Friedmann equations describing the dynamics of the universe with any spatial curvature; using the entropy formula for the static spherically symmetric black holes in Gauss-Bonnet gravity and in more general Lovelock gravity, where the entropy is not proportional to the horizon area, one can also obtain the corresponding Friedmann equations in each gravity. In this note we extend the study of hep-th/0501055 to the cases of scalar-tensor gravity and $f(R)$ gravity, and discuss the implication of results.

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The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,149 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.10 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.07 · compares with prior results 0.84 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

In the paper, hep-th/0501055 (R.G. Cai and S.P. Kim, JHEP {\bf 0502}, 050 (2005)), it is shown that by applying the first law of thermodynamics to the apparent horizon of an FRW universe and assuming the geometric entropy given by a quarter of the apparent horizon area, one can derive the Friedmann equations describing the dynamics of the universe with any spatial curvature; using the entropy formula for the static spherically symmetric black holes in Gauss-Bonnet gravity and in more general Lovelock gravity, where the entropy is not proportional to the horizon area, one can also obtain the corresponding Friedmann equations in each gravity. In this note we extend the study of hep-th/0501055 to the cases of scalar-tensor gravity and $f(R)$ gravity, and discuss the implication of results.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	In the paper, hep-th/0501055 (R.G. Cai and S.P.	background 0.99	0.04	0.02	unclear 0.98
s1	Kim, JHEP {\bf 0502}, 050 (2005)), it is shown that by applying the first law of thermodynamics to the apparent horizon of an FRW universe and assuming the geometric entropy given by a quarter of the apparent horizon area, one can derive the Friedmann equations describing the dynamics of the universe with any spatial curvature; using the entropy formula for the static spherically symmetric black holes in Gauss-Bonnet gravity and in more general Lovelock gravity, where the entropy is not proportional to the horizon area, one can also obtain the corresponding Friedmann equations in each gravity.	background 0.99	0.07	0.28	asserted 1.00
s2	In this note we extend the study of hep-th/0501055 to the cases of scalar-tensor gravity and $f(R)$ gravity, and discuss the implication of results.	contribution 1.00	0.83	0.16	asserted 0.93

What kind of paper the classifier reports this is: theoretical result 0.88 · evidence from proof or derivation 0.94.

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sources read

arXiv listing abstract v2

not examined

full text
figures and tables
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relations to other papers

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (16)

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How the literature uses it

59 arXiv papers cite this in 78 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 66 sentences without a confident label

Contests it 1 paper

It is shown that the dynamical horizon entropy of $f(R)$ gravity and scalar-tensor gravity have different forms with their stationary cases, contrary to the assumption given in many references [ref] .

Dynamical horizon entropy and equilibrium thermodynamics of generalized gravity theories · Conclusion and discussion · contest 0.76

Compares with it 1 paper

Such thermodynamics-gravity correspondence within $f(R)$ gravity has been investigated before in [ref] with different setups for the quantities ${M ,ρ_(MG) ,P_(MG)⋯}$ and thus ${ψ ,W⋯}$; compared with these earlier works, we have revised the thermodynamic setups and improved the result of entropy production.

Friedmann equations from nonequilibrium thermodynamics of the Universe: A unified formulation for modified gravity · $f(R)$ gravity · 0.93

Cites it as background 10 papers

The thermodynamic properties of the apparent horizon of FRW universe has been studied in various theories of gravity by many authors (see for examples [ref] ).

Generalized Second Law of Thermodynamics in Extended Theories of Gravity · Introduction · 0.98
The thermodynamic properties associated with the apparent horizon of FRW universe filled with the perfect fluid has been studied by many authors (see for examples [ref] ).

Viscous Cosmology and Thermodynamics of Apparent Horizon · Introduction · 0.91
The deep connection between the first law of thermodynamics and Friedmann equations has been investigated in gravity with Gauss-Bonnet term [ref] , Lovelock gravity theory [ref] , the braneworld scenarios [ref] , rainbow gravity [ref] , scalar-tensor gravity and $f(R)$ gravity [ref] .

Thermodynamics of the FRW universe at the event horizon in Palatini f(R) gravity · Introduction · 0.80

Discussion

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