Reexamining $f(R,T)$ gravity

Eric D. Carlson; Sarah B. Fisher

REVIEW 3 major objections 5 minor 2 cited by

Reexamining $f(R,T)$ gravity

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read In separable f(R,T) gravity, the f2(T) term can be absorbed into the matter Lagrangian and therefore has no physical significance.

desk verdict A clean demonstration that separable f(R,T) gravity's f2(T) term is just a matter-sector redefinition, with the main soft spot being an over-broad claim for nonlinear cases and an unperformed check on real EOS constraints. read the letter →

arxiv 1908.05306 v1 pith:ZXHIEZVC submitted 2019-08-14 gr-qc

Sarah B. Fisher , Eric D. Carlson This is my paper · ORCID

classification gr-qc

keywords f(RT)gravityseparablematterLagrangianperfectfluidconservedstress-energytensormodifiedf2(T)constraintsconstrainedperfect-fluidaction

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 argues that in $f(R,T)$ gravity with $f(R,T)=f_1(R)+f_2(T)$, the term $f_2(T)$ does not belong in the gravitational action: it can be moved into the matter Lagrangian, where it only redefines matter fields and their equation of state. The authors show this explicitly for a free scalar field and for a general perfect fluid, constructing a physical stress-energy tensor $T^{\prime\mu\nu}$ that is conserved even though the bare one is not. If correct, published limits on $f_2(T)$ parameters are limits on unphysical bare quantities, and cosmological uses of $f_2(T)$ are not probing new gravity. The deeper point is that assigning a term to 'gravity' rather than 'matter' is not physically meaningful when the same term can be put on either side of the split.

What carries the argument

The load-bearing object is the constrained perfect-fluid action of eq. (16), which uses Lagrange multipliers to enforce current conservation, entropy conservation, and flow-line labeling; this is what lets the paper show that a nonzero $f_2(T)$ conserves the rescaled current $J^{\prime\mu}$, not the original current. The identity defining the physical stress-energy tensor, eq. (23), plus the on-shell Lagrangian identity of eq. (32), together carry the equivalence: because the difference between the original and redefined Lagrangians vanishes on shell, the $f_2(T)$ term can be absorbed into the matter Lagrangian with no change in equations of motion. The modified density and pressure in eqs. (26)-(29) satisfy the standard thermodynamic relation $n'\partial\rho'/\partial n' = \rho'+p'$, so the redefined fluid is a genuine perfect fluid.

What would settle it

Take a system whose physics is governed by a known equation of state, compute a measurable observable (for example, the mass-radius curve of a white dwarf) twice: once from the bare $\rho,p$ equations with a chosen $f_2(T)$, and once from the $f_1(R)$ theory with the primed $\rho',p'$ and rescaled mass. If the two predictions differ in a regime where the constrained perfect-fluid action is uncontroversial, and observation selects the bare-$f_2(T)$ prediction, then $f_2(T)$ is not purely a matter redefinition.

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Extended reading notes

Core claim

The central claim is that for separable $f(R,T)$ gravity, $f_2(T)$ is not a distinct gravitational interaction. Redefining the fluid current as $J^{\prime\mu} = [1+2\kappa^{-2}f'_2(T)]J^{\mu}$, with number density $n'=[1+2\kappa^{-2}f'_2(T)]n$, and defining the physical stress-energy tensor $T^{\prime\mu\nu} = T^{\mu\nu} + (1/\kappa^2\sqrt{-g})\delta(\sqrt{-g}f_2(T))/\delta g_{\mu\nu}$, yields a conserved tensor and equations of motion identical to an $f_1(R)$ theory whose matter sector is described by primed variables. On shell, the difference between the original action and the redefined one vanishes, which is why the two descriptions are physically equivalent. The same absorption works for a free scalar field by rescaling the field and mass, and the paper argues analogously for fermions and vector fields. The authors conclude that the bare field, mass, density, pressure, and stress-energy tensor are unmeasurable, so $f_2(T)$ has no physical significance; consequently, all attempts to constrain its parameters are misguided.

Load-bearing premise

The argument assumes the constrained perfect-fluid action of eq. (16) is an accurate variational description of every perfect fluid that has been used to constrain $f_2(T)$, so that the redefined primed density and pressure are the physical ones; if a real fluid, such as a degenerate electron gas, is not captured by that action, the absorption proof may not apply to it.

Editorial extensions

If this is right

Constraints on the parameter $\chi$ in $f_2(T)=-\kappa^2\chi T/4\pi$ from white dwarfs, strange stars, and Earth's atmosphere are not limits on new physics, because they are computed from bare density and pressure rather than the physical primed quantities.
Cosmological models that invoke $f_2(T)$ to drive accelerated expansion are re-labeling a matter-sector rescaling as a gravitational effect; the acceleration should be interpreted in the $f_1(R)$ theory with modified matter.
In separable $f(R,T)$ gravity, the bare mass and field of a free scalar are unobservable: a linear $f_2(T)$ simply renormalizes both, so any measurement of the mass already includes the $f_2(T)$ effect.
For non-separable $f(R,T)$ gravity, mixed $R$-$T$ terms do not share this absorption and could produce genuinely new physics, while $f(0,T)$ should be treated as part of the matter Lagrangian.

Reading between the lines

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

The absorption argument suggests a consistency test: recompute the white-dwarf mass-radius relation using the primed density and pressure with the rescaled electron mass, and compare with standard general relativity; if the two disagree, the fluid is not captured by the constrained perfect-fluid action rather than $f_2(T)$ being physical.
By the same logic, any observational constraint cast directly in terms of bare $\rho$ and $p$ in a separable $f(R,T)$ model is suspect, which may apply to other modified-gravity parameter bounds beyond $f_2(T)$ when the fluid action is not explicitly specified.
If the construction extends beyond perfect fluids, $f_2(T)$ could generate derivative or self-interaction terms in the matter Lagrangian, effectively changing the matter sector while leaving gravity untouched; this would make 'modified gravity' limits on such terms a disguised choice of matter model.

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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

Carl, you should know this paper actually does what it claims. The central result—that in separable f(R,T)=f1(R)+f2(T), the f2 term can be absorbed into the matter Lagrangian, making the theory equivalent to f1(R) with a rescaled matter sector—is demonstrated carefully, not just asserted. The free-scalar case is elementary, but the perfect-fluid treatment is the real contribution. Using Brown's Lagrangian, the authors derive on-shell relations that let them define a rescaled current and a conserved stress-energy tensor T', then construct a modified matter Lagrangian L' whose difference from the original vanishes on shell to first order. That is a clean, convincing local equivalence.

The paper also deserves credit for taking the field's sloppy assumption L_m = p to task. Showing that this only holds in the absence of f2, and tracing it back to a sign error in Harko et al. that propagated, is useful. The conclusion that published constraints on the linear parameter χ (white dwarfs, strange stars, atmosphere) were constraints on redefined matter variables rather than on new gravity follows from the equivalence. For the linear case, the mass-rescaling argument is straightforward, so the specific constraints criticized are probably indeed misguided.

The soft spots are real but manageable. First, the general proof for arbitrary f2(T) is local. The map n' = [1 + 2κ^{-2} f2'(T)]n may not be globally invertible or positive for strong or nonlinear f2. The paper doesn't discuss this, and the universal claim 'all uses are misguided' is stronger than what is proven. For the existing linear constraints this doesn't matter, but the authors should hedge or address it. Second, the application to the degenerate-electron-gas EOS is asserted: they say the EOS is unchanged with rescaled masses, but they do not actually recompute the TOV equations for a white dwarf to show the constraint disappears. A referee should ask for that explicit calculation, or at least a careful statement of how the physical ρ' and p' enter the stellar structure equations. Third, identifying T' as the physical stress-energy tensor is a sensible interpretive choice, but it is not a theorem; the paper would benefit from a clearer operational definition of 'physical.' The identification rests on the argument that bare T is unmeasurable, which is strong but not airtight.

Overall, the core result stands. The paper is a useful correction to a literature that has been over-interpreting f2(T). It deserves peer review and likely publication after the authors tighten the global claims and add the explicit EOS transformation. I'd bring it to reading group.

Referee Report

3 major / 5 minor

Summary. The manuscript reexamines separable f(R,T)=f1(R)+f2(T) gravity and argues that the f2(T) term should be absorbed into the matter Lagrangian, leaving ordinary f1(R) gravity. For a free scalar field, a linear f2(T) is shown to rescale the field and mass. For a general perfect fluid described by the Brown action, the authors define rescaled current, energy density, and pressure, and construct an alternative matter Lagrangian L'_m whose action differs from the original one only by terms that vanish on shell (Eqs. 32-33). They conclude that f2(T) has no physical significance and that existing constraints on the parameter chi from white dwarfs, strange stars, and atmospheric experiments are misguided.

Significance. If the conclusions are accepted, the paper would substantially reframe the f(R,T) literature: separable models would be reinterpreted as f1(R) gravity plus a renormalized matter sector, invalidating direct parameter limits. The derivation is careful, parameter-free, and correctly identifies the common Lm=p assumption as generally invalid. The explicit on-shell action comparison (Eqs. 32-33) is a useful technique. However, the universal 'no physical significance' conclusion relies on local equivalence and on an unverified step connecting the fluid-level renormalization to the specific equations of state used in the constraints. These gaps need to be closed before the strong conclusions are fully supported.

major comments (3)

[Section IV, Eq. (21b)] The relation n' = [1+2kappa^-2 f2'(T)]n is used to define L'_m as a function of n', but the paper does not establish that this map is single-valued and invertible for a generic nonlinear f2(T). Because T is an on-shell function of n and s, f2'(T) depends on n; without monotonicity of n'(n), the density rho'(n',s) obtained by inverting Eq. (21b) may not be globally well-defined, and the equivalence proved in Eqs. (32)-(33) is only local to a given stationary branch. The authors should prove invertibility on the density range of interest or explicitly restrict the 'no physical significance' claim to linear f2 (where the factor is constant).
[Section V, second paragraph] The claim that for the linear coupling (9) the degenerate-electron-gas equation of state is unchanged once rescaled masses are used is asserted, not derived. The fluid calculation in Section IV establishes on-shell equivalence of the actions and defines renormalized variables, but it does not compute the transformed EOS rho'(n') for a degenerate electron gas or show that it equals the standard Chandrasekhar EOS with the renormalized mass. The transformation in Eqs. (26)-(29) generally changes the functional form of rho(n), for example through the (4+n d/dn) operator, so a direct check is required before dismissing the white-dwarf constraints of Ref. [10]; the same applies to the ideal-gas-law argument against Ref. [12].
[Abstract and Section V, first paragraph] The conclusion that f2(T) 'has no physical significance' is stronger than the demonstrated result. The paper shows that f2(T) can be moved from the gravitational part into the matter Lagrangian; for the linear coupling this is a renormalization, but for a nonlinear f2 the resulting L'_m is a different matter theory (the paper itself notes that free fields become interacting). Such a term remains physically meaningful as a parametrization of matter even if it is not a distinct gravitational interaction. The authors should either soften this conclusion or replace 'no physical significance' with a precise statement such as 'no independent gravitational significance beyond a redefinition of the matter Lagrangian.'

minor comments (5)

[Section I] There is a typo: 'generlazation' should be 'generalization'.
[Equations (5)-(6)] The notation partial T / partial g^{mu nu} should be defined explicitly; it is only later (Eq. 24) that T is specified as a functional of the matter fields and metric through n = sqrt(g J J).
[Equation (20)] The barred notation for on-shell quantities is introduced in Eq. (20), but equations (21)-(22) rely on it; the definition should appear before first use.
[Reference [9]] The author name 'Carams' appears to be a misspelling (likely 'Caramês'); please verify it against the published paper.
[Section III] The statement that the same reasoning applies to free fermion or vector fields would be more convincing with the explicit T trace for those cases, since fermion traces involve sign conventions.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the f2(T) absorption proof is a direct computation, and the sole self-citation is criticized rather than load-bearing.

full rationale

The paper's derivation is self-contained and does not reduce to its inputs. The free-field case (Eqs. 7-12) is a direct algebraic identity showing that f2(T) = -κ^2 χ T / 4π rescales the field and mass; the conclusion follows from the explicit field redefinition, with no fitted parameter or circular definition. The perfect-fluid argument is a constructive equivalence proof: the authors define a conserved T'^μν (Eq. 23) as the original stress-energy tensor plus the metric variation of f2(T), define rescaled current and density (Eq. 21), and then show explicitly that the Lagrangian L' built from these variables differs from L only by an on-shell vanishing term (Eqs. 31-33). The algebraic reduction is carried out, not assumed: the equivalence is verified through the equations of motion and the on-shell identity for Lm. Although the labeling of T' as 'physical' and T as 'bare' involves an interpretive choice, that choice is supported by the explicit construction, and the paper does not rely on a self-citation to enforce it. The only self-citation in the relevant chain is Ref. [12], by one of the present authors, and it is used as the target of the paper's criticism—an example of the erroneous approach being corrected—not as a supporting premise. Potential scope limitations, such as whether the Brown action or the rescaled equation of state applies to degenerate electron gases, are correctness concerns rather than evidence of circularity. No step in the derivation is equivalent by construction to the conclusion that f2(T) is physically meaningless.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or new entities. Its argument relies on standard variational calculus and the Brown fluid action, and it explicitly limits the conclusion to separable f(R,T) gravity.

assumptions (4)

standard math The variational principle and the definition of the stress-energy tensor T^{μν} = -2/√-g δI_m/δg_{μν} (Eq. 2).
The entire derivation is built on this definition, which the paper states explicitly.
domain assumption The Brown action (Eq. 16) is a valid Lagrangian for a perfect fluid, with constraints enforced by Lagrange multipliers.
This is the starting point for the fluid analysis; the paper cites Brown [14] but does not prove its generality.
domain assumption The theory is restricted to separable f(R,T) = f1(R) + f2(T) (Eq. 4).
The conclusion applies to this subcase; the paper explicitly restricts to it and notes cross terms may yield new physics.
standard math Lagrangians that differ by a term vanishing on shell yield identical equations of motion.
Used in Eq. (32) to claim equivalence between L and L'.

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Cite this review

Pith. "Pith review of Reexamining $f(R,T)$ gravity." pith.science (2026). https://pith.science/paper/ZXHIEZVC

@misc{pith2026190805306,
  author       = {Pith},
  title        = {Pith review of: Reexamining $f(R,T)$ gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZXHIEZVC}},
  note         = {Machine review of arXiv:1908.05306}
}

abstract

We study $f(R,T)$ gravity, in which the curvature $R$ appearing in the gravitational Lagrangian is replaced by an arbitrary function of the curvature and the trace $T$ of the stress-energy tensor. We focus primarily on situations where $f$ is separable, so that $f(R,T) = f_1(R) + f_2(T)$. We argue that the term $f_2(T)$ should be included in the matter Lagrangian ${\cal L}_m$, and therefore has no physical significance. We demonstrate explicitly how this can be done for the cases of free fields and for perfect fluids. We argue that all uses of $f_2(T)$ for cosmological modeling and all attempts to place limits on parameters describing $f_2(T)$ are misguided.

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

22 extracted references · 16 canonical work pages · cited by 2 Pith papers

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

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Perlmutter, G

S. Perlmutter, G. Aldering, M. D. Valle, S. Deustua, R. S. Ellis, S. Fabbro, A. Fruchter, G. Goldhaber, D. E. Groom, I. M. Hook, A. G. Kim, M. Y. Kim, R. A. Knop, C. Lidman, R. G. McMahon, P. Nugent, R. Pain, N. Panagia, C. R. Pennypacker, P. Ruiz-Lapuente, B. Schaefer, and N. Walton, Nature 391, 51 (1998)

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on shell

will be T µν = (ρ + p)uµ uν − gµν Lm , (18a) T = ρ + p − 4Lm . (18b) The equations of motion resulting from demanding sta- tionarity of the Lagrangian with respect to all the ﬁelds (other than the metric) are then 0 = ∇µ {[ 1 + 2κ −2f ′ 2(T ) ] J µ } , (19a) 0 = ∇µ { s [ 1 + 2κ −2f ′ 2(T ) ] J µ } , (19b) 0 = [ 1 + 2κ −2f ′ 2(T ) ] J µ ∇µ α A , (19c) 0 = ...

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simply rescales the ﬁeld and mass for a fermion, and hence the equation of state (once rescaled masses are used) for the electron gas will be unchanged. Simi- larly, in [ 12] the ideal gas law is used, but this ideal gas law should be applied to the physical energy density and pressure, not the “bare” energy density and pressure. 5 Can we generalize these...

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