Linear perturbations of symmetric teleparallel gravity on Minkowski background

Dehao Zhao

REVIEW 3 major objections 4 minor 1 cited by

Linear perturbations of symmetric teleparallel gravity on Minkowski background

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read This paper aims to show that the most general symmetric teleparallel gravity action, linearized on a Minkowski background, carries between 2 and 10 degrees of freedom, with one parameter branch signaling strong coupling.

desk verdict Table I's 9-DOF row is internally inconsistent, but the perturbation framework and symmetry lemma are solid enough to warrant a revise-and-resubmit. read the letter →

arxiv 2412.20696 v1 pith:MZYCMF25 submitted 2024-12-30 gr-qc

Dehao Zhao This is my paper · ORCID

classification gr-qc MSC 83D0583C25 PACS 04.50.Kd

keywords symmetricteleparallelgravitynonmetricitytensorlinearperturbationsdegreesoffreedomMinkowskibackgroundstrongcouplingscalarcouplingstransversediffeomorphisms

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

Symmetric teleparallel gravity (STG) is a modified-gravity theory whose gravitational field is the nonmetricity tensor, the quantity describing how the metric changes under parallel transport. On a flat Minkowski background, this paper derives the most general quadratic action for linear perturbations and counts how many propagating degrees of freedom each parameter choice carries. The counts form lower bounds for the full nonlinear theory: at least 2 or 3 degrees of freedom in the transverse-diffeomorphism branch, and 9, at least 7, or 10 in the generic branch. The paper also finds that adding a scalar field with derivative couplings generically adds one degree of freedom, and it identifies a branch where the linear count (7) disagrees with the known nonlinear count (10), a strong-coupling warning. These results matter because STG models are widely used in modified gravity and cosmology, yet their degree-of-freedom content has been disputed.

What carries the argument

The central object is the gauge-invariant combination $H_{\mu\nu}=\delta g_{\mu\nu}-\partial_\mu u_\nu-\partial_\nu u_\mu$, where $u^\mu$ are the four scalar fields that encode the STG affine connection. Because the quadratic action (19) is the most general kinetic term built from $H_{\mu\nu}$, the paper's counting reduces to a scalar-vector-tensor decomposition under spatial rotations: tensor modes contribute two degrees of freedom whenever $c_1\neq0$, vector modes contribute four or zero depending on $2c_1+c_2$, and scalar modes contribute the remainder through a kinetic matrix whose determinant decides how many constraint equations appear. The transverse-diffeomorphism branch $2c_1+c_2=0$ removes the vector kinetic terms and thereby changes the scalar constraint structure.

What would settle it

Run a complete Hamiltonian constraint analysis of the full action (8) for the parameter branch $2c_3+c_4=0$, $c_1+3c_3=0$: if the nonlinear theory really has ten propagating degrees of freedom while the linearized action has seven, the proposed lower-bound interpretation fails for that branch; if it has seven, the strong-coupling signal is resolved.

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Extended reading notes

Core claim

The paper claims that on a Minkowski background the most general quadratic-in-nonmetricity STG action (8) reduces at linear order to the spin-2 kinetic action (19), so the degree-of-freedom content of each model can be read off from that quadratic action. The count, with $c_1\neq0$, is: in the transverse-diffeomorphism branch $2c_1+c_2=0$, at least two degrees of freedom in two special subcases and at least three otherwise; in the generic branch $2c_1+c_2\neq0$, ten degrees of freedom except in two subcases, where it is nine (when $2c_3+c_4=0$, $c_1+c_2+c_3+c_4=0$, and $c_1+c_3=0$) or at least seven (when $2c_3+c_4=0$, $c_1+c_2+c_3+c_4=0$, and $c_1+c_3\neq0$). It also claims that adding a cosmological constant does not make the standard dS/AdS metrics solutions with a vanishing affine connection, and that coupling a scalar field generically adds one scalar degree of freedom to the pure-gravity count. In the branch $2c_3+c_4=0$, $c_1+3c_3=0$, the linearized count is seven while the known nonlinear count is ten; the paper reads this mismatch as strong coupling on flat space.

Load-bearing premise

The load-bearing premise is that counting dynamical fields in the linearized quadratic action gives a valid lower bound on the number of degrees of freedom of the full nonlinear STG theory; the paper itself reports a branch where the linear count (seven) is lower than the known nonlinear count (ten), so this premise can fail.

Editorial extensions

If this is right

In the transverse-diffeomorphism branch ($2c_1+c_2=0$), vector perturbations carry no propagating degrees of freedom, making these STG models dynamically simpler than generic ones.
Whenever $c_1\neq0$, tensor perturbations are the two usual gravitational-wave polarizations, and their propagation speed equals the speed of light, tying viable STG models to gravitational-wave observations.
For generic parameter choices the linearized theory has ten degrees of freedom, matching the count from nonlinear primary constraints, so the quadratic action gives a trustworthy lower bound in those cases.
Adding a scalar field with derivative coupling to the nonmetricity tensor generically adds exactly one scalar degree of freedom to the pure-gravity count, with special parameter choices lowering that number.
The branch where the linear count (seven) falls below the nonlinear count (ten) is a strong-coupling signal: flat-space perturbation theory there misses degrees of freedom that appear at nonlinear order.

Reading between the lines

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

Extension: the same linearized counting could be repeated on a Friedmann-Lemaître-Robertson-Walker background for general STG actions, where the f(Q) case is already known to give seven degrees of freedom.
Extension: the failure of the linear count in the strong-coupling branch suggests that, for other branches, the lower bound should be checked against a full nonlinear constraint analysis before being used to rule models in or out.
Extension: the paper's proof that a connection respecting background symmetries forces the metric equations of motion to respect them is transferable, offering a criterion for choosing consistent connection ansätze in other teleparallel and metric-affine models.
Extension: because the dS/AdS search failed only for vanishing or maximally symmetric connections, a broader scan over more general connections might still uncover cosmological backgrounds in this theory.

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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

Dear Colleague,

The headline: the paper's central Table I has a branch misclassification that invalidates the 9-DOF row as stated, but the perturbation framework around it is still worth engaging.

What is actually new: the author takes the most general quadratic-in-nonmetricity STG action, expands on Minkowski, and derives the quadratic actions for scalar, vector, and tensor perturbations, writing everything in gauge-invariant variables. The symmetry lemma in Sec. II.C is genuinely useful: if the connection respects the background spacetime symmetries, so does the metric EOM. The extension to scalar couplings in Sec. IV is a natural step and mostly correct.

The soft spot is load-bearing. In Table I, the row under 2c1+c2≠0 with conditions 2c3+c4=0, c1+c2+c3+c4=0, c1+c3=0 is claimed to give 9 DOFs. Those three conditions force 2c1+c2=0: c1+c3=0 gives c3=-c1; 2c3+c4=0 gives c4=2c1; and the third condition gives c2=-2c1. So the row belongs in the TDiFF branch, where the paper's own analysis gives 2 DOFs. The text near Eq. (57) counts three scalar DOFs by retaining (2c1+c2) in the kinetic matrix, which is zero under those conditions. The central table cannot be accepted as written.

There is also a more general caveat: counting fields with nonzero kinetic terms gives a lower bound only if strong coupling does not mask DOFs at nonlinear order. The author acknowledges this in the branch 2c3+c4=0, c1+3c3=0, where the linear count is 7 but Ref. [42] gives no primary constraints and 10 DOFs. That is honest, but it undercuts the 'lower bound' language elsewhere. The table also lists 'otherwise 10' as a lower bound even though it comes from the external Hamiltonian analysis, not from the linear perturbation count. Typos and a garbled table make verification harder.

The citation pattern looks fine; the key external result [42] is independent of the author's own work.

Who this is for: people working on STG model viability and DOF counting. The quadratic action and gauge-invariant variables are worth citing even after the table is fixed.

Recommendation: I would not desk-reject this. A serious referee can verify the algebra and the author can correct the branch structure. If the 9-DOF row is replaced by the correct TDiFF result, the remaining systematic scan has real value. So: accept for peer review, with the expectation of heavy revision.

Referee Report

3 major / 4 minor

Summary. The paper studies the most general action quadratic in the nonmetricity tensor (Eq. (8)) on a Minkowski background. It derives the quadratic action for linear perturbations around Minkowski, decomposes the perturbations into scalar, vector, and tensor sectors, and counts propagating degrees of freedom from the rank of the kinetic matrix. A table (Table I) summarizes claimed lower bounds on the number of DOFs for the gravity-only model, and an extension to scalar couplings is considered in Section IV. The paper also argues that dS/AdS backgrounds with the coincident-gauge connection are not vacuum solutions, and it proves a symmetry property of the equations of motion. The central claim is that the quadratic action (19) is the most general linear-perturbation action and that the DOF counts in Table I are lower bounds for the full theory.

Significance. If the claims were fully established, the paper would provide a systematic route to lower bounds on DOF counts for a broad class of symmetric teleparallel gravity models, complementing Hamiltonian analyses that are difficult to perform in this framework. The derivation of the quadratic action (19) is clean, and the GR limit correctly reproduces the absence of scalar DOFs. The explicit scalar/vector/tensor decomposition and the cross-check with Ref. [42] for the number of primary constraints are useful contributions. However, the central table contains a logical inconsistency, and the interpretation of the tabulated numbers as lower bounds is not fully justified; these issues affect the paper's main result and require revision.

major comments (3)

[III.B.2, Table I] The 9-DOF row of Table I is internally inconsistent. The row is placed under the assumption 2c1+c2≠0 and is specified by 2c3+c4=0, c1+c2+c3+c4=0, and c1+c3=0. These relations imply c3=-c1, c4=2c1, and c2=-2c1, so that 2c1+c2=0. The parameter point therefore belongs to the TDiFF branch, not the generic branch. At this point, c2+c4=0 and c2-2c3=0, so Eq. (52) has no scalar kinetic term; only the two tensor modes propagate at linear order. The claimed count of 9 DOFs is not supported by the paper's own formulas, and the row should be removed or reclassified.
[III.B.2, Table I] Table I's 'otherwise 10' entry under 2c1+c2≠0 is not a lower bound obtained from the linear analysis. The text after Eq. (62) explicitly states that the branch 2c3+c4=0, c1+3c3=0 has 7 DOFs at the linear perturbation level and 10 DOFs at the nonlinear level according to Ref. [42], signalling strong coupling. This branch falls under 'otherwise' in the table, so the table conflates the nonlinear Hamiltonian count with the linear lower-bound count. The table should either list '≥7' for this branch or clearly distinguish the nonlinear count that is imported from Ref. [42].
[III.B and Table I] The paper does not justify why the number of fields with nonzero quadratic kinetic terms is a lower bound on the number of physical DOFs of the full nonlinear theory. Nonlinearities can introduce constraints that remove fields that appear kinetic at quadratic order, so a rank-count of the kinetic matrix K in Eq. (54) is not automatically a lower bound. The strong-coupling example in Section III.B.2 shows that the linear analysis can undercount (7 vs 10), but the opposite direction, where the linear analysis overcounts, is not addressed. Since the abstract and conclusion claim lower bounds, this is a load-bearing gap; the paper should either provide an argument for the lower-bound interpretation or soften the claim.

minor comments (4)

[III.B.2] The sentence before Eq. (56) refers to cases 'c3+c4=0 and c1+c2+c3+c4=0' and 'c3+c4=0 and c1+3c3', but the subsequent cases and the determinant condition in Eq. (55) require '2c3+c4=0' instead of 'c3+c4=0'; this should be corrected.
[Table I and Eq. (51)] The vanishing condition in the TDiFF row of Table I is written as c2^2-4c2c3+4c2c4+3c4^2=0, whereas Eq. (51) gives c2^2-4c2c3+2c2c4+3c4^2=0; the two expressions should be reconciled.
[II.B and III.B] There are several typographical errors, including 'Firze-Pauli' for 'Fierz-Pauli' in Section II.B, 'lowed and uped' for 'lowered and raised' in Section II.B, and 'c‘4' for 'c4' in Eq. (47).
[References] References [14] and [43] appear to list the same paper (V. Gakis et al., Phys. Rev. D 101 (2020) 064024); one of the entries should be removed or replaced with the intended distinct reference.

Circularity Check

1 steps flagged · score 6.0 of 10

Table I's 9-DOF row is self-inconsistent: its conditions force 2c1+c2=0, placing it in the TDiFF branch where the paper's own formulas give 2 DOFs; no other load-bearing circularity.

self definitional [Section III.B.2, Table I (row '2c1+c2≠0' with '2c3+c4=0, c1+c2+c3+c4=0, c1+c3=0') and Eq. (57).]
"2c1 + c2 ⁄= 0 | 2c3 + c4 = 0, c1 + c2 + c3 + c4 = 0 | c1 + c3 = 0 | 9 ... K = ... [0 0 0 (2 c1 +c2)]"

The row's own three conditions imply c3=-c1, c4=2c1 and c2=-2c1, hence 2c1+c2=0. This contradicts the row heading '2c1+c2≠0', so the parameter point is in the TDiFF branch. In that branch the paper's formulas give c2+c4=0 and c2-2c3=0; Eq. (52)'s scalar kinetic term vanishes, vector modes are non-dynamical, and only the two tensor modes of Eq. (41) propagate. The 9-DOF count is instead read off from Eq. (57), whose bottom-right kinetic entry (2c1+c2) is treated as nonzero to make B dynamical, but that entry is forced to zero by the same conditions. Thus the 9-DOF entry reduces, by the row's own definitions, to the TDiFF result rather than being an independent generic-branch count.

full rationale

Apart from Table I's 9-DOF row, the paper's DOF counting is a self-contained algebraic exercise: the quadratic action (19) is expanded in scalar/vector/tensor perturbations, kinetic matrices are computed, and DOF counts follow from ranks and constraint equations; no parameters are fitted to data and no external result is assumed as input aside from the independent primary-constraint count of Ref. [42], which has no author overlap with the present paper. The minor self-citations used in the dS/AdS discussion are not load-bearing for the main lower-bound claim. The 9-DOF entry, however, is circular in a definitional sense: its branch conditions force the very coefficient (2c1+c2) to vanish that the accompanying kinetic-matrix count assumes to be nonzero, collapsing the claimed 9 DOFs to the 2 tensor DOFs of the TDiFF branch. This is a central Table-I entry, so the score reflects that partial by-construction reduction rather than any fitted-input or self-citation-chain circularity.

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

The paper introduces no new particles or forces; it works entirely with the coupling constants already present in the general STG action and the scalar-field action. The only extraneous assumptions are the usual STG gauge choices and the counting rule for DOFs from kinetic matrices, which the paper itself partially undermines by identifying a strong-coupling branch.

assumptions (5)

domain assumption The action (8) is the most general STG action quadratic in the nonmetricity tensor, and it fully determines linear perturbations on Minkowski background.
Sec. II.A: the authors state that since the nonmetricity tensor is first order on this background, (8) is the most general action influencing linear perturbations.
standard math The coincident gauge can always be chosen, so the perturbed connection is written as ∂μ∂ν uλ and the physical variables are Hμν = δgμν - ∂μuν - ∂νuμ.
Sec. II.A and III.A: standard STG result that curvature and torsion vanish, allowing Γ=0 coordinate choice.
domain assumption c1 ≠ 0 so that tensor perturbations have a kinetic term (gravitational waves propagate).
Sec. III.B: 'we impose the condition that c1 ≠ 0.'
domain assumption For a scalar field on the flat background, φ is constant with P=0 and Pφ=0, so φ itself is a first-order perturbation.
Sec. IV: 'On a flat background, from the EOMs, we have the relations that P = 0 and Pφ = 0.'
domain assumption The number of DOFs in the linear theory equals the number of dynamical variables in the quadratic action, read from the rank of the kinetic matrix.
Sec. III.B: DOF counts are inferred from which fields have kinetic terms and from solving constraint equations; this is the paper's counting rule and is stated only implicitly.

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Cite this review

Pith. "Pith review of Linear perturbations of symmetric teleparallel gravity on Minkowski background." pith.science (2026). https://pith.science/paper/MZYCMF25

@misc{pith2026241220696,
  author       = {Pith},
  title        = {Pith review of: Linear perturbations of symmetric teleparallel gravity on Minkowski background},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MZYCMF25}},
  note         = {Machine review of arXiv:2412.20696}
}

read the original abstract

Symmetric teleparallel gravity (STG) can be regarded as a modified gravity theory that lacks diffeomorphism symmetries, which complicates the calculation of its degrees of freedom. In this study, we analyze the linear perturbations of general STG models on a Minkowski background, considering both scenarios with and without scalar couplings. Furthermore, we provide lower bounds for the number of degrees of freedom associated with each model.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Weak Gravity Limit in Newer General Relativity
gr-qc 2024-12 conditional novelty 6.0 of 10

In Newer GR, stable tensor and vector sectors force one coefficient combination to its GR value, and the STEGR-plus-gradient-squared model carries 3/2 new dynamical modes, not 1.

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does not inﬂuence the tensor and vector perturbations. And th e quadratic action for scalar perturbations of Eq.( 63) is S(2) φ = 1 2 ∫ dtd3x [ −2(b1 +b2)∂tA∂tφ + 6b1∂tψ∂tφ + 2b1k2∂tE∂tφ −PX∂tφ∂tφ +2b1k2Aφ − 2(3b1 +b2)k2ψφ − 2(b1 +b2)k4Eφ − 2b2k2B∂tφ + (PXk2 + 1 2Pφφ )φ 2 ] . ...
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(71) The determinant of kinetic matrix is proportional to −b2 1c2 − 2b1b2c2 + 1 2b2 2(c2 − 6c3) − 2c2(c2 − 2c3)PX

becomes 4c4ψ =b2φ (70) 12 then the kinetic term of quadratic action of scalar perturbations is S(2) Skin = 1 2 ∫ dtd3k 2(c2 − 2c3)(A −k2E) ′2 − (2b1 −b2 + 6b2c3 c2 )(A −k2E)′φ ′ − (PX + 12b1b2c2 − 3b2 2c2 + 18b2 2c3 8c2 2 )φ ′2. (71) The determinant of kinetic matrix is propor...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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Transverse diﬀeomorphisms For the case of transverse diﬀeomorphisms where 2 c1 +c2 = 0, one can ﬁnd that B now is not a dynamical ﬁeld and it gives us the constraint equation (c2 +c4)k2E = (c2 +c4)A − (c2 + 3c4)ψ. (50) Then if c2 +c4 ⁄= 0, replacing E with A and ψ in the action ( 47), we can get the ﬁnal form of quadratic action of scalar perturbations S(...

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