Weak Field Limit of the Nonminimally Coupled Weyl Connection Gravity

Cl\'audio Gomes

REVIEW 3 major objections 3 minor 1 cited by

Weak Field Limit of the Nonminimally Coupled Weyl Connection Gravity

T0 review · 3 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read For a Weyl vector whose spatial components are all equal, the Newtonian limit of nonminimally coupled Weyl connection gravity is a closed modified-Poisson system: both Bardeen potentials and the Weyl vector are fixed by matter density.

desk verdict The first-ansatz part of the paper is consistent but derivative; the second-ansatz result, which is the new claim, fails because Eq. (43) misses a factor Z and the spatial ansatz is not spherically symmetric, so the final Poisson system does not follow. read the letter →

arxiv 2506.16532 v1 pith:SZKZFINJ submitted 2025-06-19 gr-qc

Cláudio Gomes This is my paper · ORCID

classification gr-qc PACS 04.50.Kd

keywords Weylconnectiongravitynonminimalcouplingweak-fieldlimitNewtonianBardeenpotentialsvectornon-metricitymodified

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

This paper tries to establish the Newtonian limit of a modified-gravity model in which matter couples nonminimally to the curvature scalar built from a Weyl connection, so spacetime geometry carries a non-metricity vector field. The central claim is that, for a Weyl vector of the form $A_\mu=(A_0,A_1,A_1,A_1)$, the weak-field equations close into a pair of density-sourced Poisson equations for the Bardeen potentials together with a relation $Z\nabla_\lambda\rho=-A_\lambda$ fixing the Weyl vector from the density gradient. If this is right, the model becomes directly solvable for given matter distributions and gives a modified-Poisson framework for studying gravitational collapse and large-scale structure. The paper also finds that a purely temporal Weyl vector forces the field to vanish at first order, reducing that branch to the standard nonminimal matter-curvature coupling model.

What carries the argument

The central object is the Weyl vector $A_\lambda$, which defines non-metricity through $D_\lambda g_{\mu\nu}=A_\lambda g_{\mu\nu}$, and the action $S=\int(f_1(\bar R)+f_2(\bar R)L)\sqrt{-g}\,d^4x$. The load-bearing mechanism is the order-$c^{-2}$ expansion of the resulting field equations around Minkowski space, with $g_{\mu\nu}=\mathrm{diag}(-1-2\Phi,1-2\Psi,1-2\Psi,1-2\Psi)$, curvature perturbation $\delta\bar R=2\nabla^2(2\Psi-\Phi)$, and the constraint $\nabla_\lambda\bar\Theta=-A_\lambda\bar\Theta$. Combining the trace equation with that constraint is what converts the field equations into Poisson-type equations for the Bardeen potentials and an algebraic relation between the Weyl vector and the density gradient.

What would settle it

Take a density profile of the form $\rho=f(x+y+z)$ with $f''\neq0$. The relation $Z\nabla_j\rho=-A_1$ fixes $A_1=-Zf'$, so $\nabla_\lambda A^\lambda=-Z\nabla^2\rho$. Substituting this into $\nabla^2(\Phi-\Psi)=\nabla_\lambda A^\lambda$ gives $Z\nabla^2(\Psi-\Phi)=Z^2\nabla^2\rho$, which matches the paper's Eq. (43) only if $Z^2=1$. For generic $Z$, the claimed equations (47)-(50) cannot follow from (39)-(42); this algebraic check settles the central claim.

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Extended reading notes

Core claim

The paper's central claim is that, in the order-$c^{-2}$ expansion around Minkowski spacetime, the ansatz $A_\mu=(A_0,A_1,A_1,A_1)$ with $|A_0|,|A_1|\ll1$ yields $A_0=0$ and the system (47)-(50): $\nabla^2\Psi=(\gamma/2+\nabla^2 Z)\rho$, $\nabla^2\Phi=\nabla^2 Z\,\rho$, and $Z\nabla_\lambda\rho=-A_\lambda$, where $Z=(\alpha\gamma/2-\beta)$ and $\alpha,\beta,\gamma$ are Taylor coefficients of $f_1,f_2$ at vanishing curvature. The paper reads these equations as a complete weak-field description: the Bardeen potentials $\Phi,\Psi$ and the Weyl vector are all fixed once the matter density $\rho$ is known, so one can integrate the system numerically for an arbitrary density profile. It further claims that the alternative ansatz $A_\mu=(A_0,0,0,0)$ has only the trivial first-order solution $A_0=0$, meaning that branch reproduces the known weak-field limit of ordinary nonminimal matter-curvature coupling.

Load-bearing premise

The load-bearing step is the substitution of $Z\nabla_j\rho=-A_1$ into $\nabla^2(\Phi-\Psi)=\nabla_\lambda A^\lambda$, which the paper reads as $Z\nabla^2(\Psi-\Phi)=\nabla^2\rho$ although the constraint actually yields an extra factor of $Z$ and also forces the density gradient to be equal in all three spatial directions; if that step gives way, the closed system (47)-(50) collapses.

Editorial extensions

If this is right

Given a density profile, the two Bardeen potentials are separately determined, so the model predicts a non-vanishing gravitational slip $\Phi-\Psi$ sourced by density gradients.
In the weak-field regime the Weyl vector is no longer an independent input; it is slaved to the matter distribution through $Z\nabla_\lambda\rho=-A_\lambda$.
A purely time-like Weyl vector ansatz forces $A_0=0$ at first order, so that branch of the theory recovers the known Newtonian limit of nonminimal matter-curvature coupling.
The derived system is ready for numerical integration, which the paper identifies as the route to astrophysical predictions such as Jeans-type instability analysis.
A post-Minkowskian expansion may bring Yukawa-like corrections and extra-force contributions, as the paper notes in its conclusion.

Reading between the lines

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

Since $A_\lambda=-Z\nabla_\lambda\rho$ at leading order, the Weyl vector is irrotational in the weak-field regime, so all non-metricity information is carried by density gradients and no propagating vector degrees of freedom survive at this order.
A clean null test suggested by the relation: in a region where the matter density is homogeneous (for instance, inside a large cosmic void), the model predicts that the Weyl vector vanishes and ordinary Newtonian Poisson behavior is recovered.
Because the ansatz forces $\partial_x\rho=\partial_y\rho=\partial_z\rho$, applying the derived equations to realistic galaxy or cluster profiles requires relaxing the equal-spatial-components assumption; the generic-profile version of the weak-field limit is not derived in the paper.
At the next, post-Newtonian order the time component $A_0$ may reappear; the paper leaves that sector open, so gravitomagnetic effects are the natural place to look for a distinctive signature.

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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

Quick take: the part of the paper that is actually new—the weak-field system for the spatial Weyl vector ansatz—does not hold up. Eq. (43) is missing a factor Z, and the ansatz A_x=A_y=A_z is not spherically symmetric in the way the paper claims. The first-ansatz section is fine, but it just recovers the known non-minimal matter-curvature coupling limit.

I checked the substitution. Equation (38) says Z∂_jρ=−A_1 for each spatial j. With A_0=0, ∇_λA^λ=∂_xA_1+∂_yA_1+∂_zA_1=−Z∇²ρ. Substituting into (39) gives ∇²(Φ−Ψ)=−Z∇²ρ, so Z∇²(Ψ−Φ)=Z²∇²ρ, not Z∇²(Ψ−Φ)=∇²ρ. The missing factor changes the final Poisson-type system. There is also a second, independent problem: the same A_1 in all three directions forces ∂_xρ=∂_yρ=∂_zρ at every point, which a generic spherical density does not satisfy. And the printed equations (47)–(48) do not solve (43)–(44); they would only under special conditions on ρ. So the central new equations are not derivable from the stated model.

Credit where due: the paper is clearly written, the weak-field expansion is standard, and the first-ansatz calculation is internally consistent. It correctly shows that A_0 vanishes at Newtonian order and the theory reduces to the known non-minimal matter-curvature coupling model—a useful sanity check for anyone working in this specific framework. The references are appropriate; the reliance on the author's earlier papers is legitimate because those papers define the model.

Who is this for? Only someone actively working on this particular Weyl-connection gravity program. A general relativity reader will not find a trustworthy Newtonian limit here. If the author fixes the algebra and replaces the Cartesian A_x=A_y=A_z assumption with a genuine radial ansatz, the second part could become a small valid note. As it stands, I would not send it to a referee. Desk reject with an invitation to resubmit after correcting Section 4.2.

Referee Report

3 major / 3 minor

Summary. The paper studies the weak-field Newtonian limit of a nonminimally coupled Weyl connection gravity model, expanding the metric, curvature, and matter fields around Minkowski spacetime to order c^-2. The field equations are taken from the author's earlier work [11]. Two Weyl-vector ansätze are considered. For the purely temporal ansatz A_mu=(A0,0,0,0), the analysis forces A0 to vanish at first order, so the model reduces to the nonminimal matter-curvature coupling gravity model. For the mixed ansatz A_mu=(A0,A1,A1,A1), the paper claims that the time component vanishes and that the two Bardeen potentials and the spatial Weyl vector are determined by the matter density through the modified Poisson system (47)-(50), with parameter Z=alpha*gamma/2 - beta. The conclusion proposes this system for numerical astrophysical applications.

Significance. If correct, the second-ansatz result would give a concrete, falsifiable prediction of the model: a closed local relation between the Bardeen potentials, the Weyl vector, and the matter density. The first-ansatz part is internally coherent and provides a useful comparison with the nonminimal matter-curvature coupling model. The parameters alpha, beta, gamma are Taylor coefficients of the model's free functions, and no data fitting is involved, so there is no circularity concern of that kind. However, the central derivation in Section 4.2 contains a clear algebraic error and an ansatz inconsistency; the printed equations (47)-(50) do not follow from equations (35)-(46). The claimed main result is therefore not established in the present form.

major comments (3)

[§4.2, Eqs. (38)-(43)] The derivation of Eq. (43) is algebraically incorrect. From Eq. (42), Z∇_jρ=-A1 for each spatial index j, so A_j=-Z∂_jρ and hence ∇_λA^λ=-Z∇²ρ in the weak-field flat background. Substituting this into Eq. (39), ∇²(Φ-Ψ)=∇_λA^λ, gives ∇²(Φ-Ψ)=-Z∇²ρ, equivalently Z∇²(Ψ-Φ)=Z²∇²ρ. Equation (43) instead states Z∇²(Ψ-Φ)=∇²ρ, missing one factor of Z. Since Eqs. (47)-(48) are presented as solutions of (43)-(44), the missing factor propagates into the final modified Poisson system.
[§4.2, Eqs. (30)-(38)] The ansatz A_μ=(A0,A1,A1,A1) with a single spatial function A1 is not compatible with a generic spherically symmetric density. Equations (38) or (42) applied to j=x,y,z force Z∂_xρ=Z∂_yρ=Z∂_zρ pointwise, so ρ would have to be simultaneously linear in all three Cartesian coordinates. A spherically symmetric density ρ(r) satisfies this only at isolated points where ρ'(r)=0. The claim in §4.2 that this ansatz is "the most general expression compatible with a spherically symmetric physical system" is therefore incorrect, and the final relation A_j=-Z∂_jρ in Eq. (50) is a different, gradient-type ansatz from the one introduced in Eq. (30).
[§4.2, Eqs. (43)-(48)] Even accepting Eq. (43), the printed equations (47)-(48) do not follow from (43)-(44) and are not a valid Poisson system. Since Z=(αγ/2-β) is a constant, the term ∇²Z in Eqs. (47)-(48) vanishes identically; if it is intended as an operator acting on ρ, the notation is undefined. Solving the corrected system ∇²(Ψ+Φ)=γρ/2 and ∇²(Ψ-Φ)=Z∇²ρ gives ∇²Ψ=γρ/4+(Z/2)∇²ρ and ∇²Φ=γρ/4-(Z/2)∇²ρ. These equations, not Eqs. (47)-(48), are the ones consistent with (35)-(36), and they differ from the printed result. Therefore the final modified Poisson system is unsupported as written.

minor comments (3)

[Throughout] Typos should be corrected: "Shappiro" should be "Shapiro", "Minskowski" should be "Minkowski", and "Ostragradsky" should be "Ostrogradsky".
[§4.2, Eqs. (39)-(47)] The sentence "the fourth equation can be put into the first one" appears twice in consecutive derivations, before Eq. (43) and before Eq. (47), making the logical flow difficult to follow; the equations being combined should be identified explicitly at each step.
[§4.2, Eqs. (47)-(48)] The expression ∇²Z in the final equations is undefined for the constant Z; if it is meant to be a differential operator acting on ρ, it must be defined explicitly and its dimensions checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the weak-field Poisson system is a new derivation from an assumed action, not a restatement of its inputs.

full rationale

The paper starts from the nonminimally coupled Weyl connection action (6) and the field equations of [11], where [11] is prior work by the same author and collaborator. This is an assumed starting model, not a result claimed to be derived in the present paper, so citing it is not circular. The weak-field expansion (15)-(21) is then computed in a self-contained way: α, β, γ are defined as Taylor coefficients of the free functions at R̄=0, not adjusted to reproduce the final equations. The final modified Poisson system (47)-(50) is not obtained by fitting data or by renaming inputs; it is presented as the outcome of algebraic manipulation of the expanded field equations. The reader's concern at Eq. (43) — that substituting Z∇_jρ=−A_1 into ∇_λA^λ yields Z∇²(Ψ−Φ)=Z²∇²ρ rather than Z∇²(Ψ−Φ)=∇²ρ — identifies a possible algebraic error, but an incorrect derivation is not a circular derivation. Likewise the ansatz A_μ=(A_0,A_1,A_1,A_1) is an input hypothesis, not something the paper claims to predict from its outputs. No fitted parameter is relabeled as a prediction, no uniqueness theorem is imported from the authors' prior work, and no known result is repackaged under new coordinates. The circularity burden is therefore not met; any defects are correctness risks, not circularity.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The central weak-field equations depend on the free Taylor coefficients α, β, γ from the model's functions f1 and f2, on the background condition f1(0)=0, and on the adopted Weyl vector ansätze. No new entities are introduced; the Weyl vector and coupling functions pre-exist in [11]. The paper supplies no external benchmarks or data for the parameters.

free parameters (3)

α = F'_1(0)/F_1(0)
Taylor coefficient of f_1 in the weak-field expansion; not determined by the paper, enters the final equations via Z.
β = F_2(0)/F_1(0)
Value of f_2 derivative at zero relative to f_1; not fixed, enters Z.
γ = f_2(0)/F_1(0)
Effective coupling constant multiplying matter density; not fixed by the paper.

assumptions (5)

domain assumption Action S = ∫(f1(¥barR)+f2(¥barR)L)√-g d4x is the correct starting point.
Adopted from [11] without re-derivation; all subsequent equations inherit it.
domain assumption Field equations (7)-(10) and constraint (7) from [11] are correct.
The weak-field expansion starts from these equations; the paper does not re-derive them.
domain assumption Background is Minkowski with f1(0)=0.
Stated in §4 after Eq. (19): the lowest order of the expansion implies f1(0)=0; if this fails, the expansion changes.
domain assumption Matter Lagrangian satisfies δT_00=ρ and δT=-ρ (dust-like perfect fluid).
Used in Eqs. (18)-(21) with only a brief 'for a perfect fluid' justification.
domain assumption Weyl vector ansätze (13)-(14), (22), and (30) preserve homogeneity/isotropy or spherical symmetry.
The paper assumes these forms, citing [15,16] and [11]; for the second ansatz the form is inconsistent with Eq. (38) for generic density.

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Cite this review

Pith. "Pith review of Weak Field Limit of the Nonminimally Coupled Weyl Connection Gravity." pith.science (2026). https://pith.science/paper/SZKZFINJ

@misc{pith2026250616532,
  author       = {Pith},
  title        = {Pith review of: Weak Field Limit of the Nonminimally Coupled Weyl Connection Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SZKZFINJ}},
  note         = {Machine review of arXiv:2506.16532}
}

read the original abstract

The true nature of gravity is a remarkable open problem in Gravitation. Theoretical and observational motivations open the avenue of alternative theories of gravity. One possibility resorts to nonminimal couplings and non-metricity properties of spacetime, and is dubbed as nonminimally coupled Weyl connection gravity. It has the advantage of leading to metric field equations of second order together with a constraint equation for the Weyl vector, and has well behaved space-form. We analyse this model by exploring its weak regime and its implications for astrophysics and cosmology.

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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

Atomic clocks and gravitational waves as probes of non-metricity
gr-qc 2026-01 reject novelty 5.0 of 10

The paper claims existing gravitational-wave data already bound Weyl non-metricity, α²ω̄0<10⁻⁶⁹ GeV, via backreaction of a Planck-scale Weyl field, but a dropped kinetic term numerically exceeds the assumed sensitivity.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [11]

Gomes C and Bertolami O 2019 Nonminimally coupled Weyl gravity Class. Quant. Grav. 36 235016

work page 2019

[2] [1]

Sotiriou T P and Faraoni V 2010 f(R) theories of gravity Rev. Mod. Phys. 82 451

work page 2010

[3] [2]

De Felice A and Tsujikawa S 2010 f(R) theories Living Rev. Relativ. bf 13 3

work page 2010

[4] [3]

Bertolami O, Boehmer C, Harko T and Lobo F S N 2007 Extra force in f(R) modified theories of gravity Phys. Rev. D 75 104016

work page 2007

[5] [4]

Olmo G Z and Rubiero-Garcia D 2015 Brane-world and loop cosmology from a gravity–matter coupling perspective Phys. Lett. B 740 73

work page 2015

[6] [5]

Ortiz-Ba˜ nos M, Bouhmadi-L´ opez M, Lazkoz R and Salzano V 2021 ΛCDM suitably embedded in f(R) with a non-minimal coupling to matter Eur. Phys. J. C 81 237

work page 2021

[7] [6]

Jim´ enez J B, Heisenberg L and Koivisto T 2018 Coincident general relativity Phys. Rev. D 98 044048

work page 2018

[8] [7]

Harko T, Koivisto T S, Lobo F S N, Olmo G J and Rubiera-Garcia D 2018 Coupling matter in modified Q gravity Phys. Rev. D 98 084043

work page 2018