Higher-Order Newton-Cartan Gravity

Biel Cardona; Luca Romano

REVIEW 3 major objections 6 minor 1 cited by

Higher-Order Newton-Cartan Gravity

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Quadratic gravity theories admit a finite magnetic non-relativistic limit, and the equations-of-motion limit of Gauss-Bonnet and Ricci-squared gravity yields corrected Poisson equations under on-shell constraints.

desk verdict Solid action-level construction of non-relativistic quadratic gravity; the Poisson-equation results are real but conditional on constraints the paper does not fully justify. read the letter →

arxiv 2507.05489 v1 pith:7EF2Q7NA submitted 2025-07-07 hep-th gr-qc

Biel Cardona , Luca Romano This is my paper

classification hep-thgr-qc

keywords Newton-Cartangravitynon-relativisticlimitquadraticGauss-BonnetRicci-squaredPoissonequationBargmannalgebramagnetic

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

Higher-order corrections to general relativity are expected on general grounds, and this paper asks whether they survive the non-relativistic limit in which relativistic gravity becomes Newton-Cartan gravity, namely the speed-of-light-to-infinity (magnetic) limit. The authors claim that the answer is yes: any Lagrangian built from the Ricci scalar, the Ricci-squared term, and the Riemann-squared term can be given a finite non-relativistic action by adding a single 1-form gauge field and specific higher-derivative couplings that cancel all divergent powers of $c$. They then take the limit at the level of the equations of motion for two concrete theories and show that it is possible to recover the Poisson equation with higher-derivative corrections, for Einstein-Gauss-Bonnet gravity by deforming the model with a scalar field, and for quadratic-Ricci-scalar gravity by imposing an extra on-shell constraint. The resulting full sets of equations transform covariantly under Galilean boosts and define what the authors call zero-torsion Gauss-Bonnet and quadratic-Ricci Newton-Cartan gravity. A sympathetic reader would care because this provides a non-relativistic corner of higher-order gravity that could serve as a testing ground for corrections coming from quantum or string-theoretic settings.

What carries the argument

The central object is the magnetic limit, the version of the non-relativistic limit in which a 1-form gauge field with ansatz $A_\mu=c\tau_\mu+c^{-1}a_\mu$ is introduced so that the divergent leading orders of the action cancel and the surviving order is $c^0$. At the action level the machinery is the set of tuned four-derivative combinations (21), fixed by requiring all powers of $c$ above $c^0$ to cancel. At the level of the equations of motion the machinery is the Poisson combination $[P]=A E^a_\mu E^{a\nu}[G]_{\mu\nu}+C E^0_\mu E^0_\nu[G]^{\mu\nu}-B E^0_\mu[A]^\mu$ (with an extra $D[\Phi]$ term in the Gauss-Bonnet case), whose leading orders are successively cancelled by the zero-torsion constraint and the scalar-field or on-shell conditions. The output is expressed in the Newton-Cartan curvatures $R(H)_{\mu\nu}$, $R(G)_{\mu\nu}{}^{a}$, $R(J)_{\mu\nu}{}^{ab}$ and their trace $\mathrm{Ric}(J)$, which carry the boost-covariant equations of the two new theories.

What would settle it

Take the pure (undeformed) Einstein-Gauss-Bonnet equations of motion (42), substitute the vielbein and gauge-field ansätze (2) and (9), and form the most general combination $[P]=A E^a_\mu E^{a\nu}[G]_{\mu\nu}+C E^0_\mu E^0_\nu[G]^{\mu\nu}-B E^0_\mu[A]^\mu$ with arbitrary $A,B,C$. The paper's claim predicts that no choice makes both the $c^2$ and $c^0$ coefficients of the expansion vanish simultaneously, because the $c^0$ coefficient (44b) contains two independent curvature combinations with different coefficients; if such a choice exists, the scalar-field deformation and the equivalent constraint (55) are unnecessary for obtaining the Poisson equation.

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Extended reading notes

Core claim

The paper establishes that the magnetic non-relativistic limit of quadratic gravity is well defined at the action level: starting from $S=\int d^D x\,\sqrt{-g}\left(R+\alpha R^2+\beta R_{\mu\nu}R^{\mu\nu}+\gamma R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}\right)$ and supplementing the Einstein-Hilbert-Maxwell action with the three combinations (21), all powers of $c$ above order $c^0$ cancel and the surviving action is expressed in Newton-Cartan curvatures. The recipe extends to any Lagrangian that is a function of the three curvature-squared scalars by the substitution (23), and the tuned couplings turn out to match those obtained from Kaluza-Klein and null reductions of pure higher-order gravity in one dimension higher. At the level of the equations of motion, the two-derivative procedure that yields the Poisson equation at order $c^{-2}$ under the zero-torsion constraint $F_{\mu\nu}=0$ is promoted to higher order: for Einstein-Gauss-Bonnet gravity, a scalar-field deformation with $\partial_\mu\Phi=0$ (equivalently the on-shell condition (55)) produces the corrected Poisson equation (54), and for quadratic-Ricci-scalar gravity the supplementary on-shell constraint $\langle C\rangle=R^2+2\nabla_\mu\nabla^\mu R=0$ produces (64c). In both cases the complete set of equations, supplemented by its constraints, closes under Galilean boosts and is presented as defining zero-torsion Gauss-Bonnet Newton-Cartan gravity and zero-torsion quadratic-Ricci Newton-Cartan gravity.

Load-bearing premise

The load-bearing premise is that the constraints used to delete the unwanted leading terms, zero torsion $F_{\mu\nu}=0$, the constant scalar field $\partial_\mu\Phi=0$ in the Gauss-Bonnet case, and the supplementary condition (62) in the quadratic-Ricci case, are admissible and satisfiable on real backgrounds; the paper offers no independent physical justification for them, so if they cannot be imposed, the corrected Poisson equations (54) and (64c) are not limits of the stated theories.

Editorial extensions

If this is right

Any theory with Lagrangian $f(R,R_{\mu\nu}R^{\mu\nu},R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})$ acquires a finite non-relativistic action at order $c^0$ through the substitution (23), using only one extra 1-form gauge field and dimension-independent cancellation coefficients.
The tuned relativistic actions (21) and the resulting non-relativistic actions both descend from pure higher-order gravity in $D+1$ dimensions via Kaluza-Klein and null reductions, so the divergence-cancelling couplings are not arbitrary.
The Gauss-Bonnet corrected Poisson equation (54) is the non-relativistic counterpart of a theory whose equations of motion stay second order; it trivialises in $D=4$, where Gauss-Bonnet is topological, and the full multiplet (56) closes under Galilean boosts.
The quadratic-Ricci corrected Poisson equation (64c) is controlled by the non-relativistic Ricci scalar $\mathrm{Ric}(J)$, and the system (64d)-(64g) with constraints (65) also forms a closed boost multiplet.
In both theories all equations except the Poisson equation follow from the corresponding non-relativistic action; the Poisson equation is the piece that requires the limiting combination of relativistic equations and the extra constraints.

Reading between the lines

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

If the constraint machinery is accepted, the same scalar-field-and-constraint recipe should extend to the full $f(R,R_{\mu\nu}R^{\mu\nu},R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma})$ family, with the correcting terms assembled from the action-level results (25); checking this would be a direct continuation of the paper's method.
The appearance of the same tuned coefficients from Kaluza-Klein reduction suggests that the divergence-cancelling terms (21) are effectively unique: any alternative set of four-derivative couplings that cancels the $c^2$ and $c^4$ divergences while preserving the two-derivative $c^0$ sector would have to coincide with (21).
The ad hoc constraints (55) and (62) may be hints of an emergent symmetry in the non-relativistic limit; if a local dilatation symmetry could be identified, the constraints might follow from it rather than being imposed, a possibility the paper itself raises in its outlook.
A testable extension is to apply the scalar-field trick to Lovelock or quasi-topological gravities, whose equations of motion remain second order, to see whether the corrected Poisson equation is the only consistent non-relativistic corner; the triviality of Gauss-Bonnet in $D=4$ already predicts a dimension-dependent pattern.

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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

Colleague,

The paper should be read as two separate packages. The action-level package is the stronger one. Section 2 gives explicit Maxwell-type completions (21) that cancel the c^4 and c^2 divergences in the magnetic limit for R^2, R_{\mu\nu}R^{\mu\nu}, and Riemann^2, plus a recipe for any function of those three scalars. Those combinations are new relative to the cited literature, concrete enough to check, and the KK/null-reduction remark gives an independent reason the coefficients are not arbitrary. I have no serious concern with this part.

The equations-of-motion package is where I agree with the stress-test note. The advertised Poisson equations (54) and (64c) are not limits of the pure parent theories as stated. The Gauss-Bonnet derivation goes through the scalar-deformed Lagrangian (45), with \phi=0 and \partial\phi=0, and the statement that pure EGB plus the alternative on-shell constraint (55) reproduces the same result is asserted rather than demonstrated. The quadratic Ricci scalar derivation requires (62), which is reverse-engineered to kill the c^0 terms and is not checked for consistency with (60) or for existence of nontrivial solutions. So the corrected Poisson equations are best described as conditional outputs of constrained or deformed systems until the admissibility of (55) and (62) is settled. That is a genuine gap, not a nitpick.

Two things make the gap wider than it needs to be. The multi-line expansions (17), (24), (34), (44) are presented without intermediate steps, uniqueness claims rest on the asserted completeness of basis (19), and no computer algebra notebook or ancillary file is shipped. Cadabra and xAct are mentioned, but the reader cannot reproduce the heavy parts. Also, the centrality of the scalar-field trick in Section 3.2.1 is acknowledged only in passing; a less careful reader could mistake the deformed model for the original theory.

Credit where due: the paper is transparent about every constraint it imposes, flags the Ostrogradsky caveat, and the boost-closure checks and Bianchi identities are nontrivial internal evidence that the algebra is coherent. The multiplet structure is a useful advance. The citation pattern is fine, including the immediate Tadros-Kolar and Bergshoeff-Giorgi-Romano context.

Who it is for: anyone working on non-Lorentzian gravity, especially the Newton-Cartan program. I would send it to a serious referee. The action-level results and the multiplets deserve to appear; the Poisson claims need proof of admissibility of the constraints, or rewriting as theorems about the constrained theories, plus the missing computational artifacts.

Recommendation: accept for peer review, with the constraints issue as the main referee question.

Referee Report

3 major / 6 minor

Summary. The paper studies the non-relativistic (c → ∞) 'magnetic' limit of higher-order gravity theories. In Section 2 it proposes a general recipe: supplement the Einstein-Hilbert-Maxwell action with uniquely determined four-derivative Maxwell and curvature-Maxwell couplings, given in (21), so that the c^4 and c^2 divergences cancel and the limit is finite at the action level. The authors verify this for R^2, R_{μν}R^{μν}, and R_{μνρσ}R^{μνρσ}, and observe a higher-dimensional Kaluza-Klein/null-reduction origin. In Section 3 they turn to equations of motion, reviewing the two-derivative Poisson equation, then analyzing Einstein-Gauss-Bonnet gravity and quadratic Ricci scalar gravity. For EGB they introduce a scalar field deformation (45) and the frozen-field constraint ∂_μΦ=0 to extract the corrected Poisson equation (54); for quadratic Ricci theory they impose an additional on-shell constraint (62) and obtain (64c). They then present full sets of non-relativistic equations and their boost transformations, defining 'zero-torsion Gauss-Bonnet Newton-Cartan gravity' and 'zero-torsion quadratic Ricci Newton-Cartan gravity'.

Significance. If the action-level claims are correct, the paper provides a valuable, systematic extension of the Bergshoeff-Rossel-Zojer magnetic limit technique to higher-derivative gravity, with explicit non-relativistic actions in geometric form and a nice higher-dimensional oxidation picture. The boost multiplet structures are interesting and the paper is generally well organized. However, the central equations-of-motion results are more conditional than the abstract suggests: the Poisson equations (54) and (64c) are derived for a deformed/constrained system, with the consistency and admissibility of the constraints not established. The computational results are extensive and plausible, but without the computer-algebra files or more derivation details, independent verification is difficult. These issues do not necessarily invalidate the main ideas but do require attention before the Poisson-equation results can be stated as limits of the parent theories.

major comments (3)

[§3.2.1, Eqs. (45)–(55)] The Poisson equation (54) is derived from the scalar-deformed Lagrangian L_new = L_GB + e^{aΦ}(L_EHM + b∂_μΦ∂^μΦ) with Φ=0 and ∂_μΦ=0 imposed, so the scalar equation [Φ]_new = a L_2 = 0 is an extra field equation not present in Einstein-Gauss-Bonnet gravity. The claim in (55) that the same result follows from pure EGB with the supplementary on-shell condition R²−4R_{μν}R^{μν}+R_{μνρσ}R^{μνρσ}=0 is not demonstrated. As written, Eq. (54) is the limit of a deformed system, not of the stated parent theory, and the abstract's 'we prove' overstates the result. Please either prove that (55) follows from the EGB field equations (42) together with F_{μν}=0, or explicitly reframe the Poisson result as applying to the deformed model.
[§3.3, Eqs. (62)–(64)] The supplementary constraint ⟨C⟩ := R²+2∇_μ∇^μR=0 is introduced solely to cancel the c⁰ terms in (61b), with no argument that it is consistent with the field equations (60) or that it admits non-trivial solutions. The post-limit form (64b) is then used to close the system (64d)–(64g), but its compatibility with the other equations is not checked. Since the constraint is reverse-engineered for the cancellation, the Poisson equation (64c) is conditional on an unproven on-shell condition. Please provide a consistency analysis (e.g., check that ⟨C⟩=0 propagates under the equations of motion and does not force the trivial vacuum) or weaken the claim accordingly.
[§2.2, Eq. (19)] The basis B in (19) is asserted to be complete for the divergence-cancellation analysis, with the statement that terms of the form F∇∇F 'can be proved' to play no role, but no proof is given. The subsequent claim that each of the three quadratic terms admits a unique cancellation combination relies on this completeness. Without the proof (or a reference), the uniqueness statements in Section 2.2 are not fully established. Please supply the missing argument or adjust the claim to be conditional on the stated basis.

minor comments (6)

[§2.2.1] The symbol ∇̃_m appears in (24c) without definition in Appendix A; please define it (or avoid the notation) for the reader.
[Abstract and §3.2] The sentence 'We prove that, in the first case, it is possible to obtain the Poisson equation by introducing a scalar field and imposing an appropriate constraint' should specify that the constraint is an extra field equation of the deformed model, to avoid overclaiming.
[Throughout] There are several typos and minor grammatical issues; for example, 'In between the two-derivative case and the four-derivative theory' and the repeated phrase 'we show that, in the first case' in the abstract. A careful proofread is recommended.
[Fig. 1] Figure 1 is informative but the arrows to and from 'Others' are not clearly explained in the caption; please clarify the logical relationship.
[§3.2.2, Eq. (56)] The equations (56d) and (56e) are given after dropping the coupling α and overall factors; the text should state the precise rescalings so that the reader can reproduce the original limit.
[§2/§3 expansions] Given the length of expansions (17), (24), (34), (44), providing the Cadabra/xAct notebooks as supplementary material would greatly help referees and readers verify the cancellation conditions; at present the intermediate steps are not reproducible from the text alone.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the corrected Poisson equations are computed outputs of explicitly constrained systems; the reverse-engineered constraints are added assumptions, not hidden inputs.

full rationale

The action-level construction in Section 2 is self-contained: the higher-derivative Maxwell terms in (21) are fixed by requiring cancellation of c-divergences (Section 2.2), and the finite c^0 results (24)-(25) are then computed rather than assumed. The Section 3 Poisson derivations are also non-circular in structure. For Einstein-Gauss-Bonnet gravity, the deformed Lagrangian (45) and the constraints Phi=0, dPhi=0 are stated explicitly; after tuning C and D to kill the c^2 and c^0 orders, the surviving c^{-2} term (54) is an actual output, not a pre-imposed equation. For quadratic Ricci scalar gravity, the constraint (62) and the choice C=DA-3A are used to make the c^0 terms (61b) vanish before the c^{-2} Poisson equation (64c) is computed. No equation is defined in terms of the target Poisson equation, so the derivations are not self-definitional. Two caveats should be weighed but they are not circularity: the extra constraints (55) and (62) are reverse-engineered to cancel unwanted orders and are not derived from the parent theories, and the claimed equivalence of the scalar-field trick to the pure on-shell constraint (55) is asserted without proof (Section 3.2.1). These are correctness/completeness concerns about whether the results are unconditional limits of pure EGB or quadratic-Ricci gravity. The self-citation to [57] (Bergshoeff-Giorgi-Romano, one of the present authors) supplies the scalar-field/combination method, but the method is also described in this paper and the new higher-order results do not reduce to that citation. Hence no circular step is exhibited, and the circularity score is 0.

Assumptions & free parameters 2 free parameters · 6 assumptions · 2 invented entities

The ledger shows that the paper's results are not self-contained: the divergence-cancellation recipe depends on the asserted completeness of the basis (19); the Poisson-equation extractions require the zero-torsion constraint plus, for Gauss-Bonnet, a scalar-field deformation or the equivalent on-shell condition (55), and, for quadratic-Ricci theory, the reverse-engineered constraint (62). The free parameters are the combination coefficients A, B, C, D and the scalar coupling a, all fixed to make the divergent orders cancel; the output equations (54) and (64c) are then computed, not assumed. No new physical entities with independent evidence are introduced; the scalar field is a bookkeeping device and the two new theories are the paper's output.

free parameters (2)

Poisson-combination coefficients A, B, C = GB: C = (D-5)A, B drops out when Fµν = 0; R²: C = (D-3)A
In the combination [P] (30), the constants are tuned so the c² and c⁰ orders of the expansion vanish, exposing the c^{-2} Poisson term; the surviving overall factor A is then dropped by rescaling. This is hand-tuning of the extraction procedure, not a fit to data.
Scalar-field couplings a, b and constant value ϕ = Da = A (so a = A/D), b unconstrained, ϕ = 0
In the deformed action (45), a is fixed by (53) so the scalar equation [Φ] cancels the c⁰ term (52); ϕ is set to zero by hand, absorbing its value in a rescaling. These choices are necessary for the Gauss-Bonnet Poisson equation (54).

assumptions (6)

ad hoc to paper Completeness of the four-derivative basis B (19): the seven listed terms span all terms relevant for divergence cancellation; terms of the form F∇∇F can be dropped.
Section 2.2 states 'we have considered all the independent terms up to Bianchi identities and terms of the type F∇∇F, since it can be proved that these do not play any role in the cancellation of divergences.' The proof is not shown, and the claimed uniqueness of the three solutions (21) depends on this basis.
domain assumption The magnetic-limit ansatz (2), (9) with the given c-scaling of vielbein and gauge field, and the rule that the limit selects the c⁰ (action) or lowest surviving c-order (equations of motion) piece.
Section 2.1, following Bergshoeff-Rosseele-Zojer [9]. The entire construction inherits this choice; other scalings (electric limits, p-brane generalizations cited in the introduction) are excluded.
domain assumption The zero-torsion on-shell constraint Fµν = 0 (equivalently tµν = 0 in the limit) is an admissible supplement that shifts divergent orders to c^{-2}.
Eq. (36)-(39) in section 3.1, carried into sections 3.2 and 3.3. Without it the c² and c⁰ orders cannot be removed, so the Poisson extraction fails.
ad hoc to paper For Gauss-Bonnet gravity, the scalar-field deformation (45) with the frozen-field constraint ∂µΦ = 0 (48), or equivalently the on-shell condition (55) R² - 4RµνRµν + RµνρσRµνρσ = 0, is an admissible addition to the theory.
Section 3.2.1. The claim that the Poisson equation follows from Einstein-Gauss-Bonnet gravity is only true for this deformed or further-constrained system; the equivalence of the two routes is asserted, not demonstrated.
ad hoc to paper For quadratic-Ricci-scalar gravity, the supplementary on-shell constraint (62) ⟨C⟩ := R² + 2∇µ∇µR = 0 is admissible.
Section 3.3. The constraint is introduced specifically so the c⁰ terms (61b) vanish, and no independent physical or geometric justification is provided.
domain assumption The (D+1)-dimensional oxidation claims: the adjusted D-dimensional actions (21) match a Kaluza-Klein reduction, and the non-relativistic actions match a null reduction of the same (D+1)-dimensional pure gravity theory.
Section 2.3, figure 1. Presented as a remark; the term-by-term matching is asserted rather than shown.

invented entities (2)

Auxiliary scalar field Φ with constant on-shell value
purpose: Supplies the extra equation [Φ] = 0 whose c⁰ expansion cancels the unwanted higher-curvature terms in the Poisson combination for Einstein-Gauss-Bonnet gravity (45)-(53).
Bookkeeping device, not a physical field: the constraint ∂µΦ = 0 freezes it to a constant, so it carries no degrees of freedom and makes no falsifiable prediction. Its introduction changes the theory whose non-relativistic limit is computed.
Zero-torsion Gauss-Bonnet Newton-Cartan gravity and zero-torsion quadratic-Ricci Newton-Cartan gravity
purpose: The two new non-relativistic theories defined by the equation sets (56) and (64), closed under Galilean boosts up to the imposed constraints.
These theories are defined in this paper and have no analyzed solutions, stability properties, or observational handles; the conclusion explicitly defers solution spectra to future work.

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Pith. "Pith review of Higher-Order Newton-Cartan Gravity." pith.science (2026). https://pith.science/paper/7EF2Q7NA

@misc{pith2026250705489,
  author       = {Pith},
  title        = {Pith review of: Higher-Order Newton-Cartan Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7EF2Q7NA}},
  note         = {Machine review of arXiv:2507.05489}
}

read the original abstract

We study the non-relativistic Newton-Cartan limit of higher-order gravity theories in arbitrary dimensions. We first study it at the level of the action by introducing an additional 1-form gauge field and coupling it appropriately to the gravity sector. We extend this procedure to any theory whose Lagrangian is a function of the Ricci scalar, quadratic Ricci tensor and quadratic Riemann tensor. We also study the limit of the equations of motion for two models, Einstein-Gauss-Bonnet gravity and quadratic Ricci scalar theory. We prove that, in the first case, it is possible to obtain the Poisson equation by introducing a scalar field and imposing an appropriate constraint. In the latter case, we show that it is possible to get the Poisson equation from the limit of the equations of motion as long as the on-shell constraint used in the two-derivative theory is supplemented with a further condition. We give the expressions of the two higher-order corrected Poisson equations in terms of curvatures of Newton-Cartan geometry. In both cases, we derive the full set of non-relativistic equations and study their boost transformations. The two sets of equations of motion define zero-torsion Gauss-Bonnet Newton-Cartan gravity and zero-torsion quadratic Ricci scalar Newton-Cartan gravity.

Figures

Figures reproduced from arXiv: 2507.05489 by the authors.

**Figure 2.** We show the multiplet structure of the non-relativistic equations of motion of Gauss-Bonnet [PITH_FULL_IMAGE:figures/full_fig_p020_2.png] view at source ↗

**Figure 3.** We show how the multiplet of non-relativistic equations of motion describing the quadratic Ricci [PITH_FULL_IMAGE:figures/full_fig_p022_3.png] view at source ↗

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

Curvatures and Non-metricities in the Non-Relativistic Limit of Bosonic Supergravity
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A metric-only, torsionless-connection formulation with fixed non-metricities rewrites the non-relativistic limit of bosonic supergravity covariantly and matches the vielbein string Newton-Cartan action at Lagrangian level.

Reference graph

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