Affine connections for Galilean and Carrollian structures: a unified perspective

Hamed Barzegar; James Read; Quentin Vigneron

REVIEW 3 major objections 4 minor 27 references

Affine connections for Galilean and Carrollian structures: a unified perspective

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper gives the first general classification of Carrollian affine connections with torsion and non-metricity, and uses it to build an ultra-relativistic geometric trinity.

desk verdict The Carrollian classification is a genuine gap-filler worth having, but the advertised 'ultra-relativistic geometric trinity' is a kinematic diagram, not a construction of theories with dynamics. read the letter →

arxiv 2506.03936 v2 pith:ZFM7YRMA submitted 2025-06-04 gr-qc hep-th

Quentin Vigneron , Hamed Barzegar , James Read This is my paper

classification gr-qchep-th

keywords CarrollianstructureGalileanaffineconnectiontorsionnon-metricitygeometrictrinityglobalhyperbolicitylimitsofLorentziangeometry

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 aims to give the first complete classification of affine connections on Carrollian spacetimes when the connection is allowed to have both torsion and non-metricity, and to set that classification beside the recently completed Galilean case. It shows that both families arise as the $c\to 0$ and $c\to\infty$ limits of the most general Lorentzian connection, and that global hyperbolicity of the Lorentzian parent imposes concrete constraints on the limiting structures. On top of the classification, the paper constructs an ultra-relativistic geometric trinity, the Carrollian analogue of the trio of gravitational theories in which curvature is traded for torsion or non-metricity, and shows it descends from the relativistic trinity. A further result is that a torsion-free connection compatible with both a Galilean and a Carrollian structure must be quite special, with only a static spatial curvature remaining.

What carries the argument

The load-bearing object is the decomposition identity for a general affine connection with respect to a Carrollian structure, $$\Gamma^\$\alpha${}_{\mu\nu}-{}^{v,\tau}\Gamma^\$\alpha${}_{\mu\nu} = -\check Q_{(\mu\nu)}{}^\$\alpha$+\tfrac12\check Q^\$\alpha${}_{\mu\nu}-T_{(\mu\nu)}{}^\$\alpha$+\tfrac12 T^\$\alpha${}_{\mu\nu}-v^\$\alpha$\chi_{\mu\nu},$$ together with its Galilean counterpart. This identity turns the problem of classifying connections into the problem of classifying the independent reduced data $\check Q$, $T$, and $\chi$, with the identities (9) showing which combinations are forced by the intrinsic expansion tensor $\Theta_{\mu\nu}$. The same scaffolding carries the limit analysis: writing Lorentzian metrics and connections as Taylor series in $\lambda=1/c$ or $\epsilon=c$ and reading off the leading orders produces the Galilean and Carrollian structures, while global hyperbolicity selects the global constraints $\tau=N\,dt$ and completeness of $v^\mu$.

What would settle it

Exhibit a smooth one-parameter family of globally hyperbolic Lorentzian metrics whose $c\to 0$ (or $c\to\infty$) limit is smooth but not analytic, in which the leading-order Carrollian vector $v^\mu$ fails to be complete (or the leading-order Galilean 1-form fails to be globally $\tau=N\,dt$); this would refute the paper's claim that global hyperbolicity alone forces those constraints.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the general affine connection on a Carrollian structure $(v^\mu,\gamma_{\mu\nu})$ is not an arbitrary object: once a dual Galilean structure $(\tau_\mu,h_{\mu\nu})$ is chosen, the difference between any connection and a background connection ${}^{v,\tau}\Gamma^\alpha{}_{\mu\nu}$ is fixed by the Carrollian non-metricities $\check Q^\mu{}_\nu=\nabla_\mu v^\nu$ and $\check Q^\alpha{}_{\mu\nu}=\nabla_\alpha\gamma_{\mu\nu}$, the torsion $T^\alpha{}_{\mu\nu}$, and a symmetric tensor $\chi_{\mu\nu}=\nabla_{(\mu}\tau_{\nu)}$, subject to identities that tie torsion to non-metricity and to the intrinsic expansion $\Theta_{\mu\nu}=\tfrac12\mathcal L_v\gamma_{\mu\nu}$. The paper defines reduced torsion and reduced non-metricities that remove the parts forced by $\Theta_{\mu\nu}$, and shows these reduced variables are precisely what survives in the $c\to 0$ limit. It then proves that global hyperbolicity makes $\tau=N\,dt$ hold globally in the Galilean limit and makes $v^\mu$ complete in the Carrollian limit, and uses the framework to build the first ultra-relativistic geometric trinity of gravitational theories.

Load-bearing premise

The global-hyperbolicity constraints assume that the limiting Lorentzian data are analytic, so the Taylor expansions can be taken order by order and each order inherits global properties such as foliation-forming and completeness; for arbitrary smooth metrics this need not hold.

Editorial extensions

If this is right

Every affine connection on a Carrollian spacetime, including those with both torsion and non-metricity, can be written in the paper's reduced variables, so future metric-affine Carrollian gravity theories have a ready-made kinematical vocabulary.
The ultra-relativistic geometric trinity exists: the $c\to 0$ limits of GR, TEGR, and STEGR are respectively a curvature-based Carrollian theory, a flat metric torsionful theory, and a flat torsion-free non-metric theory, with the same structure as the Galilean trinity.
Global hyperbolicity of the Lorentzian parent spacetime forces the Newtonian time 1-form to be globally exact, $\tau=N\,dt$, in the Galilean limit and forces the Carrollian vector field $v^\mu$ to be complete, without needing assumptions about matter actions.
A torsion-free connection compatible with both a Galilean and a Carrollian structure has only a static spatial curvature; such connections are also compatible with a Riemannian or Lorentzian metric, making $v^\mu$ a Killing vector.
The Galilean limit of STEGR is flat, torsion-free, and non-metric in reduced variables, and the same characterization holds for the Carrollian limit, so the two trinities are genuinely the two limits of the relativistic one.

Reading between the lines

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

Inference: Because the reduced torsion and reduced non-metricities depend on a choice of observer or Ehresmann connection, any Carrollian gravitational theory built from them will carry an additional gauge freedom; pinning down how physical quantities transform under that choice is a necessary next step.
Inference: The symmetry of $\chi_{\mu\nu}$ suggests that a potential-based Carrollian gravity, if it exists, will not be a direct copy of Newtonian gravity; one testable route is to impose a gauge condition analogous to vanishing spatial torsion and see whether a Carrollian potential emerges.
Inference: Following the Galilean pattern, the three members of the ultra-relativistic trinity may share a common 'Maxwell-type' core theory; checking whether the three Carrollian actions reduce to a single common structure would tell whether the trinity is physically meaningful or merely formal.
Inference: The simultaneously Galilean- and Carrollian-compatible connections, with their static spatial curvature, could serve as reference connections for cosmological models with non-Euclidean spatial topology, extending a known Galilean construction to the Carrollian side.

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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 Carrollian classification is real and fills a gap; the trinity is not. The paper gives a clean algebraic decomposition of general affine connections on Carrollian spacetimes, with torsion and non-metricity, which nobody had written down in full. Compared with Schwartz's Galilean classification and the torsionful metric Carrollian work of Hartong and Bekaert–Morand, this is a genuine addition. Proposition 2.1 and Table 1 are the core, and the appendix is worth a careful read because it works out the non-uniqueness of the reduced variables explicitly—six free parameters in the Galilean case, five in the Carrollian—and shows why the Carrollian reduced torsion is uniquely fixed under metricity. That is honest, reproducible algebra, and it will be a useful reference. Now the soft spots, in rough order of importance. Most significant: the paper claims to 'construct for the first time an ultra-relativistic geometric trinity of gravitational theories.' It doesn't. Section 4 identifies which Carrollian connections arise as limits of STEGR and TEGR—flat, torsion-free with reduced non-metricities, and flat, metric with reduced torsion—but it gives no action, no equations of motion, and explicitly says it will not. A trinity of theories without dynamics is a kinematical classification. The authors are aware of this, since the 'remark on variational principles' admits they eschew Lagrangians, so the abstract and conclusion overstate the content. That should be toned down or the actions supplied. Second, there is a concrete inconsistency in Section 2.3. Equations (20)–(21) define the Carrollian reduced non-metricities using ωμν, but Table 2 and Appendix A.2 use Θμν. This is almost certainly a typo—ω belongs in the Galilean definitions—but it sits in the key definition, and a referee will hit it immediately. It should be fixed. Third, the global hyperbolicity argument in Section 3.3.2 assumes analyticity to conclude that the leading-order pieces of the limit structures inherit global properties such as foliation-forming and completeness. No justification is given, and for smooth-but-not-analytic Lorentzian metrics the Taylor expansion step fails. This makes the claimed constraints from global hyperbolicity conditional. It is an aside rather than load-bearing for the classification, but the claim should be flagged as such. Bottom line: the central algebraic framework holds up. The typo is fixable, the analyticity point needs a caveat, and the trinity claim needs rephrasing. The paper is worth refereeing, and with those revisions it is citable for the classification. Send it out.

Referee Report

3 major / 4 minor

Summary. The paper develops a systematic classification of affine connections on Carrollian structures (a vector field v and a positive semidefinite spatial metric gamma) that may carry both torsion and non-metricity, and compares it with the analogous Galilean classification of Schwartz. It introduces 'reduced' torsion and non-metricity variables, discusses how Carrollian and Galilean structures arise as limits of Lorentzian structures, argues that global hyperbolicity imposes certain constraints on the limiting data, and uses the classification to characterize the Carrollian limits of STEGR and TEGR connections. It also analyzes connections compatible simultaneously with Galilean and Carrollian structures, and closes with open questions. The central algebraic claims are the general form of a Carrollian connection with torsion and non-metricity (Proposition 2.1, Table 1) and the definitions of reduced variables (Table 2).

Significance. If the classification is correct, the paper fills a genuine gap in the literature: the general Carrollian connection with both torsion and non-metricity had not been presented before. The unified treatment of Galilean and Carrollian structures, the explicit reduced variables, and the systematic derivation of both as limits of the most general Lorentzian connection are useful contributions that will likely serve as a reference for metric-affine Carrollian gravity and for further work on Carrollian boundaries. The connections compatible with both structures are also an interesting mathematical byproduct. However, the advertised 'ultra-relativistic geometric trinity' is not actually constructed at the level of actions or field equations, and the reported global-hyperbolicity constraints rest on an unstated analyticity assumption; these weaken the paper's headline claims even though the underlying algebraic framework appears sound.

major comments (3)

[Section 2.3, Eqs. (20)-(21)] The displayed definitions of the Carrollian reduced non-metricities contain the Galilean vorticity omega_mu_nu rather than the Carrollian expansion tensor Theta_mu_nu: Eq. (20) reads Qcheck_mu_nu = Qcheck_mu_nu - omega_mu_nu - 2 v^sigma omega_{sigma(mu} tau_nu) and Eq. (21) is analogous. This contradicts Table 2 and Appendix A.2, where the corresponding definitions use Theta_mu_nu. As printed, Eq. (20)-(21) define quantities that would not vanish for the Carrollian limit of the Levi-Civita connection, whose non-metricities are Qcheck_mu_nu = Theta_mu_nu and Qcheck_alpha_mu_nu = -2 tau_(mu Theta_nu)_alpha (Eq. (33)), contrary to the stated requirement that the reduced non-metricities vanish in that limit. These equations must be corrected to use Theta_mu_nu. Relatedly, Eq. (18) appears to omit the projector h^alpha_sigma in the contraction 2 tau_[mu Theta_nu]_sigma h^alpha_sigma that is present in Table 2.
[Abstract, Conclusion, and Section 4] The paper claims to 'construct for the first time an ultra-relativistic geometric trinity of gravitational theories' (abstract and conclusion), but Section 4 does not construct any gravitational theories in the sense used for the relativistic trinity or for the non-relativistic trinity of Ref. [9]. Section 4 only identifies the zero-th order connections arising in the Carrollian limits of STEGR and TEGR; there are no action principles and no equations of motion. The subsection 'A remark on variational principles' explicitly declines to present the Lagrangians ('we eschew an explicit presentation here'), and no field equations appear anywhere in the section. The non-relativistic trinity in [9] was at least presented at the level of equations of motion, but the Carrollian case reaches neither the action level nor the equation-of-motion level. As it stands, Figure 1 is a diagram of connection types, not a trinity of gravitational theories. Please either supply the missing field equations/actions or revise the abstract and conclusion to claim only the geometric groundwork for an ultra-relativistic trinity.
[Section 3.3.2] The claim that global hyperbolicity forces tau = N dt globally and v to be complete is derived under an unstated and non-general assumption. The text says 'Because we assume analyticity, then each order of lambda n_mu and epsilon n_mu has to be a complete vector field...' but no justification is given for assuming analyticity of the limit expansions. For a general smooth globally hyperbolic Lorentzian metric, the limit data need not be analytic, and the conclusion that each order inherits global properties such as global N dt form or completeness is not established. Thus the summary statement that global hyperbolicity 'already leads' to these constraints overstates the result as proved. Please either state analyticity as an explicit extra hypothesis in the proposition/table, or supply a proof that works for smooth data.

minor comments (4)

[Section 2.3] There is a duplicated word in the sentence 'we define the the Carrollian reduced torsion' before Eq. (18).
[Notation, Section 2.1] The paper would benefit from a short index-convention reminder when objects such as h_nu^mu and gamma_mu_nu appear in the same equation, since raised and lowered indices are structure-dependent and the dual structures are gauge choices.
[Table 3] In the Carrollian row of Table 3, the entry for chi_mu_nu contains omega through the term -2 v^sigma tau_(mu omega_nu)_sigma; this is correct only after using the relation tau_mu partial_[nu tau_sigma] = tau_mu omega_nu_sigma, so a parenthetical note in the table would improve readability.
[Section 5] The sentence 'This means that the Carrollian and Galilean time metrics are either dual if tau_mu v^nu != 0 or orthogonal with tau_mu v^nu = 0' is slightly confusing because tau_mu v^nu is a tensor with mixed indices; please clarify the intended scalar contraction.

Circularity Check

0 steps flagged · score 1.0 of 10

Core classification is self-contained algebra; the advertised ultra-relativistic trinity is asserted only at the level of connection types (a scope overreach), and the self-citations used are comparative rather than load-bearing.

full rationale

No circular step is exhibited. The central novelty, the general Carrollian connection with torsion and non-metricity, is derived in-paper: Proposition 2.1 follows from a direct computation using the identities (9), and Appendix A.2 solves the reduced-variable freedom algebraically (e.g., the metric Carrollian reduced torsion is uniquely fixed with 'no degrees of freedom left'). The reduced variables are defined by a stated convention rather than a hidden fit: the paper says 'we define the non-metricities such that the limit of the Levi-Civita connection of a Lorentzian metric is metric in the reduced variables,' so the later statement that Levi-Civita limits have zero reduced non-metricity is true by construction, but this is an acknowledged convention, not a prediction dressed as a result. The claimed 'ultra-relativistic geometric trinity' is, however, not actually constructed at the level of theories: Section 4 gives only flatness plus reduced torsion/non-metricity characterizations of connection types, and the 'A remark on variational principles' passage explicitly says 'we eschew an explicit presentation here' of Lagrangians and gives no equations of motion, whereas the relativistic trinity is defined by actions and the non-relativistic trinity by equations of motion. This is a claim-content mismatch (flagged per the reviewing rule), not a circular reduction. Self-citations appear but are not load-bearing: [9] (Wolf, Read, Vigneron, with two of the present authors) supplies the comparative Galilean STEGR/TEGR baseline, and [26] (Vigneron) is an application remark in Section 5; the new Carrollian limits are derived from the paper's own Section 3 computations. The weakest technical premise, the global-hyperbolicity argument in Section 3.3.2, depends on 'Because we assume analyticity' to transfer global properties order by order; for general smooth metrics this is an unproven assumption and a correctness risk, not a circularity. Overall, the derivation chain is independent of its conclusions, so the circularity score is low.

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

The classification depends on standard differential-geometric facts plus a few explicit choices. The most fragile input is the analyticity assumption used to transfer global-hyperbolicity constraints to the limits. The reduced-variable gauges are conventions chosen to match the limits, not fitted to data.

free parameters (3)

Carrollian reduced non-metricity gauge parameters = a1=1, a2=0, a3=1/3
In Appendix A.2, these values are selected so that the Carrollian limit of the Levi-Civita connection has zero reduced non-metricities. They are conventions, not derived from data.
Galilean reduced torsion gauge parameters = c1=2, c3=-4
In Appendix A.1, metricity imposes c1=2 and c3=-4; the remaining freedom (c2, c4) is left unconstrained. The choice matches the non-relativistic limit and prior work by Bekaert-Morand.
Galilean reduced non-metricity gauge parameters = a2=-2, b4=0
Imposed to match the non-relativistic limit from the Levi-Civita connection under tau wedge d tau = 0, as stated in Appendix A.1.

assumptions (5)

standard math Existence of a Riemannian metric on M to construct dual structures
Used in Section 2.1 to show that a Carrollian structure admits a dual Galilean structure; guaranteed on paracompact smooth manifolds, as cited from Wald [14].
standard math Global hyperbolicity implies a global time function tau=N dt and a complete timelike vector field
Used in Section 3.3 to transfer topological constraints to the limiting structures; standard result of Lorentzian geometry, stated without proof in Section 3.3.2.
ad hoc to paper Analyticity of the limit expansions of the Lorentzian metric and fields
Stated in Section 3.3.2 ('Because we assume analyticity') to justify that each order of the Taylor series inherits foliation-forming and completeness properties. This is not implied by global hyperbolicity and is not justified for arbitrary smooth metrics.
domain assumption Frobenius condition tau wedge d tau = 0 holds in the Galilean limit
Needed in Section 2.3 to uniquely define reduced non-metricities; the paper shows global hyperbolicity implies it in Section 3.3, but it is imposed earlier as a working assumption.
domain assumption tau_mu v^nu is non-zero in Section 5
The paper assumes the two time metrics are dual rather than orthogonal in order to classify compatible connections; the orthogonal case is set aside as unusual.

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Pith. "Pith review of Affine connections for Galilean and Carrollian structures: a unified perspective." pith.science (2026). https://pith.science/paper/ZFM7YRMA

@misc{pith2026250603936,
  author       = {Pith},
  title        = {Pith review of: Affine connections for Galilean and Carrollian structures: a unified perspective},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZFM7YRMA}},
  note         = {Machine review of arXiv:2506.03936}
}

read the original abstract

We develop a classification of general Carrollian structures, permitting affine connections with both torsion and non-metricity. We compare with a recent classification of general Galilean structures in order to present a unified perspective on both. Moreover, we demonstrate how both sets of structures emerge from the most general possible Lorentzian structures in their respective limits, and we highlight the role of global hyperbolicity in constraining both structures. We then leverage this work in order to construct for the first time an ultra-relativistic geometric trinity of gravitational theories, and consider connections which are simultaneously compatible with Galilean and Carrollian structures. We close by outlining a number of open questions and future prospects.

Figures

Figures reproduced from arXiv: 2506.03936 by the authors.

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