Trinity of Carrollian Gravity: Dynamically Equivalent Formulations of Magnetic Carrollian Gravity

Aliasghar Parvizi; Mehdi Ahmadi-Jahmani

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Trinity of Carrollian Gravity: Dynamically Equivalent Formulations of Magnetic Carrollian Gravity

High Energy Physics - Theory · Submitted 1 Oct 2026

Pith's verdict: A genuinely intrinsic Carrollian trinity with correct curvature algebra, but the equivalences are on-shell identities at hand-picked constants, and Sec. 6 concedes the missing Lagrange-multiplier enforcement.

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arXiv ID2610.01976 (version 2)

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CitationAliasghar Parvizi, Mehdi Ahmadi-Jahmani. "Trinity of Carrollian Gravity: Dynamically Equivalent Formulations of Magnetic Carrollian Gravity." arXiv:2610.01976 (2026). https://pith.science/paper/VINOIIKE

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Editor's note

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Read this one over coffee. The genuinely new thing is that the paper works intrinsically in Carrollian geometry — no c→0 limit of a Lorentzian parent — and produces explicit equivalent Lagrangians for Magnetic Carrollian gravity, which is what Refs. [27] and [28] left open. The general metric-compatible connection (4.14), the torsion decomposition (4.27), and the curvature shifts (4.56–4.60, 4.67, 4.81, 4.103, 5.55–5.58) are correct formal algebra as far as I can check; the F-term cancellations at (4.59), for instance, are legitimate. And the observation that a Carrollian trinity can exit the gravitational sector entirely — into Maxwell, fracton, and dilaton-type theories, where the Lorentzian trinity stays inside gravity — is a genuine extension of the program.

The soft spot is structural and it is the whole game. Every equivalence runs the same play: relax rigidity or metric compatibility, shift the scalar curvature, impose flatness, and read off an isolated kinetic term. Each identification leans on a hand-picked fixing of the clock-form ambiguity. Theory I locks X_{(μν)} = αK_{μν} with α tuned to match the known ECG action (4.47, 4.61); Theory V postulates Υ = λK with λ entirely free, which writes the answer ∫−λK² into the action (5.60, 5.61). Neither fixing is forced by geometry — X_{μν} is arbitrary subject to X_{μν}v^ν = 0. So the headline claim, that a curved rigid universe is dynamically indistinguishable from a flat shearing one, reduces to an on-shell identity at a tuned constant. The fracton and Maxwell mappings are kinetic analogies the authors themselves flag as lacking the required gauge symmetry (b311, b313, b326). And in Sec. 6 they concede that the flatness constraint and these gauge fixings must be enforced with Lagrange multipliers for genuine dynamical equivalence — work they do not actually do. The concession is honest; the missing enforcement is exactly the load-bearing part.

Bottom line: a construction worth refereeing, with claims not yet established off-shell. A referee should push either for the multiplier enforcement or for an explicit downgrade to on-shell equivalence at special parameter values. I'd bring it to reading group to argue about what "dynamical equivalence" should mean here, but I wouldn't cite the equivalences as established in my own work.

Referee report

Significance

If the equivalences held as genuine dynamical ones, this would be a substantial contribution: a Carrollian analog of the GR trinity with five target theories, two non-gravitational, and an explicit geometric dictionary connecting magnetic curvature to gauge and scalar sectors. The algebra is a real asset: the shifted Ricci identities (4.56–4.60, 4.67, 4.81, 4.103, 5.55–5.58) and the torsion decomposition (4.27) are internally consistent and derivable, and the paper is transparent about where gauge symmetries are absent (the fracton and Kalb-Ramond identifications, §4.2.3) and where dynamical enforcement is missing (§6). The paper would be a useful toolkit for Carrollian holography and condensed-matter analogies even at the level of on-shell algebraic relations. However, the central claim as advertised — dynamical equivalence — is not yet established: the target actions arise at tuned values of free gauge constants and under a stationarity restriction (L_v τ = 0), and the constrained variations that would make the equivalences dynamical are sketched but absent. This gap is acknowledged and, in principle, closable.

Major objections (4)

[§4.2.2 and §5.2.1, Eqs. (4.47), (4.60)–(4.61), (5.60)–(5.61)] The Theory I and Theory V equivalences are established only at tuned values of free gauge constants. The sole geometric constraint on the clock-form ambiguity is X_μν v^ν = 0 (§4.1); the symmetric part X_(μν) is otherwise free. The ansatz X_(μν) = αK_μν (4.47) is a postulate, and the match to the standard ECG Lagrangian ∫e(K_μνK^μν − K²) in (4.60)–(4.61) fixes α = −1 by hand. Similarly, Υ = λK (5.60) is introduced as a postulate, and the resulting STECG action (5.61) inherits the completely free parameter λ, so any λ is allowed. As written, these are algebraic identities at special parameter values rather than dynamical equivalences forced by the geometry. The paper should either derive these fixings from a geometric criterion (e.g., the embedding argument of §5.1) or explicitly rescope the claims as equivalences that hold only at tuned parameter values.
[§6, closing paragraph] The authors concede that the constraints defining each equivalence — global flatness, the torsion/non-metricity restrictions, and the fixings (4.47) and (5.60) — 'must be dynamically enforced' via Lagrange multipliers and that this is not done in the paper. This concession is the load-bearing gap: the flatness condition is precisely the mechanism that trades curvature for the kinetic sectors, so without the constrained variations one cannot determine whether the constrained theories share the solution space, symmetries, and degrees of freedom of the baseline ∫eR̃. A necessary addition is the explicit Lagrange-multiplier action for at least one representative (e.g., Theory I) with the multipliers included, followed by a comparison of the equations of motion, in particular a counting of propagating degrees of freedom before and after enforcement. Without this, 'dynamical equivalence' is an announcement rather than a demonstrated property.
[§4.2.2, Eqs. (4.47)–(4.50); §4.2.4, Eq. (4.96)] Combining the ansatz X_μν = αK_μν − ½F_μν with the transversality condition X_μν v^ν = 0 forces L_v τ_μ = 0, i.e., a stationary clock form with vanishing Carrollian gravitoelectric field. This is a substantive restriction on the headline duality that a curved rigid universe is indistinguishable from a globally flat expanding/shearing one: the equivalence holds only on configurations with no gravitoelectric field. The text acknowledges this under 'Stationary time', but the condition is part of the hypotheses of the result and should be stated as such in the statement of the equivalence, along with the analogous L_v φ = 0 condition in (4.96) for the dilatonic sector.
[§4.2.3, Eqs. (4.68)–(4.72)] The identifications with Kalb-Ramond and scalar-charge fracton theories are made at the level of kinetic terms only, and the author's own text states the required gauge symmetries are absent (δB = dΛ for the axial sector; the full fracton gauge freedom for V⊕q). The degree-of-freedom argument (9 − 1 = 8) used to match the traceless fracton magnetic field rests on discarding the axial sector by ansatz, not by a geometric constraint. This is an acknowledged limitation, but the 'dictionary' claim would be strengthened if the paper stated explicitly which geometric conditions select the V⊕q sector, or alternatively if it presented the identifications as structural analogies rather than equivalences. The authors acknowledge this: “However,A still lacks the gauge symmetry required for a complete realization of the scalar-charge Fracton theory. 4.2.4 Equivalent Theory III: Temporal Torsion and Dilatonic Scaling To explore another sector of equivalent theories, in the following we will isolate the torsion terms originating from the clock formFµν and the tensorXµν while eliminating all others.”

Minor objections (5)

[Abstract and §1] 'stracturally similar' is misspelled in the abstract and in §1; also 'we prove that the curvature of a Magnetic Carrollian universe can be dynamically traded' overstates what §6 demonstrates, so the wording 'show' or 'argue' would better match the body of the paper.
[§4.2.4, Eq. (4.87) and following text] The claim that ̃∇_μs^μ + s_μs^μ reproduces the transformation of the Ricci scalar under a conformal Weyl rescaling is asserted without giving the explicit transformation identity; adding it would let the reader verify the dilaton mapping in (4.94).
[Appendix A, Eq. (A.38)] The factor-2 discrepancy between the first-order gauged Ricci scalar (A.38) and the second-order scalar (4.42) used throughout the main text is flagged but not reconciled. Since both are presented as the baseline MCG action, the paper should state which one is the operative baseline or justify the relative weight of the boost-curvature sector.
[§4.2.3, Eq. (4.69) and Table 1] The coefficients c1, c2, c3 in (4.69) are left as 'specific dimensional constants'; they are fixed numerical coefficients of the SO(d) orthogonal decomposition of A_σμν and should be computed explicitly, since the fracton and Kalb-Ramond identifications depend on their relative values.
[§5.2.1, Eqs. (5.57)–(5.61)] The suppression of the shear term K_μνK^μν in Theory V follows directly from the trace-gauge Υ = λK, since only the trace Υ couples at (5.57); this should be flagged as a consequence of the gauge choice rather than a general property of minimal non-metricity.

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The paper

The paper claims

Magnetic Carrollian gravity is shown to be dynamically equivalent to flat-space gauge and scalar theories by trading curvature for torsion or non-metricity.

The reading

This paper tries to establish that the geometric trinity of gravity, well known for general relativity, has a Carrollian analogue with a richer structure. The central claim is that the curvature of a magnetic, rigid Carrollian universe can be dynamically traded, under a global flatness condition, for the kinetic energy densities of torsion (in several isolated sectors) or of minimal non-metricity. Each trade yields a distinct equivalent theory: electric Carroll gravity, a fracton-like tensor gauge theory, a Maxwell electric sector, a dilaton scalar, and an isotropic electric Carroll gravity. The equivalence is stated as an algebraic substitution of the scalar curvature $\tilde{R}$ by these kinetic densities, valid once the connection ambiguities are fixed by chosen gauge conditions with free constants.

What carries the argument

The central mechanism is the decomposition of the most general metric-compatible Carrollian connection (Equation 4.14) into geometric pieces: the clock-form ambiguity $X_{\mu\nu}$, spatial contorsion $A^\lambda{}_{\mu\nu}$, extrinsic curvature $K_{\mu\nu}$, and the clock-form field strength $F_{\mu\nu}$. By selectively setting some of these to zero, different pure torsion sectors are isolated, and by imposing global flatness (Riemann$=0$), the Magnetic Ricci scalar is algebraically equated to the quadratic kinetic density of the surviving sector. The same logic applies to the non-metric connection where minimal non-metricity is fixed by $Q_{\rho\mu\nu}=-\tau_\mu K_{\rho\nu}-\tau_\nu K_{\rho\mu}$.

Extended reading notes

Core claim

The paper claims that the Ricci scalar $\tilde{R}$ of Magnetic Carrollian gravity, defined on a torsion-free metric-compatible connection with rigid spatial geometry ($K_{\mu\nu}=0$), can be exactly replaced by specific kinetic terms for torsion or non-metricity sectors of alternative connections, provided those connections are globally flat (Riemann$=0$). In the teleparallel sector, isolating intrinsic torsion maps $\tilde{R}$ to $\alpha(K_{\mu\nu}K^{\mu\nu}-K^2)$ (Equation 4.61), giving Electric Carroll gravity; spatial torsion maps it to $A_{\alpha\beta\gamma}A^{\gamma\alpha\beta}$ (4.68), resembling tensor gauge theories; temporal vector torsion gives $-s_\mu s^\mu$ (4.82), a dilatonic scalar; and clock-form torsion yields $-\tfrac{1}{4}E_\mu E^\mu$ (4.105), a Maxwell electric sector. In the symmetric teleparallel sector, minimal non-metricity coupled with gauge fixing $\Upsilon=\lambda K$ gives $-\lambda K^2$ (5.61), an isotropic electric Carroll gravity. The cleanest instance claimed is that a curved rigid spacetime is dynamically indistinguishable from a globally flat, expanding and shearing spacetime once the clock-form ambiguity is fixed as $X_{(\mu\nu)}=\alpha K_{\mu\nu}$.

Load-bearing premise

Each equivalence relies on a manually chosen gauge fixing with a free constant (e.g., $X_{(\mu\nu)}=\alpha K_{\mu\nu}$ or $\Upsilon=\lambda K$), and the global flatness of the modified connection is imposed without proving these constraints are dynamically forced rather than merely convenient algebraic choices.

Editorial extensions

If this is right

If correct, the magnetic and electric Carrollian sectors are not independent regimes; a single flat connection can describe phenomena that appear as either a curved rigid spacetime or an expanding shearing spacetime.
Carrollian gravity can be translated into non-gravitational languages: the fracton magnetic Hamiltonian, Maxwell electrodynamics, and dilaton (Newtonian-like) scalar theory become geometric sectors of a single underlying connection.
The symmetric teleparallel construction shows that an isotropic, shear-free electric Carroll model, with action $\int -\lambda K^2$, is a valid equivalent form of magnetic Carroll gravity under minimal non-metricity.
The dictionary suggests a broader pattern: any isolated torsion or non-metricity sector of a globally flat Carrollian connection can be viewed as an equivalent formulation of the curvature-based magnetic theory, potentially extending to further matter couplings.

Reading between the lines

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

A natural testable extension is to enforce the gauge fixings (e.g., $X_{(\mu\nu)}=\alpha K_{\mu\nu}$ or $\Upsilon=\lambda K$) through explicit Lagrange multipliers in the action; the paper acknowledges this is required but does not carry it out, leaving open whether the equivalences survive as dynamical constraints rather than on-shell algebraic identities.
The constant parameters $\alpha$ and $\lambda$ in the gauge fixings appear free; if they are truly undetermined, the equivalences hold only at special tuned values, which would weaken the claim of general dynamical equivalence. A concrete calculation of the reduced phase space could settle whether the free constants are pure gauge or physical.
The mapping of spatial torsion to fracton theories stops short of reproducing the required gauge symmetries (e.g., $\delta B=d\Lambda$ for the Kalb-Ramond sector). An editorial extension would be to investigate whether the connection ambiguities can be restricted so that these symmetries emerge dynamically, connecting Carrollian geometry directly to fracton phases.
The paper's framework suggests a computational rule: any kinetic term quadratic in a purely spatial or temporal Carrollian tensor can be realized as an equivalent formulation of magnetic Carroll gravity by an appropriate flat connection, which could guide future searches for new dualities in condensed matter and holography.

Original abstract

In this paper, we extend the paradigm of the Geometrical Trinity of Gravity to the Carrollian geometry to construct a comprehensive framework of dynamically equivalent theories for Magnetic Carrollian gravity. Utilizing the inherent ambiguities and freedoms of affine connections on degenerate metrics, we systematically relax the metric-compatibility and zero-torsion constraints of the rigid magnetic sector. By imposing global flatness on these modified connections, we prove that the curvature of a Magnetic Carrollian universe can be dynamically traded for isolated kinetic energy densities of torsion and non-metricity. Unlike the Lorentzian Trinity which maps gravity to alternative gravitational frameworks, the Carrollian equivalences can also map spacetime geometry to non-gravitational theories. Specifically, we demonstrate that intrinsic torsion recovers Electric Carrollian gravity as the Teleparallel equivalent of Magnetic Carroll gravity; purely spatial torsion reproduces kinetic terms stracturally similar to the magnetic Hamiltonian of scalar-charge Fracton gauge theory; and temporal torsion sectors map independently to $U(1)$ Maxwell electric fields and Carroll-Weyl conformal dilaton theories. Finally, by formulating a Symmetric Teleparallel Carroll connection with minimal non-metricity, we show that the Magnetic Carrollian Lagrangian is dynamically equivalent to an ultralocal, isotropic Electric Carrollian gravity upon exploiting the present freedoms in connection. These results establish a geometric dictionary translating Carrollian gravitational dynamics into the language of tensor gauge theories, electromagnetism, and dilaton theory.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-02: 1 request, 5,903 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.83 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In this paper, we extend the paradigm of the Geometrical Trinity of Gravity to the Carrollian geometry to construct a comprehensive framework of dynamically equivalent theories for Magnetic Carrollian gravity. Utilizing the inherent ambiguities and freedoms of affine connections on degenerate metrics, we systematically relax the metric-compatibility and zero-torsion constraints of the rigid magnetic sector. By imposing global flatness on these modified connections, we prove that the curvature of a Magnetic Carrollian universe can be dynamically traded for isolated kinetic energy densities of torsion and non-metricity. Unlike the Lorentzian Trinity which maps gravity to alternative gravitational frameworks, the Carrollian equivalences can also map spacetime geometry to non-gravitational theories. Specifically, we demonstrate that intrinsic torsion recovers Electric Carrollian gravity as the Teleparallel equivalent of Magnetic Carroll gravity; purely spatial torsion reproduces kinetic terms stracturally similar to the magnetic Hamiltonian of scalar-charge Fracton gauge theory; and temporal torsion sectors map independently to $U(1)$ Maxwell electric fields and Carroll-Weyl conformal dilaton theories. Finally, by formulating a Symmetric Teleparallel Carroll connection with minimal non-metricity, we show that the Magnetic Carrollian Lagrangian is dynamically equivalent to an ultralocal, isotropic Electric Carrollian gravity upon exploiting the present freedoms in connection. These results establish a geometric dictionary translating Carrollian gravitational dynamics into the language of tensor gauge theories, electromagnetism, and dilaton theory.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper, we extend the paradigm of the Geometrical Trinity of Gravity to the Carrollian geometry to construct a comprehensive framework of dynamically equivalent theories for Magnetic Carrollian gravity.	contribution 0.99	0.90	0.13	asserted 0.97
s1	Utilizing the inherent ambiguities and freedoms of affine connections on degenerate metrics, we systematically relax the metric-compatibility and zero-torsion constraints of the rigid magnetic sector.	method 0.96	0.87	0.04	asserted 0.95
s2	By imposing global flatness on these modified connections, we prove that the curvature of a Magnetic Carrollian universe can be dynamically traded for isolated kinetic energy densities of torsion and non-metricity.	result 0.90	0.93	0.28	asserted 1.00
s3	Unlike the Lorentzian Trinity which maps gravity to alternative gravitational frameworks, the Carrollian equivalences can also map spacetime geometry to non-gravitational theories.	interpretation 0.56	0.50	0.14	asserted 0.96
s4	Specifically, we demonstrate that intrinsic torsion recovers Electric Carrollian gravity as the Teleparallel equivalent of Magnetic Carroll gravity; purely spatial torsion reproduces kinetic terms stracturally similar to the magnetic Hamiltonian of scalar-charge Fracton gauge theory; and temporal torsion sectors map independently to $U(1)$ Maxwell electric fields and Carroll-Weyl conformal dilaton theories.	result 0.98	0.92	0.33	asserted 1.00
s5	Finally, by formulating a Symmetric Teleparallel Carroll connection with minimal non-metricity, we show that the Magnetic Carrollian Lagrangian is dynamically equivalent to an ultralocal, isotropic Electric Carrollian gravity upon exploiting the present freedoms in connection.	result 0.85	0.95	0.22	asserted 1.00
s6	These results establish a geometric dictionary translating Carrollian gravitational dynamics into the language of tensor gauge theories, electromagnetism, and dilaton theory.	interpretation 0.99	0.85	0.64	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Schematic representation of the Geometric Trinity of Gravity. The equivalence among these formulations is most transparent at the level of their actions. In TEGR (2.9), the Ricci scalar constructed from the Levi–Civita connection can be expressed – 6 –

**Figure 2.** The Extended Geometrical Trinity of Carroll Gravity. The baseline Magnetic Carroll Gravity (top) can be dynamically mapped to flat-space equivalent theories by shifting the curvature into Non-Metricity (left) or Torsion (right). Unlike Lorentzian GR, the degenerate Carrollian geometry splits the Torsion sector into three distinct physical theories, explicitly linking gravity to scalar, tensor and gauge theories. Our results give more perspective to the standard Geometrical Trinity of General Relativity. In the Lorentzian manifold, trading curvature for torsion or non-metricity…

Figures reproduced from arXiv:2610.01976 by the authors.

Discussion

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