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REVIEW 3 major objections 5 minor 1 cited by

Revisiting Coincident GR in Internal STEGR Formulation

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Coincident General Relativity, derived as the coincident-gauge sector of internal STEGR, has vacuum field equations identical to Einstein's, while massive test scalar particles would follow a norm-flow equation instead of geodesics.

desk verdict A useful kinematic result, an unsupported central claim: the route from internal STEGR to CGR breaks on constraint (16). read the letter →

arxiv 2506.22158 v3 pith:7ZI2KHBT submitted 2025-06-27 gr-qc hep-th

Kyosuke Tomonari , Taishi Katsuragawa , Shin'ichi Nojiri This is my paper
classification gr-qchep-th MSC 83D0583C05 PACS 04.20.-q04.50.Kd
keywords coincidentgeneralrelativitySTEGRsymmetricteleparallelgravityinternal-spaceformulationStückelbergfieldsnonmetricitynorm-flowequation
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 claims that Coincident General Relativity (CGR), the formulation of symmetric teleparallel gravity in which the connection is set to vanish, has a correct gauge-theoretic foundation and that, once derived properly, its field equations coincide with those of general relativity up to an overall sign. The derivation runs through the internal-space formulation of STEGR known as Formalism 3, where the co-frame is written as gradients of four Stückelberg scalar fields and non-metricity is carried by an internal metric. Setting the Stückelberg fields equal to the coordinate functions is the coincident gauge, and it produces CGR as a well-posed sector of internal STEGR rather than of ordinary Palatini STEGR. The payoff is that in vacuum CGR and GR are the same equations living on different geometric backgrounds, and the kinematics of matter changes: a massive test scalar particle obeys a norm-flow equation instead of the geodesic equation.

What carries the argument

The load-bearing object is internal STEGR in Formalism 3: the co-frame is decomposed as $\theta^I_\mu = \partial_\mu \xi^I$ using four Stückelberg scalar fields, and the condition $\xi^I \partial_\mu \eta_{IJ} = 0$ is imposed so that torsion automatically vanishes while non-metricity survives in the internal metric. The action built from these ingredients is diffeomorphism-invariant, and its field equations contain second-order derivative terms that may signal Ostrogradski ghosts unless a degenerate condition is imposed. Imposing the coincident gauge $\xi^I = \delta^I_\mu x^\mu$ collapses the configuration space to the spacetime metric alone, produces the CGR action, and reduces the internal field equations to CGR's field equation. The equivalence with GR is carried by Einstein's 1916 Lagrangian, which equals the CGR Lagrangian exactly and equals $-\sqrt{-g}\,\mathring{R}$ up to a boundary term. The kinematic result rests on the non-metricity scalar $q$, defined as half the rate of change of the squared length of the velocity vector, which converts the geodesic equation into the norm-flow equation for massive test particles.

What would settle it

A Dirac–Bergmann analysis of internal STEGR in Formalism 3 would settle the central claim: if the constraints $\phi^A$ and $\phi^{IJ}$ do not restrict the configuration space down to the spacetime metric alone, or if no degenerate condition removes the higher-derivative ghost terms in the field equations, then the coincident-gauge sector is not a well-posed theory of gravity and the claimed CGR derivation loses its footing. Alternatively, evaluate the CGR action on a manifold that cannot be covered by a single chart without coordinate singularities: if the field equations there deviate from Einstein's, the vacuum equivalence holds only under the single-chart assumption.

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Extended reading notes

Core claim

The paper's central proposition is that “the field equations of CGR coincide with those of GR up to an overall sign, taking into account the freedom in choosing boundary terms in the variational principle,” with the vacuum case being completely equivalent. This is shown by deriving CGR from the internal STEGR action: imposing the coincident gauge $\xi^I = \delta^I_\mu x^\mu$ on the Formalism-3 action turns it into the standard CGR action, whose variation reproduces CGR's field equations. A direct calculation then identifies the CGR Lagrangian with Einstein's 1916 Lagrangian $\mathcal{L}_E = 2\sqrt{-g}\,g^{\mu\nu}\mathring{\Gamma}^\rho_{\lambda[\mu}\mathring{\Gamma}^\lambda_{\rho]\nu}$, which differs from $-\sqrt{-g}\,\mathring{R}$ only by a boundary term; including the appropriate boundary term with the opposite sign yields $-\mathring{G}_{\mu\nu} = 0$ in vacuum. The authors emphasize that the equivalence is formal: GR lives on a pseudo-Riemannian manifold, while CGR lives on a flat and torsion-free manifold, so the same equations carry different geometric meaning. They further derive the motion of a test scalar particle in a flat, torsion-free spacetime, concluding that massive particles satisfy the norm-flow equation $du^\alpha/d\lambda = q\,u^\alpha$ while massless particles follow geodesics.

Load-bearing premise

The derivation stands on the imposed decomposition $\theta^I_\mu = \partial_\mu \xi^I$ together with the condition $\xi^I \partial_\mu \eta_{IJ} = 0$, taken from earlier work without being derived from a symmetry principle, and on the assumption that a single coordinate chart covers the whole spacetime with no coordinate singularities.

Editorial extensions

If this is right

  • CGR acquires a well-posed derivation as the coincident-gauge sector of internal STEGR in Formalism 3, resolving the status of its formulation.
  • In vacuum, every GR solution formally solves CGR's field equations and vice versa, up to the sign from boundary terms — but on a flat, torsion-free geometric background, so the equivalence is formal, not physical.
  • Massive test scalar particles in flat, torsion-free spacetimes obey the norm-flow equation $du^\alpha/d\lambda = q\,u^\alpha$ instead of the geodesic equation; massless particles follow ordinary geodesics.
  • Because the non-metricity correction in the norm-flow equation is of higher order in the velocity, the paper expects Solar System constraints not to strongly exclude non-metricity, pending a quantitative check.
  • Internal STEGR faces a possible Ostrogradski ghost that would require a degenerate condition; its diffeomorphism invariance makes the ADM-foliation Dirac–Bergmann prescription applicable, unlike gauge-fixed CGR.

Reading between the lines

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

  • If the vacuum equivalence is only formal, the observables that could distinguish CGR from GR must come from matter couplings to non-metricity; the norm-flow equation gives a concrete velocity-dependent force that could be probed at higher post-Newtonian order in Solar System or binary-pulsar data.
  • The single-chart assumption is the natural fault line of the construction: on manifolds with nontrivial topology the coincident gauge $\xi^I = \delta^I_\mu x^\mu$ cannot be imposed globally, so the CGR action would be globally ill-defined and the equivalence to GR should break — a consequence testable within the paper's own framework.
  • The condition $\xi^I \partial_\mu \eta_{IJ} = 0$ might turn out to be a secondary constraint generated by the Dirac–Bergmann analysis rather than an ad hoc input; if so, the entire construction would follow from the action alone and the paper's weakest premise would be eliminated.
  • The same internal-space construction suggests a gauge-theoretic route to coincident $f(Q)$ gravity: applying the coincident gauge inside Formalism 3 would yield a well-posed starting point for re-examining the cosmological ghosts and strong-coupling pathologies reported for $f(Q)$ theories.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. This paper revisits Coincident General Relativity (CGR) using the internal-space STEGR formulation of Ref. [19]. It first reviews the Palatini formulation of STEGR with Lagrange multipliers, then presents the internal STEGR action, derives field equations, and discusses possible Ostrogradski ghosts. Imposing the coincident gauge, the authors claim to derive the CGR action and field equations, and they argue that the CGR field equations are equivalent to vacuum GR up to an overall sign. The final section studies the kinematics of a test scalar particle and derives a norm-flow equation for massive particles.

Significance. The paper addresses a genuine conceptual issue: whether CGR can be obtained as a well-posed gauge-theoretic construction rather than as a gauge-fixing of Palatini STEGR. If the route through internal STEGR Formalism 3 were sound, it would clarify the status of the coincident gauge and the bi-metric structure of the theory. The manuscript is honest about several limitations, and it provides explicit coefficient formulas in the appendices. However, the central reduction from the internal STEGR action (19) to the CGR action (30) rests on an ad hoc constraint that appears inconsistent with the coincident gauge for generic metrics, and the global-coordinate assumption is nontrivial. The standard result that Eq. (30) is equivalent to the Einstein-Hilbert Lagrangian up to boundary terms is correct, but the paper's claimed derivation of this result from internal STEGR is not established.

major comments (3)
  1. [Section III B, Eqs. (16), (26), (27)] The central reduction of the internal STEGR action (19) to the CGR action (30) is inconsistent with the constraint (16). In the coincident gauge, ξ^I = δ^I_μ x^μ and η_IJ = δ^μ_I δ^ν_J g_μν, so Eq. (16) becomes x^ρ ∂_μ g_{ρσ} = 0 for every μ and σ, after contracting with the Kronecker deltas. Generic CGR metrics, for example Schwarzschild in Cartesian coordinates, do not satisfy this condition. If Eq. (16) is imposed, the nonmetricity (29) vanishes and CGR is trivial; if it is dropped, the identity (17) and the derivative replacements (27) do not follow, so the derivation of Eq. (30) from Eq. (19) is unsupported. The paper needs to state explicitly whether Eq. (16) is a field equation, a gauge condition, or an identity at each step; as written, the claimed route to CGR has a consistency gap.
  2. [Section III B, after Eq. (26)] The coincident gauge with ξ^I = δ^I_μ x^μ requires a single global coordinate chart covering the entire spacetime manifold. This is not possible for generic manifolds: for example, the two-sphere requires an atlas of at least two charts, and Schwarzschild spacetime has coordinate singularities in the usual Cartesian coordinates. The paper acknowledges this by saying 'we assume such a simple case,' but the claimed vacuum equivalence to GR is a global statement about field equations. The manuscript should specify the class of spacetimes for which the CGR formulation is valid and explain how the formulation is to be continued across chart boundaries if the coincident gauge is not globally available.
  3. [Section III A, Eqs. (24) and (25)] The discussion of Ostrogradski ghosts is not conclusive. The argument assumes the existence of the inverse coefficients ~F^{(0)A} and ~G^{(1)AB}, and then asserts that the reduced equations contain third-order derivatives of ξ^A. No explicit mode analysis, Hamiltonian argument, or concrete counterexample is provided. The abstract's statement that the theory 'may require a degenerate condition' is therefore a conjecture. If this is intended as one of the paper's results, the claim should be supported by a proper Dirac-Bergmann or phase-space analysis; otherwise, it should be labeled as a preliminary observation.
minor comments (5)
  1. [Section III A, Eq. (17)] The quantity ξ_B appearing in Eq. (17) is not defined before use; the paper should define how internal indices are lowered and raised for the Stueckelberg fields.
  2. [Section II B, text after Eq. (3)] There is a typo: 'telepalalleism' should be 'teleparallelism'.
  3. [Section III B, text before Eq. (34)] The reference to 'Eq. (5) and Eq. (7)' leading to the Lagrangian (34) appears to be a numbering error; the relevant equations are likely the disformation definitions in Eqs. (7)-(8).
  4. [Section IV, Eq. (42)] The quantity q defined in Eq. (42) has a denominator g_μν u^μ u^ν that vanishes on null curves; the paper treats the null case separately, but this special-case structure should be stated when q is introduced.
  5. [Table I and Table II] The entry 'unknown g_μν' in Table I is unclear because the configuration variable for CGR is stated to be g_μν; the table would benefit from a footnote explaining the intended meaning of 'unknown'.

Circularity Check

1 steps flagged · score 4.0 of 10

The CGR=GR equivalence itself is independent direct algebra, but the claimed derivation of CGR from internal STEGR imports the load-bearing constraint (16) from self-cited Ref. [19], making the route partially definitional.

  1. ansatz smuggled in via citation [Sec. III A, Eq. (16); Sec. III B, Eqs. (26)-(30)]
    "In Ref. [19], one of the authors reformulated the theories of STEGR by generalizing the internal-space bundle ... Consequently, the internal-space metric alters the Minkowskian metric to a generic one. ... ξ^I ∂_μ η_IJ := impose 0. (16) This condition leads to the convenient formula given as follows: ∂_μ ξ_A = η_AB ∂_μ ξ^B. (17)"

    Eq. (16) is imported from the self-cited Ref. [19] and is not derived; it is what makes the non-metricity sector non-vanishing. In the coincident gauge, (16) becomes x^ρ ∂_μ g_{ρJ}=0 and is precisely what is needed for the identification ∂_α ξ_A = δ^β_A g_{αβ} in Eq. (27). Feeding Eq. (27) into Eq. (19) yields Eq. (30), so the advertised reduction of internal STEGR to CGR is the constraint (16) restated in gauge-fixed variables; the route to Eq. (30) is by construction. The equivalence of Eq. (30) with Einstein's Lagrangian, Eq. (34), is independent direct algebra.

full rationale

The paper's substantive claim — that the CGR action (30) and its vacuum field equations coincide with GR's up to sign and boundary terms — is not circular: it is demonstrated by direct calculation in Eqs. (30)-(36), where (30) is identified with Einstein's 1916 Lagrangian (34) and then with -√−g R plus a boundary term. That part is self-contained and externally checkable. The circularity concern lies only in the claimed route to (30): the internal STEGR Formalism 3 is taken wholesale from the same author's Ref. [19], and the key constraint (16) is imposed, not derived. Since (16) is exactly the identity needed in the coincident gauge to identify the lower-index Stückelberg derivatives with g_{αβ} and hence to reduce (19) to (30), the 'derivation' of CGR from internal STEGR is an unpacking of the ansatz rather than an independent derivation. However, this does not make the physical equivalence CGR=GR circular, because that equivalence is verified directly and the CGR action itself is an existing result (Ref. [21]). I therefore assign 4: some load-bearing self-citation and ansatz importation, but the central equivalence has independent content. A residual consistency issue — that (16) in the coincident gauge demands x^ρ ∂_μ g_{ρJ}=0, which generic CGR solutions do not satisfy — is a correctness risk rather than a circularity and is noted separately.

Assumptions & free parameters 0 free parameters · 7 assumptions · 3 invented entities

The central derivation introduces no fitted numerical constants. The cost is paid in structural axioms: the internal STEGR action is imported from the self-cited Ref [19], with an imposed coframe decomposition and constraint (Eq. 16), a dynamical internal metric, and a global-coordinate assumption for the coincident gauge. The kinematics section additionally assumes minimal coupling of a scalar test particle. These are modeling assumptions, not data-derived parameters.

assumptions (7)
  • domain assumption Teleparallel condition R^ρ_{λμν}=0 and torsion-free condition T^ρ_{μν}=0 are imposed as defining constraints for STEGR (Eqs. 1-2).
    Standard definition of STEGR; used throughout to restrict metric-affine gravity.
  • ad hoc to paper Co-frame field decomposes as θ^I_μ = ∂_μ ξ^I with Stueckelberg fields, and the internal metric obeys ξ^I ∂_μ η_IJ = 0 (Eq. 16).
    This decomposition is what turns off torsion and keeps nonmetricity nonzero when the internal metric is generic; it is imposed, not derived from a symmetry, and is taken from Ref [19].
  • ad hoc to paper The internal-space metric η_IJ is a dynamical field rather than the fixed Minkowski metric.
    Necessary for Q=dη to survive in the Weitzenbock gauge; this is the key generalization of the internal STEGR formalism and has no independent empirical support.
  • domain assumption A single regular coordinate chart is assumed to cover the whole spacetime manifold so the coincident gauge ξ^I = δ^I_μ x^μ is globally valid.
    Stated in Sec. III B after Eq. (26); without it the CGR action is only valid on an open patch U and global equivalence to GR is not established.
  • domain assumption Variation is performed with Dirichlet boundary conditions δξ^A|∂M = 0 and δη^AB|∂M = 0, but no metric boundary condition is imposed.
    This boundary-term convention is used to derive Eq. (22); it affects which field equations are obtained and the later GR equivalence up to boundary terms.
  • domain assumption Test scalar particle dynamics is governed by the minimally coupled action (37) with no direct coupling to nonmetricity or torsion.
    The norm-flow equation is a consequence of this minimal action; adding couplings, as in Ref [55], would change the trajectory.
  • ad hoc to paper The inverse coefficients F̃(0)A and G̃(1)AB are assumed to exist and the field equations are assumed to have solutions in some domain.
    Used in Eq. (24) to reduce second-order field equations to first-order form for the ghost-count heuristic; the conditions are not proven.
invented entities (3)
  • Generic internal-space metric η_IJ treated as a dynamical field
    purpose: Generates nonzero nonmetricity in the Weitzenbock gauge, which is the mechanism that defines internal STEGR.
    No observational handle is given; it is a formal device from the internal-bundle construction of Ref [19].
  • C-gauged internal space
    purpose: Names the internal space equipped with the global coincident gauge, used to globalize CGR.
    A bookkeeping concept, not a physical entity; no observable consequences beyond the field equations.
  • Stueckelberg fields ξ^I as coordinate-valued scalar fields
    purpose: Used to decompose the co-frame and to define the coincident gauge.
    Standard Stueckelberg trick, but here they are treated as global coordinate maps on a generic internal bundle; no independent evidence is provided.

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Cite this review

Pith. "Pith review of Revisiting Coincident GR in Internal STEGR Formulation." pith.science (2026). https://pith.science/paper/7ZI2KHBT

@misc{pith2026250622158,
  author       = {Pith},
  title        = {Pith review of: Revisiting Coincident GR in Internal STEGR Formulation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZI2KHBT}},
  note         = {Machine review of arXiv:2506.22158}
}
read the original abstract

We revisit Coincident General Relativity (CGR) in the gauge approach to gravity based on Symmetric Teleparallel Equivalent to General Relativity (STEGR) in the \textit{internal-space formulation}, which one of the authors recently proposed in Ref.~[J. Math. Phys. 66 (2025) 5, 052505]. First, we review the standard formulation of STEGR theories in the Palatini approach to gravity, in which formulation we impose the teleparallel and torsion-free conditions by using Lagrange multipliers. Second, we introduce the STEGR theories in the gauge approach to gravity, which is formulated in the internal space, and derive its field equations. We briefly discuss whether the Ostrogradski ghost instability exists and find that the theory may require a degenerate condition to be imposed. Finally, assuming the coincident gauge, we derive CGR in both terms of the action integral and the field equation. Discussing the possible kinematics of the STEGR theory, we formulate a motion of a test scalar particle in the spacetime with non-metricity.

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