REVIEW 4 major objections 5 minor 25 references
Spinning and Spinning Deviation Equations of Bi-metric Type Theories
T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A single Lagrangian recipe gives every bimetric gravity theory its own Papapetrou spin equations.
desk verdict The equations don't stand up: index errors and unsupported commutation steps leave every claimed deviation equation unproven. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The machinery is the modified Bazanski Lagrangian, whose variation in a deviation vector yields an equation of motion and whose variation in a deviation tensor yields the spin transport law, together with the commutation relations (11)-(14): the Ricci-type identity $A^{\mu}{}_{;\nu\rho}-A^{\mu}{}_{;\rho\nu}=R^{\mu}{}_{\beta\nu\rho}A^{\beta}$ and the matching conditions equating derivatives of the deviation and tangent vectors. These identities convert the transport equations into deviation equations. Different affine structures enter through different curvatures: a single amended Riemann curvature for combined metrics, and separate curved and flat connections with difference $\Delta^{\mu}_{\nu\rho}=\Gamma^{\mu}_{\nu\rho}-\hat{\Gamma}^{\mu}_{\nu\rho}$ for Rosen-type theories.
What would settle it
In Rosen's theory, compute the commutator of the flat-space covariant derivative directly: if $A_{\mu|\nu\rho} - A_{\mu|\rho\nu}$ does not vanish or does not match identity (11), the $\cdot_{|\rho}$ terms in equations (25) and (26) are artifacts rather than physical effects. Equivalently, take a concrete bimetric solution, write out the spin-deviation equation with and without the flat-derivative terms, and check whether the difference changes a measurable precession.
Extended reading notes
Core claim
The central claim is that the Papapetrou equations of spin motion and their deviation counterparts are not peculiar to general relativity: they can be reproduced in every bimetric type theory, provided each theory is given its own action-based Lagrangian of Bazanski form. For each theory the paper writes down such a Lagrangian, varies with respect to the deviation vector and deviation tensor, and obtains a pair of transport equations: one for momentum or velocity, the spin equation, and one for the spin tensor, the precession equation. Applying commutation identities to the transported quantities then yields the corresponding deviation equations. The distinctive physical result is that when the extended-body momentum is used, terms involving the flat-space, or second-metric, covariant derivative appear in Rosen-type theories, so the second metric affects spinning deviations despite being flat.
Load-bearing premise
The load-bearing premise is that the flat-space covariant derivative defined by the second metric obeys the same commutation and deviation identities as the Riemannian one; if that fails, the flat-space correction terms in the Rosen deviation equations are not justified.
Editorial extensions
If this is right
- Each bimetric theory listed acquires its own Papapetrou equation and matching deviation equation from a single Lagrangian per theory.
- In Rosen's theory, the flat-space covariant derivative appears in the deviation equations even though its curvature is zero, so the second metric is not inert.
- Moffat's combined metric makes the Papapetrou-like equations resemble general relativity's, but with an amended curvature built from $\hat{g}_{\mu\nu}$.
- In BIMOND, the difference of the two curvatures governs spin motion, so the MOND acceleration scale enters through the connection difference $C^{\alpha}_{\beta\gamma}$.
- For Hassan-Rosen bigravity, the derivation yields coupled spin equations for ordinary and twin matter, with combined quasi-metric versions reducing to Moffat-type forms.
Reading between the lines
- A direct next step would be to write the Rosen spin-deviation equations for a concrete solution, such as Schwarzschild geometry with a flat background metric, and compare the gamma-dependent terms with the general-relativity prediction; the paper does not perform that comparison.
- If the extra flat-space terms survive, spin-precession or tidal measurements around compact objects could in principle distinguish bimetric gravity from general relativity, since the terms have no general-relativity counterpart.
- The same variational recipe could be applied to Finsler-type or other non-Riemannian geometries, yielding Papapetrou-like equations whose curvature terms are not purely Riemannian.
- In bigravity, the paired equations point toward a 'twin spin' sector; testing it would require specifying how twin matter interacts with ordinary matter, which the paper leaves open.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proposes to derive Papapetrou-type spinning equations of motion and their corresponding deviation equations for five families of bimetric gravity theories: Rosen's bimetric theory, Moffat's variable-speed-of-light theory, BIMOND, Hassan-Rosen-type bi-gravity, and Verozub's geodesic-invariant gravity. For each theory a Bazanski-type Lagrangian is written down, variation with respect to the deviation variables is asserted to give a spinning equation, and a set of commutation relations (Eqs. (11)-(14)) is then applied to convert the spinning equation into a deviation equation. The paper contains no numerical results and no comparison with observations; its intended contribution is a formal framework for spinning test bodies in bimetric theories.
Significance. If the derivations were correct, the paper would provide a unified formalism for spinning test bodies in several bimetric theories and would identify a new contribution from the flat-space or second covariant derivative in the deviation equations. The breadth of coverage, with five distinct bimetric constructions and explicit Lagrangians for each, is a strength, and the author is clear about which curvature is intended in each case. However, the central mathematical step, the commutation of covariant derivatives, is not justified for any of the non-Riemannian derivatives used, and the displayed equations contain index errors that prevent verification. As a result, the claimed results are not currently established, and the formal framework cannot be used as a reliable foundation for further work.
major comments (4)
- [Section 3, Eqs. (11)-(14), (20), (25), (26)] The deviation equations are obtained by 'applying the commutation relations (11), (12), (13), and (14)' to the spinning equations. Those identities are the standard Riemannian commutation formulas for the Levi-Civita connection, and the paper gives no proof that any of the operators to which they are applied, namely the nabla operator defined with Delta^mu_{nu rho} = Gamma^mu_{nu rho} - hat-Gamma^mu_{nu rho} in Rosen's theory, the hat-D operator in Moffat's theory, the bar-D operator in the BIMOND and Verozub sections, and the flat derivative |, satisfies the same identities. For a general affine connection the commutator of two covariant derivatives contains torsion and connection-difference terms, and for the directional derivative built from the tensor Delta the Riemannian formula does not apply at all. In addition, the auxiliary conditions (13) and (14), which relate derivatives of the deviation vector to derivatives of the tangent vector, are not automatic for these modified operators. Since every deviation equation in the paper, and in particular the |_rho terms in Eqs. (25) and (26) that are presented as a new physical effect, depends on this step, the central derivation is incomplete. A proof of the commutation relations for each connection, or an alternative derivation of the deviation equations, is required before the results can be accepted.
- [Section 3, Eqs. (18) and (23)] Several displayed equations are not well-formed, which makes the derivation impossible to check. In Eq. (18), the left-hand side is nabla U^mu / nabla S with one free upper index mu, while the right-hand side is (1/2m) R^alpha_{.mu nu rho} S^nu U^mu; the index rho is left uncontracted and the free index alpha is not matched to the left-hand side. In Eq. (23), the right-hand side R^alpha_{.mu nu rho} S^{rho nu} U^mu U^nu has a free upper alpha that does not appear on the left, and the left-hand free index mu is contracted on the right, so the factor U^nu is an extra vector. These are not merely typographical problems: with this index structure the equations are ambiguous, and the Euler-Lagrange steps of Section 3 cannot be verified. The same index inconsistencies reappear in the hatted and barred equations of later sections.
- [Section 3, Eqs. (17), (18), (22), (23)] The text repeatedly says that the equations are obtained by 'taking the variation' of the displayed Lagrangians, but the Euler-Lagrange computation is never shown. This is a substantive omission because the Lagrangians already contain the Papapetrou-type force term, for example the term (1/2m) R_{alpha beta gamma sigma} U^alpha Psi^beta S^{gamma sigma} in Eq. (17); varying with respect to Psi^beta therefore returns the very force term that was put into the Lagrangian. The paper should state explicitly that this is the Bazanski variational ansatz and should justify the presence of the force term independently; otherwise the claim that the spinning equations have been 'derived' overstates what the calculation establishes. The same concern applies to the Lagrangians in Sections 4-8, where no intermediate variation steps are provided.
- [Section 6, Eqs. (62)-(70)] In the generalized two-metric case, the distinction between case (i) and case (ii) is not realized in the displayed equations: Eqs. (63)-(66) repeat Eqs. (54)-(57) even though the momentum in case (ii) is supposed to include the spin term U^nu DS^{mu nu}/DS, and Eq. (70) mixes the two sectors by using R, U, Psi, P, and a semicolon derivative where the twin sector should use S, V, Phi, bar-P, and a | derivative, if the advertised symmetry between the two sectors is to hold. As written, the two cases in Section 6 are either redundant or inconsistent, and the generalized equations do not support the conclusion that two independent sets of spinning equations and their deviations have been obtained from one Lagrangian.
minor comments (5)
- [Section 4, Eq. (29)] The term S^{mu nu} hat-D Psi_{nu nu} / hat-D S should almost certainly be S^{mu nu} hat-D Psi_{mu nu} / hat-D S; the repeated index nu nu appears to be a typo.
- [Section 3, Eqs. (20) and (25)] The derivative index rho in expressions such as (R^alpha_{.mu nu rho} S^{rho nu} U^mu U^nu)_{;rho} duplicates a dummy index already used inside the parentheses. This creates index clashes and makes the equations very hard to read; the dummy and derivative indices should be renamed consistently.
- [Section 8, Eq. (96)] The key transformation equation (96) is attributed to 'Verozub, private communication 2019' in footnote 3; a citable published derivation should be supplied for this central step.
- [Conclusions] The Conclusions cite equations (95), (96), (100), and (101) as spinning equations, but in the text those numbers refer to the geodesic/transformation equations and to the deviation equations; the equation numbering and cross-references should be checked throughout.
- [General notation] The paper uses nabla, hat-D, bar-D, D, semicolon, and | for covariant derivatives without a unified definition or a table of notation, and the placement of indices in the curvature tensors is not consistent (e.g., R^alpha_{.mu nu rho} versus R^alpha_{mu nu rho}). A consistent notation would substantially improve readability and verifiability.
Circularity Check
Spinning equations reduce to the Papapetrou force term inserted in each proposed Lagrangian; variation recovers the input, so the central derivation is partially circular.
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self definitional
[Section 3(i), Eqs. (17)-(18); same construction repeated in Eqs. (22), (29), (35), (41), (46), (53), (62), (98)]
"L = (gαβ − γαβ)U α ∇Ψ β /∇S + S μν ∇Ψ μν /∇S + 1/2m(R αβγσ )U α Ψ β S γσ (17) ... ∇U μ /∇S = 1/2m R α . μνρ S ν U μ , (18)"
The last term of L is linear in Ψ and is exactly the Papapetrou force term that appears on the RHS of Eq. (18). Varying the Bazanski Lagrangian with respect to Ψ selects that term and returns it as the equation of motion, so the claimed 'derived' spinning equation is an Euler-Lagrange repackaging of the interaction term put into L, not a consequence obtained from the bimetric theory. Every subsequent section repeats this construction.
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self definitional
[Section 6(i), Eq. (53) and Eqs. (54)-(55)]
"L = g μν U μ Ψ ;ν U ν + h μν V μ Φ |ν V μ + S μν Ψ μν ;ρ U ρ + ¯S μν Φ μν |ρ V ρ + 1/2m R αβγδ U β S γδ Ψ α + 1/2 ¯m S αβγδ V β ¯S γδ Φ α, (53) ... DU α /DS = 1/2m R αβγδ U β S γδ , (54)"
The last two terms of Eq. (53) are exactly the force terms that appear as the RHSs of Eqs. (54) and (55), multiplied by the variation fields Ψ^α and Φ^α. Varying with respect to those fields unpacks the inserted force terms; the twin spinning equations are therefore built into the Lagrangian by hand rather than derived from an independent dynamical principle.
full rationale
The central spinning equations are not independently derived: each proposed Bazanski Lagrangian contains the Papapetrou force term (1/2m)R U Ψ S or its analogue as an interaction, and the Euler-Lagrange variation returns exactly that term. This is the main circularity, scored as 6 because the 'prediction' reduces by construction, though the paper is transparent about proposing the Lagrangians. The deviation equations are not separately circular: they are commuted consequences of the assumed equations of motion. The unsupported use of Riemannian commutation relations (11)-(14) for flat-, bar-, and tilde-derivatives is a correctness risk but not a circularity finding. There is no load-bearing self-citation: Eq. (6) is displayed and Bazanski's method is external, and the prior-work citations provide context rather than an unverified premise. No data fitting or uniqueness-import argument is present. The score is not higher because the paper openly states the Lagrangians are proposed; it is not claiming a first-principles derivation of the force law itself, and the bimetric-connection variants give the equations some formal content, even if unsupported at the commutation step.
Assumptions & free parameters
assumptions (4)
- domain assumption The Bazanski Lagrangian variational principle yields the correct equations of motion for spinning particles in curved spacetime.
- ad hoc to paper The commutation relations (11)-(14) apply to covariant derivatives with respect to the bimetric connections, including the flat-space connection.
- domain assumption The spin tensor can be written as S^{αβ} = σ(U^α Ψ^β - U^β Ψ^α), with σ constant.
- domain assumption In Rosen's theory, the flat-space connection has zero curvature, so its associated covariant derivative behaves like a partial derivative in commutation relations.
invented entities (1)
-
Twin spin tensor S̄^{αβ}
Cite this review
Pith. "Pith review of Spinning and Spinning Deviation Equations of Bi-metric Type Theories." pith.science (2026). https://pith.science/paper/YBFNEVX3
@misc{pith2026190806501,
author = {Pith},
title = {Pith review of: Spinning and Spinning Deviation Equations of Bi-metric Type Theories},
year = {2026},
howpublished = {\url{https://pith.science/paper/YBFNEVX3}},
note = {Machine review of arXiv:1908.06501}
}
read the original abstract
Spinning equations of bi-metric types theories of gravity, the counterpart of the Papapetrou spinning equations of motion have been derived as well as their corresponding spinning deviation equations. Due to introducing different types of bi-metric theories, the influence of different curvatures based upon different affine connections , have been examined. A specific Lagrangian function for each type theory has been proposed, in order to derive the set of spinning motions and their corresponding spinning deviation equations.
Reference graph
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Reviewed August 14, 2026 · model on record in the stance chip above.
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