Free fall in modified symmetric teleparallel gravity

Nivaldo A. Lemos

REVIEW 2 major objections 5 minor 21 references

Free fall in modified symmetric teleparallel gravity

T0 review · 2 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read In f(Q) gravity, free-falling particles follow metric geodesics, not autoparallels

desk verdict A short, likely-correct argument that test particles in f(Q) gravity follow Levi-Civita metric geodesics, with a fixable typo in the quoted Bianchi identity. read the letter →

arxiv 2412.15805 v1 pith:XJFKBQEB submitted 2024-12-20 gr-qc hep-th

Nivaldo A. Lemos This is my paper · ORCID

classification gr-qchep-th MSC 83D0583C10 PACS 04.50.Kd

keywords modifiedsymmetricteleparallelgravityf(Q)equivalenceprinciplemetricgeodesicsautoparallelnonmetricityLevi-Civitaconnectionfree-fallmotion

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

Modified symmetric teleparallel gravity, or f(Q) gravity, replaces general relativity's curvature with nonmetricity, which makes the shortest paths through spacetime and the straightest paths under parallel transport genuinely different. The paper asks which of the two a freely falling test particle actually follows, and answers: the metric geodesic, governed by the Levi-Civita connection. The standard equivalence-principle argument that picks the Levi-Civita connection in general relativity fails here, because the nonmetricity tensor prevents the connection and the first derivatives of the metric from vanishing at the same point. Instead, the paper reaches the same conclusion from the theory's own structure, via the generalized Bianchi identity and conservation of energy-momentum. If the argument is right, free-fall motion in f(Q) gravity matches general relativity even when the field equations differ, with full physical equivalence whenever f is linear in Q or Q is constant.

What carries the argument

The load-bearing mechanism is the generalized Bianchi identity $\nabla_\mu M^{\mu}{}_{\nu} + C_\nu = 0$, Eq. (22), in which the covariant derivative is the Levi-Civita one, $M^{\mu}{}_{\nu}$ is the left-hand side of the metric field equations and $C_\nu$ is the left-hand side of the connection field equations. It converts the two sets of field equations into the energy-momentum conservation law $\nabla_\mu T^{\mu}{}_{\nu} = 0$. The second ingredient is the dust model $T^{\mu\nu} = \rho_0 u^\mu u^\nu$, which lets the conservation law imply the metric geodesic equation for each particle, following Papapetrou's argument. The identity is quoted from the literature rather than derived, and the claim that its derivative is the Levi-Civita derivative is essential: it is the step that selects shortest paths over straightest ones.

What would settle it

Take any explicit f(Q) solution with Q not constant, compute $\nabla_\mu T^{\mu}{}_{\nu}$ from the metric field equation (11) and compare it with the connection-field-equation term $C_\nu$ from the generalized Bianchi identity; if the two do not cancel, the conservation law that forces metric geodesics does not hold. Alternatively, a precision experiment that reconstructed free-fall worldlines in a regime where f(Q) dynamics differ from general relativity and found them to be autoparallels of the dynamically determined connection would directly contradict the paper's conclusion.

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Extended reading notes

Core claim

On its own terms, the paper establishes that in modified symmetric teleparallel gravity a freely falling structureless test particle follows a shortest path: the metric geodesic $d^2x^\mu/d\tau^2 + \Gamma^\mu{}_{\alpha\beta}(dx^\alpha/d\tau)(dx^\beta/d\tau)=0$, with $\Gamma$ the Levi-Civita connection, and not the autoparallel $d^2x^\mu/d\tau^2 + \tilde\Gamma^\mu{}_{\alpha\beta}(dx^\alpha/d\tau)(dx^\beta/d\tau)=0$ built from the dynamically determined connection. The route is not the usual equivalence-principle shortcut, which is blocked because EP3 cannot hold when nonmetricity is nonzero. The route is instead the generalized Bianchi identity written with the Levi-Civita derivative, which combines with the metric and connection field equations to yield $\nabla_\mu T^{\mu}{}_{\nu} = 0$; once the energy-momentum tensor is taken to describe a dust speck, that conservation law implies metric geodesics. The conclusion also yields a clean statement about equivalence: when $f(Q)=AQ+B$ or $Q$ is constant, f(Q) gravity agrees with general relativity not only in the field equations but in the equations of motion of freely falling test particles.

Load-bearing premise

The paper's conclusion rests on the generalized Bianchi identity quoted from earlier work, specifically on the assertion that the divergence it involves is the Levi-Civita covariant derivative; that identity is not derived in the paper, and if it failed or used a different derivative the dust argument would not force metric geodesics.

Editorial extensions

If this is right

Free-fall trajectories in f(Q) gravity are governed by the Levi-Civita connection, so the shortest and the physically realized paths coincide even though they differ from the straightest paths.
Weinberg's EP3 is sufficient but not necessary for singling out the Levi-Civita connection: f(Q) gravity does it through its Bianchi structure.
Wherever $f(Q)=AQ+B$, or $Q$ is constant on shell, f(Q) gravity and general relativity are physically equivalent at the level of field equations and test-particle motion alike.
The standard equivalence-principle argument cannot decide between the two geodesic classes in this theory, because EP3 fails whenever nonmetricity is nonzero.

Reading between the lines

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

Editorial inference: a likely broader moral is that any metric-affine theory with diffeomorphism-invariant action and a generalized Bianchi identity of the same form will route free fall through the metric connection, regardless of which connection solves the connection field equations.
Editorial inference: one can test the logic's scope by replacing dust with a spinning or extended test body, since Papapetrou-type multipole arguments typically add a spin-curvature force that may introduce deviations from pure metric geodesics which the paper's monopole dust model cannot see.
Editorial inference: in observational terms, the claim suggests that weak-field tests of free-fall trajectories alone cannot distinguish f(Q) gravity from general relativity, so distinguishing the theories would require probes of the field equations, such as cosmology or gravitational radiation.

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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

Let me give you the short version: this is a solid, concise note that makes a new argument for an old assumption in f(Q) gravity. It shows that Weinberg's EP3 fails because nonmetricity prevents the connection and the metric derivatives from vanishing simultaneously, so the standard GR derivation of metric geodesics doesn't work. But then, instead of falling back on autoparallels, the paper uses the generalized Bianchi identity and the connection field equations to show ∇_μ T^μ_ν = 0 with the Levi-Civita derivative. The standard dust argument then gives metric geodesics. This is a clean synthesis of known results, and as far as I can tell, the conclusion is right.

What's good: the paper is short, well written, and doesn't introduce any free parameters or exotic objects. It cites the relevant literature, including Heisenberg's review and Papapetrou's longstanding theorem. The self-citation is just an illustrative example of constant Q. The logical chain is transparent: field equations imply M_μν = T_μν and C_α = 0; the Bianchi identity turns that into conservation; conservation plus dust gives geodesics.

The soft spots: Eq. (22) as printed has a dimensional inconsistency. M_μν is a tensor, C_ν contains √-g and behaves like a tensor density. The displayed identity ∇_μ M^μ_ν + C_ν = 0 can't be literally correct. The fix is presumably ∇_μ M^μ_ν + (1/√-g) C_ν = 0, or equivalently ∇_μ(√-g M^μ_ν) + C_ν = 0. Since C_ν = 0 on-shell, the extra factor doesn't affect the conclusion, so this is a presentational bug, not a fatal one. Still, the authors should fix it and either derive the identity or state it with proper weights. The identity is quoted from Refs. [2,17]; a derivation would make the paper self-contained, but for a short note citation is acceptable.

Who's this for: the f(Q) / modified-gravity crowd. It settles a conceptual ambiguity and justifies using metric geodesics in phenomenological studies. I'd send it out for review; I expect a minor revision, not a rejection.

Referee Report

2 major / 5 minor

Summary. The manuscript examines which geodesic governs the free fall of test particles in modified symmetric teleparallel gravity, i.e., f(Q) gravity. It notes that in this theory metric (shortest-path) geodesics and autoparallel (straightest-path) geodesics differ, and that Weinberg's formulation of the equivalence principle does not single out the Levi-Civita connection because the additional condition EP3 fails when nonmetricity is nonvanishing. The paper then invokes a generalized Bianchi identity, together with the metric and connection field equations, to derive the energy-momentum conservation law ∇_μ T^μ_ν = 0, and uses the dust (Papapetrou) argument to conclude that a freely falling test particle follows a Levi-Civita metric geodesic rather than an autoparallel geodesic of the dynamical connection. It concludes that f(Q) gravity becomes physically equivalent to general relativity, both at the level of field equations and particle motion, when f(Q) is linear in Q or Q is constant.

Significance. If the central argument is correct, the paper settles an often-discussed ambiguity in f(Q) gravity: the physical free-fall trajectories are metric geodesics, not autoparallels of the flat dynamical connection. The reasoning is transparent and uses standard tools, namely the Noether/Bianchi identity for diffeomorphism invariance and Papapetrou's conservation-law argument, and it makes a sharp, falsifiable physical claim. The paper contains no fitted parameters and no numerical machinery; its value is conceptual. Its main weakness is that the crucial generalized Bianchi identity is quoted from the literature rather than derived, and the version printed in Eq. (22) is not literally correct as written. This issue is load-bearing and must be addressed before the conclusion can be regarded as fully verified.

major comments (2)

[Section V, between Eqs. (21) and (23)] The generalized Bianchi identity is the load-bearing step of the paper, but Eq. (22) as printed cannot be correct. M^μ_ν built from Eq. (20) is a tensor, while C_ν defined in Eq. (21) contains √-g and two covariant derivatives and is a tensor density of a different weight and dimension. The identity should read, up to conventional normalizations, ∇_μ M^μ_ν + (1/√-g) C_ν = 0, or equivalently ∇_μ(√-g M^μ_ν) + C_ν = 0. Because the connection field equations set C_ν = 0, this correction does not by itself destroy the conclusion, but as printed Eq. (22) cannot be used to verify Eq. (23). Please correct the identity and either derive it in an appendix or quote the exact normalized statement from Refs. [2,17].
[Section V, between Eqs. (21) and (23)] The paper asserts that it is a 'crucial fact' that the covariant derivative in Eq. (22) is the Levi-Civita derivative, but this is not demonstrated. Since C_ν in Eq. (21) is defined with Ṽ∇ derivatives, the generalized Bianchi identity is not the ordinary Riemannian identity, and the absence of nonmetricity terms in the term ∇_μ M^μ_ν is precisely what makes the dust argument select metric rather than autoparallel geodesics. A derivation, or an exact quotation from the literature showing which connection appears in each term of the identity, is needed for the central claim to be independently checkable.

minor comments (5)

[Eqs. (6) and (20)] The first term in Eqs. (6) and (20) is typeset as 2√-g; it should be 2/√-g to match the standard f(Q) field equations and to make M_μν a tensor.
[Eq. (19)] The last product in Eq. (19) has the same index μ twice; it should read d x̂^μ/dτ d x̂^ν/dτ.
[Abstract and Section III] The term 'shortest paths' is used for solutions of δ∫ds = 0; in Lorentzian signature these are stationary-length curves, not necessarily shortest. Consider using 'stationary-length geodesics' or 'metric geodesics' throughout.
[Eqs. (21) and (22)] In Eq. (22) the index on C is ν, whereas Eq. (21) defines C_α; either define C_ν = g_{να} C_α or write Eq. (22) with matching indices.
[Reference [9]] Reference [9] is cited as 'in press' without volume, page numbers, or DOI; this should be updated before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the central geodesic conclusion follows from external Bianchi-identity and geodesic theorems, not from fitted inputs or self-citation.

full rationale

The paper's derivation chain is: metric field equations give M_mu_nu = T_mu_nu; connection field equations give C_alpha = 0; the generalized Bianchi identity (22), nabla_mu M^mu_nu + C_nu = 0, is quoted from Refs. [2,17], which are external works; combining these yields nabla_mu T^mu_nu = 0; and Papapetrou's theorem [18] (or the dust argument) converts this conservation law into metric geodesics. No parameter is fitted and no quantity in the argument is defined in terms of the conclusion. The only self-citation, Ref. [9], is a parenthetical example involving constant Q in Godel-type metrics; the equivalence claim for constant Q follows directly from Eq. (11) and does not rest on that citation. The possible dimensional inconsistency in Eq. (22) as printed is a correctness or verification concern, not a circularity concern, because the identity is imported from external sources rather than manufactured from the paper's own assumptions.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters or invented entities. The central claim rests on the standard formulation of f(Q) gravity, the cited generalized Bianchi identity, and the standard dust-to-geodesic theorem. The axioms listed are the load-bearing background inputs.

assumptions (4)

domain assumption The generalized Bianchi identity ∇_μ M^μ_ν + C_ν = 0 holds, with the covariant derivative being the Levi-Civita one (Eq. 22).
Quoted from Refs. [2,17]; the paper calls it a crucial fact but does not derive it. The conclusion that T is conserved with the Levi-Civita connection rests entirely on this identity.
domain assumption Matter is minimally coupled to the metric, so the energy-momentum tensor is defined by metric variation and satisfies the conservation law when the field equations hold.
Standard in f(Q) gravity; the paper's dust model assumes this. If matter coupled to the connection, the conservation law would differ.
domain assumption A structureless, non-interacting test particle can be modeled as dust, and conservation of T with the Levi-Civita connection implies geodesic motion (Papapetrou's theorem).
The paper cites Ref. [18] and sketches the dust argument; it does not prove the theorem.
domain assumption The standard f(Q) action and field equations (Eqs. 5-7) are the correct starting point.
The paper takes the theory from the literature (Refs. [2,3,4]) without re-deriving it.

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Cite this review

Pith. "Pith review of Free fall in modified symmetric teleparallel gravity." pith.science (2026). https://pith.science/paper/XJFKBQEB

@misc{pith2026241215805,
  author       = {Pith},
  title        = {Pith review of: Free fall in modified symmetric teleparallel gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XJFKBQEB}},
  note         = {Machine review of arXiv:2412.15805}
}

read the original abstract

The status of the equivalence principle in modified symmetric teleparallel gravity is examined. In this theory, minimum length geodesics are distinct from autoparallel geodesics, that is, the ``shortest'' paths are not the ``straightest'' paths. We show that a standard argument that singles out metric geodesics in general relativity does not apply in modified symmetric teleparallel gravity. This is because the latter theory does not obey the equivalence principle in the sense of Weinberg. We argue, however, that the structure of the theory makes it inevitable that a freely falling test particle follows a shortest path, a geodesic of the metric. The geodesic equation that governs the motion of a freely falling test particle involves the Levi-Civita connection, not some other connection obtained by solving the connection field equations of the theory. This also has bearing on whether, under appropriate conditions, modified symmetric teleparallel gravity is fully equivalent to general relativity.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

J. M. Nester and H.-J. Yo, Chinese J. Phys. 37, 113 (1999), arXiv:gr-qc/9809049

arXiv 1999

[2] [2]

( 7): Mµν def = 2√−g ~∇α ( √−gfQP α µν ) − 1 2 f gµν + fQ ( P(µ |αβ Q αβ ν) − 2Pαβ (µ Qαβ ν) ) , (20) Cα def = ~∇µ ~∇ν ( √−gfQP µν α )

and deﬁne Cα as the left-hand side of Eq. ( 7): Mµν def = 2√−g ~∇α ( √−gfQP α µν ) − 1 2 f gµν + fQ ( P(µ |αβ Q αβ ν) − 2Pαβ (µ Qαβ ν) ) , (20) Cα def = ~∇µ ~∇ν ( √−gfQP µν α ) . (21) The generalized Bianchi identity ∇µ Mµ ν + Cν = 0 (22) holds [ 2, 17] as a consequence of the diﬀeomorphism invariance of the action ( 5). It is a 7 crucial fact that the co...

work page

[3] [3]

J. B. Jim´ enez, L. Heisenberg and T. Koivisto, Phys. Rev. D 98, 044048 (2018), arXiv:1710.03116 [gr-qc]

arXiv 2018

[4] [4]

Heisenberg, Phys

L. Heisenberg, Phys. Rep. 1066, 1 (2024), arXiv:2309.15958 [gr-qc]

arXiv 2024

[5] [5]

J. B. Jim´ enez, L. Heisenberg and T. S. Koivisto, J. Cosmo l. Astropart. Phys. 08, 039 (2018), arXiv:1803.10185v1 [gr-qc]

arXiv 2018

[6] [6]

is [ 7, 8] fQ o Gµν + 1 2 gµν ( QfQ − f ) + 2fQQ ( ∂λ Q ) P λ µν = Tµν (11) where o Gµν is the Einstein tensor associated with the Levi-Civita connection. In this form, it becomes clear that if fQQ = 0 then f (Q) = AQ+B and at the level of the ﬁeld equations the 4 theory is equivalent to general relativity (if B = 0) or general relativity with a cosmologi...

work page 1907

[7] [7]

J. B. Jim´ enez, L. Heisenberg, T. S. Koivisto and S. Pekar , Phys. Rev. D 101, 103507 (2020), arXiv:1906.10027 [gr-qc]

arXiv 2020

[8] [8]

Blixt, A

D. Blixt, A. Golovnev, M. J. Guzman and R. Maksyutov, Phys . Rev. D 109, 044061 (2024), arXiv:2306.09289v3 [gr-qc]

arXiv 2024