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Solar system tests in covariant f(Q) gravity

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Pith's one-line read Solar System tests bound the $f(Q)$ gravity parameter $\alpha$, with gravitational redshift giving the tightest window: for $n=2$ in Case II, $-5.55\times10^{-5}\,\mathrm{km}^2<\alpha<9.52\times10^{-6}\,\mathrm{km}^2$.

desk verdict A useful two-branch f(Q) constraint paper with new alpha bounds, but the Case I Cassini coefficients are off by factors of 3 and 35 and the final n=3 bound needs fixing. read the letter →

arxiv 2412.17463 v1 pith:SKUPQNJN submitted 2024-12-23 gr-qc hep-th

Wenyi Wang , Kun Hu , Taishi Katsuragawa This is my paper · ORCID
classification gr-qchep-th MSC 83D0583C2583C10 PACS 04.50.Kd04.80.Cc
keywords f(Q)gravitynonmetricitysolarsystemtestsperihelionprecessionlightdeflectionShapirotimedelaygravitationalredshiftparameterconstraints
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 asks how strongly Solar System experiments can constrain covariant $f(Q)$ gravity, a modified theory in which gravity is described by spacetime nonmetricity instead of curvature. For the model $f(Q)=Q+\alpha Q^n-2\Lambda$ with $n=2,3$, the authors derive static, spherically symmetric metric corrections for two allowed choices of the affine connection, then compute five observable effects: perihelion precession, light deflection, Shapiro time delay, the Cassini frequency shift, and gravitational redshift. The central result is a set of allowed intervals for the deviation parameter $\alpha$. The tightest windows come from the connection branch called Case II, where gravitational redshift near the Earth gives $-5.55\times10^{-5}\,\mathrm{km}^2<\alpha<9.52\times10^{-6}\,\mathrm{km}^2$ for $n=2$ and $-4.21\times10^{2}\,\mathrm{km}^4<\alpha<9.85\times10^{2}\,\mathrm{km}^4$ for $n=3$. For Case I the same data leave $\alpha$ essentially unconstrained, with bounds as large as $10^{40}\,\mathrm{km}^4$ in the $n=3$ case, so the choice of connection dramatically changes how strongly the theory is tested.

What carries the argument

The argument is carried by the nonmetricity scalar $Q$ and the symmetry-reduced field equations of covariant $f(Q)$ gravity, together with the connection ansatz $\Gamma^r_{\theta\theta}=\pm r/\sqrt{B(r)}$ for the static spherically symmetric case. This ansatz selects two branches of solutions, Case I and Case II, whose metric corrections have different radial powers and different mass dependences; that difference is what makes the observable predictions (and hence the $\alpha$ bounds) branch-dependent. For the observables, the paper uses the standard geodesic equations for the generalized metric $ds^2=-A(r)dt^2+B(r)dr^2+r^2d\Omega^2$, the Rindler\textendash Ishak invariant-cosine method for light deflection, and the Bodenner\textendash Will iterative solution of the null-geodesic equation. The final constraints combine these predictions with five sets of observational data, with the Pound\textendash Rebka gravitational redshift experiment giving the tightest Case II windows because it probes small $r$ where the $1/r^n$ corrections are largest.

What would settle it

One concrete check: compute the full set of torsion-free, curvature-free static spherically symmetric connections; if a branch other than Eq. (12) exists, the bounds do not cover the theory. Observationally, a redshift experiment with fractional precision better than the Pound\textendash Rebka uncertainty that measures a residual outside the window $-5.55\times10^{-5}\,\mathrm{km}^2<\alpha<9.52\times10^{-6}\,\mathrm{km}^2$ (for $n=2$, Case II) would falsify that branch's prediction.

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Extended reading notes

Core claim

For the action $f(Q)=Q+\alpha Q^n-2\Lambda$ in covariant $f(Q)$ gravity, the paper establishes that Solar System observations place quantitative limits on $\alpha$ for $n=2,3$, and that the limits depend strongly on which torsion-free, curvature-free affine connection is chosen. Starting from the symmetry-reduced field equations and the ansatz $\Gamma^r_{\theta\theta}=\pm r/\sqrt{B(r)}$, the authors obtain perturbative corrections to the Schwarzschild\textendash de Sitter metric: in Case I (the $-$ sign) the correction scales as $r^{3-4n}$, and in Case II (the $+$ sign) as $r^{2-2n}$. These differing radial falloffs make Case II deviate from general relativity at lower order in $M/b$ for light deflection (order $(M/b)^3$ versus $(M/b)^5$), so Case II is far more strongly constrained. Combining Mercury perihelion precession, solar light deflection, Viking Shapiro delay, Cassini ranging, and the Pound\textendash Rebka gravitational redshift measurement, the tightest resulting intervals are Eqs. (124)\textendash(125): $-5.55\times10^{-5}\,\mathrm{km}^2<\alpha<9.52\times10^{-6}\,\mathrm{km}^2$ for $n=2$ and $-4.21\times10^{2}\,\mathrm{km}^4<\alpha<9.85\times10^{2}\,\mathrm{km}^4$ for $n=3$, both in Case II. The paper also verifies that within these windows the linear nonmetricity term $Q$ dominates over $\alpha Q^n$ in the Lagrangian, so the perturbative treatment is self-consistent.

Load-bearing premise

The load-bearing premise is that the ansatz $\Gamma^r_{\theta\theta}=\pm r/\sqrt{B(r)}$ from Ref. [28] covers all relevant static spherically symmetric connections, and that matter follows metric geodesics; if a different connection branch exists or the geodesic motion is altered in covariant $f(Q)$ gravity, the metric corrections (17)\textendash(22) and every derived bound on $\alpha$ would change.

Editorial extensions

If this is right

  • If the Case II branch is the physically realized one, an $f(Q)$ theory with a quadratic correction must have $|\alpha|\lesssim 10^{-5}\,\mathrm{km}^2$, making the correction negligible at Solar System curvatures.
  • The two connection branches are observationally distinguishable in principle: the same $\alpha$ produces much larger precession, deflection, and delay signatures in Case II than in Case I.
  • For Case I, current Solar System data leave $\alpha$ almost unconstrained (bounds up to $10^{40}\,\mathrm{km}^4$ for $n=3$), so that branch is not tested by these observations.
  • Gravitational redshift is the most sensitive probe among the five tests for Case II, because it is measured near the Earth's surface where corrections decay less; perihelion precession of Mercury is the most sensitive precession probe.
  • Within the allowed windows the perturbative expansion is consistent, since $|Q|\gg|\alpha Q^n|$ for both $n=2$ and $n=3$, so the bounds are not artifacts of a broken approximation.

Reading between the lines

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

  • The stark contrast between the Case I and Case II bounds suggests that testing $f(Q)$ gravity is incomplete without fixing the connection sector; future work could seek observables that discriminate the branches rather than averaging over them.
  • Because the tightest bound comes from a ground-based redshift experiment, a next-generation clock or redshift measurement near the Earth, or at a Sun\textendash Earth Lagrange point, could plausibly tighten the Case II window by orders of magnitude.
  • The Case I precession correction grows with source mass $M$, so stellar-mass or supermassive objects (pulsar timing, S-star orbits) may probe the same $\alpha$ more efficiently than the Sun, something the paper's Solar System data cannot do.
  • A complete classification of static spherically symmetric torsion-free, curvature-free connections would settle whether the two branches in Eq. (12) are exhaustive; if additional branches exist, the reported bounds would need to be re-derived for those branches.
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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

2 major / 6 minor

Summary. This paper studies solar-system constraints on the covariant f(Q) gravity model f(Q)=Q+αQ^n-2Λ for n=2,3. Adopting the two static spherically symmetric affine-connection branches from Ref. [28], the authors derive first-order-in-α metric corrections (Eqs. 17–22). They then compute five observables—perihelion precession, light deflection, Shapiro delay, Cassini frequency shift, and gravitational redshift—for both branches, using the geodesic equation and standard post-Newtonian techniques. Comparing with data from EPM2011, Viking, Cassini, and Pound–Rebka, they place bounds on α; the headline results are the combined ranges in Eqs. (122)–(125), with the tightest constraints coming from perihelion precession for Case I n=2, from (an incorrect) Cassini bound for Case I n=3, and from gravitational redshift for Case II.

Significance. If the bounds are correct, this would be the first comprehensive solar-system test of covariant f(Q) gravity that goes beyond the coincident-gauge choice and includes a cosmological constant. The explicit analytic expressions for the connection-dependent corrections are a useful contribution, and the paper correctly notes the strong suppression of Case I corrections at higher order. However, the Cassini frequency-shift calculation contains an arithmetic inconsistency that invalidates the Case I n=3 bound in Eq. (123). The derivation framework remains sound, so a corrected Cassini analysis would restore the paper's utility.

major comments (2)
  1. [III D, Eqs. (92) and (93)] The Cassini frequency shift y=dΔt/dt is not consistent with the Shapiro delay expressions in Eqs. (85) and (87). Differentiating Δt_I^{n=2} = (64/3)αM^3/b^4 with db/dt=v_e gives y_I^{n=2} = -(256/3)αM^3v_e/b^5, not −256αM^3v_e/b^5 as in Eq. (92); differentiating Δt_I^{n=3} = -(1536/35)αM^5/b^8 gives y_I^{n=3} = +(12288/35)αM^5v_e/b^9, not +12288αM^5v_e/b^9 as in Eq. (93). Consequently the bounds in Eqs. (114) and (115) are too tight by factors of 3 and 35, respectively. Since Eq. (123) adopts the n=3 Cassini bound as the overall Case I n=3 constraint, the quoted range −4.43×10^40 < α < 4.43×10^40 km^4 is not correct; with the corrected Cassini bound the tighter limit would come from light deflection, Eq. (106).
  2. [IV D] The unit conventions in the Cassini analysis are inconsistent. The formulas for y are written in geometric units, where time and length have the same dimension, but the inputs ve = 30 km/s and c = 3×10^8 m/s are introduced without showing where c enters. The term −8M/b ve has dimensions km/s if M and b are in km, so it cannot equal the dimensionless y_GR = 10^{-10} unless ve is reinterpreted as v_e/c or an explicit factor 1/c is inserted. This ambiguity affects the numerical values of all Cassini bounds (Eqs. 114–117) and should be clarified with a consistent derivation.
minor comments (6)
  1. [Eq. (117)] The text reads 'Case Ii' and should be 'Case II'.
  2. [IV D] The sentence 'inputting these values into Eqs. (92) and (95)' should refer to Eqs. (92)–(95), since the Case I constraints use both Eqs. (92) and (93).
  3. [III A, Eq. (25)] The word 'Timeline' in 'For a timeline (η=1) test particle' should be 'timelike'.
  4. [III B] The phrase 'the deflation angle' should be 'the deflection angle'.
  5. [Abstract/Introduction] The data sets used are from EPM2011 (2013), Viking (1979), Cassini (2003), and Pound–Rebka (1959); describing them as 'latest observational data' is an overstatement, and a more recent ephemeris or a rephrasing is recommended.
  6. [V] The consistency check that |Q| ≫ |αQ^n| is valuable and should be shown for all four bound sets (Case I and II, n=2,3) rather than only the two examples presented.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the solar-system bounds on alpha are obtained by comparing independently derived observables with external data, not by feeding fitted values back into the derivation.

full rationale

The derivation chain is self-contained. The free parameter alpha enters the perturbative static spherically symmetric solutions (17)-(22), and each observable (perihelion precession, light deflection, Shapiro delay, Cassini frequency shift, gravitational redshift) is computed from those metric solutions and the standard geodesic and Killing-vector formalism (Eqs. (28), (53), (78)-(80), (96)). The bounds on alpha in Eqs. (100)-(121) are obtained by comparing these explicit alpha-dependent expressions with independent external measurements (EPM2011 residuals, gamma constraints, Viking, Cassini, Pound-Rebka); no observable is first used to fit alpha and then re-predicted from the same fit. The connection ansatz Gamma^r_theta_theta = +/- r/sqrt(B) is adopted from the external work Ref. [28] and is stated as an assumption, not derived from the data, so it is a model premise rather than a circular input-output relation. The authors' own prior works (Refs. [25], [26], [30]) are cited only for context and degree-of-freedom discussions and are not load-bearing for the constraints. The concluding check that |Q| is much larger than |alpha Q^n| uses the already-derived bounds only to validate the perturbative expansion and does not feed back into the derivation. The apparent factor discrepancies in the Cassini conversion (Eqs. (92)-(93) versus differentiating Eqs. (85) and (87)) would be an arithmetic or correctness issue, not circularity. No equation reduces to its own input and no fitted parameter is renamed as a prediction.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central derivation rests on the affine connection ansatz and the metric geodesic assumption inherited from the f(Q) literature, plus the first-order perturbative expansion in α. No new entities are introduced.

free parameters (1)
  • α = Case I n=2: -9.07e18 to 1.01e18 km^2; Case I n=3: -4.43e40 to 4.43e40 km^4; Case II n=2: -5.55e-5 to 9.52e-6 km^2…
    The model parameter controlling the nonmetricity correction; the paper derives allowed ranges from solar system data. It is not fixed by the theory.
assumptions (3)
  • domain assumption Test particles and photons follow metric geodesics in f(Q) gravity
    Used in Sec III to derive perihelion precession, light deflection, and Shapiro delay. Standard minimal coupling, but not proven in the paper.
  • domain assumption The connection ansatz Γ^r_{θθ} = ± r/√B(r) spans the admissible static spherically symmetric solutions
    Taken from Ref [28] and stated in Eq (12); the paper does not derive this from the connection field equations, so the two cases may not be exhaustive.
  • domain assumption The perturbative expansion g = g0 + α g1 is valid for the Solar System
    Solutions (13)-(14) are first order in α; the authors check a posteriori that |Q| >> |αQ^n| for the constrained values, but the expansion is an assumption.

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Pith. "Pith review of Solar system tests in covariant f(Q) gravity." pith.science (2026). https://pith.science/paper/SKUPQNJN

@misc{pith2026241217463,
  author       = {Pith},
  title        = {Pith review of: Solar system tests in covariant f(Q) gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SKUPQNJN}},
  note         = {Machine review of arXiv:2412.17463}
}
abstract

We study the Solar System constraints on covariant $f(Q)$ gravity. The covariant $f(Q)$ theory is described by the metric and affine connection, where both the torsion and curvature vanish. Considering a model including a higher nonmetricity-scalar correction, $f(Q)= Q +\alpha Q^{n} - 2\Lambda$, we derive static and spherically symmetric solutions, which represent the Schwarzschild-de Sitter solution with higher-order corrections, for two different ansatz of the affine connection. On the obtained spacetime solutions, we investigate the perihelion precession, light deflection, Shapiro delay, Cassini constraint, and gravitational redshift in the $f(Q)$ gravity. We place bounds on the parameter $\alpha$ with $n=2, 3$ in our model of $f(Q)$ gravity, using various observational data in the Solar System.

Figures

Figures reproduced from arXiv: 2412.17463 by the authors.

Figure 1
Figure 1. FIG. 1. The plane graph represents the orbit with the one-sided deflection angle [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗

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Pith tools

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