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Polarization modes of gravitational waves in general symmetric teleparallel gravity

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Pith's one-line read In the most general symmetric teleparallel gravity, gravitational waves always include two shear modes and a longitudinal mode, all travelling at light speed, when test particles carry hypermomentum — a polarization signature absent from…

desk verdict A useful but incomplete polarization-mode catalog for general symmetric teleparallel gravity; the universal-mode and f(Q) pathology claims rest on an imported detector equation the paper does not derive. read the letter →

arxiv 2505.13298 v2 pith:NAQ6FEG3 submitted 2025-05-19 gr-qc hep-th

Yu-Qi Dong , Xiao-Bin Lai , Yu-Zhi Fan , Yu-Xiao Liu This is my paper · ORCID
classification gr-qchep-th PACS 04.30.-w04.50.Kd
keywords gravitationalwavespolarizationmodessymmetricteleparallelgravityhypermomentumnon-metricityf(Q)shearlongitudinalmode
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 maps out the complete set of gravitational-wave polarization modes allowed by the most general symmetric teleparallel gravity — the family of theories that attributes gravity to non-metricity of the connection rather than to spacetime curvature — and treats that set as a fingerprint for distinguishing the family from general relativity. Its central statement is parameter-independent: when test particles carry hypermomentum, a charge that couples them to the connection, two shear modes and one longitudinal mode are always present and always travel at the speed of light, regardless of the values of the theory's coefficients. Without hypermomentum the generic prediction is just the two tensor modes, with light-speed vector modes appearing only on a finely tuned parameter surface. The same analysis shows that the two most studied special theories, f(Q) gravity and quadratic non-metricity gravity, cannot admit connection-coupled matter without producing unconstrained, physically unreasonable modes. Since polarization is read directly from the motion of test particles, a universal shear-plus-longitudinal pattern would be an observational discriminator between symmetric teleparallel gravity and the Riemannian framework.

What carries the argument

Three pieces of machinery carry the argument. A gauge-invariant decomposition of the metric and connection perturbations on a flat background separates the fields into transverse-traceless tensors $h^{TT}_{ij}$, transverse vectors $\Xi_i$ and $K_i$, and scalars $\Theta$, $\phi$, $K$, $L$, which decouples the linearized field equations sector by sector. The symmetric-teleparallel constraint of zero curvature, written linearly as $\partial_\rho\Sigma^\mu_{\nu\lambda}=\partial_\lambda\Sigma^\mu_{\nu\rho}$, forces every gauge-invariant perturbation built purely from the connection to vanish, leaving only $h^{TT}_{ij}$, $\Xi_i$, $K_i$, $\Theta$, $\phi$, $K$, and $L$ as possible carriers of radiation. The observable modes are then read off from the test-particle deviation equation: the metric part $\hat{R}^i_{0j0}$ gives the six standard modes P1–P6, while the hypermomentum term $-\partial_j N^i_{00}$ makes the deviation matrix asymmetric and generates the shear modes P7 and P8 from the transverse vector $K_i$; the longitudinal mode P1 is sourced by the scalar $L$ through the solution $L+K=0$, $\varphi=\Theta=0$. All of the parameter dependence sits in a six-coefficient quadratic action whose coefficients $A^{(1)}$, $B^{(1)}$, $C^{(1)}$, $D^{(1)}$, $E^{(1)}$, and $A^{(2)}$ decide which sectors propagate.

What would settle it

For the universality claim, substitute the plane-wave branch $L+K=0$, $\varphi=\Theta=0$ into the six scalar equations (A1)–(A6) at a generic parameter point and inspect the dispersion relation: a parameter choice for which this branch fails to solve all six equations, or for which some other scalar mode acquires a non-lightlike speed, would refute the claim that the longitudinal mode is universal. For the detector model, deriving Eq. (29) from an explicit hypermomentum matter action and comparing the resulting deviation equation would settle whether the shear modes are the modes such matter actually exhibits.

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Extended reading notes

Core claim

The paper claims that in the most general symmetric teleparallel gravity theory with second-order field equations, the gravitational-wave polarization content is nearly rigid. The tensor sector always produces the + and $\times$ modes at light speed, and a viable theory must have $A^{(2)}\neq 0$ so these modes actually propagate. When test particles carry hypermomentum, the connection perturbation enters the particle-deviation equation, the deviation matrix becomes asymmetric, and two shear modes, P7 and P8, appear; they propagate at light speed whenever $C^{(1)}+E^{(1)}\neq 0$, while in the degenerate case $C^{(1)}+E^{(1)}=0$ they are left unconstrained by the field equations, which the paper counts as a pathology. A longitudinal mode at light speed always exists as well, carried by the scalar-sector solution $L+K=0$, $\varphi=\Theta=0$. The paper further argues that f(Q) gravity is viable only if matter is independent of the connection, since any hypermomentum coupling yields unconstrained shear and longitudinal modes, and that the standard parameter conditions of quadratic non-metricity gravity likewise force the no-hypermomentum choice, leaving only tensor modes. The distinguishing claim is universality: with hypermomentum, shear and longitudinal modes are unavoidable companions of any gravitational wave, a pattern absent from Riemannian gravity.

Load-bearing premise

The load-bearing premise is the detector model of Eq. (29), adopted from the authors' earlier work rather than derived from a matter action: a hypermomentum-charged test particle is assumed to respond to a wave through $d^2\eta^i/dt^2=-\hat{R}^i_{0j0}\eta^j-\partial_j N^i_{00}\eta^j$, and this specific response rule is what makes the connection scalar $L$ and the transverse vector $K_i$ show up as shear and longitudinal modes; a different coupling between matter and the connection would change which modes are observable even if the gravitational field equations stay the same.

Editorial extensions

If this is right

  • A detector whose test masses carry hypermomentum should see two shear modes (P7 and P8) and a longitudinal mode arriving at light speed in every symmetric teleparallel theory, a pattern no Riemannian theory predicts; observing that pattern would select this family over general relativity.
  • f(Q) gravity cannot admit any coupling of matter to the connection: any such coupling produces unconstrained shear and longitudinal modes, a loss of predictability. This closes off connection–matter coupling as a cure for the ghost and strong-coupling problems found in cosmological f(Q) perturbations.
  • In the generic theory, vector-x and vector-y modes exist only under the fine-tuned condition $4A^{(2)}+C^{(1)}+E^{(1)}=0$, so a detection of light-speed vector modes would constrain the theory's parameter space rather than generically confirm the framework.
  • f(Q) gravity with metric-only matter and quadratic non-metricity gravity under its two standard parameter conditions both predict only the + and $\times$ tensor modes at light speed, so polarization alone cannot distinguish those versions from general relativity without hypermomentum-sensitive detectors.
  • The most general linear field equations contain six free coefficients while quadratic non-metricity gravity has only five, so analyses restricted to that theory miss part of the symmetric teleparallel landscape; adding a Ricci-scalar term restores full linear coverage.

Reading between the lines

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

  • Reading the paper's formulas literally, the universal longitudinal branch ($L+K=0$, $\varphi=\Theta=0$) has vanishing metric scalars: a detector following metric geodesics sees no scalar radiation, while a hypermomentum-charged detector sees a pure longitudinal signal carried entirely by the connection scalar $L$. The discriminating observation therefore needs purpose-built detectors with connecti
  • The paper's criterion for health — no polarization mode may be left unconstrained by the field equations — singles out the subspace $C^{(1)}+E^{(1)}\neq 0$ and rules out hypermomentum in f(Q) and quadratic non-metricity gravity. The same criterion could serve as a general diagnostic for strong coupling in other metric-affine theories: an unconstrained field appearing in the observable modes is a s
  • The six-versus-five parameter gap implies that results obtained within quadratic non-metricity gravity, for example in cosmological perturbation theory, need not represent the full symmetric teleparallel landscape even at linear order; revisiting those results in the Ricci-scalar-extended action (49) would show which conclusions survive.
  • A testable extension would be to model a concrete hypermomentum-carrying test particle, such as matter with intrinsic spin probing a torsion-free connection background, and compute its response to the predicted plane waves; if the response differs from Eq. (29), the shear modes would be modified or absent, giving a laboratory-scale check of the paper's detector model.
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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. The paper studies gravitational-wave polarization modes in the most general symmetric teleparallel gravity action that yields second-order field equations, using a gauge-invariant decomposition. It claims that, when test particles carry hypermomentum, two shear modes and a longitudinal mode are universally present and propagate at the speed of light, while vector-x/y modes exist only in a special parameter region. The paper also analyzes f(Q) gravity and quadratic non-metricity gravity as concrete examples, concluding that f(Q) gravity is physically unreasonable if matter couples to the connection, and that in the parameter conditions considered both theories only have tensor modes in the absence of hypermomentum.

Significance. If the results hold, the paper provides a concrete, falsifiable distinction between symmetric teleparallel gravity and Riemannian theories: the universal presence of shear and longitudinal modes for hypermomentum-charged test particles. The field-equation analysis is explicit and checkable, and the parameter mappings for f(Q) and quadratic non-metricity gravity are clearly stated, which is useful for the community. However, the observable predictions rest on a detector model that is imported from a companion paper without derivation, and one of the two universal scalar-mode conclusions is asserted rather than demonstrated against the six scalar equations in Appendix A. The significance is therefore conditional on closing these gaps.

major comments (3)
  1. [Sec. III, Eq. (29)] The central observable claims for the hypermomentum case depend on the relative-acceleration equation d^2 eta^i/dt^2 = -R^hat_i_0j0 eta^j - partial_j N^i_00 eta^j, which is imported from Ref. [29] without derivation. The paper does not specify the matter action that defines the hypermomentum charge, nor does it show that Eq. (29) is the unique or correct detector response for matter coupled to the connection. Since the definitions of P7/P8 in Eq. (32), the identification of the longitudinal mode P1, and the f(Q) pathology conclusion in Sec. V.A all read off from this equation, a different hypermomentum-matter coupling could change which field perturbations are observable. The authors should derive Eq. (29) from a concrete action, or state explicitly the minimal assumptions under which it holds.
  2. [Sec. IV.C and Appendix A] The claim that there always exists a plane-wave solution L+K=0, phi=Theta=0 propagating at the speed of light is asserted but not verified against the six scalar equations (A1)-(A6). These equations are coupled and contain non-standard terms such as Delta/partial_0 and partial_0^3/Delta; the proposed ansatz should be substituted explicitly or shown by a linear-algebra argument to solve the full system for all allowed parameters. Without this, the universal existence of the longitudinal mode is not established. The companion claim that all scalar modes propagate at light speed also needs a concrete derivation from the scalar equations, since the stated reasons of homogeneity and formal Lorentz symmetry are not self-evident for these momentum-space equations.
  3. [Sec. IV.B, Eqs. (35)-(37)] The vector-mode analysis states, after Eqs. (35)-(37), that when C(1)+E(1) is nonzero there exists a solution for K_i propagating at the speed of light, and that when C(1)+E(1)=0 the K_i are unconstrained. The latter is evident, but the former propagation claim is not demonstrated from the given equations. This is load-bearing for the conclusion that shear modes always exist (either as physical modes or as unconstrained pathological modes). The authors should show the elimination of Xi_i and the resulting dispersion relation for K_i, or provide the algebraic step that leads to this conclusion.
minor comments (5)
  1. [Global] There are several typos and OCR-like artifacts: 'carameter' in Sec. V.B should be 'parameter'; Eq. (A5) contains '1/2 C(1) 1/partial_0 K' and similar expressions that appear to be incorrect typesetting; and the second sentence of the abstract is missing a period after 'field equations'.
  2. [Sec. IV.A, Eq. (34)] The sentence 'this implies that symmetric teleparallel gravity necessarily requires A(2) != 0' is unclear: if A(2)=0, Eq. (34) imposes no constraint on hTT_ij, so the tensor modes would be unconstrained rather than nonexistent. The phrasing should be sharpened to say that a well-defined propagation of tensor modes requires A(2) != 0.
  3. [Sec. III, Eqs. (28) and (32)] The expression for P1 changes between the no-hypermomentum case and the hypermomentum case, with additional terms involving Pi, Pi-bar, and L appearing in Eq. (32). The text should explain how these terms arise from the connection contributions in Eq. (29), since P1 is read off from the same A_ij matrix.
  4. [Sec. III, Fig. 2] Figure 2, which illustrates the shear modes, is referenced but does not appear to be included in the manuscript text; please ensure the figure is present or remove the reference.
  5. [Sec. V.B] The statement that conditions (a) and (b) endow the second-order action with 'the gauge symmetry and the Weyl Transverse Diffeomorphism (WTDiff) symmetry' is vague; the specific gauge symmetry should be named.

Circularity Check

2 steps flagged · score 4.0 of 10

No fitted-input circularity in the mode calculation itself, but the central observable predictions (shear modes, universal longitudinal mode, f(Q) pathology) rest on a load-bearing self-citation: the hypermomentum deviation equation (29) and the general action (13) are imported from the authors' own companion papers rather than derived here.

  1. self citation load bearing [Sec. III, Eq. (29); used in Secs. IV.B, IV.C, and V.A]
    "For the case with hypermomentum, the equation of relative motion for the test particles is [29] d2ηi dt2 = −Ai jηj := −bRi 0j0ηj−∂jNi 00ηj."

    The paper's distinctive conclusions — the two shear modes P7/P8, the always-present longitudinal mode, and the claim that f(Q) with hypermomentum has unconstrained pathological modes — are all read off from this deviation equation via Eq. (32). The equation is not derived in this paper from a matter action and is not shown to be the unique response of hypermomentum-carrying test particles; it is simply cited from the authors' own Ref. [29]. A different hypermomentum-matter coupling would change Aij and therefore which connection perturbations are observable, even though the field equations in Secs. IV.A-IV.C would be unchanged.

  2. self citation load bearing [Sec. II, Eq. (13)]
    "In order to study the polarization modes of gravitational waves in the most general symmetric teleparallel gravity theory that can derive second-order field equations, we need to provide the most general second-order perturbation action S(2) g for Sg. This has already been presented in our previous paper [29]."

    The parameter space that generates all subsequent tensor, vector, and scalar mode conditions — including the special condition 4A(2)+C(1)+E(1)=0 for vector modes and the f(Q) parameter set — is inherited from the action (13), which is imported wholesale from the authors' own previous paper. The paper does not rederive the 'most general' claim here; it relies on a same-author citation. If that action were incomplete or not actually the most general second-order action, the whole mode catalogue and the f(Q) pathology argument would change. This is load-bearing self-citation, though the subsequent linear algebra and mode classification are internally consistent given that action.

full rationale

There is no numerical fitting in this paper and no fitted parameter that is later renamed as a prediction. The tensor, vector, and scalar equations in Secs. IV.A-IV.C are obtained by solving the stated linearized field equations, so those parts are self-contained calculations conditional on the imported general action. The circularity concern is concentrated at the detector and framework stage: Eq. (29), which defines how hypermomentum-charged test particles respond to connection perturbations, is taken directly from the authors' own Ref. [29]. All of the paper's novel observable claims — the universal shear modes P7/P8, the universal longitudinal mode, and the conclusion that f(Q) gravity must have matter independent of the connection — are read off through Eq. (32) from that imported deviation equation. Because Eq. (29) is not derived from a matter action here and is not shown to be the unique hypermomentum detector response, the central observable predictions inherit a load-bearing assumption from a same-author paper rather than being established independently in this work. This is genuine load-bearing self-citation, but it does not reduce the results to their inputs by definition; the field-equation analysis still contains substantial independent content. Accordingly, the circularity score is moderate rather than high.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the flat-background linearization, the general second-order action imported from Ref. [29], and the hypermomentum test-particle equation imported from Ref. [29]. No empirical data or fitted parameters are involved; the action coefficients are undetermined theory parameters whose values are scanned. No new particles, forces, or geometric entities are postulated beyond the existing hypermomentum and nonmetricity concepts.

free parameters (6)
  • A^(1) = not fitted
    Coefficient of metric-connection one-derivative terms in the general second-order action (13). The polarization results are conditional on combinations of these coefficients.
  • B^(1) = not fitted
    Coefficient in the general second-order action (13), part of the most general symmetric teleparallel parameter space scanned in Section IV.
  • C^(1) = not fitted
    Coefficient in the general second-order action (13). The combination C^(1)+E^(1) controls the vector and shear mode behavior.
  • D^(1) = not fitted
    Coefficient in the general second-order action (13), appearing in scalar mode equations and in the parameter maps for example theories.
  • E^(1) = not fitted
    Coefficient in the general second-order action (13), appearing with C^(1) in the vector and shear mode conditions.
  • A^(2) = not fitted
    Coefficient of the h^2 and Sigma^2 kinetic terms in the general second-order action (13). The tensor mode condition A^(2) box h^TT_ij = 0 depends on it.
assumptions (4)
  • domain assumption The flat vacuum background with g = eta, Gamma = 0, and partial_rho \bar p^{(mu nu)rho}_lambda = 0 can be imposed without loss of generality.
    Sec. II, Eqs. (4)-(5). This is the standard weak-field linearization assumption; if the background Lagrange multiplier field has nontrivial structure, the linearized equations would differ.
  • domain assumption Equation (13) is the most general second-order symmetric teleparallel perturbation action.
    The action is imported from the authors' Ref. [29] without re-derivation here. All subsequent parameter scans and mode conclusions depend on this being the correct general action.
  • domain assumption Test particles carrying hypermomentum obey the deviation equation d^2 eta^i/dt^2 = -R^i_0j0 eta^j - partial_j N^i_00 eta^j.
    Sec. III, Eq. (29), taken from Ref. [29]. This measurement model is what makes the shear modes P7 and P8 observable in the hypermomentum case.
  • standard math Zero curvature and zero torsion are enforced as constraints, so the action with Lagrange multipliers in Eq. (2) is equivalent to the torsionless form in Eq. (3).
    Sec. II, Eqs. (2)-(3). The later claim that a c0 R term can still contribute despite R = 0 depends on a subtle point about where the constraint acts, so this equivalence is load-bearing.

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Pith. "Pith review of Polarization modes of gravitational waves in general symmetric teleparallel gravity." pith.science (2026). https://pith.science/paper/NAQ6FEG3

@misc{pith2026250513298,
  author       = {Pith},
  title        = {Pith review of: Polarization modes of gravitational waves in general symmetric teleparallel gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NAQ6FEG3}},
  note         = {Machine review of arXiv:2505.13298}
}
abstract

In this paper, we investigate the polarization modes of gravitational waves within the most general symmetric teleparallel gravity theory that allows for second-order field equations We consider both scenarios where test particles either carry or do not carry a hypermomentum charge. Our findings reveal the existence of tensor, vector, and scalar modes of gravitational waves. Firstly, the theory supports the + and $\times$ tensor modes propagating at the speed of light. Secondly, in the case where particles do not carry hypermomentum, vector modes propagating at the speed of light exist only within a very specific parameter space. However, when particles do carry hypermomentum, there are two shear modes that propagate at the speed of light, while the vector-$x$ and vector-$y$ modes emerge only under very specific conditions. Thirdly, in the presence of hypermomentum, there is always a longitudinal mode propagating at the speed of light. The universal existence of the shear modes and the longitudinal mode in the presence of hypermomentum is a key feature of symmetric teleparallel gravity, distinguishing it from the Riemannian framework through gravitational wave polarization detection. We also analyze the polarization modes in two widely studied special theories: $f(Q)$ theory and quadratic non-metricity theory. Our study reveals that, within the $f(Q)$ gravity framework, it is crucial to assume that matter fields are independent of the connection, as any dependence would lead to unphysical results.

Figures

Figures reproduced from arXiv: 2505.13298 by the authors.

Figure 1
Figure 1. FIG. 1: The six polarization modes of gravitational waves [28]. Here, the gravitational wave [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The two new shear modes [29] [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.