Propagating Gravitational Waves in Teleparallel Gauss-Bonnet Gravity

B. Mishra; Jackson Levi Said; Shivam Kumar Mishra

REVIEW 2 major objections 4 minor 79 references

Propagating Gravitational Waves in Teleparallel Gauss-Bonnet Gravity

T0 review · 2 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Tensor perturbations of flat FLRW in F(T,T_G) teleparallel Gauss-Bonnet gravity give alpha_T = 0: gravitational waves travel at the speed of light.

desk verdict First F(T,T_G) tensor perturbation analysis gives a plausible luminal-speed result, but the central action is unshown and the appendix determinant has a factor-two error. read the letter →

arxiv 2505.18192 v2 pith:V2YGAWLZ submitted 2025-05-20 gr-qc hep-th

Shivam Kumar Mishra , Jackson Levi Said , B. Mishra This is my paper · ORCID

classification gr-qchep-th MSC 83C3583D0583F05 PACS 04.30.-w04.50.Kd

keywords teleparallelgravityGauss-BonnettensorperturbationsgravitationalwavespeedGW170817FLRWcosmologyluminositydistancemodified

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 the gravitational-wave sector behaves in $F(T,T_G)$ teleparallel Gauss-Bonnet gravity, a class of modified gravity in which the geometry is carried by torsion rather than curvature and the action is a function of the torsion scalar and its Gauss-Bonnet counterpart. By perturbing a spatially flat FLRW universe with a transverse, traceless tensor mode and expanding the action to second order, the authors derive a propagation equation in which the $k^2/a^2$ term has coefficient exactly one, so the tensor excess speed is zero and gravitational waves travel at the speed of light. The amplitude of the waves is not standard: the effective Planck mass becomes $M_*^2 = \kappa^{-2}(-F_T + 4H\dot{F}_{T_G})$, and the ratio of gravitational-wave to electromagnetic luminosity distance is set by the running of this mass. The paper's main point of interest is that the whole class of $F(T,T_G)$ cosmologies is therefore consistent with the GW170817 bound on the gravitational-wave speed, which has ruled out many other modified-gravity proposals. It also notes that avoiding ghost instabilities requires the tensor coefficient $C_{\rm tensor} = -F_T + 4H\dot{F}_{T_G}$ to be positive.

What carries the argument

The central object is the quadratic action of the tensor mode, obtained by perturbing the vierbein as $\delta e^{\hat{k}}_\alpha = \frac{a}{2}\delta^{\hat{i}}_\alpha \delta^{\hat{k}\hat{j}} h_{\hat{i}\hat{j}}$ and expanding the torsion scalar $T$ and its teleparallel Gauss-Bonnet counterpart $T_G$ (the torsion object equal to the Gauss-Bonnet invariant up to a total divergence) to second order in $h_{ij}$. The transverse-traceless gauge conditions remove unwanted terms, integration by parts brings the action into the form (33), and variation with respect to $h_{ij}$ produces the propagation equation. The load-bearing mechanism is a set of cancellations in the second-order expansion that leave the gradient term with the same $a^{-2}$ coefficient as in general relativity, rather than a rescaled speed or an extra $k^2 h^2$ mass term; this cancellation is what forces $\alpha_T = 0$.

What would settle it

A direct symbolic computation of $T^{(2)}$ and $T_G^{(2)}$ from the perturbed vierbein (31), followed by integration of the action (24), would settle the claim: if any term $h_{ij}\nabla^2 h^{ij}$ or $h_{ij}h^{ij}$ survives with a coefficient other than $a^{-2}C_{\rm tensor}$, then $\alpha_T \neq 0$ or a mass term is present.

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Extended reading notes

Core claim

The central claim is that tensor perturbations of flat FLRW in $F(T,T_G)$ teleparallel gravity yield the second-order action (33), $S_T^{(2)} = \frac{1}{2\kappa^2}\int dt\,d^3x\, a^3 \frac{1}{2}C_{\rm tensor}[\dot{h}_{ij}^2 - a^{-2}(\nabla h_{ij})^2]$, with $C_{\rm tensor} = -F_T + 4H\dot{F}_{T_G}$. Varying this action gives the gravitational-wave propagation equation (34), whose momentum term is exactly $k^2/a^2$ with no extra mass term, so comparison with the standard parametrization gives $\alpha_T = 0$ and $M_*^2 = \kappa^{-2}(-F_T + 4H\dot{F}_{T_G})$. Thus in these theories gravitational waves propagate at the speed of light, while the waveform amplitude is damped or enhanced depending on the time evolution of $M_*$; stability against ghosts requires $C_{\rm tensor} > 0$. The authors also connect the amplitude modification to the gravitational-wave versus electromagnetic luminosity-distance ratio through the standard-siren relation for GW luminosity distance. In the curvature-based analogue $f(R,G)$, by contrast, the propagation speed deviates from $c$, which is what makes the teleparallel result distinctive.

Load-bearing premise

The load-bearing premise is that $T$ and $T_G$ expand to second order in the tensor perturbation into exactly the quadratic action (33) with coefficient $C_{\rm tensor}$ and no additional $k^2 h^2$ terms; the paper states this outcome but the appendix lists only first-order torsion components, so the expansion is not shown.

Editorial extensions

If this is right

Every $F(T,T_G)$ model built on flat FLRW automatically satisfies the GW170817-GRB170817A limit on gravitational-wave speed, because $\alpha_T = 0$ at the level of this calculation.
Gravitational-wave amplitudes in these theories carry a fingerprint: the running effective Planck mass $M_*^2 = \kappa^{-2}(-F_T + 4H\dot{F}_{T_G})$ rescales the waveform and makes the GW luminosity distance differ from the electromagnetic one by $\exp\!\left(\tfrac{1}{2}\int \frac{\alpha_M}{1+z}\,dz\right)$.
Requiring $C_{\rm tensor} > 0$ to avoid ghost-like tensor modes translates into a concrete inequality on $F$ and its derivatives, which can be used to reject candidate functional forms of $F(T,T_G)$.
A measured gravitational-wave speed exactly equal to $c$ cannot, by itself, distinguish $F(T,T_G)$ from general relativity; distinguishing signals must come from the amplitude channel or from scalar and vector perturbations.

Reading between the lines

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

A natural next step, not taken here, is to perturb the same action at scalar and vector order; tensor ghosts may be absent while scalar instabilities still rule out many $F(T,T_G)$ functional forms, so the ghost-free condition is necessary but not sufficient.
Because the gradient coefficient was checked only on a flat background, the same second-order computation on an open or closed FLRW metric could produce a curvature-dependent contribution to $\alpha_T$; testing that would show whether the luminal result is tied to flatness.
Standard-siren datasets from future detectors could turn the amplitude modification into a direct test: a measured GW/EM distance ratio that tracks the predicted exponential integral would favour $F(T,T_G)$, while a null result would constrain the allowed running of $M_*$.
The method used here, perturbing the action rather than the field equations, can be ported directly to other teleparallel extensions such as $f(T,B)$, where the same cancellation question decides whether their tensor speed is also protected.

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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

First, the new thing: this is the first derivation of the cosmological tensor perturbation equation for F(T,T_G) teleparallel gravity. The result is that tensor modes propagate at luminal speed, with the amplitude modified through M_*^2 = kappa^{-2}(-F_T + 4H dot(F_TG)). That is a useful step for the teleparallel modified-gravity community, and the approach—perturbing the action rather than the field equations—is the right one. The GR limit (F=-T) and the linear Gauss-Bonnet limit (F=-T+alpha T_G) both give C_tensor=1, so the speed claim passes those checks.

The soft spot is load-bearing. Equation (33), the second-order action from which everything follows, is stated without showing the second-order expansion of T and T_G. Appendix A lists only first-order torsion components, and the one second-order quantity it does give—the determinant e = a^3[1 - (1/4)Tr(h^2)]—is wrong by a factor of two. From the paper's own perturbed vierbein, the determinant is a^3[1 - (1/8)Tr(h^2)]. Because e multiplies the background F in the action, an incorrect e^(2) leaves an uncancelled h^2 term unless T^(2) and T_G^(2) compensate exactly, and the paper gives no way to check that. So the coefficient C_tensor = -F_T + 4H dot(F_TG) and the claimed absence of a mass term are, as things stand, unverified. I want to be clear that this is not a refutation: the consistency checks pass and the structure of the claim is plausible. But a reader cannot reproduce the central computation from the text, and the mistaken determinant in the appendix suggests the perturbative data need care.

Two secondary points. The abstract says the paper derives the distance-duality relationship, but in the text Eq. (36) is simply quoted from Ezquiaga and Zumalacárregui. And the paper would benefit from a sentence explaining why the speed stays luminal in the teleparallel Gauss-Bonnet setting when the curvature-based f(R,G) case in Ref. [79] deviates; that contrast is the kind of thing the reader will immediately wonder about.

The citation pattern is fine, with proper credit to [40] for the Minkowski polarizations and [78] for f(T) gravity. No fitted parameters, no circular steps.

This paper is for the modified-gravity and teleparallel-cosmology subfield. It deserves a serious referee, but the referee should demand a major revision: the full second-order expansion of T and T_G, or a supplementary file, showing how (33) follows from (11) and (15), plus a corrected appendix. If the computation holds up, the alpha_T = 0 result is worth having.

Referee Report

2 major / 4 minor

Summary. The manuscript studies tensor perturbations of a spatially flat FLRW background in F(T,T_G) teleparallel gravity. Its central claim is that the quadratic action for transverse-traceless tensor modes takes the form (33) with coefficient Ctensor = -F_T + 4H Fdot_TG. Varying this action gives the gravitational-wave propagation equation (34), whose k^2/a^2 term has coefficient unity, implying alpha_T = 0 and luminal gravitational-wave speed for the entire class of F(T,T_G) theories. The paper also relates the modified friction term to the GW/EM luminosity-distance ratio and concludes that the class is observationally consistent with the GW170817 speed bound. The derivation is based on a perturbed vierbein (31), but the crucial second-order action is asserted rather than derived; Appendix A supplies only partial perturbation data.

Significance. If the central calculation is correct, the result is significant: it would place the broad F(T,T_G) teleparallel family on the observationally safe side of the GW170817 constraint, in contrast to many curvature-based modified-gravity theories. The paper also gives a clear stability condition, Ctensor > 0, and a standard-siren relation for the modified amplitude, which are testable in principle. The result passes the consistency check of reducing to GR for F = -T and to the linear Gauss-Bonnet limit, both of which give Ctensor = 1. The main concern is that the load-bearing second-order action is not verified from the text, and one displayed intermediate expression appears to contain a determinant error. The contribution is timely and potentially important, but its validity cannot currently be assessed from the manuscript as written.

major comments (2)

[III, Eq. (33), and Appendix A] The derivation of the second-order action (33) is the load-bearing step and is not shown. The text states that inserting the perturbed relations into (11) and (15) and expanding the action yields (33), but Appendix A lists only torsion components up to first order (with some second-order corrections) and does not give T^(2) or T_G^(2). Since the action integral contains e F(T,T_G), the second-order part e^(2) multiplies the background F and must be cancelled by contributions from F_T T^(2) and F_TG T_G^(2). Without an explicit expansion of T and T_G to second order, the claimed coefficient Ctensor = -F_T + 4H Fdot_TG and the absence of a mass term are unsupported. Please provide the complete second-order expansion of e, T, and T_G, including the boundary terms discarded after integration by parts.
[Appendix A] The stated determinant of the perturbed vierbein appears to be incorrect. For the spatial block a(delta_ij + (1/2)h_ij) with h traceless, direct evaluation gives e = a^3[1 - (1/8)h_ij h^ij + O(h^3)], not a^3[1 - (1/4)h^2_ij] as written. Since e^(2) contributes to the quadratic action, this factor-of-two discrepancy could change the h^2 term in the action. The paper needs to correct this expression and demonstrate explicitly how the h^2 term is cancelled in the passage to Eq. (33), or explain if a different convention for h^2_ij is being used.

minor comments (4)

[Abstract and Section III] The abstract says the distance-duality relationship is derived, but Eq. (36) is quoted from Ref. [66] rather than derived in this work; please rephrase to 'employed' or 'applied'.
[Appendix A] The notation h^2_ij is ambiguous; if it means h_i^k h_kj, the trace should be written explicitly as h_ij h^ij or h_i^k h_k^i.
[Section III, Eq. (34)] The phrase 'after making sure that on shell conditions are satisfied' is vague; the only condition needed to obtain (34) from (33) is the Euler-Lagrange equation, and any additional assumptions should be stated explicitly.
[Eq. (25)] The name 'Friedmann-Lemaitre-Robertson-Walker' contains a character-encoding artifact ('LemaÃ®tre'); please correct it.

Circularity Check

0 steps flagged · score 1.0 of 10

No circular derivation: the luminal-speed result follows from an explicit second-order action; self-citations are not load-bearing.

full rationale

The central claim that tensor perturbations in F(T,T_G) teleparallel gravity propagate at luminal speed follows from the second-order action (33), whose coefficient C_tensor = -F_T + 4H dot(F_TG) is stated as the result of expanding (11) and (15) with the perturbed vierbein (31). No parameter is fitted to observational data; alpha_T = 0 is read off from the k^2/a^2 term in the resulting GWPE (34), which inherits coefficient unity directly from the gradient term in (33). The derivation is self-contained with respect to the action, and the result is not equivalent to an input by construction: the coefficient C_tensor could in principle have produced a non-canonical gradient term, but the expansion yields a canonical one. The only self-referential elements are citations to the authors' own prior work: [40] for the two polarization modes on a Minkowski background and [34] for the perturbed vierbein remaining in Weitzenböck gauge. Neither feeds back into the luminal-speed claim; the polarization count is not used in the derivation, and the Weitzenböck-gauge statement is a standard property of the explicitly given tetrad. The paper does contain an omitted proof: the passage from the perturbed tetrad to Eq. (33) is not shown, and Appendix A only lists first-order torsion components and a determinant e = a^3(1 - (1/4)h^2) that appears inconsistent with the given vierbein (a direct determinant of (31) gives a^3(1 - (1/8)Tr(h^2))). This is a correctness risk, not a circularity, because the reduction is a direct computation rather than an imposition of the desired result.

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

No parameters are fitted and no new entities are introduced. The theory function F(T,T_G) is arbitrary, not determined by the paper. The main input beyond standard teleparallel geometry is the unproved second-order action coefficient, which is the critical assumption.

assumptions (4)

domain assumption The diagonal vierbein (26) and Weitzenböck gauge are valid for the flat FLRW background.
Used to derive T and T_G in Eqs. (27) and (28); standard in teleparallel cosmology.
domain assumption The tensor perturbation ansatz (31) preserves the Weitzenböck gauge at perturbative order.
Invoked before Eq. (33); relies on Ref. [34]. If the spin connection perturbation cannot be set to zero consistently, the second-order action would change.
ad hoc to paper The second-order expansion of the action under (31) yields Eq. (33) with coefficient C_tensor = -F_T + 4H ẍ F_{T_G}.
This is the load-bearing unproved step; the paper does not show the expansion of T and T_G to second order.
domain assumption Matter is a perfect fluid with no anisotropic stress, so it does not source tensor perturbations.
Stated in Section III; standard for deriving the source-free gravitational wave equation.

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Pith. "Pith review of Propagating Gravitational Waves in Teleparallel Gauss-Bonnet Gravity." pith.science (2026). https://pith.science/paper/V2YGAWLZ

@misc{pith2026250518192,
  author       = {Pith},
  title        = {Pith review of: Propagating Gravitational Waves in Teleparallel Gauss-Bonnet Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/V2YGAWLZ}},
  note         = {Machine review of arXiv:2505.18192}
}

read the original abstract

Gravitational waves offer a key insight into the viability of classes of gravitational theories beyond general relativity. The observational constraints on their speed of propagation can provide strong constraints on generalized classes of broader gravitational frameworks. In this work, we reconsider the general class of Gauss-Bonnet theories in the context of teleparallel gravity, where the background geometry is expressed through torsion. We perform tensor perturbations on a flat FLRW background, and derive the gravitational wave propagation equation. We find that gravitational waves propagate at the speed of light in these classes of theories. We also derive the distance-duality relationship for radiation propagating in the gravitational wave and electromagnetic domains.

Discussion (0). Continue with ORCID to comment.

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