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Gravitational Waves Emission in Quadratic Gravity: longitudinal modes, angular momentum emission, and positivity of the radiated power

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper argues that in quadratic gravity the radiated power and angular momentum become positive-definite when the massive spin-2 field is restricted to its transverse-traceless modes, whereas including its longitudinal modes makes the…

desk verdict The real result is the negative longitudinal-mode power; the advertised positivity holds only for an imposed TT truncation, so the abstract oversells the theory-level conclusion. read the letter →

arxiv 2411.10098 v2 pith:AX5YONGC submitted 2024-11-15 gr-qc

Matheus F. S. Alves , R.R. Cuzinatto , C.A.M. de Melo , L.G. Medeiros , P.J. Pompeia This is my paper · ORCID
classification gr-qc
keywords gravitationalwavesquadraticgravityOstrogradskyinstabilitymassivespin-2modesangularmomentumemissionquadrupoleapproximationtransverse-tracelessfreeprecession
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

Quadratic gravity adds $R^2$ and Weyl-squared terms to Einstein's action, producing a massless spin-2 wave, a massive spin-2 wave, and a scalar wave. The paper derives the wave equations and the conserved energy and angular momentum currents for these waves, then computes how much energy and angular momentum a source radiates. It finds that the longitudinal modes of the massive spin-2 field would make the radiated power negative, a signature of the Ostrogradsky instability, so the full theory is pathological in the quadrupole regime. Imposing that only the transverse-traceless part of the massive field radiates, $\Psi_{ij}\to\Lambda_{ij,kl}\Psi_{kl}$, removes the negative terms and makes both $dE/dt$ and $dJ/dt$ positive-definite. This matters because quadratic gravity is among the few renormalizable extensions of general relativity, and the same projection is then used to show that a freely precessing ellipsoid loses precession frequency and wobble angle more slowly than in general relativity.

What carries the argument

The load-bearing device is the transverse-traceless projection operator $\Lambda_{ij,kl}$, applied to the massive spin-2 perturbation $\Psi_{ij}$, together with the decomposition of the metric perturbation into a massless tensor $\tilde{h}_{\mu\nu}$, a massive tensor $\Psi_{\mu\nu}$, and a massive scalar $\Phi$. The argument runs through second-order Noether currents: because the quadratic Lagrangian depends on second derivatives of the metric, the usual first-order current formulas are replaced by a generalized conserved current from which the gravitational-wave energy-momentum tensor and the angular momentum current are built. In the quadrupole approximation the modes are organized into $+$, $\times$, $B$, $C$, and $D$ polarizations; $B$, $C$, and $D$ are the longitudinal modes whose negative contributions to the radiated power are eliminated by the TT prescription.

What would settle it

Take the linearized field equations (84)-(86) with a generic matter source and ask whether the longitudinal components of $\Psi_{ij}$ can be set to zero consistently; if the sourced solution necessarily contains non-TT components, the projection $\Psi_{ij}\to\Lambda_{ij,kl}\Psi_{kl}$ is not dynamically preserved and Eqs. (150)-(151) do not describe the theory's actual radiation. An observational check is to search for the $\Psi_B$, $\Psi_C$, and $\Psi_D$ longitudinal polarization pattern in a precessing-ellipsoid waveform, since the TT-restricted theory predicts their complete absence.

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Extended reading notes

Core claim

The central discovery is that the apparent Ostrogradsky instability of quadratic gravity is confined, at quadrupole order, to the three longitudinal polarizations of the massive spin-2 field. When those polarizations are kept, the radiated power in Eqs. (123)-(131) acquires negative contributions proportional to $(m_\Psi c/\omega)^2$ and $(m_\Psi c/\omega)^4$; for a circular binary this makes $P<0$ in the interval $0<m_\Psi c/(2\omega_s)<0.87$. If one imposes the transverse-traceless condition $\Psi_{ij}\to\Lambda_{ij,kl}\Psi_{kl}$, all such terms vanish and the loss equations (150) and (151) become positive-definite combinations of the massless TT modes, the massive TT modes, and the scalar mode. The paper also derives the angular momentum flux, including orbital and spin contributions, and shows that under the TT projection the precessing ellipsoid satisfies $dE/dt=\Omega\,dJ/dt$, with the massive fields softening the secular decay of both precession frequency and wobble angle.

Load-bearing premise

The massive spin-2 perturbation is assumed to contain only transverse-traceless modes, an assumption imposed by hand to remove the negative-energy longitudinal waves; if that projection is not a genuine dynamically closed sector of quadratic gravity, the positive-energy and positive-angular-momentum results do not apply to the full theory.

Editorial extensions

If this is right

  • In the TT-restricted sector, a circular binary avoids the negative-power window: the orbital frequency asymptotes to $\omega_s^{\rm max}\simeq 1.145\,\omega_0$ as the total radiated power approaches zero, instead of diverging in finite time as in general relativity.
  • Including the longitudinal massive modes makes the binary's radiated power negative for $0<m_\Psi c/(2\omega_s)<0.87$, which the paper identifies as a physical inconsistency of the full quadrupole theory.
  • For a freely precessing ellipsoid, the massive fields slow the decrease of both precession frequency and wobble angle relative to general relativity, and the effect grows as $m_\Psi$ decreases.
  • Under the TT projection the energy and angular momentum losses satisfy $dE/dt=\Omega\,dJ/dt$, so the precession frequency and the rotational angular momentum of the source decay in step.
  • A vanishing total power does not imply the absence of waves: at the saturation frequency an observer on the rotation axis still detects oscillatory $h_+ + \Psi_+$ and $h_\times + \Psi_\times$ signals with nonzero amplitude because $q_\Psi^{\rm max}\ne 1$.

Reading between the lines

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

  • Editorial inference: nothing in the action singles out the TT subspace, so if the linearized constraint algebra shows that longitudinal components of $\Psi_{ij}$ are inevitably sourced by matter, the positivity results describe a truncated model rather than quadratic gravity itself.
  • Editorial inference: the TT-restricted theory predicts a distinctive observational signature, namely massive spin-2 radiation appearing only as $+$ and $\times$ distortion of the general-relativity waveform, with no $B$, $C$, or $D$ longitudinal components, which a detector network could in principle test.
  • Editorial inference: the paper's finding that $P=0$ can coexist with nonzero wave amplitude breaks the standard balance between observed strain and orbital decay, offering a testable anomaly in the inspiral signal of a binary.
  • Editorial inference: the same Noether-current construction and projection strategy could be applied to other higher-derivative gravity theories, potentially providing a general way to quarantine ghost degrees of freedom in radiative sectors.
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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 / 4 minor

Summary. This paper studies gravitational-wave emission in quadratic gravity with action (1), linearized around a Minkowski background. The metric perturbation is decomposed into a massless spin-2 field h_μν, a massive spin-2 field Ψ_μν, and a massive scalar Φ (Eq. 3). The authors derive the wave equations (13)-(15), construct Noether currents for a second-order Lagrangian, and obtain the radiated energy and angular momentum fluxes (Eqs. 75 and 83). In the quadrupole approximation they compute all polarization modes for a generic propagation direction and find that the total radiated power (Eq. 131) contains negative contributions from the longitudinal massive modes Ψ_B, Ψ_C, Ψ_D; for a circular binary this yields P < 0 in a finite parameter range (Fig. 1). To cure this, Section III.D imposes the transverse-traceless projection Ψ_ij → Λ_ij,kl Ψ_kl (Eq. 149), under which Eqs. (150)-(151) give positive-definite dE/dt and dJ/dt. The paper then applies this restricted model to a freely precessing rotating ellipsoid and shows that the massive modes soften the GR decrease of precession frequency and wobble angle.

Significance. If the negative-power result for the full linearized theory is correct, it is a concrete and useful demonstration that the Ostrogradsky instability manifests in quadrupole gravitational-wave emission through the longitudinal massive modes; this genuinely extends Ref. [45]. The Noether-current formalism for second-order Lagrangians is developed carefully, and the paper is transparent that the TT restriction in Eq. (149) is imposed rather than derived. However, because that projection is inconsistent with the sourced field equations, as detailed below, the advertised positivity results apply to a truncated model rather than to the quadratic-gravity theory defined by Eq. (1). The abstract's claim that 'the theory avoids the issues generated by the Ostrogradsky instabilities' therefore overstates the scope of what is established; the paper would be much stronger if it explicitly framed Eqs. (150)-(151) as defining a restricted, constrained sector and discussed whether that sector is dynamically consistent.

major comments (3)
  1. [Section III.D, Eq. (149)] The TT projection Ψ_ij → Λ_ij,kl Ψ_kl is imposed by hand, not derived from the action or the field equations. The sourced field equation (85) generates the longitudinal modes: for a planar binary in the xy-plane, Eq. (112) gives Ψ_B proportional to 2 M̈_11 sin θ sin 2φ + 2 M̈_12 sin θ cos 2φ, which is generically nonzero, and similarly for Ψ_C and Ψ_D (Eqs. 113-114). Thus the full linearized theory does not satisfy Eq. (149). Consequently, Eqs. (150) and (151) are statements about a different, truncated model, not about the theory defined by Eq. (1). The authors should either derive the TT sector from a constrained action (e.g., adding a Lagrange multiplier term that enforces the transversality and tracelessness conditions) or explicitly state in the abstract and conclusions that the positivity results characterize only this restricted sector.
  2. [Eq. (85) and trace consistency] There is a trace-level inconsistency in the TT restriction. Taking the trace of Eq. (85) gives (□ − m_Ψ^2) Ψ = 2κ T. If Ψ is transverse-traceless, then Ψ = 0, and this equation forces T = 0 in the source region. Nonrelativistic sources have T ≠ 0, so the condition Ψ_ij = Λ_ij,kl Ψ_kl cannot hold in the presence of such sources. This is not a gauge choice for a massive field. The manuscript should address this obstruction explicitly; otherwise the TT sector cannot be regarded as a consistent subsector of quadratic gravity even at the linearized level.
  3. [Section III.D, Eqs. (150)-(151) and Abstract] The positive-definiteness of dE/dt and dJ/dt after imposing Eq. (149) follows by construction: the negative longitudinal-mode terms in Eq. (123) and Eq. (131) are deleted by the projection. As a resolution of the Ostrogradsky instability, this is circular. The paper itself demonstrates in Section III.B that the generic power is not positive (Eq. 131) and in Section III.C that P < 0 occurs for a circular binary. The claim in the Abstract that 'the theory avoids the issues generated by the Ostrogradsky instabilities' should therefore be replaced by a statement that a TT-restricted sector of the theory has positive energy and angular momentum fluxes, while the full theory does not.
minor comments (4)
  1. [Section III.A, Eqs. (95)-(96)] Equation (96) is labeled h+ but is the expression for h× (it contains M̈′_12); the label should be corrected.
  2. [Section II, Eqs. (8)-(9)] The factor 1/(2κ) appears both in the definition of ¯S in Eq. (8) and inside the expression for L in Eq. (9); this double counting of the κ normalization should be checked and fixed, or at least explained.
  3. [Section III.A, paragraph after Eq. (104)] The sentence describing the spin-0 contribution as 'produced by the single component n_i n_j M̈_ij = −n_i n_j M̈^ij' is confusing because M_ij and M^ij differ by a sign by Eq. (93); clarifying the notation would help the reader track the quadrupole formulas.
  4. [Section IV] The mode expressions in Eqs. (163)-(169) mix damping and oscillatory regimes through step-like conditions; it would be helpful to state explicitly that the Heaviside factors from Eqs. (75) and (83) are understood to apply in the integrated formulas (170)-(181), so the piecewise structure is consistent with the general flux formulas.

Circularity Check

1 steps flagged · score 6.0 of 10

The advertised positive-energy/momentum result is obtained by imposing the TT projection Eq. (149), which by construction deletes the only negative modes; the full sourced field equations generate those modes, so the central claim characterizes a truncated sector rather than the full quadratic-gravity theory.

  1. self definitional [Section III.D, Eq. (149) leading to Eqs. (150)-(151); Abstract]
    "Another way to deal with this problem is to impose that the transverse-traceless massive spin-2 modes are the only modes present in the theory. In other words, there should be Ψij → Λij,klΨkl = ΨT T ij . (149) Under this assumption, the longitudinal modes ΨB, ΨC, and ΨD are disregarded ... all the terms in the second lines of Eqs. (123), (124), and (131) vanish, thus leading to a positive-definite power."

    The advertised positivity is not extracted from the action; it is put in by projection (149), which deletes exactly the modes that carry negative power. In Eqs. (123)/(131) the only negative terms are the Ψ_B, Ψ_C, Ψ_D pieces; setting these modes to zero by fiat is equivalent to assuming the conclusion. The projection is not a gauge condition: the sourced massive equation (□−m_Ψ²)Ψ_μν=2κT_μν (Eq. 85) generates nonzero Ψ_B, Ψ_C, Ψ_D even for binary sources (Eq. 112), and its trace shows Ψ=0 would force T=0. Hence Eqs. (150)-(151) prove positivity for a truncated TT-only model, not for the full QG theory of Eq. (1).

full rationale

The paper is transparent: Eq. (149) is an imposed restriction, and the text explicitly says the longitudinal modes are disregarded in order to obtain positive-definite power. That makes the core positivity claim a conditional statement about a truncated TT-only sector. It is nevertheless circular in the specific sense that the advertised 'theory avoids the Ostrogradsky issues' conclusion is built into the assumption used to reach it: the negative contributions in Eq. (123) and Eq. (131) are precisely the longitudinal modes that Eq. (149) removes. The paper does contain independent, non-circular content: the derivation that the full quadrupole theory, including longitudinal modes, has negative radiated power for a binary system (Eq. 134, Fig. 1) and the angular-momentum calculation are genuine derivations from the field equations, not imported fits. There is no numerical fitting, and the citations to Ref. [45] are not load-bearing for the circularity issue because the key currents and mode decomposition are re-derived in this paper. On the other hand, the Abstract's phrasing that 'the theory avoids the issues generated by the Ostrogradsky instabilities and achieves positive energy and angular momentum emissions' overstates the scope: it is true only after the hand-imposed TT projection, which conflicts with the sourced field equations for generic nonrelativistic sources. The central advertised prediction therefore reduces, by construction, to its own input, while the surrounding analysis retains independent value. Score 6 reflects partial circularity: the positivity result is forced by the projection, but a substantial part of the paper's derivation is self-contained and independent of that projection.

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

The central positive-emission result depends on two unproven modeling choices: the gauge-fixed action before variation and, above all, the TT projection of the massive spin-2 field. The paper's numerical constants (m_Psi, m_Phi) are inputs from the action, not fitted, so no free numerical parameters are introduced. No new particles or forces are postulated.

assumptions (4)
  • domain assumption The linearized action can be gauge-fixed by imposing partial_mu h_tilde^{mu nu} = partial_mu Psi^{mu nu} = 0, h_tilde = Psi = 0 before varying (Eq. 7 and following), and the neglected terms do not affect the conserved currents.
    Invoked in Section II to simplify the Lagrangian (9) and to derive the field equations (13)-(15); standard in Ref. [45] but subtle for the massive sector.
  • domain assumption Quadrupole approximation and non-relativistic source motion; scalar dipole emission is neglected because total momentum is conserved.
    Used in Section III to write solutions (87)-(89) and to set T = -T00 so that M_ij = -M_ij (Eq. 93).
  • ad hoc to paper The massive spin-2 field is restricted to transverse-traceless modes: Psi_ij -> Lambda_ij,kl Psi_kl (Eq. 149).
    This assumption is the entire basis for the positive-energy result; it is not a consequence of the field equations and removes the longitudinal modes that cause negative power.
  • domain assumption The angular momentum flux is computed by multiplying each mode's angular momentum density by its group velocity (Eqs. 82-83).
    Heuristic construction stated in Section II A 3; standard in GR but not derived from the Noether current in this paper.

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Pith. "Pith review of Gravitational Waves Emission in Quadratic Gravity: longitudinal modes, angular momentum emission, and positivity of the radiated power." pith.science (2026). https://pith.science/paper/AX5YONGC

@misc{pith2026241110098,
  author       = {Pith},
  title        = {Pith review of: Gravitational Waves Emission in Quadratic Gravity: longitudinal modes, angular momentum emission, and positivity of the radiated power},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AX5YONGC}},
  note         = {Machine review of arXiv:2411.10098}
}
read the original abstract

In this paper, the emission of gravitational waves in quadratic gravity theory is examined. The wave equations for massless and massive perturbations are derived, followed by the calculation of the energy and angular momentum radiated. In the quadrupole approximation, and taking into account only the transverse-traceless modes, it is shown that the theory avoids the issues generated by the Ostrogradsky instabilities and achieves positive energy and angular momentum emissions. As an example, a rotating ellipsoid with free precession is analyzed, and the effects of the massive perturbations on its rotation are highlighted.

Figures

Figures reproduced from arXiv: 2411.10098 by the authors.

Figure 1
Figure 1. FIG. 1. Binary system radiated power [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Orbital angular frequency as a function of time for [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Dimensionless precession velocity as a function of [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Wobble angle as a function of time (in [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

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