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Gravitational quasinormal modes of black holes in quadratic gravity

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Quadratic gravity adds massive spin-2 quasinormal modes to black hole ringdowns and leaves both black hole families stable.

desk verdict The genuinely new result is the axial QNM spectrum for the hairy branch in quadratic gravity, but the hairy-branch frequencies rest on an approximate background with no independent cross-check; the Schwarzschild massive modes and the qualitative stability result are on firmer ground. read the letter →

arxiv 2412.15037 v3 pith:Q2PAH4QN submitted 2024-12-19 gr-qc hep-th

Georgios Antoniou , Leonardo Gualtieri , Paolo Pani This is my paper · ORCID
classification gr-qchep-th MSC 83C5783C3583D05 PACS 04.30.-w04.70.-s04.50.Kd
keywords quasinormalmodesquadraticgravityEinstein-Weylblackholestabilitymassivespin-2ringdownhairyholesSchwarzschild
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

The paper studies how black holes ring in Einstein-Weyl quadratic gravity, where the Einstein-Hilbert action gains a Weyl-squared term that adds a massive spin-2 field. It computes the axial quasinormal modes of both known static black hole families: the Schwarzschild solution and a 'hairy' solution whose metric differs from Schwarzschild, with the hair here being purely gravitational. The central result is that the perturbation spectrum decomposes into a massless and a massive spin-2 sector: Schwarzschild keeps the usual general-relativity modes and gains new massive vector and tensor modes, while the hairy black hole has massless-led and massive-led modes that all differ from those of general relativity. No unstable axial modes were found in the radially stable parameter range, providing numerical evidence that both families are stable. The authors argue that the massive modes can be excited during ringdown, making them a possible observational signature of quadratic gravity.

What carries the argument

The argument is carried by the auxiliary tensor field fμν, which rewrites the quadratic-gravity action in a form with second-order field equations while keeping the massive spin-2 degree of freedom explicit. Perturbations are expanded in axial tensor spherical harmonics, giving metric functions h0, h1 and massive-field functions F0, F1, F2; after imposing gauge and constraint conditions the system reduces to a coupled set of Schrödinger-type equations for h1, F1, F2. Quasinormal-mode frequencies are found by imposing ingoing waves at the horizon and outgoing waves at infinity, using direct forward integration for both backgrounds and a continued-fraction method as an independent benchmark for the Schwarzschild case. The hairy background itself is represented by a semi-analytical continued-fraction parametrization, which is what makes the full computation feasible.

What would settle it

Compute the ℓ = 2 axial quasinormal-mode frequencies of the hairy branch on a fully numerical background, rather than the continued-fraction fit, at several values of p in [0.876, 1.143] and compare with the paper's results; disagreement at the few-percent level would indicate that the reported hairy spectrum is an artifact of the approximate background. Alternatively, any found mode with Im(ω) > 0 in this parameter range would refute the stability claim.

Watch

Extended reading notes

Core claim

In Einstein-Weyl quadratic gravity, axial gravitational perturbations of static, spherically symmetric black holes are governed by a massless spin-2 field, the usual graviton, coupled to a massive spin-2 field. On a Schwarzschild background the massive field decouples, so the spectrum consists of the standard general-relativity quasinormal modes plus two new series of massive modes, classified as vector and tensor. On the hairy background the fields remain coupled, producing massless-led and massive-led modes, and within the explored parameter range all of these frequencies differ from the corresponding general-relativity values. Searching for modes with positive imaginary frequency in the domain where both backgrounds are radially stable, 0.876 ≲ p ≲ 1.143, the authors found none, which they take as strong numerical evidence of axial stability for both families.

Load-bearing premise

The hairy-background results assume the semi-analytical continued-fraction metric accurately represents the true solution for every p between 0.876 and 1.143, and this assumption is not cross-checked with an independent method.

Editorial extensions

If this is right

  • Schwarzschild black holes in quadratic gravity, even with the same stationary metric as in general relativity, should emit massive spin-2 modes during ringdown, giving a direct observational handle on the theory.
  • For hairy black holes the massless-led and massive-led mode frequencies shift by up to a few percent relative to general relativity, in principle measurable by third-generation ground-based detectors if such light black holes exist.
  • Both black hole families appear stable under axial perturbations in the parameter range where they are radially stable, supporting their classical physical viability.
  • Observable stationary deviations require black hole masses below Mc ≈ 0.438/μ, so current observations favor small couplings and leave a narrow window for detecting hairy solutions.
  • The massive modes provide a qualitative signature that does not require the stationary solution to differ from general relativity, so ringdown can reveal quadratic gravity even when the spacetime is Schwarzschild.

Reading between the lines

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

  • The polar sector, not computed here, will likely show similar spectral deformation and would bring in the massive scalar degree of freedom from the βR² term, so a complete ringdown template needs both parities.
  • The heuristic estimate places the massive-mode excitation at order α in the metric, but a field-redefinition argument suggests it might appear only at order α²; computing the actual excitation by an inspiralling particle would settle which order is physical.
  • The paper notes the massive spin-2 equation also admits quasi-bound states with frequency below μ; those could source metric perturbations and produce a distinct low-frequency ringdown component that the current analysis does not cover.
  • If hairy black holes are stable, their larger horizon for a given mass may yield measurable differences in tidal deformability or accretion-disk spectra, extending tests beyond the ringdown signal.
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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 axial gravitational perturbations of static, spherically symmetric black holes in Einstein-Weyl quadratic gravity, using the auxiliary-field formulation of the theory. It derives the linearized perturbation equations for both the Schwarzschild background and the non-Schwarzschild ('hairy') background, reduces them to first-order-in-derivative systems for three perturbation functions, and computes quasinormal-mode frequencies with direct integration and, for Schwarzschild, continued fractions. For Schwarzschild, the massless axial modes are found to coincide with the Regge-Wheeler spectrum of general relativity, while additional vector and tensor modes associated with the massive spin-2 field appear; for the hairy background, the paper reports massless-led and massive-led mode families that are all different from GR, and it finds no unstable axial modes in either family. The paper closes with a phenomenological discussion of whether these extra modes could be observed in ringdown signals.

Significance. If the numerical results hold, this is a useful first computation of gravitational axial quasinormal modes for the hairy black-hole branch of quadratic gravity, and it provides concrete evidence that massive spin-2 degrees of freedom generate additional ringdown content even when the stationary solution is Schwarzschild. The Schwarzschild-sector computation is strengthened by the use of two independent methods (direct integration and continued fractions) with explicit convergence checks, and the QNM calculation has no free parameters fitted to the target frequencies. The main limitations are that the hairy-branch results rest on a single numerical method applied to an approximate background, and that the stability conclusion covers only the ω>μ QNM sector, which the paper itself notes is not the full massive-field spectrum.

major comments (3)
  1. [Sec. IV B and Appendix A] The quantitative claims for the hairy background—in particular the mode frequencies shown in Figs. 5–8 and the statement in Sec. V B that deviations of a few percent are measurable by third-generation detectors—rest entirely on the direct-integration method applied to the approximate continued-fraction background of Appendix A, with no independent verification for this branch. Since the coefficient matrices P^h and V^h in Eq. (61) contain A, B and their derivatives, any pointwise error in the fit propagates directly into the QNM frequencies, and the authors themselves note in Sec. IV B that numerical errors are expected in this computation. The few-percent deviations highlighted in the phenomenology are of the same order as that expected uncertainty, so the present calculation does not demonstrate that the hairy-branch deviations are physical rather than artifacts of the approximate background. The frequencies should be recomputed on the fully numerical background, or at least an error estimate should be obtained by varying the truncation order of the continued-fraction fit, before the quantitative phenomenological claims are made.
  2. [Secs. IV A, IV B and VI] The stability claim ('no unstable modes were found') covers only the quasinormal sector with ω > μ: the boundary conditions in Eqs. (50)–(52) and (65)–(67) use k = sqrt(ω^2 − μ^2), which restricts the search to ω > μ, while the paper itself notes in Sec. VI that the massive spin-2 equation also admits quasi-bound states with ω < μ. An instability, if present, could live in that sector, so the conclusion that the solutions are stable under axial perturbations is not established by the present calculation. The text should either extend the analysis to the quasi-bound spectrum or explicitly restrict the stability statement to the ω > μ QNM sector.
  3. [Sec. IV B, Eq. (61)] The matrix V^h in Eq. (61) is displayed with the (3,2) entry written as V^h_{23}, which duplicates the (2,3) entry; this should presumably be V^h_{32}. More importantly, the derivation leading to Eq. (61) is only sketched: the text states that the constraint equation and the (θφ) component of Eq. (29) are used to eliminate h0 and F0, but the resulting expressions for P^h and V^h are not given. Given that the entire hairy-branch computation depends on these matrices, at least the general structure of the reduction (or a brief verification) should be provided, or the explicit matrices should be included in an appendix.
minor comments (5)
  1. [Sec. IV A] The sentence 'By substituting (47)-(48) into (40)' appears to refer to the wrong equation: Eq. (40) is the two-function Schrödinger-type system for Q and Z, whereas Eqs. (47)-(49) are the horizon expansions for h1, F1 and F2; the substitution should be into the full three-function system (45).
  2. [Sec. IV B] The paragraph on boundary conditions states that the three forward integrations fix the horizon parameters '(h1^{(0)}, F1^{(0)}, F1^{(0)})'; the third entry should be F2^{(0)}, as given earlier in the same subsection.
  3. [Sec. V B] At the bifurcation point p ≈ 0.876 the hairy solution approaches the Schwarzschild solution, so the statement that 'all modes are different from those of general relativity' should be qualified, and a numerical check that the hairy modes reduce to the Schwarzschild modes as p → p_min would be a useful consistency test.
  4. [Fig. 1] Fig. 1 compares the numerical and semi-analytical metric functions for p = 1.1, but it does not show the pointwise fractional difference; plotting that difference would directly quantify the accuracy of the approximate background used in the QNM computation.
  5. [General] A table with representative QNM frequencies and estimated errors for both backgrounds and for several values of p would make the paper substantially more useful for quantitative comparisons than the figures alone.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the QNM spectra are obtained by forward numerical solution of derived perturbation equations, with self-citations serving as benchmarks rather than as fitted inputs.

full rationale

The paper's central calculation is a forward eigenvalue problem. The theory coupling α (equivalently µ) and the background parameter p are inputs, and the quasinormal-mode frequencies are computed by solving the derived perturbation systems, Eqs. (45) and (60), subject to horizon and asymptotic boundary conditions, with no free parameter fitted to the target frequencies. The Schwarzschild-sector result that the massless modes coincide with those of GR follows explicitly from the decoupling δfμν=0 in Eqs. (33)-(34), not from an imposed fit; the subsequent numerical agreement with continued-fraction results is an independent consistency check. The self-cited reference [59] supplies the known massive spin-2 equations and QNM results on Schwarzschild, which the paper uses as a baseline and benchmark, but those results are not an input that forces the new modes computed here. For the hairy branch, the background is a 4th-order continued-fraction fit from [22], presented in Appendix A and used in Sec. IV B; this introduces a possible numerical-accuracy limitation, which the authors explicitly acknowledge, but it is not a circular reduction because the QNM frequencies are not fed back into the background construction and the fit parameters do not by construction determine the frequencies. The stability conclusion (no mode with positive imaginary part) is likewise a direct numerical search result. Overall, no step in the derivation chain reduces to its own inputs, and the self-citations are not load-bearing in the sense required for a circularity finding.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central QNM computation introduces no fitted free parameters: mu is the theory coupling, p = r_h mu scans the solution space, and b1 is a shooting parameter fixing each hairy solution. The auxiliary field f_mu_nu is an equivalent second-order reformulation, not a new physical entity, and the massive spin-2 degree of freedom is already present in the theory. The key domain assumptions are the axial parity restriction, the semi-analytical background approximation, and the exclusion of quasi-bound states from the QNM spectrum.

assumptions (5)
  • standard math Auxiliary field reformulation: the action in Eq. (5) with auxiliary f_mu_nu is classically equivalent to the Einstein-Weyl action (1)-(4).
    Used in Sec. II A to obtain second-order field equations; equivalence is shown by substituting Eq. (8) back into Eq. (5).
  • domain assumption Static, asymptotically flat solutions of quadratic gravity have R=0, so the beta R^2 term does not affect backgrounds or linear perturbations.
    Invoked in Sec. II to justify setting beta=0 and restricting to Einstein-Weyl gravity; relies on results of Refs. [20,30].
  • domain assumption Hairy black hole solutions exist and are radially stable only for 0.876 ≲ p ≲ 1.143, and the continued-fraction parametrization of Appendix A accurately represents them.
    Sec. III C-D; the QNM calculation is performed on this semi-analytical background rather than the fully numerical one.
  • domain assumption Axial parity perturbations are a self-contained sector; conclusions about stability and QNM spectra are limited to this sector.
    Sec. IV restricts the analysis to axial parity; polar sector is deferred to future work (Sec. VI).
  • domain assumption QNM boundary conditions: ingoing at horizon, outgoing at infinity, with omega > mu for the massive modes; quasi-bound states with omega < mu are excluded.
    Eqs. (47)-(52) and related expansions; the paper notes quasi-bound states exist but are not part of the QNM spectrum studied.

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Pith. "Pith review of Gravitational quasinormal modes of black holes in quadratic gravity." pith.science (2026). https://pith.science/paper/Q2PAH4QN

@misc{pith2026241215037,
  author       = {Pith},
  title        = {Pith review of: Gravitational quasinormal modes of black holes in quadratic gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q2PAH4QN}},
  note         = {Machine review of arXiv:2412.15037}
}
read the original abstract

We study the gravitational perturbations of black holes in quadratic gravity, in which the Einstein-Hilbert term is supplemented by quadratic terms in the curvature tensor. In this class of theories, the Schwarzschild solution can coexist with modified black hole solutions, and both families are radially stable in a wide region of the parameter space. Here we study non-radial perturbations of both families of static, spherically symmetric black holes, computing the quasi-normal modes with axial parity and finding strong numerical evidence for the stability of these solutions under axial perturbations. The perturbation equations describe the propagation of a massless and a massive spin-two fields. We show that the Schwarzschild solution admits the same quasi-normal modes as in general relativity, together with new classes of modes corresponding to the massive spin-two degrees of freedom. The spectrum of the modified black hole solution has the same structure, but all modes are different from those of general relativity. We argue that both classes of modes can be excited in physical processes, suggesting that a characteristic signature of this theory is the presence of massive spin-two modes in the gravitational ringdown, even when the stationary solution is the same as in general relativity.

Figures

Figures reproduced from arXiv: 2412.15037 by the authors.

Figure 2
Figure 2. FIG. 2. Domain of existence of BHs in EW gravity. The [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Real (upper panel) and imaginary (lower panel) parts [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figures from the paper (4 more)
Figure 5
Figure 5. Figure 5: FIG. 5. Real (upper panel) and imaginary (lower panel) parts [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 9
Figure 9. Figure 9: FIG. 9. BH compactness as function of the BH mass, normal [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

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