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Can We Detect Deviations from Einstein's Gravity in Black Hole Ringdowns?

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

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Pith's one-line read The paper argues that causality constraints on graviton scattering force black hole ringdown quasinormal modes to obey Einstein's gravity alone, with any corrections suppressed below observability.

desk verdict A plausible, clearly argued no-go for observable higher-derivative effects in black hole ringdowns, but the key flat-space-to-curved step is asserted and Eq. (8) contains a numerical slip. read the letter →

arxiv 2411.12428 v1 pith:I7EDEWLK submitted 2024-11-19 gr-qc hep-phhep-th

A. Kehagias (NTUA) , A. Riotto (University of Geneva) This is my paper · ORCID
classification gr-qchep-phhep-th MSC 83C5783C3583D05 PACS 04.30.-w04.70.-s04.80.Cc
keywords blackholeringdownquasinormalmodeshigher-derivativegravitycausalityconstraintsgravitonscatteringspectroscopygeneralrelativitytestsgravitationalwaves
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 asks whether the ringing of a black hole after merger could reveal modifications of Einstein's gravity. Its answer is no for any weakly coupled theory: consistency conditions on how gravitons scatter in flat space force the higher-derivative corrections that would shift quasinormal-mode frequencies or quadratic-mode couplings to either vanish or be suppressed far below detector sensitivity. If the argument is right, the black hole spectroscopy program becomes a direct test of general relativity rather than a window into weakly coupled extensions. For a solar-mass black hole the suppression is of order $10^{-32}$, so no foreseeable measurement can reach it.

What carries the argument

The machinery is the causality bound on the trilinear graviton vertex in flat spacetime, derived through an eikonal high-energy scattering setup. In impact-parameter space the tree-level four-point graviton amplitude yields an eikonal phase whose associated time delay can acquire the opposite sign for one choice of helicities when the impact parameter is comparable to $\alpha_4^{1/4}$; causality then forces the coefficient to zero or requires an infinite tower of higher-spin states. The paper's application is to treat quasinormal modes, whose frequencies are governed by unstable circular null geodesics, as gravitons scattering in a flat region far from the black hole, so the same constraint applies to the vertex that sources both linear and quadratic QNMs.

What would settle it

An explicit calculation of quasinormal-mode frequencies and quadratic-mode couplings in a causal, weakly coupled higher-derivative theory that yields a relative shift larger than $\alpha_4\omega^4 \simeq 10^{-32}$ for a solar-mass black hole would falsify the claim, as would a ringdown observation whose mode frequencies or nonlinear couplings deviate from general relativity by more than that amount.

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Extended reading notes

Core claim

Stated in the paper's own terms, the discovery claim is that the spectrum and the nonlinearities of the quasinormal modes can only arise from standard Einstein's gravity. The paper reaches this by taking the known causality constraint on the on-shell three-graviton coupling in flat spacetime, where a higher-derivative term such as $\alpha_4 R^3$ allows a Shapiro time delay with the wrong sign for selected polarizations and small impact parameters, and transferring it to the ringdown, where two fundamental modes scatter as gravitons in a region far from the black hole. Consequently the higher-derivative vertex that would source quadratic QNMs must vanish or be accompanied by new higher-spin physics; in the latter case the relative deviation in QNM frequencies and nonlinearities is bounded by $\alpha_4 \omega^4 \sim 10^{-32}(\alpha_4/\mu\mathrm{m}^4)(M_\odot/M)^4$. The conclusion is that any observable ringdown behavior must come from the Einstein-Hilbert action.

Load-bearing premise

The flat-spacetime causality bound on graviton scattering must apply unchanged to the ringdown vibrations of a real black hole; the paper asserts this transfer rather than proving it.

Editorial extensions

If this is right

  • If the central claim is correct, black hole spectroscopy, which measures multiple ringdown tones, tests general relativity itself rather than weakly coupled extensions of it.
  • The recently extracted quadratic quasinormal modes, sourced by the trilinear graviton coupling, would carry no beyond-Einstein information, making them clean probes of strong-field general-relativity nonlinearities.
  • Any observed deviation from Einstein's quasinormal-mode spectrum would force new physics at macroscopic scales, such as new higher-spin particles with gravitational-strength forces, which tabletop gravity tests already disfavor.
  • Effective-field-theory corrections that shift quasinormal-mode frequencies only slightly remain possible but cannot be detected with foreseeable sensitivity for solar-mass black holes.

Reading between the lines

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

  • If the transfer of the causality bound is sound, the same suppression should apply to the full ringdown waveform, including overtones and mode-mixing coefficients, since the paper itself notes that overtones lie outside the effective-field-theory regime.
  • A natural extension is to make the spin dependence explicit for Kerr black holes, because the geometric-optics map between QNMs and null geodesics is less clean at high spin and the paper does not compute the analogue of its suppression estimate as a function of spin.
  • The argument implies that within weakly coupled gravity, a genuine observational deviation from Einstein's ringdown would point to a non-weakly-coupled or non-local completion, so future searches might focus on dispersion or birefringence signatures that evade the flat-space causality bound.
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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

4 major / 4 minor

Summary. This manuscript argues that higher-derivative corrections to the linear and quadratic quasinormal-mode (QNM) spectra of Schwarzschild and Kerr black holes are either exactly zero or suppressed by the factor alpha_4 omega^4 ~ 10^{-32} (alpha_4 / micron^4)(M_sun/M)^4, and that, within weakly coupled gravity, QNM spectroscopy can therefore only probe Einstein gravity. The argument is based on the causality bound of Camanho et al. for on-shell trilinear graviton couplings in flat space, which the authors assert applies to the operators controlling QNMs on curved backgrounds, and on an order-of-magnitude estimate in Eq. (8). The manuscript is written as a compact viewpoint/letter rather than a full technical derivation.

Significance. The question is timely and important: the field is actively constructing higher-derivative QNM templates, and a rigorous suppression argument would change how black-hole spectroscopy is interpreted. The paper is transparent about its reliance on Ref. [67], the estimate is easy to check, and the conclusions are stated in falsifiable form. However, the central step from flat-space scattering to curved-background QNM physics is asserted rather than derived, and the operator-basis loophole is not addressed. As it stands the paper is a plausible viewpoint, not a proof.

major comments (4)
  1. [The impact on the physics of QNMs] The sentence 'The causality argument in flat spacetime applies' is the load-bearing step, but no derivation is given for why the causality bound of Ref. [67], obtained for on-shell flat-space graviton scattering in the eikonal regime, constrains the operators that set QNM frequencies on a Schwarzschild or Kerr background. Linear QNM frequencies are determined by the quadratic action expanded around the curved background, whose higher-derivative part contains terms like alpha_4 (partial g)^4 (partial h)^2 in Eq. (7); the paper does not show that the flat-space bound on the three-point vertex implies a bound on such background-dependent quadratic operators. Please either supply this argument explicitly or state the suppression claim as conditional on an unproven assumption.
  2. [The impact on the physics of QNMs] The manuscript does not identify the independent operator basis for 4D cubic gravity and does not show that the flat-space causality constraint covers all combinations that contribute to QNMs. The flat-space on-shell three-point amplitude depends on one combination of the available couplings, while the quadratic action on a curved background can depend on a different combination. A theory whose causal combination is tuned to zero could still leave a large QNM-visible combination, giving observable ringdown deviations without flat-space causality violation. This operator-combination loophole is not mentioned and directly affects the central claim that all QNM corrections are suppressed.
  3. [Introduction] The paper lists Refs. [47-60], which report explicit QNM frequency corrections in higher-derivative gravity, but it never explains why those computations are invalid or how they are reconciled with the present conclusion. If the causality argument implies that all such corrections vanish or are suppressed by 10^{-32}, the authors should pinpoint which assumption in those works is incompatible with their argument; otherwise the reader cannot assess whether the claim is a theorem or a reinterpretation.
  4. [The impact on the physics of QNMs] The numerical suppression in Eq. (8) is dimensionally and logically unclear. alpha_4 omega^4 is dimensionless and equals alpha_4/(G^4 M^4) for omega ~ 1/(GM), but the preceding sentence bounds h by G^{-1/2}, which is not dimensionless, and the paper never states how the amplitude h enters the QNM frequency shift. Linear QNM frequencies are amplitude-independent, so any suppression must come from the background curvature scale rather than from h. Please clarify the chain leading to alpha_4 omega^4 and state explicitly where, if anywhere, the ringdown amplitude appears.
minor comments (4)
  1. [The causality argument] In Eq. (2), the text refers to coefficients c2 and c4, but only c4 appears in the displayed expression; additionally, as printed, 'c4 alpha_4^4/b^8' is not dimensionless if alpha_4 has dimension length^4. Please correct the formula and the surrounding notation.
  2. [The impact on the physics of QNMs] There are typographical errors in the manuscript: 'lenght' should be 'length', 'liner' in 'at the liner and nonlinear level' should be 'linear', and 'Vigo' should be 'Virgo'.
  3. [The impact on the physics of QNMs] The expansion 'g = g + h' uses the same symbol for the background and the full metric; use a barred background metric, e.g., \bar{g} + h, to avoid confusion.
  4. [Acknowledgments] The Acknowledgments section contains a garbled sentence: 'We thank E. Berti and G. Carullo for useful discussions and feedback on the V. Cardoso, G. Carullo, K. Fransen, L. Senatore and F. Serra for useful discussions and comments on the manuscript.' This should be rephrased.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the suppression conclusion is a conditional consequence of the external flat-space causality bound, and the self-citations are background only.

full rationale

The paper's central chain is: (i) a causality bound from Ref. [67] constrains trilinear graviton couplings in flat space; (ii) the paper assumes this bound applies to the couplings that source linear and quadratic QNMs; (iii) if the couplings must vanish, QNM spectra are Einstein's; if new physics intervenes, the correction is suppressed by the dimensional factor alpha4*omega^4. Step (ii) is an external assumption from Camanho et al., not a result derived inside the paper, and step (iii) is ordinary dimensional analysis once (i) and (ii) are granted. The conclusion 'spectrum and nonlinearities of QNMs can only arise from standard Einstein's gravity' is therefore a logical consequence of the stated premises, not an input disguised as a prediction. The paper's own self-citations (Refs. [32], [33], [41]) appear in the introduction and in background remarks about quadratic QNM amplitudes; they are not load-bearing for the causality argument or the suppression estimate. The weakest point, the transfer of the flat-space causality constraint to curved-background QNM operators, is a substantive physics assumption about which one may disagree, but it is not an instance of circular reasoning. No fitted parameter is relabeled as a prediction, and no uniqueness theorem from the authors' own prior work is invoked to force the conclusion. The paper is transparent that everything hinges on the causality argument: 'Our conclusions hinge on the well-known causality violation argument concerning the trilinear graviton coupling.' That is a conditional derivation, not a circular one.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The paper rests on external causality bounds, the geodesic-QNM correspondence, and the transferability of flat-space scattering constraints to curved QNM physics. The only free parameter is the EFT coefficient α4, which is constrained rather than fitted. No new entities are introduced.

free parameters (1)
  • α4 (coefficient of the R^3 term) = not fitted; bounded by tabletop tests (α4^{1/4} ≲ 10 μm)
    The central suppression estimate in Eq (8) scales with α4. The paper does not fit it; it uses experimental bounds and the causality argument to constrain it.
assumptions (5)
  • domain assumption Causality forbids large on-shell three-graviton couplings from higher-derivative operators in flat spacetime.
    This is the external result from Ref [67] on which the entire argument hinges. It is summarized in the section 'The causality argument' and not rederived.
  • domain assumption The quasinormal mode spectrum is determined by unstable circular null geodesics.
    Used to identify QNM frequencies via Eq (6) and to argue that QNM energies are set by the photon ring. Cited to Ref [69].
  • domain assumption Flat-space causality constraints apply to quasinormal modes on curved Schwarzschild and Kerr backgrounds.
    This transferability is asserted in the section 'The impact on the physics of QNMs' but not rigorously derived. It is the main fragile step and is challenged by Ref [72].
  • domain assumption Tabletop tests at tens of microns constrain α4 and rule out new forces with gravitational strength at larger scales.
    Used in Eq (8) to convert experimental bounds into a suppression estimate for solar-mass black hole ringdowns.
  • domain assumption The theory is weakly coupled and treated at tree level.
    Stated at the start of the causality argument. The conclusions do not apply to strongly coupled theories or loop-level effects.

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Cite this review

Pith. "Pith review of Can We Detect Deviations from Einstein's Gravity in Black Hole Ringdowns?." pith.science (2026). https://pith.science/paper/I7EDEWLK

@misc{pith2026241112428,
  author       = {Pith},
  title        = {Pith review of: Can We Detect Deviations from Einstein's Gravity in Black Hole Ringdowns?},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I7EDEWLK}},
  note         = {Machine review of arXiv:2411.12428}
}
read the original abstract

The quasinormal mode spectrum of gravitational waves emitted during the black hole ringdown relaxation phase, following the merger of a black hole binary, is a crucial target of gravitational wave astronomy. By considering causality constraints on the on-shell graviton three-point couplings within a weakly coupled gravity theory, we present arguments indicating that the contributions to the physics of linear and quadratic quasinormal modes from higher derivative gravity theories are either negligible or vastly suppressed for Schwarzschild and Kerr black holes. Their spectrum and interactions are dictated solely by Einstein's gravity.

Figures

Figures reproduced from arXiv: 2411.12428 by the authors.

Figure 1
Figure 1. FIG. 1. Tree-level four-point graviton amplitude. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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Forward citations

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.