Quasinormal modes of nonlocal gravity black holes

Rocco D'Agostino; Vittorio De Falco

REVIEW 3 major objections 5 minor 85 references

Quasinormal modes of nonlocal gravity black holes

T0 review · 3 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that the combined relative deviations of the fundamental quasinormal modes of a nonlocal-gravity black hole from Schwarzschild can reach about 12%, and that a 1–10% detector sensitivity would constrain the model…

desk verdict A new but unverified QNM spectrum for a nonlocal-gravity black hole metric; the headline deviations likely mix physical effects with a WKB-order baseline mismatch. read the letter →

arxiv 2507.01698 v2 pith:Z76DQWOD submitted 2025-07-02 gr-qc astro-ph.HEhep-th

Rocco D'Agostino , Vittorio De Falco This is my paper

classification gr-qcastro-ph.HEhep-th MSC 83C5783C3583C2583D05 PACS 04.70.-s04.30.-w04.50.Kd

keywords quasinormalmodesnonlocalgravityDeser-WoodardtheoryblackholeperturbationsRegge-WheelerequationWKBmethodPadéapproximantsgravitationalwaveringdown

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 studies the ringdown frequencies of a new class of static, spherically symmetric black hole solutions in the revised Deser-Woodard nonlocal gravity theory, where the metric is a linear perturbation of Schwarzschild with lapse $A(r)=1-2/r-\alpha/r^k$. It derives the effective potentials for scalar, electromagnetic, and axial gravitational perturbations and computes the complex quasinormal-mode frequencies using third-order WKB theory with Padé approximants. The paper's central result is that the combined relative deviation of the fundamental modes from their Schwarzschild values can reach about 12%, for $\alpha=0.1$ and $k=1$, and that the deviations fall toward zero as $k$ approaches 5. From this it concludes that a future gravitational-wave detector able to measure ringdown deviations in the 1–10% range would force the model parameters to satisfy $\alpha\gtrsim 0.015$ and $k\lesssim 2.84$. A sympathetic reader should care because, if this is right, the ringdown spectrum becomes a direct observational probe of a purely metric-based modification of general relativity.

What carries the argument

The machinery is the standard quasinormal-mode reduction of a spherically symmetric black hole: decompose the perturbation into scalar, vector, and tensor harmonics, write the radial part as a time-independent Schrödinger-like equation $\mathrm{d}^2\Psi/\mathrm{d}r_*^2 + [\omega^2 - V(r)]\Psi = 0$ in the tortoise coordinate $r_*$, and impose ingoing waves at the horizon and outgoing waves at infinity. The effective potentials $V^{(0)}$, $V^{(1)}$, and $V^{(2)}$ for the three spin channels are computed from the modified metric (7), and the complex frequencies are obtained from the third-order WKB condition using derivatives of the potential at its peak, with $(2,2)$ Padé approximants around $r=3$ used to stabilize the higher derivatives. The parity argument that the auxiliary scalar fields are even under parity, so that their linear perturbations vanish in the axial sector, is what licenses the use of the standard Regge-Wheeler equation with the nonlocal background metric. The combined deviation $\Gamma$, the average over the fundamental modes of each spin of the Euclidean sum of relative real and imaginary deviations from Schwarzschild, is the quantity that converts the frequency shifts into parameter constraints.

What would settle it

Compute the axial gravitational quasinormal modes of the metric (7) by solving the linearized nonlocal field equations while keeping all first-order perturbations of the auxiliary scalar fields, without imposing the parity-decoupling argument, and compare the lowest multipole frequencies with the Regge-Wheeler results of this paper. A shift larger than the WKB/Padé truncation error would falsify the axial-sector claim, and a high-precision ringdown observation that resolves sub-percent frequency deviations at a parameter point predicted to deviate by more than 10% would test the quoted constraints.

Watch

Extended reading notes

Core claim

The paper's central claim is that nonlocal gravity leaves a small but measurable imprint on black hole ringdown. For each perturbation channel the quasinormal frequencies differ from the Schwarzschild values by an amount controlled by the small parameter $\alpha$ and the power-law index $k$; at $\alpha=0.1$, $k=1$ the largest tabulated deviations are 13.3% for scalar modes, 11.5% for electromagnetic modes, and 11.8% for gravitational modes, while the combined fundamental-mode measure $\Gamma$ reaches about 11.9%. A second claim, argued from parity, is that the axial gravitational sector is untouched by the nonlocal scalar fields at linear order: the auxiliary fields $X$, $Y$, $U$, $V$, and $W$ are even under parity, so axial perturbations of the nonlocal field equations reduce to the standard Regge-Wheeler equation on the modified background metric. The paper therefore presents the reported gravitational deviations as coming entirely from the deformed background, not from new axial dynamics, and all nonlocal effects are confined to the polar sector. The observable statement at the end is that a detector sensitive to 1–10% ringdown deviations would constrain $\alpha\gtrsim 0.015$ and $k\lesssim 2.84$, leaving the region $k\gtrsim 5$ effectively indistinguishable from Schwarzschild.

Load-bearing premise

The load-bearing premise is that the nonlocal scalar fields $X$, $Y$, $U$, $V$, and $W$ are even under parity and therefore contribute nothing to the axial gravitational perturbations at linear order, so the axial ringdown obeys the standard Regge-Wheeler equation in the modified background; if that decoupling fails, the quoted gravitational frequencies and the 12% combined deviation would change.

Editorial extensions

If this is right

Requiring $1\%\le \Gamma\le 10\%$ selects $\alpha\gtrsim 0.015$ and $k\lesssim 2.84$ within the consistency range $\alpha\le 0.1$, and tightening detector precision shrinks this allowed region.
For fixed $\alpha$ the deviations drop rapidly with $k$ and are effectively zero by $k\simeq 5$, so a detected large deviation would favor small $k$.
The three perturbation channels deviate by similar amounts, so the combined measure $\Gamma$ yields tighter constraints than any single channel and none of the channels is observationally preferred.
The axial gravitational QNM spectrum coincides with the GR one at linear order, so the nonlocal imprint must be sought in polar modes or in the modified background potential rather than in axial ringdown.
Higher overtones can be included in a generalized $\Gamma$; since some of them show larger percentage deviations, their inclusion could strengthen the bounds.

Reading between the lines

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

Because the axial sector is argued to be GR-like, the first decisive check of this model is the polar gravitational spectrum, which the paper leaves for future work and which may show larger or parity-breaking deviations than the quoted axial number.
A direct numerical integration of the full linearized nonlocal field equations without the parity decoupling assumption would test whether the 12% maximum and the $k\le 2.84$ bound survive; the paper does not perform that integration.
The appendix itself cautions that the precise accuracy of the Padé regularization is hard to assess, so an independent computation of the same potentials by higher-order WKB or time-domain methods would be the natural check on the reported frequencies.
The bounds $\alpha\gtrsim 0.015$, $k\lesssim 2.84$ are tied to the chosen 1–10% detector window; a detector able to resolve sub-percent deviations would push these boundaries and could exclude the $k=1$ corner entirely.

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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

Quick read: this is a one-calculation paper. The authors take the static, spherically symmetric black hole solution they built earlier in Deser–Woodard nonlocal gravity (metric 7), derive the scalar, electromagnetic and axial-gravitational effective potentials, push them through standard third-order WKB with (2,2) Padé smoothing, and tabulate quasinormal frequencies with percentage deviations from Schwarzschild. No one has computed QNMs for this specific metric before, so the table is new. The analytical part is the strongest piece: full potential expressions in Appendix A, clean α→0 reduction to Schwarzschild, finite limits at r=3 computed rather than assumed. That part is reproducible and I see no problem in it.

The soft spot is the baseline comparison, and it is load-bearing. The nonlocal frequencies come out of a third-order WKB pipeline; the Schwarzschild reference values are lifted from Konoplya (2003), produced with a higher-order WKB implementation. The relative deviations in Eqs. (51)–(52), and everything built on them—the Γ bounds α ≳ 0.015, k ≲ 2.84—therefore mix physical differences with a systematic WKB-order mismatch. For the low-ℓ fundamental modes this systematic error can be the same size as the reported 1–5% deviations, which is exactly the regime where the headline claim lives. There is no convergence study, no Leaver or time-domain cross-check, and no released code. Appendix B is refreshingly candid: 'the precise degree of approximation is difficult to assess.' So the numbers are plausible, not verified.

Two smaller notes. The parity decoupling argument for the axial sector (Sec. III.C) is standard and, on the page, coherent—the auxiliary scalar fields are parity-even, so their first-order perturbations should not source odd-parity fluctuations. I don't think that is the weak link; the baseline mismatch is. And the Γ bounds are explicitly labeled indicative, which is fair.

Who should read this: people doing modified-gravity ringdown phenomenology or benchmarking WKB against more exact methods. It deserves a serious referee: the computation is new, the fix is straightforward (recompute the Schwarzschild baseline in the same pipeline, add a higher-order WKB or Leaver check, quantify the Padé error, release code), and the current conclusions should be treated as provisional until then. I would not cite the numbers in my own work yet.

Referee Report

3 major / 5 minor

Summary. This paper studies quasinormal modes of static, spherically symmetric black holes in the revised Deser-Woodard nonlocal gravity model. Building on the authors' prior first-order-in-alpha solutions, it derives effective potentials for scalar, electromagnetic, and axial gravitational perturbations (Appendix A), argues on parity grounds that axial gravitational perturbations decouple from the auxiliary scalars and obey the standard Regge-Wheeler equation, and computes fundamental and overtone QNM frequencies with third-order WKB supplemented by (2,2) Pade approximants. The central quantitative claims are that combined relative deviations from Schwarzschild can reach about 11.9% (for alpha = 0.1, k = 1) and that requiring 1% <= Gamma <= 10% gives alpha >= 0.015 and k <= 2.84.

Significance. The analytical potential formulas in Appendix A and the manifest reduction to Schwarzschild as alpha -> 0 are useful contributions, and the parity-based decoupling argument for the axial sector is plausible. If the numerical comparison is made robust, the result provides a falsifiable, testable deviation from general relativity in the ringdown band. However, the paper does not ship code or independent numerical verification, and the Schwarzschild baseline is not computed with the same pipeline as the nonlocal frequencies; the reported deviations, and especially the Gamma bounds, rest on this comparison.

major comments (3)

[Sec. V, Table I and Eqs. (51)-(52)] The relative deviations that drive the paper's conclusions are computed against Schwarzschild frequencies taken from Table II of Konoplya [67], which were obtained with a higher-order WKB implementation, while the nonlocal frequencies are obtained with the third-order WKB method plus (2,2) Pade approximants. This is an inconsistent baseline: delta_omega^(R,I) mixes the physical alpha,k dependence with the systematic difference between the third-order and higher-order WKB approximations. The effect is largest precisely for the low-ell modes that enter Gamma in Eq. (54), namely the scalar l=0 and l=1 modes. Please recompute the Schwarzschild reference with the identical pipeline, including the same Pade treatment, and ideally also verify with an independent method such as Leaver's continued fraction or time-domain integration, before the deviations and parameter bounds can be considered robust.
[Sec. IV.B and Appendix B] The (2,2) Pade approximant is used to replace the effective potential and its derivatives V1 through V6 near r=3, and these derivatives enter the WKB corrections Lambda_j at the order used. Appendix B states that 'the precise degree of approximation is difficult to assess,' and no convergence test is provided. Because the headline deviations are at the 1-5% level, the error introduced by the Pade truncation is not obviously negligible. Please include a convergence study, for example comparing (2,2) with (3,3) and (4,4) approximants or comparing against direct numerical derivatives where stable, and quantify the resulting spread in omega for representative modes.
[Sec. V, general] No independent numerical check is given for any nonlocal QNM frequency. The manuscript reports only the third-order WKB plus Pade values, and no code is released. Given that the claimed signal is a few percent deviation from Schwarzschild, a confirmation of at least a few representative modes by a different method, such as time-domain evolution, the continued-fraction method, or higher-order WKB without the Pade regularization, is needed. Without such a check, it is not possible to distinguish physical deviations from numerical artifact.

minor comments (5)

[Sec. I] The phrase 'dumping timescales' should read 'damping timescales.'
[Sec. VI] The word 'constrains' in 'yields the constrains' should be 'constraints.'
[Sec. V and Fig. 3] The coordinate order is inconsistent: the text gives the star as (alpha,k) = (0.07,1.70), but the vertices A-D appear to be (k, alpha) pairs such as A=(1.000,0.015). Please label the axes and unify the notation.
[References] Reference [83] is incomplete ('36, 143001 (2019)') and appears to duplicate Ref. [9]; please correct it.
[Sec. II, Eq. (12)] For alpha=0.1 and k=5, the first-order horizon shift alpha 2^(k-1) = 1.6 is not small, so the linearized solution may be outside its regime of validity for some of the tabulated parameter values. A brief comment on this point would be helpful, even though the inferred constraints favor smaller alpha 2^(k-1).

Circularity Check

0 steps flagged · score 2.0 of 10

No substantive circularity: the quasinormal-mode deviations are genuine outputs of perturbation equations on an input metric; the Schwarzschild baseline is external, and the main caveats (WKB-order mismatch, Padé accuracy, parity decoupling) are correctness risks, not circular reductions.

full rationale

The derivation chain is linear and non-circular. The nonlocal black hole metric (Eqs. 7a, 7b) is an input taken from the authors' prior derivation (Ref. [52], note 1, as a first-order-in-α perturbation of Schwarzschild). The perturbation potentials V^(0), V^(1), V^(2) (Eqs. 24, 28, 39; Appendix A) are derived in-text from the Klein-Gordon, Maxwell, and Regge-Wheeler equations on that background. Frequencies follow from the standard third-order WKB formula (Eq. 44) with a (2,2) Padé regularizer (Eqs. 46-50), and the deviations δω (Eqs. 51-52) and Γ (Eq. 54) are arithmetic combinations of those computed frequencies. The Schwarzschild baseline values are quoted from the external tables of Ref. [67] (Konoplya 2003), not computed by the authors, so the comparison is anchored outside the paper. The model parameters α and k are chosen by hand (α = 0.01, 0.05, 0.1; k = 1-5), never fitted to the QNM output; the constraints α ≳ 0.015 and k ≲ 2.84 are read off the Γ = 1-10% contours (Fig. 3), not imposed. No quantity here equals its input by construction: the frequencies are not defined as 'Schwarzschild plus a fitted shift,' and the deviations are not defined in terms of the claimed result. The self-citation to Ref. [52] is the model substrate, but it is independent support by the stated rule: its assumptions (static, spherically symmetric, linear in α) do not include the target result (QNM deviations), and the QNM calculation would stand or fall on its own math given the metric. The parity-based decoupling of Section III.C (Eqs. 30-37) is argued in-text rather than imported; if wrong (auxiliary fields contributing to the axial sector), the frequencies would change, but that is a physics correctness risk. The numerically significant caveats are likewise correctness risks, not circularity: the nonlocal frequencies use third-order WKB while the external Schwarzschild baselines come from a higher-order implementation, so the quoted deviations may mix in systematic method error; no independent check (Leaver, time domain, higher-order WKB) is given; and Appendix B concedes the Padé 'precise degree of approximation is difficult to assess.' None of these makes a predicted quantity equivalent to an input. Verdict: no significant circularity.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the authors' prior black hole solution, the parity-based decoupling of axial perturbations, the accuracy of WKB and Pade approximations, and an assumed detector sensitivity window. No new particles or fields are introduced by this paper; the parameters α and k are model inputs, not fit results.

free parameters (3)

α (perturbative parameter) = 0.1, 0.05, 0.01; upper bound 0.1 chosen for plots
Dimensionless parameter measuring the deviation from Schwarzschild in the metric (7); values are chosen by hand for numerical illustration and for the parameter-space constraints.
k (power-law index) = 1, 2, 3, 4, 5
Power-law index in the lapse function; the text states k > 1 but the tables include k = 1. It is a model parameter, not fitted to data.
Observational window for Γ = 1% to 10%
The lower and upper bounds in Eq. (55) are selected by hand based on projected detector sensitivities; the resulting allowed region in Fig. 3 depends directly on this choice.

assumptions (4)

domain assumption The metric functions (7a)-(7b) describe a valid first-order solution of the revised Deser-Woodard field equations.
The paper takes the black hole solution from Ref. [52] without re-deriving it; all subsequent quasinormal mode calculations inherit this assumption.
domain assumption Nonlocal auxiliary scalar fields couple only to even-parity perturbations, so axial gravitational perturbations obey the GR Regge-Wheeler equation (Eq. 38).
This parity decoupling, argued in Section III.C, is load-bearing for the gravitational quasinormal mode frequencies.
domain assumption Third-order WKB theory gives accurate quasinormal mode frequencies for the low multipoles considered, and the (2,2) Pade approximants preserve the true potential while removing numerical instabilities near r = 3.
The paper uses WKB at N = 3 and Pade smoothing (Sections IV.A and IV.B) but provides no convergence or error study for the final frequencies.
domain assumption Future gravitational wave detectors will measure ringdown deviations from GR with precision between 1% and 10%.
The parameter-space constraints in Section V are derived from the chosen interval 1% ≤ Γ ≤ 10%, citing projected sensitivities but not a single-detector error budget.

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Cite this review

Pith. "Pith review of Quasinormal modes of nonlocal gravity black holes." pith.science (2026). https://pith.science/paper/Z76DQWOD

@misc{pith2026250701698,
  author       = {Pith},
  title        = {Pith review of: Quasinormal modes of nonlocal gravity black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z76DQWOD}},
  note         = {Machine review of arXiv:2507.01698}
}

abstract

We present a comprehensive study of the quasinormal modes of a new class of nonlocal static and spherically symmetric black hole (BH) solutions within the framework of the revised Deser-Woodard theory of gravity. These solutions are constructed as linear perturbations of the Schwarzschild spacetime and are characterized by an inverse power-law behavior of the lapse metric function. We derive the radial profiles of the effective potentials corresponding to scalar, electromagnetic and axial gravitational fluctuations on the BH background. Using the WKB method, complemented by Pad\'e approximants to regularize the trend of the effective potential near its peak, we compute the complex quasinormal mode frequencies associated with each type of perturbation. Our results are examined from both mathematical and physical perspectives, and are substantiated with references to existing literature. In particular, we compare the numerical outcomes with the predictions of the Schwarzschild metric to quantify deviations from the framework of general relativity. When all types of perturbations are combined, the relative deviations of the fundamental modes can reach up to $\sim 12\%$. Finally, we discuss the possibility to place observational bounds in the BH parameter space, based on the predicted sensitivities of future gravitational wave detectors.

Figures

Figures reproduced from arXiv: 2507.01698 by the authors.

**Figure 2.** FIG. 2. Average percentage deviations from the Schwarzschild predictions of the QNM frequencies for scalar, electromagnetic, [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** FIG. 3. Allowed region in the BH parameter space obtained [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. Pad´e approximants of the derivatives of the effective potential for scalar perturbations for [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

**Figure 5.** Figure 5: FIG. 5. Pad´e approximants of the derivatives of the effective potential for gravitational perturbations for [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]

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