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REVIEW 3 major objections 4 minor 78 references

Test-Field vs Physical Quasi-Normal Modes in Scalar-Tensor Theories

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For the BCL scalar-tensor black hole, spin-2 test-field quasi-normal modes diverge from physical gravitational modes at high damping while the fundamental modes stay close.

desk verdict Solid test-field QNM computation for the BCL black hole with a clean log(2) monodromy formula, but the headline contrast with physical QNMs is only as strong as the external [47] data it leans on. read the letter →

arxiv 2505.16883 v2 pith:2TDKBT6M submitted 2025-05-22 gr-qc

Alexandre Arbey , Etera R. Livine , Clara Montagnon This is my paper · ORCID
classification gr-qc MSC 83C5783C3583D05 PACS 04.70.-s04.30.-w04.50.Kd
keywords quasi-normalmodesscalar-tensortheoriesBCLblackholetest-fieldperturbationsTeukolskyequationLeavercontinuedfractionmonodromymethodspectroscopy
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 test-field perturbations—waves that propagate on a fixed black-hole background without back-reaction—can stand in for the true gravitational perturbations of a black hole in a modified theory of gravity. It answers this for the BCL solution of quadratic shift-symmetric Horndeski scalar-tensor theories, where both the physical axial gravitational quasi-normal modes and the metric are already known. The authors compute the spin-0 and spin-2 test-field quasi-normal spectra with Leaver's continued-fraction method and analytic monodromy techniques, then compare the spin-2 test-field spectrum with the physical one. They find that the fundamental modes agree to within about one to three percent, but that the higher overtones differ sharply: the test-field spectrum keeps the Schwarzschild-like shape with an imaginary-axis crossing and a vertical asymptote, while the physical spectrum has neither. This matters because test-field calculations are the only option for many effective black-hole models that lack explicit modified Einstein equations, and the result shows such calculations can be systematically misleading outside the low-damping sector.

What carries the argument

The argument runs on the BCL metric, $ds^2 = -f(r)dt^2 + f(r)^{-1}dr^2 + r^2d\Omega^2$ with $f(r) = (1 - r_+/r)(1 + r_-/r)$, where $r_-$ is a single deformation parameter away from Schwarzschild. Both test-field and physical perturbations are reduced to Schr\"odinger-like equations, $d^2\Psi/dx^2 + [\omega^2 - V(r)]\Psi = 0$, with effective potentials that agree to third order at infinity but differ near the horizon. The numerical engine is Leaver's continued-fraction method, here applied to a five-term recursion reduced by Gaussian elimination; the analytic engine is the monodromy technique, which tracks solutions around the complex-plane singularities and yields the asymptotic formula $4\pi R\omega = \log 2 - i(2n+1)\pi$, plus the near-horizon $sl_2$ symmetry that reproduces the equally spaced imaginary gap $\Delta_{\rm Im} = 1/(2R)$. To make the physical and test-field problems directly comparable, the propagation speed of the physical gravitational perturbations is normalized to $c = 1$ by the rescaled tortoise coordinate $d\tilde{x} = (1/c(r))dx$ with $c(r) = r/\sqrt{r^2 + 2r_- r_+}$.

What would settle it

Independently recompute the physical axial QNM spectrum of the BCL black hole from the first-order system [46,76] using a code that reaches $\mathrm{Im}(\omega)$ beyond roughly $-10$ and check the asymptote's slope and whether any mode crosses the imaginary axis; if the physical spectrum becomes vertical or crosses, the claimed contrast with test-field modes fails.

Watch

Extended reading notes

Core claim

The paper's central claim is that, on the BCL black hole of quadratic shift-symmetric Horndeski theories, the spin-2 test-field quasi-normal spectrum is not a reliable proxy for the physical axial gravitational spectrum outside the low-damping sector. The fundamental modes stay close—the test-field value differs from the physical one by about 0.78% at $r_- = 0.1$ and 2.5% at $r_- = 0.5$—but the overtones diverge systematically. The test-field spectrum crosses the imaginary axis and approaches a vertical asymptote whose first-order analytic law is $4\pi R\omega = \log 2 - i(2n+1)\pi$ with $R = r_+^2/(r_+ + r_-)$, while the physical spectrum shows no imaginary-axis crossing and a non-vertical asymptote that the prior numerical analysis [47] parametrizes by $\mathrm{Im}(\omega) = a\,\mathrm{Re}(\omega) + b$. As the authors phrase it, as long as that prior numerical analysis can be trusted, the test-field spin-2 QNMs are substantially different from the physical gravitational ones, especially for highly damped modes.

Load-bearing premise

The comparison's headline result rests on trusting the prior numerical data [47] for the physical axial QNMs and on the speed-normalizing rescaling of the tortoise coordinate, so if that data or that normalization is wrong, the claimed test-field/physical difference could be an artifact.

Editorial extensions

If this is right

  • For the BCL black hole, the fundamental spin-2 test-field QNM tracks the physical one within 1–3% for $r_-$ between 0.1 and 0.5, so test fields remain useful for the dominant ringdown mode.
  • At higher overtones the two spectra part ways: the test-field spectrum has a vertical asymptote and an imaginary-axis crossing, while the physical spectrum has neither.
  • The asymptotic imaginary gap of the test-field spectrum is $\Delta_{\rm Im} = 1/(2R)$ with $R = r_+^2/(r_+ + r_-)$, matching the monodromy prediction to sub-percent accuracy for the imaginary part.
  • The physical asymptote is instead non-vertical, with $1/a \approx -0.14\,r_- + 0.04\,r_-^2$ from the prior data [47], which a future analytical treatment would need to explain.

Reading between the lines

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

  • If this pattern is generic, spin-2 test-field QNMs should only be trusted for the fundamental and lowest overtones of effective black-hole models that lack explicit Einstein equations; high-overtone predictions from such models would carry an unquantified systematic error.
  • The monodromy discontinuity ($\log 2$ for BCL versus $\log 3$ for Schwarzschild as $r_- \to 0$) suggests a boundary-layer regime at small $r_-$; a next-to-leading-order monodromy computation could explain the observed bouncing of the real part with $r_-$.
  • A monodromy-style analysis of the physical perturbation equations, which the authors did not perform, would test whether the non-vertical asymptote is a genuine feature of the scalar-tensor dynamics or an artifact of the numerical scheme in [47].
  • The eikonal-limit WKB comparison hints that real parts of physical and test-field QNMs converge for large $l$ while imaginary parts do not; an analytic proof of this $l \to \infty$ split could isolate which potential terms control damping.
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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. The paper computes test-field (spin-0 and spin-2) quasi-normal mode (QNM) spectra for the BCL black hole in scalar-tensor theories, using Leaver's continued fraction method, the monodromy technique, near-horizon sl2 symmetry, and WKB cross-checks. It derives an asymptotic formula (4πRω = log 2 − i(2n+1)π, with R = r_+^2/(r_+ + r_-)) for highly damped test-field modes. The central comparison is between the spin-2 test-field spectrum and the physical axial gravitational QNM spectrum taken from Roussille et al. [47]. The paper reports that the fundamental modes are close (0.78% difference at r_- = 0.1, 2.5% at r_- = 0.5), but that the test-field spectrum crosses the imaginary axis and has a vertical asymptote, whereas the physical spectrum shows no crossing and a non-vertical asymptote, concluding that test-field spin-2 QNMs differ substantially from physical gravitational QNMs for highly damped modes.

Significance. If the comparison is valid, the paper provides a concrete, well-studied example in which test-field perturbations fail to reproduce physical gravitational perturbation spectra for a black hole in modified gravity, with the deviation growing in the highly damped regime. This is relevant for black hole spectroscopy and for effective quantum-gravity models that lack explicit modified Einstein equations. The monodromy result (74) is a new analytical prediction, and the test-field computations are cross-checked by multiple methods (WKB, Leaver, monodromy, near-horizon symmetry). However, the headline comparison relies on external data from [47] in precisely the regime where numerical control is hardest, and there is an internal inconsistency in the quoted physical effective potential. These issues must be addressed before the central claim can be regarded as fully supported.

major comments (3)
  1. [V.B, Eqs. (106) and (108)] There is an inconsistency between the displayed physical effective potential (106) and its quoted large-r asymptotic (108). Equation (106) writes the prefactor as (1 + 2 r_- r_+/r^2)^3, while the text immediately below says the factor is (1 + 2 r_- r_+/r^2)^{-3}; neither choice reproduces the 1/r^4 coefficient in Eq. (108). For example, with r_+ = 1, r_- = 0.5 and λ = 2, expanding Eq. (106) at large r gives a 1/r^4 coefficient of +7.25 for exponent +3 and −4.75 for exponent −3, whereas Eq. (108) gives −7.75. Because Vphys is the basis for the WKB checks in Appendix A and for the comparison with [47], this inconsistency must be resolved and the corrected potential used to re-verify the numerical results.
  2. [V.C and Appendix A] The central comparison of test-field and physical QNM spectra rests entirely on the physical spectrum from [47]. The independent WKB check in Table II covers only l = 2 for a few low overtones (where WKB is known to be inaccurate for n ~ l, as the authors note) and l = 10, 100 for the first few overtones; it does not probe the highly damped regime where the claimed qualitative differences (crossing of the imaginary axis vs no crossing, vertical vs non-vertical asymptote) occur. The authors should either compute the physical QNM spectrum from the corrected Vphys with the same continued fraction method used on the test-field side, or provide a quantitative convergence and completeness analysis of the [47] data. Without this, the headline claim is conditional on external numerical results in exactly the asymptotic regime where such results are most difficult to control.
  3. [V.B, Eq. (105)] The comparison presupposes that the rescaled tortoise coordinate d\tilde{x} = dx/c(r) defined in Eq. (105) is exactly the normalization used by [47] to define their physical QNM frequencies. If [47] quoted frequencies in the unrescaled tortoise coordinate, the physical spectrum would be shifted systematically and the comparison would be invalid. The paper should state explicitly how the [47] frequencies are normalized and, ideally, verify this by reproducing a few of their low-lying modes from the corrected Vphys with the authors' own continued fraction implementation.
minor comments (4)
  1. [IV.C] The statement that 'The limited data we have did not allow us to infer the coefficient B, so we only focused on the constant one A' is unclear: if B is not fitted, the fit reduces to a constant A, and the reported error bars (obtained by varying the number of removed low-lying modes) should be described accordingly; please clarify the exact fitting procedure.
  2. [IV.B, Figs. 4-8] Individual QNM frequencies are plotted without numerical error bars; a convergence study with respect to the continued-fraction truncation order (e.g., varying the number of retained terms) would strengthen the spectral plots and support the claim of 'good accuracy'.
  3. [V.C] The fitted relation 1/a = −0.14(±0.01)r− + 0.04(±0.01)r2− for the physical asymptote slope is presented without details on how it was obtained from [47]'s data; please provide the underlying data or a clear reference to the specific plot in [47].
  4. [General] There are several typographical issues, including 'obsctable' in the Discussion and the reference to 'table 9' for a figure; please proofread the manuscript.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the test-field QNM computation is self-contained and the physical-spectrum comparison rests on an independent external benchmark, with its limitations explicitly stated.

full rationale

The paper's central derivation chain is not circular. The test-field QNM frequencies are obtained by Leaver's continued-fraction method applied to the BCL Teukolsky equations (Sec. II.B), and the asymptotic prediction (74) is derived independently from the monodromy technique with no fitted constants (Sec. III.A). The numerical Leaver results are compared with this analytical prediction using a diagnostic A+B/sqrt(n) fit; the fit does not enter the derivation of (74) and no fitted parameter is renamed as a prediction. The comparison with physical gravitational QNMs (Sec. V.C) uses the external numerical data of Roussille et al. [47], with the authors explicitly conditioning the headline result on that benchmark ('As long as the previous numerical analysis of the BCL BH [47] can be trusted'). This is an external, independently produced benchmark rather than a self-citation, and the paper's own WKB check (Appendix A, Table II) is a genuine cross-check rather than a construction of the result. The authors also explicitly flag the missing monodromy analysis for the physical spectrum as an open limitation. Self-citations to [56] and [70] concern standard methods and are not load-bearing in the derivation. No uniqueness theorem, ansatz, or fitted quantity is imported from the authors' prior work to force the conclusion.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central comparison uses no fitted physical parameters; the BCL parameter r- is a background deformation parameter scanned over a range, not fitted. The analytical monodromy formula and numerical QNMs are derived independently. The main caveats are the first-order monodromy truncation and the numerically observed branch choice in the continued-fraction reduction.

free parameters (1)
  • asymptotic fit constant A = varies with r-; plotted in Figs. 10 and 11; typical values around 0.06 to 0.10 in units r+=1
    The constant A in the ansatz f(n)=A+B/sqrt(n) is fitted to the computed asymptotic QNM sequences after removing the first 15-25 modes (Eq. 90). It is a data-reduction parameter used to compare with the monodromy prediction, not an input to the derivation.
assumptions (4)
  • domain assumption The BCL metric (3)-(6) is an exact solution of the quadratic shift-symmetric Horndeski action (1) with the chosen functions (2).
    Taken from [29] and [46]; the perturbation analysis and QNM comparison are built on this background solution.
  • domain assumption The spin-0 and spin-2 test-field dynamics are correctly captured by the Teukolsky equations (10)-(19) derived in [70,71].
    The paper follows the Newman-Penrose derivation of [70]; no independent derivation of the master equation is given.
  • ad hoc to paper The monodromy computation can be truncated to first order near r=0 and assumes Im(omega) much larger than Re(omega).
    This approximation produces Eq. (74) and is explicitly acknowledged to break the r- to 0 Schwarzschild limit (log 2 vs log 3); it limits the accuracy of the analytical real-part prediction, with relative errors up to 40% in Figs. 10 and 11.
  • domain assumption In the Gaussian reduction of the five-term Leaver recursion, the physical asymptotic branch for the reduced continued-fraction coefficients is zeta to -1/4, not zeta to r-/(r-+1)^2.
    The authors state that numerical exploration shows the r--dependent branch is never realized, but no proof is given (Section II.B, Eq. (46)); the Leaver computation depends on this branch choice.

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Cite this review

Pith. "Pith review of Test-Field vs Physical Quasi-Normal Modes in Scalar-Tensor Theories." pith.science (2026). https://pith.science/paper/2TDKBT6M

@misc{pith2026250516883,
  author       = {Pith},
  title        = {Pith review of: Test-Field vs Physical Quasi-Normal Modes in Scalar-Tensor Theories},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2TDKBT6M}},
  note         = {Machine review of arXiv:2505.16883}
}
abstract

In the context of the general effort to model black hole dynamics, and in particular their return-to-equilibrium through quasi-normal modes, it is crucial to understand how much test-field perturbations deviate from physical perturbations in modified gravity scenarios. On the one hand, physical perturbations follow the modified Einstein equations of the considered extension of general relativity. The complexity of those equations can quickly escalate with extra fields and non-linear couplings. On the other hand, test-field perturbations, with negligible back-reaction on the space-time geometry, describe the propagation of both matter fields and spin $s=2$ gravitational waves on the black hole geometry. They are not subject to the intricacies of the modified Einstein equations, and only probe the background spacetime metric. If their physics were to not deviate significantly from physical perturbations, they would be especially useful to investigate predictions from quantum gravity scenarios which lack explicit detailed Einstein equations. Here we focus on a specific modified gravity solution -- BCL black holes in scalar-tensor theories -- for which physical perturbations and related QNM frequencies have already been studied and computed numerically. We compute the test-field QNM frequencies and compare the two QNM spectra. This provides a concrete example of the significant differences arising between test-fields and physical perturbations, and flags unphysical deviations related to the test-field framework.

Figures

Figures reproduced from arXiv: 2505.16883 by the authors.

Figure 1
Figure 1. FIG. 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p021_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p022_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12 [PITH_FULL_IMAGE:figures/full_fig_p025_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13 [PITH_FULL_IMAGE:figures/full_fig_p026_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p027_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p028_15.png]

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