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arxiv: 2601.09607 · v2 · submitted 2026-01-14 · 🌀 gr-qc · astro-ph.HE· hep-th

Recognition: 1 theorem link

· Lean Theorem

Confronting eikonal and post-Kerr methods with numerical evolution of scalar field perturbations in spacetimes beyond Kerr

Ciro De Simone , Sebastian H. V\"olkel , Kostas D. Kokkotas , Vittorio De Falco , Salvatore Capozziello
Authors on Pith no claims yet

Pith reviewed 2026-05-16 14:37 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.HEhep-th
keywords quasinormal modesblack hole ringdowndeformed Kerr spacetimeeikonal approximationpost-Kerr approximationnumerical evolutionscalar perturbationsgravitational wave spectroscopy
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The pith

Eikonal and post-Kerr approximations produce growing errors in scalar quasinormal mode frequencies on deformed Kerr spacetimes when checked against full numerical evolution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests how reliably two common shortcut formulas predict the ringing frequencies of scalar fields around black holes whose geometry deviates from the exact Kerr solution. It generates reference frequencies by evolving the scalar field in 2+1 dimensions across a range of spins and deformation strengths. The resulting mismatches are then compared with the frequency precision that gravitational-wave detectors are expected to achieve at different signal strengths. The work also finds that deformations near the horizon shift prograde and retrograde modes by different amounts and supplies a geometric account of that difference.

Core claim

When benchmarked against 2+1-dimensional numerical evolutions of scalar perturbations, both the eikonal and post-Kerr approximations yield quasinormal frequencies whose errors increase with black-hole spin and with the strength of the deviation from Kerr; these modeling errors become comparable to or larger than projected statistical uncertainties at high signal-to-noise ratios, while near-horizon deformations affect prograde and retrograde modes asymmetrically through distinct geometric sensitivities.

What carries the argument

The 2+1-dimensional numerical time-evolution code that evolves scalar-field perturbations on a family of deformed Kerr backgrounds and extracts their quasinormal frequencies as the reference against which the eikonal and post-Kerr formulas are tested.

If this is right

  • Approximate formulas remain usable only for small deviations from Kerr when the signal-to-noise ratio is moderate.
  • High-precision ringdown spectroscopy of deformed black holes requires either full numerical simulations or higher-order analytic methods.
  • Prograde and retrograde modes respond differently to near-horizon changes, so they must be analyzed separately in tests of modified gravity.
  • The range of validity of both approximations shrinks as black-hole spin increases or as the deformation parameter grows.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the numerical framework is reliable, events with signal-to-noise ratio above roughly 100 could already place meaningful bounds on certain near-horizon deformations using ringdown data alone.
  • The scalar-field results suggest that similar differential effects on prograde and retrograde modes may appear in gravitational-wave perturbations of other non-Kerr black-hole solutions.
  • Extending the same numerical benchmark to vector or tensor fields would test whether the reported accuracy limits apply directly to gravitational-wave ringdown.

Load-bearing premise

The 2+1-dimensional numerical evolution accurately represents the complete dynamics of scalar perturbations in the deformed spacetime for the spin and deformation values examined.

What would settle it

A single numerical run at a chosen deformation strength and angular index that extracts a frequency differing from the eikonal or post-Kerr prediction by an amount larger than the modeling error reported in the paper.

Figures

Figures reproduced from arXiv: 2601.09607 by Ciro De Simone, Kostas D. Kokkotas, Salvatore Capozziello, Sebastian H. V\"olkel, Vittorio De Falco.

Figure 1
Figure 1. Figure 1: FIG. 1. The solid lines correspond to the prograde ( [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: , which appear to be dominated by the prograde mode of a given pℓ, mq and its retrograde counterpart pℓ, ´mq. The QNM frequencies, as well as the amplitudes and phases of the modes, have been extracted using the Prony method [48], which consists of fitting the waveform with a sum of damped sinusoids. To extract accurate QNM estimates, the Prony method must be applied only in the time interval rti , tf s wh… view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. First row: estimates of the real and imaginary parts of prograde and retrograde modes as a function of the BH spin for [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Fundamental QNMs for different deviation parameters [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plots of [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: shows the corner plot obtained at ρ “ 100 for the mode pℓ “ 10, a “ 0.3, ϵ “ 1q, where the Prony estimate, as well as the eikonal and post-Kerr predictions at different orders, are shown with the 1σ and 2σ confi￾dence intervals. In this case, the eikonal prediction offers an accurate estimate of the Prony injection, especially for the imaginary part. While the first-order post-Kerr approximation is not ver… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Histograms of the SNR for real and imaginary parts applied to different methods compared to the eikonal predictions. [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Prony estimates for prograde and retrograde modes as a function of the starting time [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Same notation adopted in Fig [PITH_FULL_IMAGE:figures/full_fig_p016_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Same notation adopted in Fig [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
read the original abstract

The accurate computation of quasinormal modes from rotating black holes beyond general relativity is crucial for testing fundamental physics with gravitational waves. In this study, we assess the accuracy of the eikonal and post-Kerr approximations in predicting the quasinormal mode spectrum of a scalar field on a deformed Kerr spacetime. To obtain benchmark results and to analyze the ringdown dynamics from generic perturbations, we further employ a 2+1-dimensional numerical time-evolution framework. This approach enables a systematic quantification of theoretical uncertainties across multiple angular harmonics, a broad range of spin parameters, and progressively stronger deviations from the Kerr geometry. We then confront these modeling errors with simple projections of statistical uncertainties in quasinormal mode frequencies as a function of the signal-to-noise ratio, thereby exploring the domain of validity of approximate methods for prospective high-precision black-hole spectroscopy. We also report that near-horizon deformations can affect prograde and retrograde modes differently and provide a geometrical explanation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper assesses the accuracy of eikonal and post-Kerr approximations for scalar-field quasinormal modes on deformed Kerr spacetimes by comparing them against benchmarks obtained from 2+1-dimensional numerical time evolutions of generic initial data. It quantifies modeling errors across angular harmonics, spin parameters, and increasing deviations from Kerr, then confronts these errors with projected statistical uncertainties in QNM frequencies as a function of signal-to-noise ratio, while also reporting differential effects of near-horizon deformations on prograde versus retrograde modes together with a geometrical interpretation.

Significance. If the numerical benchmarks hold, the work supplies a concrete, systematic map of the validity domains of two widely used approximate methods for black-hole spectroscopy beyond Kerr, directly relevant to high-precision tests of general relativity with future gravitational-wave detectors. The use of time-domain evolution from generic data rather than mode-specific initial data is a methodological strength that allows assessment of the full ringdown dynamics.

major comments (2)
  1. [Numerical Evolution Framework] The central claim that the 2+1D evolutions furnish reliable benchmark QNM frequencies rests on an unverified assumption of numerical accuracy. No resolution-doubling studies, convergence tests, or independent frequency-extraction comparisons (e.g., via Prony or matrix-pencil methods) are presented for the strongest deformations considered, leaving open the possibility that truncation or boundary-reflection errors contaminate the reported spectra.
  2. [Comparison with Statistical Uncertainties] The confrontation of modeling errors with statistical uncertainties is presented only qualitatively. Specific SNR values, the precise functional form of the projected frequency uncertainties, and error bars on the eikonal/post-Kerr deviations are not shown, making it impossible to judge the quantitative domain of validity claimed in the abstract.
minor comments (2)
  1. [Introduction] Notation for the deformation parameters and the precise form of the deformed metric should be stated explicitly in the introduction rather than deferred to later sections.
  2. [Results] Figure captions for the QNM frequency comparisons should include the exact spin and deformation values used in each panel to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We appreciate the positive assessment of the work's significance for black-hole spectroscopy and address each major comment below, indicating the revisions we will implement.

read point-by-point responses

  1. Referee: [Numerical Evolution Framework] The central claim that the 2+1D evolutions furnish reliable benchmark QNM frequencies rests on an unverified assumption of numerical accuracy. No resolution-doubling studies, convergence tests, or independent frequency-extraction comparisons (e.g., via Prony or matrix-pencil methods) are presented for the strongest deformations considered, leaving open the possibility that truncation or boundary-reflection errors contaminate the reported spectra.

    Authors: We agree that explicit demonstration of numerical accuracy is required to support the benchmark results. In the revised manuscript we will add a dedicated subsection on numerical validation that includes resolution-doubling studies and convergence tests for the strongest deformations. We will also extract frequencies using both the existing Fourier-based method and an independent Prony analysis on the same waveforms, reporting the agreement (or quantifying any residual discrepancy) to confirm that truncation and boundary errors do not contaminate the spectra at the reported precision. revision: yes

  2. Referee: [Comparison with Statistical Uncertainties] The confrontation of modeling errors with statistical uncertainties is presented only qualitatively. Specific SNR values, the precise functional form of the projected frequency uncertainties, and error bars on the eikonal/post-Kerr deviations are not shown, making it impossible to judge the quantitative domain of validity claimed in the abstract.

    Authors: We acknowledge that the comparison was presented qualitatively. In the revision we will add a new figure (or panel) that displays the modeling errors together with the projected statistical uncertainties for concrete SNR values (SNR = 50, 100, 200, 500, 1000). We will state the functional form used (standard 1/SNR scaling for frequency errors in ringdown analyses, with the prefactor taken from the literature on Kerr QNM parameter estimation) and include error bars on the eikonal and post-Kerr deviations. These quantitative elements will be cross-referenced in the abstract and discussion to make the claimed domain of validity explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: numerical evolution supplies independent benchmark

full rationale

The paper's central claim compares eikonal and post-Kerr predictions against quasinormal frequencies extracted from a separate 2+1D numerical time-evolution code on deformed Kerr backgrounds. This benchmark is generated from first-principles evolution of the scalar wave equation with generic initial data and is not obtained by fitting or re-deriving the approximations under test. No self-definitional steps, fitted inputs renamed as predictions, or load-bearing self-citations appear in the methodology; the numerical results function as an external reference against which modeling errors are quantified and then contrasted with projected statistical uncertainties. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Since only the abstract is available, the ledger is limited to the high-level assumptions stated therein. No specific free parameters or invented entities are mentioned.

axioms (1)
  • domain assumption The deformed Kerr spacetime is an appropriate model for black holes beyond general relativity
    Central to both the approximations and the numerical simulations described.

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