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arxiv: 2605.24704 · v1 · pith:ERLOU3YKnew · submitted 2026-05-23 · 🌀 gr-qc

Shaping black hole resonances I. Black hole ringdown as a spectral filtering process

Alejandro Svyatkovskyy Kholyavka , Jose Antonio Le\'on Vega , Samuel G\'omez G\'omez , Xisco Jim\'enez Forteza , Sayak Datta This is my paper

Pith reviewed 2026-06-30 12:54 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasinormal modesblack hole ringdownspectral filteringexcitation coefficientsFourier transformgravitational wavesperturbation theorynumerical relativity
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The pith

Black hole ringdown excites each quasinormal mode according to the Fourier content of the perturbation at that mode's frequency.

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

The paper establishes that the ringdown of a perturbed black hole follows a spectral filtering rule rather than depending on case-by-case details of the source. The amplitude of each quasinormal mode is set directly by the Fourier transform of the initial perturbation evaluated at the mode's frequency. This creates a quantitative link between the spectrum of the perturbation and the resulting ringdown signal. A sympathetic reader would care because it replaces ad-hoc excitation modeling with a simple, controllable mechanism. The authors demonstrate the rule by building localized perturbations with tunable bandwidth and carrier frequency, then confirming it analytically through coefficient factorization and numerically through time-domain evolutions.

Core claim

QNM excitation is governed by a simple spectral rule: each mode is excited according to the Fourier content of the perturbation evaluated at its characteristic frequency. This result follows from the factorization of the excitation coefficients and establishes a direct, quantitative connection between the spectral properties of the perturbation and the resulting ringdown amplitudes. The excitation amplitude of each mode equals the weighted spatial Fourier transform of the initial data evaluated at wavenumber k ~ ω_n so that the filter selectively excites modes whose frequencies lie within the spectral support of the perturbation while suppressing others.

What carries the argument

Factorization of the excitation coefficients that isolates the Fourier transform of the initial data evaluated at the mode frequency.

If this is right

  • The filter selectively excites modes whose frequencies lie within the spectral support of the perturbation.
  • Excitation is maximized when the dominant perturbation frequency lies close to the real part of the QNM frequency.
  • Modes outside the spectral support of the perturbation are suppressed.
  • The rule is validated at the percent level with fits to time-domain numerical evolutions using sliding windows.

Where Pith is reading between the lines

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

  • The same spectral matching could predict how different astrophysical perturbations, such as those from surrounding matter, imprint on observed ringdown signals.
  • The rule suggests a way to design initial-data sets in numerical simulations that target specific modes while minimizing unwanted excitations.
  • Similar filtering behavior may appear in other wave systems with resonant modes, such as neutron-star oscillations or cavity resonances.

Load-bearing premise

The factorization of the excitation coefficients into a form that isolates the Fourier transform of the initial data at the mode frequency holds for the class of localized perturbations considered.

What would settle it

A numerical evolution or gravitational-wave observation in which measured ringdown amplitudes deviate from the values predicted by the perturbation's Fourier transform evaluated at each quasinormal-mode frequency would falsify the spectral rule.

Figures

Figures reproduced from arXiv: 2605.24704 by Alejandro Svyatkovskyy Kholyavka, Jose Antonio Le\'on Vega, Samuel G\'omez G\'omez, Sayak Datta, Xisco Jim\'enez Forteza.

Figure 1
Figure 1. Figure 1: ID and their spatial Fourier spectra. Top left: Spatial profiles Ψ(r⋆) of the localized perturbations. The Gaussian envelope, characterized by width σ and centered at r0, sets the spatial localization, while an oscillatory modulation with frequency ν introduces a controllable driving frequency. Top right: Corresponding spatial Fourier spectra |Ψ( ˜ ω)|. The bandwidth scales as ∆ω ∼ 1/σ, while the oscillato… view at source ↗
Figure 2
Figure 2. Figure 2: Spectral structure of the ID Ψ( ˜ x) as a function of α = σν and x = ω/ν. Notice the transition at α = 1 from a single maximum at x = 0 to two symmetric max￾ima at finite frequencies x = ±x∗. The central extremum changes from a maximum for α < 1 to a minimum for α > 1, with α = 1. This transition directly controls whether the BH response is tail-dominated (α < 1) or QNM-dominated (α > 1). creases, oscillat… view at source ↗
Figure 3
Figure 3. Figure 3: shows representative waveforms |Ψ| extracted at fixed r obs ⋆ = 100M for the ℓ = 2 mode and r0 = 100, illustrating the effect of varying σ (top) and ν (down). In the top panel (ν = 0), the three curves correspond to σ = {1, 5, 9}. The quantity tpeak is the time at which the numerical maximum amplitude is observed. All three waveforms exhibit an initial prompt phase, peaking at tpeak ∼ r0 + r obs ⋆ , where … view at source ↗
Figure 4
Figure 4. Figure 4: Quasinormal excitation coefficients |Cn| as a function of the carrier frequency ν for oscillatory Gaus￾sian ID with fixed width σ = 5 and ℓ = 2. Vertical dotted lines indicate ω Re n for n = 0, 1. Each mode is maximally excited when ν ≈ ω Re n , demonstrating that the BH re￾sponds as a resonant spectral filter. B. QNECs: the spectral filter in action To quantify this behavior, we now turn from the time￾dom… view at source ↗
Figure 5
Figure 5. Figure 5: Fundamental-mode QNEC |C0| as a function of the carrier frequency ν for oscillatory Gaussian ID with varying width σ (asymptotic approximation). As σ increases, the spectral bandwidth narrows (∆ω ∼ 1/σ) and the resonance at ν = ω Re 20 (vertical dash-dotted line) sharpens. The dashed black curve shows the envelope of maximum excitation as σ varies, highlighting how the optimal driving frequency approaches … view at source ↗
Figure 7
Figure 7. Figure 7: Excitation amplitude of the fundamental mode [PITH_FULL_IMAGE:figures/full_fig_p011_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Dependence of |C0| on r0 for a pure Gaussian source (ν = 0) at three widths σ ∈ {1, 3, 6}. Solid: nu￾merical median, with 1σ and 3σ credible bands (same convention as Sec. VI D). Dashed: asymptotic prediction of Eq. (23). The asymptotic value is independent of r0 by construction; its hierarchy can be deduced observing the resonance curve of [PITH_FULL_IMAGE:figures/full_fig_p012_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: Multipolar amplitude ratios Rℓℓ′ of Eq. (35) as a function of ν at fixed σ = 5, for the fundamental mode of each multipole. Solid lines: numerical median, with 1σ and 3σ credible bands. Dashed and dotted lines: Leaver and asymptotic predictions, respectively. The collapse of the three curves over four decades indicates that the near-zone weight W (ℓ) n largely cancels in the ratio, leaving the asymptotic … view at source ↗
Figure 12
Figure 12. Figure 12: Amplitude-phase space structure of QNECs [PITH_FULL_IMAGE:figures/full_fig_p014_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Time evolution of the extracted QNM coeffi [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Excitation amplitude of the fundamental mode |C0| as a function of the carrier frequency ν at fixed width σ = 5 for the higher multipoles (top) ℓ = 3 and (bottom) ℓ = 4. The conventions follow those of [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Excitation phase of the fundamental mode [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
read the original abstract

The ringdown of a perturbed black hole (BH) can be described as a superposition of quasinormal modes (QNMs), whose frequencies are determined by the spacetime geometry while their amplitudes depend also on the perturbing source. However, the physical mechanism governing mode excitation remains unclear and is typically treated on a case by case basis. In this work, we show that QNM excitation is governed by a simple spectral rule: each mode is excited according to the Fourier content of the perturbation evaluated at its characteristic frequency. This result follows from the factorization of the excitation coefficients and establishes a direct, quantitative connection between the spectral properties of the perturbation and the resulting ringdown amplitudes. To make this mechanism explicit and controllable, we construct localized perturbations with independently tunable spectral bandwidth and carrier frequency. We demonstrate analytically and numerically that BHs act as resonant spectral filters. We show analytically that the excitation amplitude of each mode equals the weighted spatial Fourier transform of the initial data evaluated at wavenumber $k\sim\omega_n$ so that the filter selectively excites modes whose frequencies lie within the spectral support of the perturbation while suppressing others. Consequently, the excitation is maximized when the dominant perturbation frequency lies close to the real part of the QNM frequency, and we validate this at the percent level with fits to time-domain numerical evolutions. To robustly perform these fits, we have developed a new fitting algorithm, $\mathtt{QNMToolkit}$, which performs ringdown fits over large ensembles of sliding time-domain windows and quantifies the resulting fitting variance.

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 claims that black hole ringdown is a spectral filtering process in which each QNM is excited according to the Fourier content of the perturbation evaluated at its characteristic frequency. This follows from an analytic factorization of the excitation coefficients into a form that isolates the weighted spatial Fourier transform of the initial data at wavenumber k ∼ ω_n (with ω_n complex). The authors construct localized perturbations with tunable spectral bandwidth and carrier frequency, derive the exact equality analytically, and validate the resulting filter behavior at the percent level via time-domain evolutions using a new sliding-window fitting algorithm QNMToolkit.

Significance. If the central derivation holds, the result supplies a general, quantitative rule linking perturbation spectra directly to ringdown amplitudes, replacing case-by-case excitation calculations with a transparent spectral criterion. The construction of controllable perturbations and the new fitting tool are concrete strengths that enable reproducible tests of the filter picture.

major comments (2)
  1. [§3] §3 (derivation of excitation coefficients): the factorization that isolates the weighted spatial Fourier transform at complex k ∼ ω_n requires an explicit justification of the analytic continuation of the weighted integral off the real axis. The manuscript must state the precise decay or support conditions on the localized perturbation that guarantee the continuation introduces no additional terms or contour contributions; without this, the claimed exact equality between excitation amplitude and the Fourier transform does not follow rigorously from the wave equation.
  2. [§4.2] §4.2 (numerical validation): the percent-level agreement between the predicted spectral-filter amplitudes and the fitted ringdown coefficients is reported for an ensemble of perturbations, but the manuscript does not detail the precise criteria used to exclude early-time or late-time windows in the QNMToolkit fits. Because the central claim is quantitative, the fitting protocol and variance quantification must be specified so that the agreement cannot be attributed to post-hoc window selection.
minor comments (2)
  1. [§2] Notation for the weighting function arising from the adjoint projection should be introduced once and used consistently; its dependence on the specific wave operator is currently introduced only in passing.
  2. [Figure 4] Figure 4 (spectral support vs. mode excitation): the vertical lines marking Re(ω_n) would be clearer if accompanied by a short inset showing the corresponding imaginary-part shift for one representative mode.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and will incorporate clarifications to strengthen the presentation.

read point-by-point responses

  1. Referee: [§3] §3 (derivation of excitation coefficients): the factorization that isolates the weighted spatial Fourier transform at complex k ∼ ω_n requires an explicit justification of the analytic continuation of the weighted integral off the real axis. The manuscript must state the precise decay or support conditions on the localized perturbation that guarantee the continuation introduces no additional terms or contour contributions; without this, the claimed exact equality between excitation amplitude and the Fourier transform does not follow rigorously from the wave equation.

    Authors: We agree that an explicit statement of the support/decay conditions is needed for full rigor. Our localized perturbations are constructed with compact spatial support (or Gaussian profiles whose parameters ensure absolute convergence of the integral in a complex strip containing the relevant QNM frequencies). For compactly supported initial data the weighted Fourier transform is an entire function, permitting analytic continuation with no additional contour contributions from the wave equation. We will add a paragraph in §3 stating these conditions, citing the relevant properties of Fourier transforms of compactly supported functions, and confirming that the factorization holds exactly under them. revision: yes

  2. Referee: [§4.2] §4.2 (numerical validation): the percent-level agreement between the predicted spectral-filter amplitudes and the fitted ringdown coefficients is reported for an ensemble of perturbations, but the manuscript does not detail the precise criteria used to exclude early-time or late-time windows in the QNMToolkit fits. Because the central claim is quantitative, the fitting protocol and variance quantification must be specified so that the agreement cannot be attributed to post-hoc window selection.

    Authors: We acknowledge that the precise window-selection criteria were not fully detailed. QNMToolkit selects early-time cutoffs once the initial perturbation has decayed below a fixed threshold (10^{-10} of peak amplitude) and late-time cutoffs before the onset of numerical noise or power-law tails (determined per simulation by monitoring the residual). Variance is computed as the standard deviation over an ensemble of at least 100 sliding windows per run. We will expand §4.2 with an explicit subsection describing these thresholds, the sliding-window parameters, and how variance is reported, ensuring the protocol is fully reproducible. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained from wave equation factorization

full rationale

The central result—that excitation amplitude equals the weighted spatial Fourier transform of initial data at k∼ω_n—follows analytically from factorization of the excitation coefficients for the class of localized perturbations. This is presented as a direct consequence of the wave equation projection rather than a fit or self-citation. Numerical validation uses independent time-domain evolutions and a new fitting tool, with no evidence that the factorization or analytic continuation step reduces to the target amplitudes by construction. No load-bearing self-citations, ansatz smuggling, or renaming of known results appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Central claim rests on factorization of excitation coefficients (not shown in abstract) and the assumption that ringdown is a linear superposition of QNMs with amplitudes set solely by initial-data Fourier content; no free parameters or invented entities are mentioned.

axioms (2)
  • domain assumption Black hole ringdown is a linear superposition of quasinormal modes whose amplitudes are determined by initial data
    Stated in first sentence of abstract as the starting description of ringdown.
  • ad hoc to paper Factorization of excitation coefficients isolates the Fourier transform of the perturbation
    Abstract states the spectral rule follows from this factorization without providing the factorization step.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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