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Probing the Unstable Spectrum of Schwarzschild-like Black Holes

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Pith's one-line read The quasinormal spectrum of a Schwarzschild-like black hole deformed by a physically motivated Rezzolla-Zhidenko deformation is unstable, with pseudospectra spreading across the complex plane rather than forming the closed contours that…

desk verdict Useful first pseudospectra for physically motivated RZ deformations, but the headline contrast with random perturbations lacks a controlled norm comparison. read the letter →

arxiv 2501.13815 v3 pith:MNQVLQJL submitted 2025-01-23 gr-qc

Pedro Henrique Croti Siqueira , Lucas Tobias de Paula , Rodrigo Panosso Macedo , Maurício Richartz This is my paper · ORCID
classification gr-qc
keywords quasinormalmodespseudospectrumspectralinstabilityRezzolla-ZhidenkoparametrizationSchwarzschild-likeblackholesholeperturbationtheoryhyperboloidalapproachnon-self-adjointoperators
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 tests a counterintuitive claim from the pseudospectrum literature: that random perturbations of the effective potential governing gravitational waves can make a black hole's quasinormal-mode spectrum stable. The authors ask whether the same stabilizing effect occurs when the deviation from Schwarzschild is a physically motivated spacetime deformation rather than an ad-hoc random potential. Using the Rezzolla-Zhidenko parametrization with deformation parameters fixed by weak-field and horizon conditions, they compute the pseudospectrum of the axial gravitational-wave operator for four parameter choices and find spreading, open contour lines---the signature of spectral instability---in every case. A sinusoidal perturbation added to the potential exacerbates the drift of overtones, and the authors point out that with several deformation sources at once, overtone instabilities may no longer identify which source caused them. If the paper is right, the stabilizing effect seen for random perturbations does not carry over to physically realized deviations from Schwarzschild.

What carries the argument

The central object is the $\varepsilon$-pseudospectrum of the non-self-adjoint wave operator $L$ obtained by rewriting the axial perturbation equation in hyperboloidal coordinates, so that quasinormal modes are eigenvalues of $L$ acting on a two-component state $u=(\bar\psi,\bar\phi)$ on a compactified radial domain. The pseudospectrum is defined through the resolvent condition $\|(s\,\mathrm{Id}-L)^{-1}\|^{-1}<\varepsilon$, with the norm induced by the energy scalar product of Eqs. (33)--(34), and it is read as a topographic map: steep closed contours around an eigenvalue mean stability, while open contours spreading across the complex plane mean instability. The numerical implementation uses Chebyshev collocation with analytical mesh refinement, and the discrete adjoint entering the singular-value computation is taken with respect to that same energy scalar product, so the entire diagnosis is tied to this choice.

What would settle it

Recompute the pseudospectrum for the configuration $(\epsilon,a_3)=(10^{-2},500)$ using the standard $L^2$ inner product on the compactified radial domain instead of the energy scalar product; if the $\varepsilon$-level sets close into concentric contours around the RZ overtones, the paper's instability claim fails, because the diagnosis is tied to the energy-norm choice.

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Extended reading notes

Core claim

The central claim is that the quasinormal spectrum of a Schwarzschild-like black hole obtained from the physically motivated Rezzolla-Zhidenko setup of Ref. [37] is unstable. Concretely, for axial $\ell=2$ perturbations and the parameter pairs $(\epsilon,a_3) = (10^{-4},20)$, $(10^{-4},500)$, $(10^{-2},20)$, and $(10^{-2},500)$, the $\varepsilon$-pseudospectrum of the wave operator does not show the closed concentric contour lines that would signal stability; the contours spread across the complex plane, including around overtones that are already displaced from their Schwarzschild values. The authors therefore conclude that the stabilizing effect reported for random perturbations in Ref. [12] is not generic: when the deviation from Schwarzschild is realized through a spacetime deformation with a physically motivated matter content, the resulting spectrum remains fragile. They corroborate this by adding sinusoidal perturbations $\delta q = \eta \sin(2\pi k \sigma)$ to the conformal potential and showing that moderate wavenumbers amplify the overtone drift, and they note that with multiple sources of deformation the overtone spectrum alone may be degenerate across different explanations.

Load-bearing premise

The load-bearing premise is that spectral instability is correctly diagnosed by the $\varepsilon$-pseudospectrum defined with the energy scalar product of Eqs. (33)--(34); the paper itself notes that pseudospectra depend on this choice, and if another natural scalar product produced closed contours around the Rezzolla-Zhidenko modes, the conclusion that the spectrum is unstable would not follow.

Editorial extensions

If this is right

  • The stabilizing behavior seen for random potential perturbations is not a generic property of operators with already-destabilized quasinormal spectra.
  • Overtones of Rezzolla-Zhidenko black holes remain sensitive to further perturbations even when the deformation parameters already displace them far from Schwarzschild.
  • Low-wavenumber sinusoidal perturbations leave the RZ spectrum nearly unchanged, but moderate wavenumbers amplify the instability.
  • When several perturbation sources act together, overtone instabilities can become degenerate, making it hard to attribute observed spectral distortions to a specific deformation.
  • Preliminary evidence indicates the qualitative instability picture extends to higher multipoles, though a full multipole analysis is left to future work.

Reading between the lines

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

  • The paper flags but does not test the scalar-product dependence of its diagnosis; recomputing the same pseudospectra under the standard $L^2$ inner product would show whether the 'unstable' verdict is an artifact of the energy norm.
  • The degeneracy the authors find implies that gravitational-wave ringdown measurements of unstable overtones should fit several deformation sources simultaneously; fitting a single RZ parameter could misattribute environmental or theory-agnostic perturbations to the spacetime deformation.
  • Because the adopted RZ matter has $p_r=-\rho$, the instability might be tied to the dark-energy-like equation of state rather than to the geometry alone; repeating the analysis for an RZ branch with different matter would separate the two.
  • A direct numerical confrontation of the two mechanisms is possible: apply the same random perturbations used in Ref. [12] to the RZ effective potential and check whether the spectrum remains unstable, as the pseudospectra computed here imply.
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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 / 5 minor

Summary. The paper studies the stability of quasinormal-mode (QNM) spectra for a Schwarzschild-like black hole described by the Rezzolla-Zhidenko (RZ) parametrization, using the physically motivated parameter choice of Cardoso, Kastha, and Panosso Macedo. The authors adopt a hyperboloidal compactification, a Chebyshev spectral discretization with analytical mesh refinement, and the energy scalar product to compute QNM spectra and ε-pseudospectra for four RZ configurations. They find that the RZ spectra, whose overtones are already displaced relative to Schwarzschild, have open pseudospectra with no sign of the 'regularization' reported in Ref. [12] for random perturbations. They corroborate this with explicit sinusoidal perturbations of the effective potential and argue that multiple perturbation sources make it difficult to identify the origin of overtone instabilities.

Significance. The numerical computation is careful and reproducible: the methods are standard in the field (hyperboloidal framework, Chebyshev collocation, AnMR), a QNM convergence criterion is stated, and the data are openly available. If the central contrast with random perturbations holds, the paper establishes that the random-perturbation stabilization discovered in Ref. [12] is not generic for physically motivated smooth deformations, which is relevant for black-hole spectroscopy and for interpreting overtone instabilities. The paper also makes a useful practical point about degeneracies among multiple perturbation sources. The main caveat, developed below, is that the comparison with random perturbations lacks a quantitative norm control, so the 'in contrast' claim is not yet fully established.

major comments (3)
  1. [Sec. IV A, Fig. 2; abstract] The central contrast with the random-perturbation results of Ref. [12] is not controlled because the energy norm of the deformation δL = L_RZ − L_Schwarzschild is never computed or reported. The ε-pseudospectrum (Eq. 32) diagnoses sensitivity to perturbations of norm ε in the chosen scalar product, and the random perturbations in Ref. [12] have a specific norm relative to the Schwarzschild operator. If ||δL||_E is much larger than the random perturbation amplitudes used in Ref. [12], open pseudospectra at comparable ε would be expected and would not establish that physical deformations behave differently from random ones. To support the abstract's 'in contrast' claim, the authors should compute ||δL||_E (or an equivalent scale) and perform a same-norm comparison; for example, normalize the RZ deformation and random perturbations to equal energy norm and compare their pseudospectra, and/or apply random perturbations to the RZ operator as a positive control.
  2. [Sec. III B and Sec. IV A] The diagnosis of spectral instability is made exclusively with the energy scalar product in Eqs. (33)-(34). The authors acknowledge the norm dependence in Sec. III B, but they do not test whether a different natural scalar product (for instance, the standard L^2 product on the compactified domain) would change the qualitative picture, such as producing closed contours around the RZ QNMs. Since the paper's main conclusion is precisely that the pseudospectra are open and unstable, a sensitivity check with at least one alternative scalar product is needed to establish robustness of the central claim.
  3. [Sec. III C and Sec. IV A] The truncation parameter N is chosen by requiring convergence of the eighth QNM overtone, but the pseudospectrum is a more sensitive object because it involves the resolvent norm. No convergence test of the pseudospectrum level sets with N (e.g., comparing contours for N = 200, 300, 400) is reported. A pseudospectrum-specific convergence check would rule out numerical artifacts as the source of the open contours in Fig. 2.
minor comments (5)
  1. [Sec. IV A, first paragraph] The enumeration of the four parameter configurations is inconsistent with Fig. 2: the text lists (ϵ, a3) = (10^-2, 500) twice for the bottom panels, whereas the caption correctly has (10^-4, 500) for the bottom-left panel.
  2. [Throughout] 'Pseudoespectrum' should be 'pseudospectrum' throughout; there are also typos such as 'ovetones' in Sec. IV A and 'result from' in the caption of Fig. 2.
  3. [Eqs. (31) and (33)] The energy norm in Eq. (33) uses q(σ), while the conformal potential was defined as qℓ(σ) in Eq. (31); the notation should be harmonized.
  4. [Sec. III C, Eqs. (44)-(45)] Calling smin a 'generalized singular value' is misleading; Eq. (45) defines the ordinary smallest singular value of the energy-weighted matrix A, not a generalized singular value in the standard sense. The wording should be adjusted.
  5. [Sec. IV, numerical setup] The optimal AnMR parameters (κ = 2.2 for a3 = 20 and κ = 3.8 for a3 = 500) are stated without showing the dependence of the results on κ; a brief convergence statement or a supplementary figure would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the RZ spectra and pseudospectra are computed from the metric, with self-citations supplying framework and comparison rather than the conclusion.

full rationale

The paper's central quantitative outputs—the RZ QNM spectrum in Fig. 1 and the ε-pseudospectra in Fig. 2—are obtained by solving the eigenvalue problem L u = s u (Eq. 29) for the operator constructed from the RZ metric functions (Eqs. 6-8) and the axial potential (Eq. 16), with no free parameter fitted to the target result. The deformation parameters (ϵ, a3) are fixed by weak-field and near-horizon regularity conditions in Sec. II B, following Ref. [37], but the instability of the RZ overtones and the openness of the pseudospectral level sets are computed outputs rather than inputs. Refs. [12] and [37] involve overlapping authors, yet they are used as a comparison benchmark and as a model-generation framework, respectively; neither citation is invoked to prove that the RZ pseudospectrum is open. The energy-norm choice (Eqs. 33-34) is an explicitly flagged modeling assumption, not a way of building the conclusion into the definition; the paper even notes pseudospectra depend on this choice (Sec. III B). The lack of a quantitatively normalized perturbation amplitude relative to Ref. [12] is a possible comparability weakness, but it is a correctness/robustness concern, not a circular reduction. Overall, no derivation step reduces to its own input.

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

The central results depend on the RZ deformation parameters (epsilon, a3) and on the energy-norm choice for the pseudospectrum. No new physical entities are introduced; the sinusoidal perturbation (47) is a deliberately ad-hoc probe.

free parameters (4)
  • epsilon (RZ deformation parameter) = 1e-4 and 1e-2
    Free metric-deformation parameter in the RZ parametrization; values chosen to represent small deviations consistent with observational bounds (epsilon=1e-4) and a stronger test (epsilon=1e-2).
  • a3 (RZ continued-fraction parameter) = 20 and 500
    Controls the shape of the deformation near the horizon; values chosen to trigger visibly different overtone branches.
  • eta (sinusoidal perturbation amplitude) = 1e-8
    Amplitude of the ad-hoc potential perturbation in Eq. (47); chosen small.
  • k (sinusoidal wavenumber) = 5, 10, 15, 20
    Wavenumber of the ad-hoc potential perturbation; varied to show k-dependence.
assumptions (4)
  • domain assumption The RZ metric (6)-(9) with the imposed conditions (18)-(20) describes a physically relevant Schwarzschild-like black hole and is a valid background for axial perturbations.
    The entire analysis builds on this spacetime; if the model is not physically representative, the conclusion about physical deformations is weakened.
  • domain assumption Axial gravitational perturbations of the RZ spacetime are governed by the Regge-Wheeler-like master equation (14)-(16) assuming a non-dissipative fluid.
    Standard perturbative framework; any extra coupling or dissipation would change the operator.
  • domain assumption The pseudospectrum defined via the energy scalar product (33)-(34) is the appropriate measure of spectral stability.
    Pseudospectra depend on the scalar product, as the authors note; the stability diagnosis is tied to this choice.
  • standard math Chebyshev spectral discretization with AnMR converges to the continuum operator for the considered parameter ranges.
    Assumed standard spectral method convergence; authors check N-convergence for the eighth overtone.

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Pith. "Pith review of Probing the Unstable Spectrum of Schwarzschild-like Black Holes." pith.science (2026). https://pith.science/paper/MNQVLQJL

@misc{pith2026250113815,
  author       = {Pith},
  title        = {Pith review of: Probing the Unstable Spectrum of Schwarzschild-like Black Holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MNQVLQJL}},
  note         = {Machine review of arXiv:2501.13815}
}
read the original abstract

We investigate the pseudospectrum of a Schwarzschild-like spacetime within the framework of black hole perturbation theory to analyze a counterintuitive assertion regarding the instability of quasinormal modes. Recent findings suggest that random perturbations to the effective potential associated with gravitational waves may enhance the stability of the underlying wave operator, thereby yielding a stable spectrum of randomly displaced quasinormal modes. Given the unphysical nature of such random perturbations, this work examines these findings within a spacetime that inherently exhibits a perturbed quasinormal spectrum. We find that, in contrast to the QNM spectrum of the Schwarzschild spacetime under random perturbations, the quasinormal spectrum of a Schwarzschild-like black hole deformed through a physically motivated implementation of the Rezzolla-Zhidenko parametrization is unstable. In particular, we show that the pseudospectra of these Schwarzschild-like black holes do not display the typical features associated with wave operators that yield stable quasinormal spectra. We corroborate our findings by computing the quasinormal spectra when additional (ad-hoc) deformations are added to the effective potential of the Rezzolla-Zhidenko black hole. We also argue that when multiple perturbation sources are present, identifying the origin of the instability may be difficult.

Figures

Figures reproduced from arXiv: 2501.13815 by the authors.

Figure 1
Figure 1. FIG. 1. Left Panel: Density distribution for the RZ spacetime parametrized by [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. QNM spectra of a Schwarzschild-like black hole de [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. QNM spectra for the Schwarzschild and RZ spacetimes, with associated black hole potentials perturbed by the same [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

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Forward citations

Cited by 3 Pith papers

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

  1. Spectrum instability and greybody factor stability for parabolic approximation of Regge-Wheeler potential

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    Replacing the Regge-Wheeler potential by piecewise parabolas makes quasinormal-mode spectra unstable, with long-lived overtones, while greybody factors stay close to the exact Schwarzschild result.

  2. Bound States of the Schwarzschild Black Hole

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    The bound states of the inverted Regge-Wheeler potential are exponentially condensed near zero energy and strongly delocalized, linking black hole overtone instability to long-range potential features.

  3. Exceptional line and pseudospectrum in black hole spectroscopy

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Reference graph

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