arxiv: 2604.24926 · v1 · submitted 2026-04-27 · 🌀 gr-qc

Recognition: unknown

Sensitivity of black hole spectral instability against perturbations of the effective potential

Ramin G. Daghigh , Michael D. Green , Guan-Ru Li , Jodin C. Morey , Wei-Liang Qian , Stefan J. Randow

Authors on Pith no claims yet

Pith reviewed 2026-05-08 01:41 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black holesquasinormal modesspectral instabilityeffective potentialgravitational wavesringdownperturbation theory

0 comments

The pith

Small perturbations to the effective potential strongly shift the fundamental quasinormal mode of black holes.

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

The paper examines how black hole quasinormal modes respond to small changes in the effective potential of the wave equation. It finds that the fundamental mode can shift substantially in frequency and damping, contrary to expectations of robustness. Analytical and numerical examples illustrate different patterns of this sensitivity. The effect depends on the unperturbed potential shape and the radial profile of the added perturbation. A reader would care because these modes determine the ringdown phase of gravitational waves from black hole mergers.

Core claim

The authors establish that black hole spectral instability appears as a strong dependence of the fundamental quasinormal mode on small perturbations of the effective potential in the linearized wave equation. These perturbations arise from small-scale modifications of the spacetime metric. The qualitative character of the resulting shifts varies with the potential shape and the way the perturbation evolves away from the black hole. Several distinct examples are worked out both analytically and numerically to show the diversity of the instability.

What carries the argument

Perturbations added to the effective potential in the radial Schrödinger-like wave equation for linearized perturbations around a black hole.

Load-bearing premise

Small modifications to the spacetime metric produce perturbations to the effective potential that remain small enough not to change the global structure or asymptotic properties of the black hole spacetime.

What would settle it

A numerical integration of the perturbed wave equation in which a concrete small bump added to the potential produces no measurable change in the lowest quasinormal mode frequency or decay rate.

Figures

Figures reproduced from arXiv: 2604.24926 by Guan-Ru Li, Jodin C. Morey, Michael D. Green, Ramin G. Daghigh, Stefan J. Randow, Wei-Liang Qian.

**Figure 1.** Figure 1: FIG. 1 view at source ↗

**Figure 2.** Figure 2: FIG. 2 view at source ↗

**Figure 3.** Figure 3: FIG. 3 view at source ↗

**Figure 4.** Figure 4: FIG. 4 view at source ↗

**Figure 5.** Figure 5: FIG. 5 view at source ↗

**Figure 6.** Figure 6: FIG. 6 view at source ↗

**Figure 7.** Figure 7: FIG. 7 view at source ↗

read the original abstract

Black hole spectral instability is counterintuitive and contradicts many plausible assumptions of the properties of black hole quasinormal modes. The present study aims to explore different types of instability phenomena. It is understood that the fundamental mode is surprisingly sensitive to small perturbations of the effective potential of the linearized wave equation. Such perturbations can be produced by small-scale modifications of the spacetime metrics. From both the analytical and numerical perspectives, we elaborate on a few qualitatively different examples illustrating the strong sensitivity and diversity in black hole spectral instability caused by such effective potential perturbations. It turns out that the qualitative way in which the fundamental mode becomes perturbed depends on many factors such as the shape of the potential and how the perturbation changes as it moves away from the central black hole.

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.

Desk Editor's Note private letter to a colleague

The paper shows through examples that small changes to the effective potential can shift the fundamental quasinormal mode in several distinct ways, but it does not derive which of those changes actually arise from allowed metric perturbations.

read the letter

The main takeaway is that the fundamental quasinormal mode reacts strongly to modest alterations in the effective potential, and the authors lay out several qualitatively different cases to illustrate how the outcome depends on the perturbation's shape and radial profile. They treat both analytical approximations and numerical computations, which helps make the sensitivity concrete rather than purely formal. That part adds useful detail to the existing literature on spectral instability by showing the range of behaviors instead of a single generic statement. The examples are straightforward to follow and highlight factors like how the perturbation decays away from the horizon. This is the sort of incremental clarification that can be handy for people running mode calculations in modified spacetimes. The soft spot is the missing step between an arbitrary delta-V and a perturbation that comes from a real metric change. The text asserts that small-scale metric modifications can generate these potential shifts, yet it does not construct an explicit map that preserves asymptotic flatness, regularity, or the linearized Einstein equations. Without that link, it is unclear whether the chosen perturbations are representative or whether they could be ruled out by physical constraints. The numerical sections would also benefit from more explicit convergence tests and error estimates, which are not summarized in enough detail to judge robustness at a glance. Overall this is aimed at specialists already working on black-hole perturbation theory and quasinormal-mode stability. A reader outside that niche will not get much new physical insight because the metric-to-potential connection stays loose. I would still send it to referees. The examples are worth having in the record, and the authors engage the topic directly; a revision could tighten the physical mapping without changing the core contribution.

Referee Report

1 major / 0 minor

Summary. The manuscript investigates black hole spectral instability, focusing on the surprising sensitivity of the fundamental quasinormal mode to small perturbations of the effective potential in the linearized wave equation. It posits that such perturbations arise from small-scale modifications to the spacetime metric and illustrates the resulting instability through several qualitatively different analytical and numerical examples, emphasizing that the qualitative behavior depends on the shape of the potential and the radial location of the perturbation away from the black hole.

Significance. If the results hold, the work would demonstrate an important fragility in black hole quasinormal mode spectra, with potential consequences for the reliability of black hole spectroscopy in gravitational-wave observations. The use of diverse analytical and numerical examples to show varying instability behaviors is a positive aspect, providing concrete illustrations of the sensitivity phenomenon.

major comments (1)

Abstract: The claim that perturbations of the effective potential 'can be produced by small-scale modifications of the spacetime metrics' is asserted without derivation. No explicit map is constructed from a concrete metric perturbation (satisfying the linearized Einstein equations, asymptotic flatness, or regularity) to the corresponding change in the effective potential, leaving open whether the chosen examples are representative of physically allowed metric changes.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thoughtful review and constructive criticism of our manuscript. We address the major comment below and indicate where revisions will be made to improve clarity.

read point-by-point responses

Referee: Abstract: The claim that perturbations of the effective potential 'can be produced by small-scale modifications of the spacetime metrics' is asserted without derivation. No explicit map is constructed from a concrete metric perturbation (satisfying the linearized Einstein equations, asymptotic flatness, or regularity) to the corresponding change in the effective potential, leaving open whether the chosen examples are representative of physically allowed metric changes.

Authors: We appreciate the referee highlighting this point. The core contribution of the manuscript is a mathematical investigation of the sensitivity of black hole quasinormal mode spectra to small perturbations in the effective potential, illustrated through several analytical and numerical examples that exhibit qualitatively distinct instability behaviors. The reference to small-scale modifications of the spacetime metric is intended solely as physical motivation for why such potential perturbations might arise in modified gravity scenarios or with additional matter fields, rather than as a derived result. We did not construct an explicit map from a metric perturbation obeying the linearized Einstein equations (with appropriate boundary conditions) to the induced change in the effective potential, as this would require a separate analysis of specific metric deformations and lies outside the present scope. Our examples are chosen to demonstrate the range of possible instability phenomena depending on potential shape and perturbation location; they are not asserted to be exhaustive or directly representative of every physically allowed metric change. We will revise the abstract and the relevant introductory paragraph to qualify the statement, emphasizing its motivational character and clarifying that the work studies the wave equation directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity; analysis is direct and self-contained

full rationale

The paper illustrates sensitivity of quasinormal modes to ad hoc perturbations of the effective potential via analytical and numerical examples. No derivation reduces to self-definition, fitted inputs renamed as predictions, or load-bearing self-citations. The central claims rest on explicit computations of mode shifts under varied perturbation shapes and supports, without tautological loops or imported uniqueness theorems from the authors' prior work. The skeptic concern about mapping to metric perturbations is a question of physical representativeness, not circularity in the mathematical steps.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Relies on established mathematical frameworks in general relativity without new parameters or entities.

axioms (2)

standard math The wave equation for perturbations around black holes can be reduced to a Schrödinger-like equation with an effective potential.
This is a standard reduction in black hole perturbation theory for Schwarzschild or similar metrics.
domain assumption Small modifications to the spacetime metric can be modeled as perturbations to the effective potential.
Assumed in the study to explore sensitivity.

pith-pipeline@v0.9.0 · 5441 in / 1155 out tokens · 65995 ms · 2026-05-08T01:41:30.339843+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 16 canonical work pages · 2 internal anchors

[1]

LIGO Scientific, Virgo, B. P. Abbottet al., Phys. Rev. Lett.116, 061102 (2016), arXiv:1602.03837

work page internal anchor Pith review arXiv 2016
[2]

Nollert, Phys

H.-P. Nollert, Phys. Rev.D53, 4397 (1996), arXiv:gr-qc/9602032

work page arXiv 1996
[3]

Quantifying excita tions of quasinormal mode systems

H.-P. Nollert and R. H. Price, J. Math. Phys.40, 980 (1999), arXiv:gr-qc/9810074

work page arXiv 1999
[4]

J. M. Aguirregabiria and C. V. Vishveshwara, Phys. Lett. A210, 251 (1996)

1996
[5]

C. V. Vishveshwara, Curr. Sci.71, 824 (1996)

1996
[6]

R. G. Daghigh, M. D. Green, and J. C. Morey, Phys. Rev.D101, 104009 (2020), arXiv:2002.07251

work page arXiv 2020
[7]

W.-L. Qian, K. Lin, C.-Y. Shao, B. Wang, and R.-H. Yue, Phys. Rev.D103, 024019 (2021), arXiv:2009.11627

work page arXiv 2021
[8]

Liuet al., Phys

H. Liuet al., Phys. Rev.D104, 044012 (2021), arXiv:2104.11912

work page arXiv 2021
[9]

J. L. Jaramillo, R. Panosso Macedo, and L. Al Sheikh, Phys. Rev. X11, 031003 (2021), arXiv:2004.06434

work page arXiv 2021
[10]

M. H.-Y. Cheung, K. Destounis, R. P. Macedo, E. Berti, and V. Cardoso, Phys. Rev. Lett.128, 111103 (2022), arXiv:2111.05415

work page arXiv 2022
[12]

L. Hui, D. Kabat, and S. S. C. Wong, JCAP12, 020 (2019), arXiv:1909.10382

work page arXiv 2019
[13]

R. G. Daghigh, G.-R. Li, W.-L. Qian, and S. J. Randow, (2025), arXiv:2502.05354

work page arXiv 2025
[14]

Qian, G.-R

W.-L. Qian, G.-R. Li, R. G. Daghigh, S. Randow, and R.-H. Yue, Phys. Rev. D111, 024047 (2025), arXiv:2409.17026

work page arXiv 2025
[15]

Quasinormal modes of black holes and black branes

E. Berti, V. Cardoso, and A. O. Starinets, Class. Quant. Grav.26, 163001 (2009), arXiv:0905.2975

work page internal anchor Pith review arXiv 2009
[16]

Wang, Braz

B. Wang, Braz. J. Phys.35, 1029 (2005), arXiv:gr-qc/0511133

work page arXiv 2005
[17]

Nollert, Class

H.-P. Nollert, Class. Quant. Grav.16, R159 (1999)

1999
[18]

Shen, W.-L

S.-F. Shen, W.-L. Qian, K. Lin, C.-G. Shao, and Y. Pan, Class. Quant. Grav.39, 225004 (2022), arXiv:2203.14320

work page arXiv 2022
[19]

Kyutoku, H

K. Kyutoku, H. Motohashi, and T. Tanaka, Phys. Rev. D107, 044012 (2023), arXiv:2206.00671

work page arXiv 2023
[20]

E. W. Leaver, Proc. Roy. Soc. Lond.A402, 285 (1985)

1985
[21]

Li, G-R., Morey, J.C., Qian, W-L., Daghigh, R.G., Green, M.D., Lin, K., and Yue, R-H., (2026), arXiv:2602.06536

work page arXiv 2026