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arxiv: 2506.16217 · v2 · submitted 2025-06-19 · 🌀 gr-qc

Quasinormal modes and grey-body factors of axial gravitational perturbations of regular black holes in asymptotically safe gravity

Qi-Long Shi , Rui Wang , Wei Xiong , Peng-Cheng Li This is my paper

Pith reviewed 2026-05-19 09:02 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasinormal modesgrey-body factorsregular black holesasymptotically safe gravityaxial gravitational perturbationsWKB approximationblack hole perturbations
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The pith

Axial perturbations of regular black holes in asymptotically safe gravity show weak effect on fundamental modes but notable deviations in higher overtones.

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

This paper investigates the quasinormal modes arising from axial gravitational perturbations around a regular black hole solution in asymptotically safe gravity. Using the Bernstein spectral method and the asymptotic iteration method, the authors compute the QNM spectrum with high precision for various overtones. They find that the fundamental mode frequencies are only weakly influenced by the deviation parameter that regularizes the black hole, while higher overtones exhibit significant departures from the standard Schwarzschild values. The study also verifies that grey-body factors computed via the sixth-order WKB approximation agree closely with those inferred from the QNMs, especially when the multipole number l is large.

Core claim

The regular black hole metric in asymptotically safe gravity admits axial gravitational perturbations whose quasinormal mode spectrum deviates from the Schwarzschild case primarily in the higher overtones, with the fundamental mode remaining largely unaffected by the regularization parameter, and with grey-body factors showing excellent correspondence to the QNM data for large angular momenta.

What carries the argument

The regular black hole background metric with a deviation parameter, solved via the Bernstein spectral method and asymptotic iteration method for the perturbation master equation to obtain the complex QNM frequencies.

If this is right

  • The fundamental quasinormal mode is robust against the regularization effects of asymptotically safe gravity.
  • Higher overtones are more sensitive probes of the deviation from classical general relativity.
  • Grey-body factors can be accurately estimated from quasinormal mode data using WKB methods for high multipoles.
  • The numerical techniques employed allow reliable computation of the spectrum even for highly damped modes.

Where Pith is reading between the lines

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

  • If ringdown observations detect higher overtones, they could constrain the deviation parameter in this model.
  • The stability under axial perturbations is preserved similar to Schwarzschild for low modes.
  • This analysis could be extended to other perturbation types or different regular black hole solutions to test universality.

Load-bearing premise

The proposed regular black hole solution serves as an accurate background metric for deriving and solving the linear axial perturbation equations without introducing errors that change the quasinormal spectrum.

What would settle it

Finding that the quasinormal frequencies for the first overtone remain identical to the Schwarzschild values for any nonzero value of the deviation parameter would contradict the reported deviations.

Figures

Figures reproduced from arXiv: 2506.16217 by Peng-Cheng Li, Qi-Long Shi, Rui Wang, Wei Xiong.

Figure 1
Figure 1. Figure 1: FIG. 1: The metric function [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The effective potential [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: Complex frequencies of the fundamental [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: Re( [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Re( [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Left panel: Grey-body factors of the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Left panel: Grey-body factors of the [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

In this paper, we present a detailed study of axial gravitational perturbations of the regular black hole solution in asymptotically safe gravity, as proposed in \cite{Bonanno:2023rzk}. We analyze the quasinormal mode (QNM) spectrum of this black hole using two numerical techniques: the Bernstein spectral method and the asymptotic iteration method (AIM). These approaches allow us to compute QNM frequencies with high accuracy, even for higher overtones. Our results show that the fundamental mode is only weakly affected by the deviation parameter, whereas notable deviations from the Schwarzschild case emerge for higher overtones. Additionally, we examine the correspondence between grey-body factors and QNMs using the sixth-order WKB approximation, finding excellent agreement, especially for larger multipole numbers l.

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 manuscript computes quasinormal modes of axial gravitational perturbations for the regular black-hole metric of Bonanno:2023rzk in asymptotically safe gravity. Two independent numerical schemes (Bernstein spectral method and asymptotic iteration method) are used to obtain the spectrum, with the central claim that the fundamental mode depends only weakly on the deviation parameter while higher overtones exhibit clear deviations from the Schwarzschild case. Grey-body factors are additionally evaluated via the sixth-order WKB approximation and reported to agree well with the QNM data, especially at large multipole number l.

Significance. If the axial master equation is correctly obtained from the linearized effective field equations rather than by direct substitution into the GR Regge-Wheeler operator, the results would supply concrete, falsifiable predictions for how asymptotically safe corrections modify ringdown signals and transmission probabilities. The dual numerical methods and the explicit WKB-QNM comparison constitute reproducible computational evidence that strengthens the work relative to single-method studies.

major comments (2)
  1. [§3 (perturbation equation derivation)] §3 (or the section deriving the perturbation equation): the axial master equation appears to be obtained by inserting the regular metric directly into the standard Regge-Wheeler form. Linearization of the full effective field equations of asymptotically safe gravity can generate additional potential terms absent from the GR operator; these terms would alter the QNM spectrum, particularly the higher overtones whose deviations are highlighted as the main result. This step is load-bearing for both the reported mode shifts and the claimed WKB agreement.
  2. [§5 (numerical results)] §5 (numerical results): no explicit convergence tests, error bars, or tabulated precision for the deviation parameter are provided, even though the abstract asserts “high accuracy” for higher overtones. Without these diagnostics it is impossible to quantify how much of the reported deviation from Schwarzschild is physical versus numerical artifact.
minor comments (2)
  1. [Abstract] Abstract: the phrase “excellent agreement” for the WKB-QNM comparison should be accompanied by a quantitative measure (e.g., relative difference for l ≥ 3) rather than a qualitative statement.
  2. [Introduction / parameter definition] Notation: the deviation parameter is introduced without an explicit range or normalization; a short table or sentence stating the interval explored would improve reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive report. The comments on the derivation of the perturbation equation and the numerical validation are well taken, and we address each point below with specific plans for revision.

read point-by-point responses

  1. Referee: §3 (or the section deriving the perturbation equation): the axial master equation appears to be obtained by inserting the regular metric directly into the standard Regge-Wheeler form. Linearization of the full effective field equations of asymptotically safe gravity can generate additional potential terms absent from the GR operator; these terms would alter the QNM spectrum, particularly the higher overtones whose deviations are highlighted as the main result. This step is load-bearing for both the reported mode shifts and the claimed WKB agreement.

    Authors: We appreciate the referee highlighting the importance of a rigorous derivation from the effective field equations. In the manuscript the axial master equation was constructed by direct substitution of the regular metric into the Regge-Wheeler operator, which is the standard approach when the background satisfies the modified equations. To address the concern directly, we have now carried out the linearization of the asymptotically safe effective equations for axial (odd-parity) perturbations. This calculation shows that no additional potential terms appear in the axial sector; the only modifications enter through the background metric function. We will insert the explicit derivation and the resulting master equation into the revised §3, together with a brief explanation of why extra terms vanish for this parity. This step confirms that the reported deviations for higher overtones remain physical rather than an artifact of an incomplete operator. revision: yes

  2. Referee: §5 (numerical results): no explicit convergence tests, error bars, or tabulated precision for the deviation parameter are provided, even though the abstract asserts “high accuracy” for higher overtones. Without these diagnostics it is impossible to quantify how much of the reported deviation from Schwarzschild is physical versus numerical artifact.

    Authors: We agree that the absence of explicit convergence diagnostics and error estimates weakens the presentation of the numerical results. In the revised manuscript we will add a dedicated subsection to §5 that reports convergence tests for both the Bernstein spectral method and the asymptotic iteration method. These will include tables of QNM frequencies versus resolution/iteration order, estimated absolute and relative errors, and error bars on the real and imaginary parts as functions of the deviation parameter. The updated figures and tables will allow readers to assess directly the accuracy of the deviations observed in the higher overtones. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external background metric and standard numerical methods

full rationale

The paper adopts the regular black hole metric from the cited reference Bonanno:2023rzk as input background, then applies standard numerical techniques (Bernstein spectral method, asymptotic iteration method, and sixth-order WKB) to solve the axial perturbation equations for QNMs and grey-body factors. No equations in the provided text reduce a claimed prediction or result to a fitted parameter or self-referential definition by construction. The central results (weak effect on fundamental mode, deviations in overtones, QNM-greybody agreement) follow from direct computation on the given metric rather than from any self-definitional loop or load-bearing self-citation chain. This is a normal, self-contained numerical study with independent content.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central results rest on the validity of the background metric from the cited 2023 paper and standard linear perturbation theory in GR; the deviation parameter is inherited from that model rather than fitted here.

free parameters (1)
  • deviation parameter
    Controls the strength of quantum correction in the regular black hole metric; introduced in the cited prior work and treated as input for the perturbation analysis.
axioms (2)
  • domain assumption The regular black hole metric satisfies the field equations of asymptotically safe gravity.
    Invoked by adopting the solution proposed in Bonanno:2023rzk as the background spacetime.
  • standard math Linear axial gravitational perturbations obey the standard wave equation derived from the Einstein equations or modified gravity action on this background.
    Standard assumption in black hole perturbation theory used to set up the QNM eigenvalue problem.

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Forward citations

Cited by 4 Pith papers

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