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High-precision spectral methods uncover the full quasinormal spectrum of the Bonanno-Reuter regular black hole, including overtones and overdamped modes missed by WKB.

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load-bearing objection Solid high-precision QNM catalogue for Bonanno-Reuter that actually fills the overdamped gap left by WKB; methodologically careful, model-dependent by construction.

arxiv 2607.10199 v1 pith:777BNECB submitted 2026-07-11 gr-qc

Quasinormal modes of Bonanno-Reuter black holes via the Spectral Method

Davide Batic , Denys Dutykh , Fabio Scardigli This is my paper
classification gr-qc
keywords quasinormal modesBonanno-Reuter black holeasymptotically safe gravityspectral methodregular black holesoverdamped modesovertones
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper computes the quasinormal modes of the Bonanno-Reuter black hole, a singularity-free geometry obtained by improving the Schwarzschild metric with a running Newton coupling from asymptotically safe gravity. The authors use a spectral method on a compactified radial interval to extract the frequencies of scalar, electromagnetic and gravitational perturbations, both away from and at extremality. The method recovers the classical Schwarzschild spectrum for large mass and, for the first time for this geometry, systematically resolves long towers of overtones together with purely imaginary overdamped modes. In the large-mass limit the spacing of those overdamped modes approaches the surface gravity; near extremality it is instead set by a finite near-horizon length scale of the double-zero throat. The work therefore shows that high-accuracy spectral techniques can reveal geometric signatures of the regular core that semi-analytic WKB approximations routinely miss, and that those signatures may become observationally relevant for light or primordial black holes.

Core claim

The spectral method applied to the Bonanno-Reuter metric yields the complete quasinormal spectrum for spin-0,1,2 perturbations, including extensive overtone towers and families of purely imaginary overdamped modes that previous WKB analyses of the same background entirely overlooked; the spacing of the high-damping modes is controlled by surface gravity at large mass and by the near-horizon throat length at extremality.

What carries the argument

The spectral method: after the radial master equation is transformed so that the quasinormal boundary conditions are built into a regular function on the compact interval [-1,1], that function is expanded in Chebyshev polynomials and collocated, converting the problem into a quadratic matrix eigenvalue problem whose stable roots are the physical frequencies.

Load-bearing premise

The whole spectrum rests on one particular phenomenological choice of how the renormalisation-group scale is identified with the radial coordinate, together with two fixed numerical values of the free parameters; different but equally motivated choices would produce different metrics and therefore different spectra.

What would settle it

Recompute the same high-overtone and overdamped frequencies with an independent high-precision method (for example continued-fraction or time-domain integration at the same multipoles and masses) and check whether the evenly spaced overdamped ladders and the extremal spacing 1/(3L) reappear.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 4 minor

Summary. The manuscript computes the quasinormal-mode spectrum of the Bonanno–Reuter (RG-improved Schwarzschild) black hole for scalar, electromagnetic and vector-type gravitational perturbations, both in the non-extremal and extremal regimes. After a careful Frobenius/Olver asymptotic analysis that extracts the correct ingoing/outgoing factors, the radial master equation is transformed into a regular Chebyshev collocation problem on [-1,1] and solved with multiprecision arithmetic. The resulting spectra recover the Schwarzschild limit at large mass, reproduce low-lying modes previously obtained by WKB/Leaver methods where those methods are reliable, and reveal extensive towers of overtones together with previously unreported, nearly equally spaced purely imaginary overdamped modes whose spacing asymptotes to the surface gravity (non-extremal) or to 1/(3L) of the near-horizon AdS2 throat (extremal).

Significance. If the numerical spectra are correct, the work supplies the first high-precision, spin-complete QNM catalogue for a canonical ASG-inspired regular black hole, including the overdamped sector that semi-analytic WKB approximations systematically miss. The geometric interpretation linking the high-damping spacing to surface gravity or to the extremal throat length L is a concrete, falsifiable diagnostic of the short-distance completion. The multiprecision Chebyshev implementation, N-stability checks and explicit Schwarzschild recovery constitute a reproducible methodological advance that can be applied to other quantum-corrected metrics. These results are of clear interest for both the theoretical study of regular black holes and for future precision ringdown analyses of light or primordial black holes.

minor comments (4)
  1. In the extremal analysis (around Eqs. (108)–(116) and the discussion in §VI.D) the factor 1/3 that converts the throat length L into the observed spacing is presented as a numerical coincidence that “should be regarded as a property of the global connection problem.” A short remark clarifying whether this factor can be derived from the monodromy or Stokes structure of the radial equation, or whether it remains an empirical observation, would strengthen the physical interpretation without altering any numerical result.
  2. Tables VI–XXIX are extremely dense. Adding a brief caption note that only a representative subset of the computed overtones is shown (and that the full data set is available upon request or in a repository) would improve readability.
  3. A few typographical slips remain: “Ultaviolet” in the reference list, occasional missing spaces around mathematical operators, and the inconsistent use of “overdamped” versus “purely imaginary.” These are easily corrected in proof.
  4. The phenomenological character of the scale-setting k(r) is correctly acknowledged in the conclusions; a single sentence in the introduction reminding the reader that the spectra are specific to the Bonanno–Reuter choice of k(r) would make the scope even clearer for non-specialists.

Circularity Check

0 steps flagged

No significant circularity: independent high-precision numerical solution of the QNM eigenvalue problem for a fixed, externally specified metric.

full rationale

The paper takes the Bonanno–Reuter metric (with literature values α=118/(15π), γ=9/2 and the phenomenological k(r) of Bonanno–Reuter) as given input, derives the radial master ODE and QNM boundary conditions from first principles via Frobenius/Olver asymptotics, regularizes the problem by extracting the known singular factors, and discretizes the resulting regular ODE on [−1,1] by Chebyshev collocation. The quadratic matrix eigenvalue problem is solved with multi-precision arithmetic and N-stability checks; the only external anchors are the large-M recovery of the known Schwarzschild spectrum and direct numerical comparison of low-lying modes against independent WKB/Leaver results of other groups. No parameter is fitted to any QNM data, no uniqueness theorem is imported from the authors’ prior work to force the spectrum, and the observed overdamped spacings (surface gravity or 1/(3L)) are post-hoc geometric interpretations of independently computed eigenvalues, not inputs. The Spectral Method itself is cited from the authors’ earlier papers, but that citation is methodological scaffolding, not a load-bearing premise that reduces the present spectra to a prior definition; the calculation for this metric stands on its own numerics. Hence the derivation chain is self-contained against external benchmarks and exhibits no circular reduction.

Axiom & Free-Parameter Ledger

2 free parameters · 3 axioms · 0 invented entities

The calculation rests on the standard ASG-improved Schwarzschild metric with a specific phenomenological cutoff identification and two fixed numerical constants; once those are granted, the rest is ordinary linear perturbation theory plus high-order spectral numerics. No new dynamical entities are postulated.

free parameters (2)
  • α = 118/(15π)
    Dimensionless RG parameter fixed by hand to the literature value 118/(15π); controls the strength of the quantum correction and the core curvature.
  • γ = 9/2
    Dimensionless parameter in the infrared cutoff identification k(r); fixed to 9/2 following Bonanno-Reuter and subsequent papers.
axioms (3)
  • domain assumption The running Newton coupling takes the form G(k)=G_N/(1+α G_N k²/ℏ) and the cutoff is identified with k(r)=ℏ(r+γ G_N M/r³)^{1/2}.
    Standard phenomenological input of the ASG black-hole literature (eqs. (1)–(3)); not derived from first principles inside the paper.
  • standard math Massless spin-s perturbations reduce to a single Schrödinger-type master equation with the effective potential U_ε(r)=F(r)[ε/r F'+ℓ(ℓ+1)/r²].
    Classic Regge-Wheeler / Zerilli reduction for static spherical backgrounds; used without re-derivation.
  • domain assumption QNM boundary conditions are purely ingoing at the horizon and purely outgoing at infinity.
    Standard definition of quasinormal modes for open black-hole systems.

pith-pipeline@v1.1.0-grok45 · 58152 in / 2508 out tokens · 27153 ms · 2026-07-14T13:33:12.163337+00:00 · methodology

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In this work, we explore the quasinormal modes (QNMs) of the Bonanno-Reuter black hole, one of the first regular black hole metric suggested by the Asymptotically Safe Gravity (ASG) program. The running parameter $\alpha$ is set to a positive value, the related running Newton coupling vanishes at high energies, fully achieving an ultraviolet fixed point and eliminating non-physical UV divergences. This yields a singularity-free geometry. Hence, we focus on the resulting renormalisation-group-improved Schwarzschild metric, which naturally produces an (Anti)deSitter non-singular core. On the basis of this background, we compute the QNM spectrum for scalar, electromagnetic, and gravitational perturbations by employing the Spectral Method (SM). This method, recognised for its enhanced precision compared to high-order WKB methods, allows the identification of fundamental modes, extensive collections of overtones, and purely imaginary overdamped modes that were entirely missed in previous analyses. These characteristics, resolved here for the first time in the Bonanno-Reuter black hole, underscore the crucial importance of high-precision spectral methods in investigating delicate signatures of black hole models inspired by quantum gravity.

Figures

Figures reproduced from arXiv: 2607.10199 by Davide Batic, Denys Dutykh, Fabio Scardigli.

Figure 1
Figure 1. Figure 1: FIG. 1: Lapse function [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

discussion (0)

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Reference graph

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    only considered the caseM= 4 using a sixth-order WKB approximation, whereas our analysis covers the entire range 3.503≤M≤9.503. For the nearly extremal caseM= 3.503, we detect only one QNM and 35 overdamped modes, a subset of which is listed in Table XVII. ForM= 4, we find good agreement with the results of [13] up toN= 4. However, the accuracy of their o...

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    22 TABLE VI: QNMs for scalar perturbations of the non-extremal Bonanno-Reuter black hole forα= 118 15π ,γ= 9/2,ℓ= 0, and different values of the mass parameterM

    We note however that considering values of the mass parameterM <1 (in Planck units) may lack physical justification within this framework, as such configurations would fall outside the regime where a semiclassical description remains valid, and the concept of a self-gravitating droplet becomes questionable. 22 TABLE VI: QNMs for scalar perturbations of th...