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Resonance of black hole quasinormal modes in coupled systems

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Coupled black hole quasinormal modes from different fields repel and amplify at avoided crossings.

desk verdict Careful, honest formalism for coupled-field QNM excitation, with a real avoided-crossing example in EMA theory—but the claimed amplification may be partly a normalization artifact and needs a time-domain check. read the letter →

arxiv 2505.03883 v2 pith:PFRY6H4I submitted 2025-05-06 gr-qc hep-th

Takuya Takahashi , Hayato Motohashi , Kazufumi Takahashi This is my paper · ORCID
classification gr-qchep-th PACS 04.70.-s04.30.-w
keywords quasinormalmodesblackholeperturbationtheoryexcitationfactorsavoidedcrossingresonanceEinstein-Maxwell-axionringdowncontinuedfractionmethod
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 claims that when black hole perturbation fields are coupled, the longest-lived quasinormal modes of different fields can repel each other in the complex-frequency plane, and that the ease with which these modes are excited is sharply amplified at the repulsion point. It proposes a basis-invariant excitation factor $B_{\alpha n}$ for coupled systems, Eq. (21), and tests it on an Einstein-Maxwell-axion black hole, where the axion-photon coupling mixes gravitational, electromagnetic, and axion perturbations. The observed amplification is a resonance signature analogous to the one previously seen between Kerr overtones, but here it occurs between fundamental modes of different degrees of freedom, which is a new regime. If the claim stands, ringdown signals from such theories would show a distinctive, parameter-dependent resonant enhancement, giving a concrete way to search for extra fields around black holes.

What carries the argument

Three ingredients carry the argument. The basis-invariant excitation factor $B_{\alpha n}=i\det A^{(out)}(2i\omega_{\alpha n})^{-N}\left[d\det A^{(in)}/d\omega\right]^{-1}$ in Eq. (21), built from the coefficient matrices of in-going and outgoing waves, generalizes the single-field excitation factor while removing the field-basis ambiguity. The coupled master equations (48)--(50) for $\psi$, $Z_+$, and $Z_-$ describe the axial sector of the Einstein-Maxwell-axion system, with coupling terms proportional to $g_{a\gamma\gamma}Q$. A matrix-valued Leaver continued fraction computes the QNM frequencies, and a two-level non-Hermitian model $\omega_{\pm}^2=E_c\pm\sqrt{E_d^2+\Delta^2}$ explains why the excitation-factor difference follows a square-root Lorentzian near the crossing.

What would settle it

Time-evolve the axial perturbation system (48)--(50) with a localized broadband source for $Q/M=0.1$ and $g_{a\gamma\gamma}$ on both sides of 28, then decompose the late-time ringdown; the claimed resonance is ruled out if the gravity and EM channel amplitudes do not peak where $|\omega_{g0}-\omega_{e0}|$ is smallest. Alternatively, fix $g_{a\gamma\gamma}Q/M\simeq 2.83$ and reduce $Q/M$; the peak of $|B_{g0}-B_{e0}|$ should grow roughly as $(Q/M)^{-1}$ until the large-coupling instability intervenes.

Watch

Extended reading notes

Core claim

On a Reissner-Nordström black hole with $l=2$ and $Q/M=0.1$, the gravity-led and EM-led fundamental quasinormal frequencies approach each other as the axion-photon coupling $g_{a\gamma\gamma}$ grows, then repel in both real and imaginary parts near $g_{a\gamma\gamma}\simeq 28$; the same repulsion appears when $Q/M$ is varied at fixed $g_{a\gamma\gamma}=10$, near $Q/M\simeq 0.26$. The paper shows that the excitation factors $B_{g0}$ and $B_{e0}$ of the participating modes are amplified at these parameter values, with $|B_{g0}-B_{e0}|\propto |\omega_{g0}-\omega_{e0}|^{-1}$, and reproduces the peak shape with a square-root Lorentzian model. This establishes, in the authors' formulation, a new resonance phenomenon specific to coupled perturbation systems, distinct from the overtone resonance found in Kerr black holes.

Load-bearing premise

The load-bearing premise is that the coupled master equations (48)--(50) faithfully represent the axial perturbations of the Einstein-Maxwell-axion black hole and that each QNM branch can be unambiguously labeled by its decoupling-limit origin; if either part fails, the resonance between fundamental modes of different fields is not established.

Editorial extensions

If this is right

  • Resonant amplification can occur between the longest-lived quasinormal modes of different fields, not only between highly damped overtones of a single field, widening the observational window for black hole spectroscopy.
  • The definition of $B_{\alpha n}$ in Eq. (21) applies to any system of $N$ coupled perturbation variables with a common horizon radius, so it provides a general diagnostic for resonance in modified-gravity and dark-matter-inspired models.
  • In the Einstein-Maxwell-axion system, the resonance location is controlled by the combination $g_{a\gamma\gamma}Q/M$ for small $Q/M$, with the spectral-repulsion width narrowing as $Q$ decreases and the peak excitation growing as $Q^{-1}$.
  • The same system becomes linearly unstable for large coupling, with a purely imaginary mode turning unstable, so the resonant behavior is confined to the stable, moderate-coupling regime.
  • Because the avoided crossing occurs at nearly the same parameter values across overtones, the resonance is not a fine-tuned single-mode effect but a robust spectral feature.

Reading between the lines

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

  • If the basis-invariant excitation factor is adopted as the standard measure, the resonance could be used to constrain axion-photon couplings from ringdown amplitude ratios, not just frequencies, once gravitational and electromagnetic channels are separately resolved.
  • The authors' two-level model suggests that the square-root-Lorentzian peak shape is generic for coupled fields, while the lemniscate shape in the Kerr case is specific to overtone coupling; a systematic survey of two-field systems could test this distinction.
  • The $Q\to 0$ divergence of the peak excitation hints that extremely small charges with large couplings could produce very narrow resonances, but the simultaneous appearance of instability sets an upper bound on the attainable amplification.
  • A direct time-domain simulation of a pulse in the axial sector around an EMA black hole would convert the frequency-domain resonance prediction into a waveform prediction, making the effect testable against template searches.
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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

4 major / 5 minor

Summary. The paper develops a basis-independent definition of excitation factors for quasinormal modes of black holes whose perturbation equations form a system of N coupled second-order ODEs. The construction is based on the Green's matrix and Wronskian formalism, and the proposed quantity B_alpha n is designed to reduce to the standard single-field excitation factor in the decoupled limit. The formalism is then applied to the axial sector of a Reissner-Nordström black hole in Einstein-Maxwell-axion theory, where three degrees of freedom (gravitational, electromagnetic, and axion) are coupled. Using a continued-fraction method, the authors report avoided crossings between gravity-led and EM-led fundamental QNMs, accompanied by amplified excitation factors, and interpret this as a resonant excitation. A two-level non-Hermitian model with aligned phases is used to reproduce the spectral trajectories and the scaling of the excitation factors.

Significance. If the claimed effect is genuine, the paper identifies a new type of QNM resonance that is distinct from the overtone avoided crossings in Kerr: it occurs between the longest-lived modes that originate from fundamental modes of different fields. The Green's matrix/Wronskian derivation is careful, the basis-independence argument is clearly presented, and the EMA system provides a concrete, well-motivated example. The explicit recurrence coefficients in Appendix B and the simple two-level model with a falsifiable square-root Lorentzian scaling are useful assets. However, the central claim is not yet established at the level of a physical observable: the excitation factor B_alpha n carries a determinant normalization whose growth near a near-degeneracy may be partly a normalization artifact, and the paper itself notes in footnote *3 that a different channel-adapted normalization gives qualitatively different behavior. No code, convergence tables, error bars, or time-domain waveforms are provided, so the numerical demonstration and its interpretation remain under-supported.

major comments (4)
  1. [Sec. II B 2, Eq. (21)]
  2. [Sec. III C and Figs. 7–8]
  3. [Sec. III C 2 and Fig. 1]
  4. [Sec. III D 2, Eq. (75)]
minor comments (5)
  1. [Eq. (19)]
  2. [Eqs. (39)–(41)]
  3. [Sec. III C 1]
  4. [Sec. III D 1]
  5. [Conclusions]

Circularity Check

1 steps flagged · score 6.0 of 10

The growth of Bαn near the avoided crossing is built into the determinant normalization of Eq. (21), so the 'resonant excitation' confirmation is partly self-definitional; the avoided-crossing spectrum itself is an independent numerical result.

  1. self definitional [Sec. II B 2, Eq. (21); Sec. III D 2, Eq. (75) and Fig. 8; footnote *3]
    "Bαn = i detA(out) / ((2iωαn)^N) (d/dω detA(in))^{-1}|_{ω=ωαn} ... Written explicitly, the extra factor takes the form of (2iω)^{-(N-1)} ∏_{β≠α} A^{(out,β)}_β / A^{(in,β)}_β. ... In Ref. [21], it was demonstrated that |B_+−B_-| ∝ |ω_+−ω_-|^{-1}. ... They show excellent agreement ... This result confirms the validity of the model and the resonant excitation."

    By construction, Bαn contains the inverse derivative (d/dω detA(in))^{-1} evaluated at the QNM pole. Since detA(in) vanishes at both ωg0 and ωe0 near an avoided crossing, one has detA(in) ≈ C(ω−ωg0)(ω−ωe0), so (d/dω detA(in))^{-1}|_{ωg0} ∝ |ωg0−ωe0|^{-1}. The amplification of B and the Fig. 8 agreement with |ωg0−ωe0|^{-1} therefore follow algebraically from the chosen determinant normalization, not from an independent calculation of an observable amplitude. The paper itself notes that in the decoupling limit Eq. (21) carries the extra factor ∏_{β≠α} A^{(out,β)}_β / A^{(in,β)}_β, which already diverges when another field's A^{(in,β)}_β vanishes at a level crossing, and footnote *3 reports that a different normalization gives qualitatively different behavior.

full rationale

The QNM frequencies and the avoided-crossing structure in the EMA system are obtained by a direct continued-fraction solution of the coupled master equations (48)-(50), which are taken from Ref. [42]; that spectral part is not circular. The circularity concerns the resonance claim: the excitation factor Bαn in Eq. (21) is defined with the inverse derivative of detA(in), so its growth at near-degenerate QNMs is a built-in property of the determinant normalization. The paper's confirmation of resonance by fitting |Bg0−Be0| to |ωg0−ωe0|^{-1} (or a square-root Lorentzian) is therefore a check of the definition rather than a prediction from an independent observable amplitude. The citation of Ref. [21] for Eq. (75) reinforces this issue, but the relation is derivable from the definition itself, so the self-citation is not the sole load-bearing element. Because the central spectral finding remains an independent numerical result, the overall circularity is partial rather than total.

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

The central numerical claims rely on the coupled master equations, the decoupling-limit labeling, and the numerical methods; the fitted model parameters are not part of the core derivation but are used for interpretation.

free parameters (4)
  • p = not quoted
    Varying real parameter in the two-level model (73), fitted to reproduce the EM-led mode trajectory; stated to be proportional to gaγγ Q.
  • q = not quoted
    Varying real parameter in the two-level model (73), inferred to be proportional to Q from the |ω_g0 - ω_e0| ∝ Q scaling in Fig. 7.
  • ω0 and ϑ = not quoted
    Real constants in the two-level model (73), chosen to match the gravity-led branch and the phase alignment of the QNM trajectories.
  • f0, x0, γ = not quoted
    Positive constants in the square-root Lorentzian fit of |B_g0 - B_e0| in Fig. 8, used to claim agreement with the model.
assumptions (4)
  • domain assumption The coupled master equations (48)-(50), taken from Ref. [42], correctly describe linear axial perturbations of the EMA black hole.
    All QNM frequencies and excitation factors in Sec. III are computed from this system; the paper does not re-derive the master equations from the action.
  • ad hoc to paper Each QNM of the coupled system can be unambiguously labeled by its decoupling-limit parent as gravity-led, EM-led, or axion-led.
    Sec. II A, footnote *2 admits modes may appear or disappear; Sec. III C 2 uses this labeling to define the branches and to assign excitation factors Bαn.
  • domain assumption The fields are massless, share the same horizon, and the potential matrix vanishes at both boundaries.
    Sec. II A states these assumptions; they underlie the boundary conditions (55) and the ansatz (56).
  • ad hoc to paper The two-level model (71)-(73) with aligned phases captures the relevant dynamics near the avoided crossing.
    Sec. III D 2 introduces E1=p e^{iϑ}, E2=ω0^2 e^{iϑ}, Δ=q e^{iϑ} and uses it to derive the square-root Lorentzian dependence that is then fitted to the numerical data.

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Pith. "Pith review of Resonance of black hole quasinormal modes in coupled systems." pith.science (2026). https://pith.science/paper/PFRY6H4I

@misc{pith2026250503883,
  author       = {Pith},
  title        = {Pith review of: Resonance of black hole quasinormal modes in coupled systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFRY6H4I}},
  note         = {Machine review of arXiv:2505.03883}
}
read the original abstract

Black hole quasinormal modes (QNMs) can exhibit resonant excitations associated with avoided crossings in their complex frequency spectrum. Such resonance phenomena can serve as novel signatures for probing new physics, where additional degrees of freedom are commonly introduced. Motivated by this possibility, we investigate QNMs in systems where multiple degrees of freedom are coupled with each other, and introduce a definition of excitation factors suitable for such systems. To demonstrate our formulation, we apply it to a black hole in the Einstein-Maxwell-axion theory, where we find that avoided crossings can appear even between longest-lived modes originating from the fundamental modes of different degrees of freedom, in contrast to the Kerr case in General Relativity. We show that the excitation factors are indeed amplified as a manifestation of resonance at parameter values corresponding to the avoided crossings.

Figures

Figures reproduced from arXiv: 2505.03883 by the authors.

Figure 1
Figure 1. FIG. 1. QNMs originating from the fundamental modes of the three degrees of freedom as functions of [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. QNMs originating from the fundamental modes of the three degrees of freedom as functions of [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. QNM frequencies for [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Imaginary part of the non-oscillatory mode frequencies as a function of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Excitation factors for the QNMs originating from the fundamental modes of the three degrees of freedom with [PITH_FULL_IMAGE:figures/full_fig_p013_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Real (left) and imaginary (middle) parts of the QNM frequencies originating from the fundamental modes of the [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Absolute values of the difference between QNM frequencies of gravity-led and EM-led modes (left) and their corre [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Absolute values of the difference between the excitation factors (solid) and the inverse of the difference between QNM [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

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

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Reference graph

Works this paper leans on

67 extracted references · 17 canonical work pages · cited by 1 Pith paper

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    For this purpose, we consider Eq

    Waveform Next, we introduce excitation factors for coupled systems, which characterize how easily each QNM is excited in response to an external source for each QNM. For this purpose, we consider Eq. (3) with an added source term, d2 dr2∗ +ω2 ⃗ˆΨ(ω,r )−V (r)⃗ˆΨ(ω,r ) = ⃗S(r) . (9) *2 Of course, it may in principle be possible that some QNMs (dis)appear at...

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    down-mode

    Definition With the setup above, let us define the excitation factors in coupled systems. In the case of a single degree of freedom, i.e., N = 1, the definition of the excitation factor is given by [22, 23] Bsingle n = A(out) 2ωn d dωA(in) −1 ω=ωn . (20) In linear perturbation theory, there is an ambiguity in the overall normalization of the wavefunction....

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    A representative calculation method is the direct integration method [44], which is widely applicable in many cases

    Continued fraction method First, we describe the computation method used in this paper. A representative calculation method is the direct integration method [44], which is widely applicable in many cases. However, it is not suitable for extracting the *5 The differences from the notation in Ref. [42] are as follows: One of our master variables, hodd, corr...

    work page

  4. [4]

    In particular, we present results only for l = 2, as it is the smallest l where all the degrees of freedom are coupled and is also considered observationally important

    Numerical results Here, we show numerical results of the QNM frequencies of the BH in the EMA system. In particular, we present results only for l = 2, as it is the smallest l where all the degrees of freedom are coupled and is also considered observationally important. This system has three degrees of freedom, which correspond to purely gravitational, EM...

    work page

  5. [5]

    in” with “out

    Instability In the regime where gaγγQ/M is relatively large, the real part of the gravity-led QNM frequency tends to decrease beyond the point where the avoided crossing occurs. In this regime, the convergence of the continued fraction method with respect to the truncation order becomes increasingly poor, especially for higher overtones. Moreover, as inve...

    work page

  6. [6]

    First, we briefly describe the numerical calculation method

    Numerical results To obtain the excitation factor, it is necessary to compute the coefficient matrix, which in turn requires solving the homogeneous solutions under each boundary condition. First, we briefly describe the numerical calculation method. As discussed in the previous subsection, the resonant excitation is expected among the QNMs originating fr...

    work page

  7. [7]

    Although QNMs are difficult to treat analytically, their features can be captured by the following simple model

    Key feature of the resonant excitation Finally, we discuss the resonance behaviors of QNM spectra and the excitation factors. Although QNMs are difficult to treat analytically, their features can be captured by the following simple model. As proposed in Ref. [21], the theory of QNMs can be well understood in terms of quantum mechanics for non-Hermitian sy...

    work page

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    7, it can be inferred that q is proportional to Q

    shown in Fig. 7, it can be inferred that q is proportional to Q. This is due to the indirect coupling between gravity-led and EM-led modes through the axion-led mode. We show in Fig. 8 the difference between excitation factors corresponding to the gravity-led and EM-led QNMs along with the inverse of the difference between QNM frequencies |ωg0−ωe0|−1. We ...

    work page

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.