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Quasinormal mode spectrum of rotating black holes in Einstein-Gauss-Bonnet-dilaton theory

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Pith's one-line read This paper computes the quasinormal-mode spectrum of rapidly rotating Einstein-Gauss-Bonnet-dilaton black holes non-perturbatively and finds that the dominant ringdown mode can change with the coupling and spin.

desk verdict First non-perturbative rotating EGBd QNM spectrum, with a credible method and honest caveats, but the headline mode-ordering reversal at j=0.8 sits exactly where the authors concede accuracy loss. read the letter →

arxiv 2412.17073 v2 pith:T3B3KR7U submitted 2024-12-22 gr-qc hep-th

Jose Luis Blázquez-Salcedo , Fech Scen Khoo , Burkhard Kleihaus , Jutta Kunz This is my paper · ORCID
classification gr-qchep-th MSC 83C5783C3583D05 PACS 04.70.-s04.30.-w04.50.Kd
keywords quasinormalmodesEinstein-Gauss-Bonnet-dilatontheoryrotatingblackholesringdownspectralmethodsisospectralitybreakingGauss-Bonnetcouplingscalar-led
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 reports a fully non-perturbative computation of the quasinormal-mode spectrum of rapidly rotating black holes in Einstein-Gauss-Bonnet-dilaton (EGBd) theory, a modified gravity in which a scalar dilaton couples to the Gauss-Bonnet invariant. The authors first construct numerically exact rotating EGBd black hole backgrounds, then linearize the coupled metric-dilaton equations and solve them with a spectral method. The central result is that the Kerr degeneracy between polar-led and axial-led modes is broken, and that as the Gauss-Bonnet coupling grows the ordering of the fundamental l=2 and l=3 modes changes: near the boundary of the region where black hole solutions exist, scalar-led modes can become the longest-lived. The dominant ringdown mode therefore depends on the coupling of the theory and on the spin of the final black hole, which matters for using gravitational-wave ringdown observations to constrain such theories.

What carries the argument

The load-bearing machinery is the spectral reduction of the coupled perturbation equations. The seven perturbation functions — the metric functions H1, T, N, L, the axial functions h0, h1, and the dilaton perturbation Φ1 — are expanded in Chebyshev polynomials in the compactified radial coordinate x and in associated Legendre functions of y=cosθ, with the outgoing and ingoing boundary behavior factored out. Evaluating the PDEs and boundary conditions on a Gauss-Lobatto and uniform grid converts the system into a quadratic eigenvalue problem (M0 + M1 ω + M2 ω²) C = 0, whose eigenvalues are the complex quasinormal frequencies. The same scheme is validated against Kerr and against second-order-in-spin, sixth-order-in-coupling perturbative results.

What would settle it

Repeat the eigenvalue calculation for the fundamental l=2 scalar-led mode at j=0.2 and ξ near the upper end of the domain (around 0.17) with doubled spectral resolution (larger Nx and Ny) and with an independent time-domain evolution of the linearized EGBd equations; if the sharp drop in MωI that makes the scalar mode longest-lived near the domain boundary shifts or disappears as the resolution changes, the claimed mode-order change is a numerical artifact rather than a property of the theory.

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Extended reading notes

Core claim

The central claim is that the quasinormal modes of rapidly rotating EGBd black holes can be extracted without treating either the Gauss-Bonnet coupling or the angular momentum as small parameters. Working with numerically exact stationary axisymmetric backgrounds, the authors linearize the metric and dilaton field equations, impose purely outgoing waves at infinity and purely ingoing waves at the horizon, and expand the seven perturbation functions in Chebyshev and Legendre series, reducing the problem to a quadratic eigenvalue problem. For azimuthal number Mz=2, l=2- and l=3-led fundamental modes, they find that polar-led and axial-led modes split once the coupling is turned on, and that the imaginary parts — the damping times — develop a strong coupling dependence. Toward the boundary of the domain of existence the scalar-led modes cross the metric-led modes and become the longest lived, so the ordering of modes, and hence which mode dominates the ringdown, changes with both the coupling and the angular momentum.

Load-bearing premise

The load-bearing premise is that the spectral expansion of the seven coupled perturbation functions is converged for the EGBd backgrounds, so the complex frequencies it returns are true quasinormal modes; the paper validates the pipeline on Kerr and against weak-coupling perturbative results, but it reports no dedicated convergence or error study for the EGBd eigenvalue problem itself, and it states that for j=0.6 and 0.8 the calculations lose accuracy toward the boundary of the domain of existence.

Editorial extensions

If this is right

  • Isospectrality is generically broken in EGBd: once the Gauss-Bonnet coupling is nonzero, polar-led and axial-led modes split, and the splitting grows with the coupling.
  • Near the boundary of the domain of existence, scalar-led modes can be the longest lived, so the dominant ringdown mode is not simply the Kerr quadrupole mode; ringdown analyses that assume Kerr isospectrality would misidentify the mode.
  • Perturbative expansions in the spin and the coupling deviate increasingly from the exact spectrum at large angular momentum and large coupling, so strong-coupling ringdown predictions require the full numerical treatment.
  • The domain of existence of rotating EGBd black holes, including solutions that slightly exceed the Kerr bound j=1, is charted by the background construction, showing where perturbative methods break down.
  • Precision ringdown observations could in principle place bounds on the Gauss-Bonnet coupling by comparing the measured frequencies and damping times with these non-perturbative spectra.

Reading between the lines

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

  • (Editorial inference) The crossing pattern suggests that the ringdown of an EGBd black hole could show a sharp transition in the longest-lived mode as a function of spin or coupling; a search that follows the longest-lived mode continuously would be a sharper test than comparing isolated frequencies.
  • (Editorial inference) The same spectral method should carry over to other scalar-tensor theories, such as shift-symmetric Einstein-Gauss-Bonnet gravity, and the tendency of scalar-led modes to dominate near the domain boundary is likely a general feature; this is a testable prediction for those theories.
  • (Editorial inference) The paper computes the fundamental modes only; overtone spectra could change which mode actually dominates a matched-filter ringdown search at early times, so the dominance claim is a statement about fundamentals, not the full signal.
  • (Editorial inference) A practical next step would be to compute excitation coefficients from binary merger initial data, since spectral dominance alone does not determine how strongly each mode is rung up in a real event.
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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 presents a numerical spectral method for computing the quasinormal mode (QNM) spectrum of rapidly rotating Einstein-Gauss-Bonnet-dilaton (EGBd) black holes without perturbative expansions in either the Gauss-Bonnet coupling or the angular momentum. The authors first obtain stationary, axially symmetric background solutions with the FIDISOL/CADSOL package, then linearize the metric and dilaton equations using a seven-function perturbation ansatz (polar, axial, and scalar sectors), apply ingoing/outgoing boundary conditions at the horizon and infinity, and solve the resulting quadratic eigenvalue problem with a Chebyshev-Legendre spectral decomposition. They report fundamental l=2- and l=3-led modes with azimuthal number Mz=2 for angular momenta j=0.2, 0.4, 0.6, 0.8 and both signs of the real frequency. The results are validated against the Kerr limit and against second-order perturbative results for small coupling and rotation, and they exhibit isospectrality breaking and a reordering of modes with increasing coupling, including a claimed dominance of scalar-led modes near the boundary of the domain of existence.

Significance. If the results are correct, this is a significant advance: it provides the first fully non-perturbative QNM spectrum for rapidly rotating black holes in a well-motivated modified gravity theory, with direct implications for ringdown-based tests of gravity. The paper's strengths include a detailed description of the background construction and perturbation method, extensive numerical tables, and genuine external checks against Kerr and second-order perturbative results in the weak-coupling regime. However, the central phenomenological claim that the dominant ringdown mode changes to the scalar-led sector—and in particular that the l=3 scalar mode can dominate at high angular momentum—rests on data in a region where the authors themselves state that accuracy is degraded, and no EGBd-specific convergence or error study is provided. As presented, the qualitative scalar-dominance trend is plausible, but the stronger quantitative claims are not yet fully established.

major comments (3)
  1. [Section 4, Tables 13 and 16] The claimed octupole dominance at j=0.8 is decided by the last tabulated point xi=0.11221: the scalar l=3 mode's M*omega_I changes from -0.06379 at xi=0.10926 to -0.05660, overtaking the polar l=2 mode (M*omega_I=-0.06271) by only about 0.006. This is precisely the regime covered by the sentence in Section 4: 'for j=0.6 and 0.8 our calculations lose accuracy towards the boundary of the domain of existence.' Since a spectral method can produce spurious eigenvalues near a boundary, the crossing at the final tabulated point cannot be distinguished from a numerical artifact without a convergence or resolution study. The conclusion in Section 5 that 'the octupole modes may also dominate over the quadrupole modes for large coupling' is therefore not secure at j=0.8.
  2. [Section 3.4, Eqs. (93)-(102)] No convergence tests or error estimates are reported for the EGBd eigenvalue problem itself. The Kerr limit (xi=0) checks the method only in the decoupled case, and the perturbative comparison in Fig. 5 explicitly covers polar-led modes only (the text notes 'the axial modes were not given in there'). The scalar-led and axial-led strong-coupling modes are therefore validated neither against an independent solver nor against the static EGBd limit. To support the central mode-ordering claim, the paper should provide, at minimum for the j=0.6 and j=0.8 cases, a convergence study in Nx and Ny for the fundamental polar, axial, and scalar modes, and report the resulting errors on M*omega_R and M*omega_I, or state explicitly which tabulated points meet a chosen accuracy threshold.
  3. [Section 4 and Fig. 4] The scalar-led modes in the strong-coupling regime are not checked against the static (j=0) EGBd QNMs of Refs. [38,41] or any other independent non-Kerr limit. Such a comparison would directly test the scalar sector and the axial/polar splitting at finite coupling, where no perturbative anchor exists. If this check cannot be performed, the claim that 'scalar modes tend to dominate' toward the boundary should be restricted to the j=0.2 and 0.4 data, where the trend is visible before the accuracy caveat becomes severe, and the stronger j=0.8 octupole-ordering statement should be deferred until the numerical accuracy is demonstrated.
minor comments (5)
  1. [Section 3.1 and Fig. 3] The classification of modes as polar-led, axial-led, or scalar-led is based on visual inspection of the parity of Re(T~), Re(h0~), and Re(Phi~) at a single point (j=0.6, xi=0.11). Because rotation and coupling mix multipoles, and the mixing increases with xi, a quantitative criterion (for instance, the relative magnitude of the dominant spectral coefficients) would make the mode labels reproducible and less subjective.
  2. [Section 4 and Appendix] The text states that for j=0.6 and 0.8 the calculations lose accuracy towards the boundary of the domain of existence and that the authors 'refrained from showing their values there,' yet Tables 9-16 do list values at the largest couplings. Please clarify which entries are considered unreliable and mark them in the tables (for example, with parentheses or italics).
  3. [Fig. 4 caption] The dotted vertical lines are said to represent the maximal value of the scaled coupling constant for which background solutions exist, but the caption does not explain how those values are obtained; a reference to Fig. 1b or to the boundary curves would help the reader.
  4. [Section 3.4, Eq. (102)] The quadratic eigenvalue problem is solved 'with the same numerical methods described previously [49],' but the present paper does not specify the linearization procedure, matrix sizes, or tolerance settings used for the EGBd runs; adding one sentence with these details would improve reproducibility.
  5. [Appendix tables] The tables report four or five significant digits without error bars; if convergence information is obtained, including error estimates in the tables would make the numerical content more useful to future comparisons.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the quasinormal-mode frequencies are outputs of an independently solved linearized eigenvalue problem, checked against Kerr and perturbative limits rather than fitted to them.

full rationale

The paper's derivation chain is self-contained as a numerical calculation: the EGBd background solutions (Section 2) are obtained by solving the coupled field equations with stated boundary conditions, and the perturbation equations (Section 3) are linearized about those backgrounds. The quasinormal-mode frequencies are the eigenvalues of the quadratic matrix problem (M0 + M1 omega + M2 omega^2) C = 0, Eq. (102), with purely ingoing and outgoing boundary conditions at the horizon and infinity. Nothing in this construction fits omega_R or omega_I to the values used for validation. The checks performed—reduction to Kerr modes at xi = 0 and comparison with second-order perturbative results of Ref. [43]—are genuine external comparisons: the perturbative data are obtained by an independent method, and the agreement is reported as confirmation rather than used as input. The mode classification into polar-led, axial-led, and scalar-led sectors is made by inspecting the parity of the perturbation eigenfunctions (Fig. 3), not by assigning labels to match known results. The paper's own caveat that for j = 0.6 and 0.8 'our calculations lose accuracy towards the boundary of the domain of existence' is a numerical-convergence concern about strong-coupling points near the boundary; it is a correctness risk, not a circularity, because the affected values are still outputs of the same eigenvalue problem rather than quantities forced by definition or by self-citation. The only self-citations are to the authors' earlier construction of the background solutions and to their earlier Kerr implementation of the same spectral method, both of which are independent support (previously published numerical results and a tested code) and are not used to define the present eigenvalues. No step reduces by construction to its own input, and no fitted parameter is later renamed as a prediction. The central claim—that mode ordering can change with coupling and angular momentum—is an emergent property of the computed spectrum, not an assumption. Therefore the paper exhibits no significant circularity and merits a score of 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no fitted physical parameters: the GB coupling alpha is scanned and the dilaton coupling gamma is fixed to 1 as a theory choice; QNM frequencies are outputs of the eigenvalue problem, not inputs. The main assumptions are the completeness of the seven-function perturbation ansatz and the convergence of the spectral method, which is validated only indirectly through the Kerr limit. No new particles, fields, or forces are postulated.

assumptions (5)
  • domain assumption The action (1) with f(phi)=exp(-gamma phi), gamma=1, is an appropriate low-energy EGBd theory.
    Section 2.1; this defines the theory under study and is not derived within the paper.
  • ad hoc to paper The axial/polar perturbation ansatz with functions H1,T,N,L,h0,h1,Phi captures all linear gravitational and scalar perturbations relevant for the QNM spectrum.
    Section 3.1; completeness and gauge fixing are assumed; the same ansatz was used and tested for Kerr in Ref. [49].
  • domain assumption Quasinormal modes are selected by purely outgoing waves at infinity and purely ingoing waves at the horizon.
    Section 3.3; standard definition of quasinormal modes.
  • standard math The double Chebyshev-Legendre spectral series converges to the true perturbation solutions on EGBd backgrounds.
    Section 3.4; convergence is assumed and was checked for Kerr in Ref. [49], but not demonstrated for EGBd here.
  • ad hoc to paper The numerical background solutions computed with FIDISOL/CADSOL with estimated error 10^-5 are accurate enough for the perturbation calculation.
    Section 2.2; the 10^-5 accuracy statement covers the background functions only, and its impact on the QNM eigenvalues is not quantified.

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Pith. "Pith review of Quasinormal mode spectrum of rotating black holes in Einstein-Gauss-Bonnet-dilaton theory." pith.science (2026). https://pith.science/paper/T3B3KR7U

@misc{pith2026241217073,
  author       = {Pith},
  title        = {Pith review of: Quasinormal mode spectrum of rotating black holes in Einstein-Gauss-Bonnet-dilaton theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T3B3KR7U}},
  note         = {Machine review of arXiv:2412.17073}
}
read the original abstract

Quasinormal modes are excited during the ringdown phase of black holes after merger. Determination of quasinormal modes of rapidly rotating black holes in alternative theories of gravity has remained a challenge for a long time. Here we discuss in detail our recently developed method to extract the quasinormal modes for rapidly rotating black holes in Einstein-Gauss-Bonnet-dilaton theory. We first obtain numerically the exact rapidly rotating background solutions, which also clarify their domain of existence. Then we solve the equations for the linear perturbations of the metric and the dilaton field by employing an appropriate set of boundary conditions and a spectral decomposition of the perturbation functions. The resulting spectrum agrees well with the known limits obtained for slow rotation and weak coupling, while it exhibits larger deviations for stronger coupling.

Figures

Figures reproduced from arXiv: 2412.17073 by the authors.

Figure 1
Figure 1. Rotating EGBd black holes (γ = 1): Scaled horizon area aH = AH/16πM2 vs scaled angular momentum j = J/M2 (a) and scaled angular momentum j = J/M2 vs scaled coupling constant ξ = α/M2 (b). The shaded areas indicate the respective domains of existence bounded by the Kerr, static, critical and extremal EGBd black holes. Curves within the domains have fixed values of the scaled horizon angular velocity ΩH √ α. The inset… view at source ↗
Figure 2
Figure 2. Example of the quasinormal mode spectrum for EGBd black [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Perturbation functions for l = Mz = 2 fundamental modes for scaled angular momentum j = 0.6 and scaled GB coupling ξ = 0.11: function Re(Te) for the polar-led mode (a) and the axial-led mode (b), function Re(eh0) for the polar-led mode (c) and the axial-led mode (d), function Re(Pe) for the polar-led mode (e) and the axial-led mode (f). In order to identify the type of a given mode, and classify it as polar-led, axi… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Fundamental EGBd quadrupole (l = 2)-led and octupole (l = 3)-led quasinor￾mal modes for Mz = 2: scaled real part MωR (left column) and scaled imaginary part MωI (right column) vs the scaled coupling strength ξ for j = 0.2, j = 0.4, j = 0.6, and j = 0.8 (from top to bot…
Figure 5
Figure 5. Figure 5: Comparison with second order approximation: Fundament [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Fundamental EGBd quadrupole (l = 2)-led and octupole (l = 3)-led quasinor￾mal modes for Mz = 2 for negative ωR: scaled real part MωR (left column) and scaled imaginary part MωI (right column) vs the scaled coupling strength ξ for j = 0.2, j = 0.4, j = 0.6, and j = 0.8 …

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Reviewed August 11, 2026 · model on record in the stance chip above.