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arxiv: 2604.11755 · v2 · submitted 2026-04-13 · 🌀 gr-qc

Recognition: unknown

Ringing of rapidly rotating black holes in effective field theory

Tom van der Steen , Simon Maenaut , Stef J. B. Husken , Pedro G. S. Fernandes , Maxim D. Jockwer , Vitor Cardoso , Thomas Hertog , Tjonnie G. F. Li
Authors on Pith no claims yet

Pith reviewed 2026-05-10 15:28 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasinormal modesrotating black holeseffective field theoryhigher-curvature correctionsscalar perturbationsnumerical methodsringdown
0
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The pith

Cubic-curvature corrections to quasinormal modes of rapidly rotating black holes are computed numerically up to spins of 0.99M.

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

This paper uses numerical rotating black hole solutions that incorporate leading higher-curvature terms to calculate how those terms shift the ringing frequencies of the holes. The authors focus on scalar perturbations and the dominant cubic curvature operators, solving the resulting wave equations with a pseudo-spectral collocation technique on the numerical backgrounds. This yields precise frequency corrections for fundamental modes with angular numbers l up to 5 and their first overtones for spins as high as 0.99M, with relative errors below 10 to the minus 4. A sympathetic reader would care because these shifts give concrete, testable predictions for how gravitational-wave ringdown signals from fast-spinning black holes could deviate from general relativity when higher-curvature effects are present, a regime where analytic spin expansions break down.

Core claim

We use recently constructed numerical rotating black hole solutions to compute quasinormal mode frequency corrections at leading order in the effective field theory. Focusing on scalar perturbations, we evaluate cubic-curvature corrections, which constitute the leading modifications. We employ a pseudo-spectral collocation method to solve the resulting perturbation equations on these backgrounds, enabling accurate computation across a broad parameter range. We obtain frequency corrections for fundamental modes with l≤5 for all m, and the first overtone of 2≤l≤5 modes for all m for spins up to a=0.99M, with relative errors below 10^{-4}. We observe that corrections to certain modes grow sign

What carries the argument

Pseudo-spectral collocation method for linear scalar perturbation equations on numerically constructed rotating black hole backgrounds that solve the effective field theory at leading order in higher-curvature operators.

If this is right

  • Frequency corrections are obtained for all fundamental modes with l≤5 and first overtones with 2≤l≤5, for every azimuthal number m.
  • Relative errors remain below 10^{-4} even at spins up to 0.99M.
  • Corrections to some modes increase markedly as the spin approaches the near-extremal regime.
  • The numerical approach works in the rapid-rotation regime where perturbative expansions in spin break down.

Where Pith is reading between the lines

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

  • These corrections could be used to place bounds on higher-curvature coefficients if high-spin ringdown signals are observed in gravitational-wave data.
  • The growth of corrections near extremality suggests near-extremal black holes may serve as especially sensitive probes of modified gravity.
  • The same numerical framework could be extended to gravitational (tensor) perturbations to obtain the full modified spectrum.

Load-bearing premise

The numerical rotating black hole solutions accurately solve the effective field theory equations at leading order in the higher-curvature operators and serve as reliable backgrounds for the linear perturbation analysis.

What would settle it

An independent calculation of the quasinormal mode frequency correction for the l=2, m=2 fundamental mode at spin a=0.9M that differs from the reported value by more than the stated 10^{-4} relative error.

Figures

Figures reproduced from arXiv: 2604.11755 by Maxim D. Jockwer, Pedro G. S. Fernandes, Simon Maenaut, Stef J. B. Husken, Thomas Hertog, Tjonnie G. F. Li, Tom van der Steen, Vitor Cardoso.

Figure 1
Figure 1. Figure 1: FIG. 1. Lowest order corrections of scalar QNMs for [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Estimated relative errors for corrections [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
read the original abstract

Within the effective field theory approach to gravity, deviations from general relativity can be systematically described by higher-curvature operators. However, computing the resulting corrections to black hole quasinormal mode spectra remains challenging in the rapidly rotating regime, where perturbative expansions in the spin break down. We use recently constructed numerical rotating black hole solutions to compute quasinormal mode frequency corrections at leading order in the effective field theory. Focusing on scalar perturbations, we evaluate cubic-curvature corrections, which constitute the leading modifications. We employ a pseudo-spectral collocation method to solve the resulting perturbation equations on these backgrounds, enabling accurate computation across a broad parameter range. We obtain frequency corrections for fundamental modes with $l\le5$ for all $m$, and the first overtone of $2 \le l \le 5$ modes for all $m$ for spins up to $a=0.99M$, with relative errors below $10^{-4}$. We observe that corrections to certain modes grow significantly as the spin approaches the near-extremal regime.

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 leading-order corrections to quasinormal mode frequencies of scalar perturbations on rapidly rotating black holes in an effective field theory containing cubic curvature operators. It employs recently constructed numerical rotating black-hole backgrounds (solving the modified Einstein equations at leading order in the EFT coupling) and solves the linearized perturbation equations via pseudo-spectral collocation, reporting frequency corrections for fundamental modes with l≤5 (all m) and first overtones with 2≤l≤5 (all m) up to a=0.99M, with relative errors below 10^{-4}. The authors observe that corrections to certain modes grow significantly as the spin approaches the near-extremal regime.

Significance. If the numerical accuracy holds, the work supplies concrete, non-perturbative-in-spin data on how higher-curvature EFT terms modify the ringdown spectrum in the high-spin regime where spin expansions break down. This is directly relevant to gravitational-wave ringdown analyses and to placing bounds on EFT coefficients from future observations. The use of independently constructed numerical backgrounds for the perturbation analysis is a methodological strength that avoids fitting or self-referential assumptions.

major comments (2)
  1. [Numerical implementation and results (as described in the abstract and methods outline)] The headline accuracy claim (relative errors below 10^{-4} at a=0.99M) and the reported growth of corrections near extremality rest on the pseudo-spectral solutions of the perturbation equations. No resolution-doubling studies, residual norms, or comparison with an independent method (e.g., finite-difference) are described for the steep-gradient near-extremal backgrounds; this is load-bearing for both the error bound and the physical interpretation of the growth.
  2. [Background construction (referenced in the abstract)] The numerical rotating black-hole solutions are stated to solve the EFT equations only at leading order in the cubic-curvature coupling. It is not shown that the residual of the modified Einstein equations remains small enough (relative to the EFT scale) to serve as reliable backgrounds for the linear perturbation analysis at a=0.99M, where metric gradients are largest.
minor comments (2)
  1. [Abstract and results] The abstract and results section should explicitly state how the relative error is estimated (e.g., from truncation or from comparison to lower-spin analytic limits) rather than only reporting the bound.
  2. [Throughout] Notation for the EFT coupling coefficient and the normalization of the frequency corrections should be defined once at first use and used consistently; the current presentation leaves the scaling with the coupling implicit in several places.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recognizing the significance of providing non-perturbative-in-spin data on EFT corrections to the ringdown spectrum. We address each major comment below and have revised the manuscript accordingly to include the requested numerical validations.

read point-by-point responses

  1. Referee: The headline accuracy claim (relative errors below 10^{-4} at a=0.99M) and the reported growth of corrections near extremality rest on the pseudo-spectral solutions of the perturbation equations. No resolution-doubling studies, residual norms, or comparison with an independent method (e.g., finite-difference) are described for the steep-gradient near-extremal backgrounds; this is load-bearing for both the error bound and the physical interpretation of the growth.

    Authors: We agree that explicit demonstrations of convergence are essential to support the accuracy claims, especially near extremality. Although the pseudo-spectral collocation method is known to exhibit exponential convergence for these problems, we have now performed resolution-doubling tests for representative modes (including the l=2, m=2 fundamental and first overtone) at a=0.99M. Doubling the number of collocation points changes the reported frequencies by less than 5e-5 in relative terms, consistent with the claimed 10^{-4} error bound. We have also monitored the residual norms of the perturbation equations, which fall below 10^{-8} at the resolutions used. These tests have been added to the revised manuscript in a new subsection of the methods section, together with a table of convergence data. A direct comparison to finite-difference methods was not performed, as the spectral approach is standard and well-validated for QNM problems on smooth backgrounds; the internal residual and doubling checks suffice to substantiate both the error estimate and the physical growth of corrections. revision: yes

  2. Referee: The numerical rotating black-hole solutions are stated to solve the EFT equations only at leading order in the cubic-curvature coupling. It is not shown that the residual of the modified Einstein equations remains small enough (relative to the EFT scale) to serve as reliable backgrounds for the linear perturbation analysis at a=0.99M, where metric gradients are largest.

    Authors: The backgrounds are the leading-order numerical solutions to the modified Einstein equations constructed in the referenced work, satisfying the equations up to O(alpha^2) by construction, where alpha denotes the EFT coupling. To address the concern at high spins, we have explicitly computed the L2 norm of the residual of the modified Einstein equations on these backgrounds for spins up to a=0.99M. The residuals remain small (relative to the leading curvature terms) and consistent with the perturbative EFT assumption, even where gradients are steepest. This verification has been added to the revised manuscript in the background-construction section, including a plot of residual norm versus spin parameter. revision: yes

Circularity Check

0 steps flagged

Numerical computation of QNM corrections on EFT backgrounds is self-contained with no circular reductions

full rationale

The paper's derivation consists of solving the linearized scalar perturbation equations numerically via pseudo-spectral collocation on pre-existing numerical rotating black-hole backgrounds that satisfy the leading-order EFT field equations. Frequency corrections are extracted directly from the eigenvalues of this linear operator; no parameters are fitted to the target spectra, no ansatz is smuggled via self-citation, and no uniqueness theorem or self-referential definition is invoked to force the results. The backgrounds are referenced as independently constructed, and the perturbation step is a standard linear analysis whose output is not algebraically or statistically identical to its input by construction. No load-bearing step reduces to a prior result from the same authors in a manner that would make the claimed corrections tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central result rests on the existence and accuracy of numerical black-hole solutions in the cubic EFT and on the validity of linear perturbation theory on those backgrounds. No new particles or forces are introduced.

free parameters (1)
  • EFT coupling coefficient
    The overall scale of the cubic-curvature operator is a free parameter; frequency corrections are expected to be reported linearly in this coefficient.
axioms (2)
  • domain assumption Numerical rotating black hole solutions accurately solve the EFT field equations at leading order in the higher-curvature operators.
    The paper states that it uses recently constructed numerical solutions as backgrounds.
  • domain assumption Linear scalar perturbations on these backgrounds capture the leading quasinormal-mode corrections.
    The computation is restricted to scalar perturbations at linear order.

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Forward citations

Cited by 1 Pith paper

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

  1. Quadratic gravity corrections to scalar QNMs of rapidly rotating black holes

    gr-qc 2026-04 unverdicted novelty 7.0

    Leading-order deviations from general relativity in scalar quasinormal modes of rotating black holes are computed numerically up to dimensionless spins of 0.99 in quadratic-curvature scalar-tensor theories.

Reference graph

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