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arxiv: 2510.05354 · v2 · submitted 2025-10-06 · 🌀 gr-qc · hep-th

Quasinormal modes of Kerr-Newman black holes: revisiting the Dudley-Finley approximation

Sagnik Saha , Hector O. Silva This is my paper

Pith reviewed 2026-05-18 08:51 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords quasinormal modesKerr-Newman black holesperturbation decouplingzero-damped modesnear-extremal limitgravitoelectromagnetic perturbationsblack hole quasinormal spectrum
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The pith

The Dudley-Finley approximation reproduces Kerr-Newman quasinormal frequencies to within 10 percent real and 1 percent imaginary for low modes while mapping zero-damped regions near extremality.

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

This paper tests how well a decoupling trick works for the ringing frequencies of charged spinning black holes. The trick freezes one of the perturbation fields and solves for the other, avoiding the full coupled equations. For the modes with angular numbers two two zero, two two one, and three three zero the results stay close to the exact coupled calculation across most of the allowed spin and charge values. In the near-extremal corner of parameter space the approximation shows that some parts support only the long-lived zero-damped modes while neighboring parts also support damped modes, and it supplies exact expressions for the dividing curves. The work also follows how the modes with many overtones move through the complex frequency plane.

Core claim

In the Dudley-Finley approximation the decoupled equations yield quasinormal frequencies that agree with the full coupled gravitoelectromagnetic system to roughly 10 percent for the real part and 1 percent for the imaginary part when applied to the (2,2,0), (2,2,1) and (3,3,0) modes throughout the subextremal domain. The near-extremal regime divides into subregions that contain exclusively zero-damped modes and subregions that contain both zero-damped modes and modes with nonzero damping; analytic formulas locate the boundaries between these subregions. The zero-damped modes in the approximation are related to the near-horizon and photon-sphere modes of the complete theory, while the quadrup

What carries the argument

The Dudley-Finley approximation obtained by freezing one of the gravitational or electromagnetic perturbation fields to its background value, thereby decoupling the otherwise coupled system of equations.

Load-bearing premise

Freezing one perturbation field to its background value continues to represent the dynamics of the full coupled gravitoelectromagnetic system for all subextremal spins and charges.

What would settle it

A numerical computation of the full coupled system at a near-extremal spin-charge value lying inside one of the predicted subregions that reveals a different number of zero-damped modes than the number expected from the analytic boundary.

Figures

Figures reproduced from arXiv: 2510.05354 by Hector O. Silva, Sagnik Saha.

Figure 1
Figure 1. Figure 1: FIG. 1. Logarithmic absolute error between the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Logarithmic absolute error between the quasinormal [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Logarithmic absolute error between the quasinormal [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: we show the variation of δ 2 and F 2 s over the ex￾tremal quarter circle, a 2 ext = 1 4 − q 2 ext, for s = −2 and (ℓ, m) = (2, 2) and (2, 1) in the top and bottom panels, respectively. In the former case, as we increase the value of aext, the functions δ 2 and F 2 −2 cross over to positive values at slightly different values of the spin. In the lat￾ter case, δ 2 and F 2 −2 remain negative for all values of… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Comparison between the [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Numerical evaluation of the logarithm of the radial continued fraction [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Numerical evaluation of the logarithm of the radial [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Numerical evaluation of the logarithm of the radial [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Numerical evaluation of the logarithm of Leaver’s radial continued fraction [PITH_FULL_IMAGE:figures/full_fig_p010_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Predictions of the matched-asymptotic expansion [PITH_FULL_IMAGE:figures/full_fig_p011_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12. Predictions of Eqs [PITH_FULL_IMAGE:figures/full_fig_p012_12.png] view at source ↗
Figure 14
Figure 14. Figure 14: FIG. 14. Trajectories of the [PITH_FULL_IMAGE:figures/full_fig_p013_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: FIG. 15. Trajectories of the [PITH_FULL_IMAGE:figures/full_fig_p014_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: FIG. 16. The real part of the quasinormal frequencies of the [PITH_FULL_IMAGE:figures/full_fig_p014_16.png] view at source ↗
Figure 18
Figure 18. Figure 18: FIG. 18. The potential [PITH_FULL_IMAGE:figures/full_fig_p016_18.png] view at source ↗
read the original abstract

We present a comprehensive study of the Kerr-Newman quasinormal mode spectrum in the Dudley-Finley approximation, where the linear gravitoelectromagnetic perturbations are decoupled by "freezing" either one of the fields to its background value. First, we reassess the accuracy of this approximation by comparing it to calculations that solve the coupled system of gravitoelectromagnetic perturbation equations across the subextremal spin-charge parameter space. We find that for the $(\ell,m,n) = (2,2,0)$, $(2,2,1)$, and $(3,3,0)$ modes, the agreement is typically within $10\%$ and $1\%$ for the real and imaginary parts of the frequencies, respectively. Next, we investigate the spectrum in the near-extremal limit, and study the family of long-lived ("zero-damped") gravitational modes. We find that the near-extremal parameter space consists of subregions containing either only zero-damped modes, or zero-damped modes alongside modes that retain nonzero damping. We derive analytic expressions for the boundaries between these regions. Moreover, we discuss the connection between the zero-damped and damped modes in the Dudley-Finley approximation and the "near-horizon/photon-sphere" modes of the full Kerr-Newman spectrum. Finally, we analyze the behavior of the quadrupolar gravitational modes with large overtone numbers $n$, and study their trajectories in the complex plane.

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 studies quasinormal modes of Kerr-Newman black holes in the Dudley-Finley approximation, which decouples linear gravitoelectromagnetic perturbations by freezing one field to its background value. It first reassesses the approximation's accuracy by direct numerical comparison to solutions of the full coupled system for the (ℓ,m,n)=(2,2,0), (2,2,1), and (3,3,0) modes across the subextremal (a,Q) domain, reporting typical agreement within 10% for real parts and 1% for imaginary parts of the frequencies. The work then examines the near-extremal limit, identifies subregions containing only zero-damped modes or zero-damped modes together with damped modes, derives analytic expressions for the boundaries between these regions, discusses connections to near-horizon/photon-sphere modes of the full spectrum, and analyzes trajectories of quadrupolar gravitational modes at large overtone numbers n in the complex plane.

Significance. If the reported numerical agreement holds under more detailed scrutiny, the results would be useful for efficient computation of Kerr-Newman quasinormal modes and for mapping the near-extremal spectrum, including the distribution of long-lived zero-damped modes. The analytic boundary expressions constitute a concrete advance, and the direct comparison to the coupled system (rather than an untested decoupling assumption) strengthens the central validation. This could inform studies of charged rotating black holes in gravitational-wave astrophysics.

major comments (2)
  1. [Abstract] Abstract: the accuracy assessment states agreement 'typically within 10% and 1%' for real and imaginary parts but supplies no error bars, convergence tests, or explicit description of the numerical method used to solve the full coupled gravitoelectromagnetic system, leaving the central validation claim only partially substantiated.
  2. [Near-extremal analysis] Near-extremal analysis section: the analytic expressions for the boundaries separating regions with only zero-damped modes from those containing both zero-damped and damped modes are derived within the Dudley-Finley approximation; the manuscript should quantify how these boundaries may shift when the full coupled system is solved in the same near-extremal regime.
minor comments (2)
  1. [Numerical comparison] The manuscript would benefit from a brief table or figure summarizing the exact (a,Q) values at which the 10%/1% agreement levels were verified for the three quoted modes.
  2. [Methods] Clarify the precise implementation of 'freezing' one field in the Dudley-Finley decoupling (e.g., which field is held fixed and at what background value) when presenting the decoupled equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the accuracy assessment states agreement 'typically within 10% and 1%' for real and imaginary parts but supplies no error bars, convergence tests, or explicit description of the numerical method used to solve the full coupled gravitoelectromagnetic system, leaving the central validation claim only partially substantiated.

    Authors: We agree that the central validation would be strengthened by additional details. In the revised manuscript we will expand the methods section to include an explicit description of the numerical scheme used to solve the coupled gravitoelectromagnetic system, report convergence tests with respect to grid resolution and truncation order, and attach error bars (or estimated uncertainties) to the quoted agreement percentages. revision: yes

  2. Referee: [Near-extremal analysis] Near-extremal analysis section: the analytic expressions for the boundaries separating regions with only zero-damped modes from those containing both zero-damped and damped modes are derived within the Dudley-Finley approximation; the manuscript should quantify how these boundaries may shift when the full coupled system is solved in the same near-extremal regime.

    Authors: The analytic boundary expressions are indeed obtained inside the Dudley-Finley approximation. While we have already compared the approximation against the coupled system for representative modes over the full subextremal domain, a systematic quantification of the boundary shifts in the near-extremal regime would require a separate, computationally intensive campaign of coupled-system calculations at high spin and charge. We regard this as a valuable direction for follow-up work rather than an extension that can be completed within the present study. In the revision we will add a short paragraph noting this limitation and the expected direction of any corrections. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's core claims rest on direct numerical comparisons between the Dudley-Finley decoupled equations and independent solutions of the full coupled gravitoelectromagnetic system for specific (ℓ,m,n) modes across the subextremal domain, plus derivations of analytic boundary expressions performed entirely within the approximation. These steps do not reduce to self-defined quantities, fitted inputs renamed as predictions, or load-bearing self-citations; the accuracy assessment uses external full-system benchmarks rather than quantities constructed from the approximation itself. The derivation chain is therefore self-contained against external numerical validation.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on standard assumptions of linear black-hole perturbation theory and the validity of the Dudley-Finley decoupling; no new free parameters or invented entities are introduced.

axioms (2)
  • domain assumption Linear perturbation theory is sufficient to describe gravitoelectromagnetic excitations of the Kerr-Newman background.
    Invoked throughout the abstract as the framework for both the approximation and the full coupled system.
  • domain assumption Freezing one field to its background value yields a useful decoupling of the perturbation equations.
    Central premise of the Dudley-Finley approximation being reassessed.

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