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arxiv: 2406.11694 · v2 · pith:MWSEWFC2new · submitted 2024-06-17 · 🌀 gr-qc

Correspondence between grey-body factors and quasinormal modes

R. A. Konoplya , A. Zhidenko This is my paper

Pith reviewed 2026-05-17 19:30 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasinormal modesgrey-body factorsblack holeseikonal approximationscatteringgravitational wavesstabilityeffective potential
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0 comments X

The pith

Grey-body factors of spherically symmetric black holes can be expressed directly in terms of their quasinormal modes, exactly in the high-frequency limit.

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

This paper establishes a link between quasinormal modes, which describe the characteristic ringing of perturbed black holes with purely outgoing waves, and grey-body factors, which quantify transmission and absorption in scattering problems that include incoming waves from the horizon. The correspondence is approximate at general frequencies but becomes exact in the high-frequency eikonal regime. A sympathetic reader would care because this relation offers a simpler way to compute scattering properties from mode data and accounts for why grey-body factors stay more stable than higher overtones when the effective potential is slightly deformed. The work also ties into connections between grey-body factors and gravitational-wave amplitudes from black holes.

Core claim

We establish an approximate correspondence between the quasinormal modes and grey-body factors, which becomes exact in the high-frequency (eikonal) regime. In this regime, the grey-body factors of spherically symmetric black holes can be remarkably simply expressed via the fundamental quasinormal mode, while at smaller ℓ the correction terms include values of the overtones.

What carries the argument

The direct mapping from the outgoing-wave boundary conditions of quasinormal modes to the scattering boundary conditions of grey-body factors, which holds without significant higher-order corrections in the eikonal limit.

If this is right

  • In the eikonal regime, grey-body factors depend only on the fundamental quasinormal mode.
  • At lower angular momentum values, overtones enter as correction terms in the grey-body factor expression.
  • The link accounts for the observed greater stability of grey-body factors than higher overtones under small deformations of the effective potential.
  • This correspondence connects to the recently noted relation between grey-body factors and amplitudes of gravitational waves emitted by black holes.

Where Pith is reading between the lines

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

  • The relation may allow direct prediction of scattering transmission coefficients from quasinormal mode spectra without separate scattering calculations.
  • Similar mappings could be tested in numerical evolutions of perturbed black holes to check accuracy outside the strict high-frequency limit.
  • If the correspondence holds more broadly, it might simplify analysis of wave propagation in other compact-object spacetimes where both quantities are computable.

Load-bearing premise

The effective potential permits a direct mapping between the outgoing-wave boundary conditions of quasinormal modes and the scattering boundary conditions of grey-body factors without large higher-order corrections outside the high-frequency limit.

What would settle it

A numerical computation of the grey-body factor for a known spherically symmetric black hole at high frequency that deviates substantially from the value predicted solely by its fundamental quasinormal mode.

read the original abstract

Quasinormal modes and grey-body factors are spectral characteristics corresponding to different boundary conditions: the former imply purely outgoing waves to the event horizon and infinity, while the latter allow for an incoming wave from the horizon, thus describing a scattering problem. Nevertheless, we show that there is a link between these two characteristics. We establish an approximate correspondence between the quasinormal modes and grey-body factors, which becomes exact in the high-frequency (eikonal) regime. We show that, in the eikonal regime, the grey-body factors of spherically symmetric black holes can be remarkably simply expressed via the fundamental quasinormal mode, while at smaller $\ell$, the correction terms include values of the overtones. This might be interesting in the context of the recently observed connection between grey-body factors and the amplitudes of gravitational waves from black holes. The correspondence might explain why grey-body factors are more stable, i.e. less sensitive, than higher overtones to small deformation of the effective potential.

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 paper claims an approximate correspondence between quasinormal modes (QNMs) and grey-body factors (GBFs) for spherically symmetric black holes, arising from their differing boundary conditions (purely outgoing waves for QNMs versus scattering with incoming waves from the horizon for GBFs). This correspondence is shown to become exact in the high-frequency eikonal regime, where GBFs can be expressed simply in terms of the fundamental QNM frequency, while overtones enter as sub-leading corrections at finite multipole number ℓ. The work suggests this link may explain the relative stability of GBFs to small deformations of the effective potential and could relate to gravitational-wave amplitudes.

Significance. If the eikonal correspondence holds, the result provides a concrete analytic bridge between two central spectral quantities in black-hole perturbation theory, allowing GBFs to be reconstructed from QNM data without additional fitting parameters. The explicit mapping in the high-frequency limit and the identification of overtone corrections at lower ℓ constitute a falsifiable prediction that could be tested against known exact solutions (e.g., Schwarzschild). Such a relation would be useful for interpreting scattering observables and for understanding why GBFs appear more robust than higher overtones under potential perturbations.

major comments (2)
  1. [Eikonal limit section] Eikonal-regime derivation: the claim that the GBF transmission coefficient is exactly determined by the fundamental QNM once the leading WKB/geodesic-orbit term is isolated requires explicit demonstration that curvature corrections and higher derivatives of the effective potential produce only O(1/ℓ) or smaller residuals that vanish in the strict eikonal limit; the skeptic note indicates this identification of boundary conditions may retain potential-shape dependence.
  2. [Corrections at finite ℓ] Finite-ℓ correction formula: the statement that overtones supply the leading corrections at smaller ℓ is asserted, but the explicit dependence of those corrections on the potential (or on the QNM spectrum) is not derived in closed form; without this, the practical utility of the correspondence outside the eikonal regime remains unclear.
minor comments (2)
  1. Notation for the effective potential and its derivatives should be unified between the QNM WKB analysis and the GBF scattering calculation to avoid reader confusion.
  2. A brief comparison table of the leading eikonal GBF expression versus known numerical values for Schwarzschild or Reissner-Nordström would strengthen the verification of the claimed exactness.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We respond to each major comment below and indicate the revisions that will be incorporated in the next version.

read point-by-point responses

  1. Referee: [Eikonal limit section] Eikonal-regime derivation: the claim that the GBF transmission coefficient is exactly determined by the fundamental QNM once the leading WKB/geodesic-orbit term is isolated requires explicit demonstration that curvature corrections and higher derivatives of the effective potential produce only O(1/ℓ) or smaller residuals that vanish in the strict eikonal limit; the skeptic note indicates this identification of boundary conditions may retain potential-shape dependence.

    Authors: We agree that an explicit demonstration of the vanishing residuals is warranted. In the revised manuscript we add a dedicated paragraph in the eikonal section that isolates the leading WKB/geodesic term (determined solely by the photon-sphere frequency) and shows, via the standard WKB expansion, that all curvature corrections and higher-derivative contributions enter at O(1/ℓ) or higher and therefore vanish in the strict ℓ → ∞ limit. Consequently the boundary-condition distinction between QNMs and GBFs affects only sub-leading terms, rendering the leading-order correspondence independent of the detailed shape of the potential. revision: yes

  2. Referee: [Corrections at finite ℓ] Finite-ℓ correction formula: the statement that overtones supply the leading corrections at smaller ℓ is asserted, but the explicit dependence of those corrections on the potential (or on the QNM spectrum) is not derived in closed form; without this, the practical utility of the correspondence outside the eikonal regime remains unclear.

    Authors: We acknowledge that a general closed-form expression for the overtone corrections is not supplied in the original text. In the revision we insert a short perturbative subsection that derives the leading finite-ℓ correction by expanding the WKB transmission coefficient around the eikonal result and retaining the first overtone contribution; the resulting formula expresses the correction explicitly in terms of the overtone frequencies and the second derivative of the effective potential at its peak. While this is not a fully non-perturbative closed form, it provides a concrete, usable approximation that can be tested against exact solutions and improves the practical utility outside the strict eikonal regime. revision: partial

Circularity Check

0 steps flagged

No circularity: QNM-GBF correspondence derived from independent boundary conditions and eikonal limit

full rationale

The paper derives the approximate correspondence by comparing the purely outgoing boundary conditions for quasinormal modes against the scattering (incoming-from-horizon) conditions for grey-body factors, then taking the high-frequency eikonal limit to obtain an exact relation expressible via the fundamental QNM frequency (with overtones entering only as sub-leading corrections). This chain uses the wave equation and effective potential directly and does not reduce any claimed prediction to a quantity defined or fitted inside the paper itself; external self-citations, if present, are not load-bearing for the central mapping.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard assumptions of linear perturbation theory around spherically symmetric black-hole backgrounds and the validity of the eikonal approximation; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)
  • domain assumption Linearized gravitational perturbations around a fixed spherically symmetric black-hole background obey the wave equation with the given effective potential.
    Invoked when mapping boundary conditions for quasinormal modes versus scattering states.
  • domain assumption The eikonal (high-frequency) limit allows a WKB-style connection between the two sets of boundary conditions.
    Central to the claim that the correspondence becomes exact.

pith-pipeline@v0.9.0 · 5470 in / 1432 out tokens · 74461 ms · 2026-05-17T19:30:37.961474+00:00 · methodology

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