Grey-body factors of black holes

Alexander Zhidenko; Roman Konoplya

REVIEW

Grey-body factors of black holes

Roman Konoplya, Alexander Zhidenko

General Relativity and Quantum Cosmology · Submitted 17 Oct 2026 · Published version

Pith's verdict: A thorough, well-organized review of grey-body factors that will be the standard reference for the next several years, though it leans heavily on the authors' own work and could be more critically synthetic.

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arXiv ID2610.24356 (version 2)

Pith IDDY5PYT4R, Pith's permanent record of this paper and its review

CitationAlexander Zhidenko, Roman Konoplya. "Grey-body factors of black holes." arXiv:2610.24356 (2026). https://pith.science/paper/DY5PYT4R

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Editor's note

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This is a useful review. Konoplya and Zhidenko are the right people to write it: they have produced much of the recent work on grey-body factors, including the correspondence with quasinormal modes that gets a long section here. The review collects an enormous amount of material—definitions, scattering setups for flat/dS/AdS, analytic approximations (WKB, eikonal, low- and high-frequency limits, Visser bound), numerical methods, public codes, the quasinormal-mode connection, and applications to specific theories from stringy black holes to regular ones. The organization is logical and the prose is clear, even for a reader new to the subject. The authors are also candid about limitations: they repeatedly state where WKB breaks down (low ℓ, double barriers, modified gravity) and where the correspondence fails (de Sitter branch, superradiance, AdS). That honesty is a real strength.

That said, the review has the weaknesses you would expect from a review written by leading practitioners. The section on the grey-body–quasinormal correspondence is almost entirely based on the authors’ own papers. Outside references are cited, but the framing gives the impression that the correspondence is more settled than it probably is, especially for realistic rotating binaries. The same self-citation pattern appears in the compilation of results for various theories (Table 1): many entries cite the authors’ own WKB calculations rather than independent checks. For a reference work that aims to be authoritative, I would have liked to see more cross-validation or at least a discussion of how reliable those single-group numbers are. Also, the review is more of a catalogue than a critical synthesis. Sections 8.1–8.4 list results without much meta-analysis: which effects are robust across models, which are artifacts of a particular approximation? The reader gets a map of the territory but not a clear sense of which paths are well-trodden and which are swampy.

Still, these are minor complaints for a review of this scope. The paper will be cited heavily and deserves to be published in a high-quality journal like Reports on Progress in Physics. I would recommend acceptance after minor revision: the authors should add a paragraph or two of critical commentary on the reliability of the methods across different theories, and they should balance the self-citation in Section 7 with more independent references or a frank statement that independent tests are needed. Once that is done, it's ready.

Referee report

Major objections (1)

[§4.4, Eq. (127) and surrounding text] The Regge variable is defined as κ = ℓ + D − 3/2, but later in the same paragraph the Poisson-summation phase factor is written as exp(2πip[κ − (D−3)/2]). If κ = ℓ + D − 3/2, then κ − (D−3)/2 = ℓ + D/2, not ℓ. The correct definition should be κ = ℓ + (D−3)/2, so that κ − (D−3)/2 = ℓ. All subsequent expressions that depend on this definition (Eqs. (129)–(135) and the associated text) must be checked and corrected consistently. This is not a merely cosmetic issue: the shift affects the argument of the exponential in the eikonal formula and therefore the high-frequency absorption cross section derived from it.

Minor objections (4)

[§2.4, Eq. (24)] The grey-body factor for de Sitter spacetime contains the factor (ω − Φ_H)/(ω − Φ_C). When ω lies between Φ_H and Φ_C this ratio is negative, which correctly signals superradiance. A brief parenthetical remark clarifying that the ratio is positive in the superradiant interval (or, more precisely, that the transmitted-energy fraction remains positive because |T|^2 compensates) would help the reader.
[§4.1, Eq. (78)] At leading order K = i U_0/√(2U_2) is purely imaginary. The expression for |T|^2 in Eq. (81) nevertheless yields a real number between 0 and 1 because e^{2πiK} becomes e^{-2π Im(K)}. A sentence noting that K acquires a real part at higher orders but that the final transmission probability remains real would prevent confusion.
[§7, Eq. (176)] The correspondence formula (176) is stated as the eikonal-limit result. Authors should add a clarifying remark that corrections beyond the eikonal regime, expressed in terms of ω_1 as in Eq. (177), are necessary when ℓ is not large.
[Acknowledgments] The authors acknowledge OpenAI assistance for language review. Authors should verify that this statement complies with the journal's policy on the use of artificial intelligence in manuscript preparation.

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The paper

The paper claims

Grey-body factors are the frequency-dependent transmission probabilities that reshape Hawking radiation, determine absorption and scattering cross sections, and provide real-frequency observables complementary to quasinormal modes, as…

The reading

This review establishes that grey-body factors are not ad hoc emissivities but mode-dependent scattering probabilities fixed by the field equations and boundary conditions. They convert the idealised thermal Hawking spectrum into the one an observer actually sees, govern absorption and scattering cross sections, and offer a real-frequency complement to quasinormal modes for black-hole ringdown. The paper unifies the formulation across asymptotically flat, de Sitter, and anti-de Sitter spacetimes, surveys analytical, semi-analytical, and numerical computation methods, and shows how these factors connect to wormhole transmission, superradiance, modified gravity, and gauge/gravity duality. A reader cares because accurate grey-body factors are essential for predicting black-hole evaporation spectra, interpreting gravitational-wave ringdown, and testing strong-field geometry.

What carries the argument

The central object is the grey-body factor Γ_{ℓm}(ω), defined as the transmitted-energy fraction in a Schrödinger-type radial equation with an effective potential barrier. The key identity is Γ_{ℓm}(ω) = 1 − |R_{ℓm}|^2 = (ω − Φ_H)/k |T_{ℓm}|^2, where R and T are reflection and transmission coefficients. The workhorse computational method is the higher-order WKB approximation, which expands the potential near its peak and matches parabolic cylinder functions. The recently established correspondence (Eq. 176) expresses Γ_{ℓm}(ω) directly in terms of the quasinormal frequency ω_0 and the first overtone ω_1, providing a bridge between scattering and ringdown.

Extended reading notes

Core claim

The paper's central claim is that grey-body factors are well-defined, mode-dependent transmission probabilities Γ_{ℓm}(ω) = 1 − |R_{ℓm}|^2 that solve a one-dimensional scattering problem with ingoing boundary conditions at the horizon. They are not phenomenological insertions but direct outputs of the wave equation in a curved background. The review demonstrates how the same object connects absorption cross sections (with a universal low-frequency limit equal to the horizon area), Hawking radiation spectra (by multiplying the thermal flux), and ringdown waveforms (through the modulus and phase of the reflection amplitude). It also establishes a WKB-based correspondence between grey-body factors and quasinormal modes, showing that the transmission probability can be expressed in terms of the fundamental quasinormal frequency and first overtone, with high accuracy for ℓ > spin.

Load-bearing premise

Many semi-analytical results (the WKB method and the grey-body factor–quasinormal-mode correspondence) assume that the effective potential has a single peak and decays slowly at the boundaries; this fails for multiple-barrier potentials, rapidly varying potentials, or certain modified-gravity backgrounds.

Editorial extensions

If this is right

Grey-body factors can be used to infer the mass and spin of a black-hole remnant directly from the frequency-domain ringdown spectrum, without relying solely on overtone fits.
In primordial-black-hole evaporation, accurate grey-body factors are necessary to compute the observable gamma-ray and particle fluxes; approximations like unit transmission can misestimate both the spectrum and the lifetime.
The correspondence between grey-body factors and quasinormal modes offers a new consistency test for black-hole spectroscopy: if the mass and spin derived from the real-frequency scattering envelope disagree with those from quasinormal frequencies, it may indicate deviations from general relativity.
For modified-gravity black holes, the frequency-dependent transmission probability often carries more diagnostic power than the Hawking temperature alone, and can reverse the expected trend of evaporation intensity.
The Visser bound provides a rigorous, non-perturbative lower limit on transmission that holds even when WKB methods fail, guaranteeing that grey-body factors are never exactly zero at any finite frequency.

Reading between the lines

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

The review implicitly suggests that grey-body factors may become a standard observable in black-hole astrophysics, on par with quasinormal modes and shadows, especially as gravitational-wave detectors improve sensitivity to the post-merger signal.
The correspondence between grey-body factors and quasinormal modes, while accurate for single-peaked potentials, is likely to break down for double-barrier potentials or in the superradiant regime of rotating black holes; future work should map the precise domain of validity.
The availability of public codes (GrayHawk, BlackHawk) that compute grey-body factors will accelerate phenomenological studies of primordial black holes and allow systematic scans over parameter spaces of modified gravity models.

Original abstract

Grey-body factors describe the frequency-dependent transmission of fields through the effective potential generated by a black hole or another compact object. They convert the idealized thermal spectrum of Hawking radiation into the spectrum measured by an observer, determine absorption and scattering cross sections, and provide real-frequency observables complementary to quasinormal modes. In this review, we present a unified account of grey-body factors in black-hole physics and related systems. We formulate the scattering problem for asymptotically flat, de Sitter, and anti-de Sitter spacetimes, and discuss the connections between transmission coefficients, absorption cross sections, Hawking radiation, and ringdown. We also review methods for their computation and survey recent developments, including the correspondence between grey-body factors and quasinormal modes, their stability under deformations of the effective potential, wormhole transmission and superradiance, and their behavior in modified theories of gravity. Finally, we discuss the appearance of grey-body factors in gauge/gravity duality, where absorption probabilities in the bulk are related to correlation functions and transport coefficients in the boundary theory.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-20: 1 request, 5,039 input tokens.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.05 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.11 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

Grey-body factors describe the frequency-dependent transmission of fields through the effective potential generated by a black hole or another compact object. They convert the idealized thermal spectrum of Hawking radiation into the spectrum measured by an observer, determine absorption and scattering cross sections, and provide real-frequency observables complementary to quasinormal modes. In this review, we present a unified account of grey-body factors in black-hole physics and related systems. We formulate the scattering problem for asymptotically flat, de Sitter, and anti-de Sitter spacetimes, and discuss the connections between transmission coefficients, absorption cross sections, Hawking radiation, and ringdown. We also review methods for their computation and survey recent developments, including the correspondence between grey-body factors and quasinormal modes, their stability under deformations of the effective potential, wormhole transmission and superradiance, and their behavior in modified theories of gravity. Finally, we discuss the appearance of grey-body factors in gauge/gravity duality, where absorption probabilities in the bulk are related to correlation functions and transport coefficients in the boundary theory.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Grey-body factors describe the frequency-dependent transmission of fields through the effective potential generated by a black hole or another compact object.	background 1.00	0.04	0.04	asserted 1.00
s1	They convert the idealized thermal spectrum of Hawking radiation into the spectrum measured by an observer, determine absorption and scattering cross sections, and provide real-frequency observables complementary to quasinormal modes.	background 1.00	0.05	0.06	asserted 1.00
s2	In this review, we present a unified account of grey-body factors in black-hole physics and related systems.	contribution 0.99	0.72	0.09	asserted 0.94
s3	We formulate the scattering problem for asymptotically flat, de Sitter, and anti-de Sitter spacetimes, and discuss the connections between transmission coefficients, absorption cross sections, Hawking radiation, and ringdown.	contribution 0.79	0.77	0.09	asserted 0.94
s4	We also review methods for their computation and survey recent developments, including the correspondence between grey-body factors and quasinormal modes, their stability under deformations of the effective potential, wormhole transmission and superradiance, and their behavior in modified theories of gravity.	contribution 0.91	0.23	0.08	asserted 0.93
s5	Finally, we discuss the appearance of grey-body factors in gauge/gravity duality, where absorption probabilities in the bulk are related to correlation functions and transport coefficients in the boundary theory.	contribution 0.97	0.17	0.10	asserted 0.98

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Schematic illustration of the WKB treatment of barrier scattering in terms of the tortoise coordinate r∗. The function U possesses a single minimum, giving rise to two turning points which separate three regions: Region I (near the horizon), Region II (near the peak), and Region III (asymptotic infinity). The boundary conditions appropriate for grey-body factors are imposed as an incident and reflected wave at spatial infinity, and a purely transmitted wave at the horizon. The WKB method matches solutions across the barrier via a uniform approximation in Region II. Here Dν(z) is…

**Figure 2.** Schematic comparison of scattering by a Schwarzschild black hole (dashed line) and an alternative blackhole model (solid line) for the same perturbation channel and multipole number. The left panel shows the effective potentials normalized by a common asymptotic mass M. The middle panel shows the corresponding grey-body factors. The shaded band indicates the intermediate-frequency transition region, where the curves generally carry the most model-dependent information. The right panel shows the signed additive difference ∆Γℓ = Γalt ℓ − Γ Schw ℓ . Parametrized geometries and…

Figures reproduced from arXiv:2610.24356 by the authors.

Discussion

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