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arxiv: 2603.15979 · v3 · pith:E2LLN7ADnew · submitted 2026-03-16 · 🌀 gr-qc

Ringdown waves from hairy black holes

Ariadna Uxue Palomino Ylla , Kosuke Makino , Akane Tanaka , Akihiro Ishibashi , Chul-Moon Yoo This is my paper

Pith reviewed 2026-05-21 10:12 UTC · model grok-4.3

classification 🌀 gr-qc
keywords quasinormal modesringdownblack hole hairanisotropic fluideikonal correspondenceSchwarzschildKerr
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The pith

Formulas allow direct reading of anisotropic fluid parameters from deviations in black hole quasinormal mode frequencies.

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

This paper shows how the ringdown signals from hairy black holes can reveal details about the matter causing the hair. By modeling the hair as a perturbative anisotropic fluid on top of Schwarzschild or Kerr black holes, the authors connect changes in quasinormal modes to shifts in the frequency and stability of photon orbits. They provide explicit formulas that translate the fluid's equation-of-state parameters into corrections to the standard ringdown spectrum. This offers a practical way to compute quasi-normal modes for many hairy black hole models without solving the full wave equations each time. The approach is independent of whether the fluid satisfies energy conditions.

Core claim

Using the eikonal correspondence between quasinormal modes and unstable null geodesics, we relate shifts in the ringdown spectrum to perturbations of the photon-orbit frequency and Lyapunov exponent. The black hole hair is treated as an anisotropic fluid perturbatively added to the vacuum black holes. In particular, we derive formulas which allow one to directly read off deviations from the Schwarzschild or Kerr QNM spectrum in terms of the corresponding equation-of-state parameters of the anisotropic fluid. Under this setting, independent of energy conditions, our formulas offer a systematic method to compute quasi-normal mode frequencies for a broad class of hairy black holes.

What carries the argument

Eikonal correspondence between quasinormal modes and unstable null geodesics, used to link ringdown shifts to perturbations of photon-orbit frequency and Lyapunov exponent in the presence of anisotropic fluid hair.

Load-bearing premise

The black-hole hair can be treated as a perturbative anisotropic fluid that preserves the applicability of the eikonal correspondence between quasinormal modes and unstable null geodesics.

What would settle it

Measure the quasinormal mode frequencies for a concrete hairy black hole solution whose anisotropic fluid parameters are known independently, then check whether the observed frequency shifts match the shifts predicted by the derived formulas.

Figures

Figures reproduced from arXiv: 2603.15979 by Akane Tanaka, Akihiro Ishibashi, Ariadna Uxue Palomino Ylla, Chul-Moon Yoo, Kosuke Makino.

Figure 1
Figure 1. Figure 1: Tangential pressure parameter wθ(r) and QNM shifts for the Bardeen black hole. In figure 1a, we observe that the tangential fluid parameter wθ becomes smaller if we increase the value of the hairy parameter q. At the asymptotic region, the value of the tangential parameter tend to 1.5 for any finite value of q. In figure 1b we can observe that the δΩ/Ω0 is positive and δλ/λ0 negative as expressed in (3.5) … view at source ↗
Figure 2
Figure 2. Figure 2: Tangential pressure parameter wθ(r) and frequency shifts for the Hayward black hole. In particular for this case, ρ + Pθ = 6q 3M 8π(q 3+r 3) 3 , then the energy conditions NEC and WEC [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: QNM shifts for different values of wq. Continuous lines refer to δΩ/Ω0 and dashed lines refer to δλ/λ0 for different values of k. The frequency components are Ω⋆ ≃ 1 3 √ 3M  1 − 3k 2(3M) 1+3wq  , λ⋆ ≃ 1 3 √ 3M  1 + 3wq(1 + wq) − 2 4(33wqM1+3wq ) k  . (3.23) Different wq values tell how fast the “hair” δf(r) decays (or grows). In its original con￾struction, the source is “quintessence-like”, giving a ne… view at source ↗
Figure 4
Figure 4. Figure 4: Representative QNM and geometric shifts for the anisotropic fluid halo Kiselev model. [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Time evolution of the ringdown waveform, Ψ( [PITH_FULL_IMAGE:figures/full_fig_p013_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Plots for rotating Bardeen black holes [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Plots for rotating Hayward black holes [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Plots for rotating Kiselev black holes [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
read the original abstract

We study how quasinormal-mode frequencies may encode information about the effective matter source responsible for black-hole hair. Using the established eikonal correspondence between quasinormal modes and unstable null geodesics, we relate shifts in the ringdown spectrum to perturbations of the photon-orbit frequency and Lyapunov exponent. The black hole hair is treated as an anisotropic fluid perturbatively added to the vacuum black holes (Schwarzschild and Kerr black holes). In particular, we derive formulas which allow one to directly read off deviations from the Schwarzschild or Kerr QNM spectrum in terms of the corresponding equation-of-state parameters of the anisotropic fluid. Under this setting, independent of energy conditions, our formulas offer a systematic method to compute quasi-normal mode frequencies for a broad class of hairy black holes.

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 to derive perturbative formulas that map small corrections from an anisotropic fluid (modeling black-hole hair) onto shifts in the eikonal quasinormal-mode quantities (photon-orbit frequency and Lyapunov exponent) for Schwarzschild and Kerr backgrounds, allowing direct readout of ringdown deviations in terms of the fluid equation-of-state parameters while remaining independent of energy conditions.

Significance. If the central derivations hold, the work supplies a practical, metric-based tool for estimating QNM shifts without solving the full linearized perturbation equations, which could aid in interpreting ringdown data from future gravitational-wave detectors as probes of effective matter sources around black holes.

major comments (2)
  1. [Derivation of geodesic perturbations] The perturbative addition of the anisotropic-fluid stress-energy to the vacuum metric: the manuscript must explicitly show (in the section deriving the first-order corrections to the effective potential for null geodesics) that the background remains a solution to the Einstein equations at zeroth order and that the fluid parameters enter only at linear order without back-reacting on the zeroth-order geometry.
  2. [Eikonal correspondence and QNM shifts] Validity of the eikonal correspondence under the fluid perturbation: while the correspondence holds for any stationary metric with an unstable light ring, the paper should demonstrate in the eikonal section that the first-order shifts in Ω and λ remain accurate when the fluid is treated perturbatively, including a brief check that the Lyapunov exponent correction does not invalidate the high-frequency limit.
minor comments (2)
  1. [Abstract and introduction] The abstract states independence from energy conditions; a short sentence in the introduction or derivation section clarifying that the geodesic calculation never invokes the null or weak energy condition would make this explicit.
  2. [Notation and definitions] Notation for the fluid equation-of-state parameters should be defined once at first use and used consistently when expressing the frequency shifts.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the clarity of the presentation. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [Derivation of geodesic perturbations] The perturbative addition of the anisotropic-fluid stress-energy to the vacuum metric: the manuscript must explicitly show (in the section deriving the first-order corrections to the effective potential for null geodesics) that the background remains a solution to the Einstein equations at zeroth order and that the fluid parameters enter only at linear order without back-reacting on the zeroth-order geometry.

    Authors: We agree that an explicit statement will improve the rigor of the derivation. The zeroth-order metric is the exact vacuum Schwarzschild or Kerr solution, which satisfies the Einstein equations with vanishing stress-energy tensor. The anisotropic fluid is introduced as a first-order perturbation, so its equation-of-state parameters contribute exclusively to the linear corrections in the effective potential for null geodesics and do not modify the background geometry. We will add a clarifying paragraph in the relevant derivation section to state this explicitly. revision: yes

  2. Referee: [Eikonal correspondence and QNM shifts] Validity of the eikonal correspondence under the fluid perturbation: while the correspondence holds for any stationary metric with an unstable light ring, the paper should demonstrate in the eikonal section that the first-order shifts in Ω and λ remain accurate when the fluid is treated perturbatively, including a brief check that the Lyapunov exponent correction does not invalidate the high-frequency limit.

    Authors: We thank the referee for highlighting this point. The eikonal correspondence is applied to the perturbed stationary metric, which retains an unstable light ring. Because the fluid is treated as a small perturbation of order ε, the shifts δΩ and δλ are likewise first-order quantities. The correction to the Lyapunov exponent is O(ε) and therefore does not alter the leading high-frequency behavior of the eikonal approximation. We will insert a short explanatory paragraph in the eikonal section to make this perturbative consistency explicit. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper applies the established eikonal correspondence (between QNMs and unstable null geodesics) to a perturbatively added anisotropic-fluid hair on Schwarzschild/Kerr backgrounds. Shifts in photon-orbit frequency and Lyapunov exponent are computed directly from the modified metric plus fluid stress-energy, with EOS parameters treated as external inputs. No step reduces by construction to a self-definition, a fitted parameter renamed as prediction, or a load-bearing self-citation chain; the eikonal relation is invoked as a standard external result rather than derived internally. The claimed independence from energy conditions follows from the geodesic calculation never invoking them. The central formulas therefore map external fluid parameters onto QNM deviations without circular reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

The derivation rests on the eikonal correspondence as a background fact and on the perturbative addition of an anisotropic fluid whose equation-of-state parameters are treated as free inputs; no new particles or forces are postulated beyond the fluid description.

free parameters (1)
  • equation-of-state parameters of the anisotropic fluid
    Deviations from the vacuum QNM spectrum are expressed directly in terms of these parameters, which are not derived but supplied as model inputs.
axioms (2)
  • standard math Eikonal correspondence between quasinormal modes and unstable null geodesics
    Invoked to relate frequency shifts to perturbations of the photon-orbit frequency and Lyapunov exponent.
  • domain assumption Hair can be modeled as a perturbative anisotropic fluid on vacuum black-hole backgrounds
    Stated in the abstract as the setting for the derivation.
invented entities (1)
  • anisotropic fluid representing black-hole hair no independent evidence
    purpose: To provide an effective matter source that generates hair while remaining perturbatively small
    Introduced as the model for the hair; no independent falsifiable signature outside the fluid parameters is given.

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Reference graph

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