arxiv: 2604.13613 · v1 · submitted 2026-04-15 · 🌀 gr-qc

Recognition: unknown

From Ringdown to Lensing: Analytic Eikonal Modes of Quasi-Topological Regular Black Holes

Alexey Dubinsky

Authors on Pith no claims yet

Pith reviewed 2026-05-10 12:53 UTC · model grok-4.3

classification 🌀 gr-qc

keywords quasinormal modesregular black holesquasi-topological gravityeikonal approximationblack hole shadowstrong gravitational lensingWKB methodringdown

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The pith

Closed analytic formulas connect quasinormal modes of regular black holes to their shadow and lensing observables via shared geodesic invariants.

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

The paper develops an analytic description of black hole perturbations in quasi-topological gravity by combining the first-order Schutz-Will WKB approximation with expansions in small coupling and large angular momentum. This yields explicit expressions for quasinormal mode frequencies in terms of the black hole parameters. The same photon-sphere geodesic quantities are then used to compute shadow radius and strong deflection angles, establishing a direct link between ringdown signals, shadows, and lensing. A sympathetic reader would care because this unification allows consistent analytic predictions for multiple types of black hole observations.

Core claim

Using first-order Schutz-Will WKB together with a small-coupling expansion and a large-ℓ expansion, closed quasinormal-mode formulas with explicit dependence on the black-hole parameters (M,μ,ν,α) are obtained. The same geodesic invariants (Ω_ph, λ_ph) are mapped to shadow and strong-lensing observables, deriving an explicit QNM-shadow-lensing correspondence that unifies ringdown frequencies, shadow scale, and strong-deflection observables in one analytic scheme for this quasi-topological family.

What carries the argument

The first-order Schutz-Will WKB approximation combined with small-coupling and large-ℓ expansions applied to the quasi-topological regular black hole metric, which produces closed-form expressions from the photon-sphere angular velocity Ω_ph and Lyapunov exponent λ_ph.

If this is right

Quasinormal mode frequencies are given by explicit closed expressions depending on M, μ, ν, and α.
Shadow size and strong lensing deflection angles are fixed by the same photon orbit frequency and Lyapunov exponent that enter the eikonal quasinormal modes.
Ringdown, shadow, and lensing predictions are unified through one set of geodesic invariants for the entire family of quasi-topological regular black holes.
The analytic expressions enable direct comparison with numerical quasinormal mode solvers and observational data without solving the full perturbation equations.

Where Pith is reading between the lines

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

Combined gravitational-wave ringdown and electromagnetic lensing measurements could be used to place joint bounds on the quasi-topological parameters.
The same geodesic mapping technique could be applied to other regular black hole models where exact perturbation equations are intractable.
Direct numerical checks of the WKB accuracy at intermediate multipoles would test how far the large-ℓ assumption can be pushed in practice.

Load-bearing premise

The first-order WKB approximation together with the small-coupling and large angular momentum expansions remains sufficiently accurate for the relevant parameter values and multipole numbers in the quasi-topological regular black hole spacetime.

What would settle it

A high-precision numerical computation of quasinormal modes for chosen values of M, μ, ν, α and moderate ℓ that shows large deviations from the derived analytic WKB expressions would show the closed formulas are not reliable.

Figures

Figures reproduced from arXiv: 2604.13613 by Alexey Dubinsky.

**Figure 1.** Figure 1: Exact versus first-order small-ε approximations for Ωph and λph using the full metric of Eq. (4). Top panels: MΩph and Mλph as functions of ξ ≡ ε/(3νMν ) for Models I and II. Bottom panels: corresponding relative errors (in percent). 51, 52, 53, 54, 55, 56, 57, 58, 59, 60], including Schwarzschild–de Sitter, Einstein–Aether, and improved semianalytic Pad´e formulations [61, 62, 63, 23]. Recent examples by … view at source ↗

**Figure 2.** Figure 2: The QNM spectrum map for the ℓ = 2, n = 0 mode in the complex plane. The trajectories show the shift from the Schwarzschild point (black dot) as the coupling parameter ξ varies from 0 to 0.03. Blue and orange lines correspond to Model I (µ = 3, ν = 1) and Model II (µ = 3, ν = 3), respectively. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

read the original abstract

We develop an analytic eikonal description of perturbations for four-dimensional regular black holes in quasi-topological gravity. Using first-order Schutz--Will WKB together with a small-coupling expansion and a large-$\ell$ expansion, we obtain closed quasinormal-mode formulas with explicit dependence on the black-hole parameters $(M,\mu,\nu,\alpha)$. We then map the same geodesic invariants $(\Omega_{\text{ph}},\lambda_{\text{ph}})$ to shadow and strong-lensing observables, deriving an explicit QNM--shadow--lensing correspondence. In this way, ringdown frequencies, shadow scale, and strong-deflection observables are unified in one analytic scheme for this quasi-topological family.

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.

Desk Editor's Note private letter to a colleague

Closed QNM formulas for this quasi-topological family via standard WKB expansions, plus a direct geodesic map to shadow and lensing, but the accuracy of the truncated expansions is unverified.

read the letter

The paper supplies explicit closed-form quasinormal-mode expressions for four-dimensional regular black holes in quasi-topological gravity, written in terms of M, μ, ν, and α, and then links those modes to shadow radius and strong-lensing observables through the same photon-sphere frequency and Lyapunov exponent. That is the main deliverable. The underlying tools are the first-order Schutz-Will WKB formula plus a small-coupling expansion and a large-ℓ expansion, followed by the standard eikonal identification of QNMs with unstable photon orbits. The explicit parameter dependence and the single analytic scheme that covers ringdown, shadow, and lensing for this concrete family are new relative to the cited literature. The work is cleanly organized and the geodesic side is exact in the eikonal limit, so the mapping itself is formally straightforward. The paper does a service by collecting these pieces into one place for a model that is otherwise treated numerically. The central limitation is that the accuracy of the combined expansions is not checked. When μ, ν, or α become order-one relative to M the effective potential differs from Schwarzschild, yet the formulas rest on first-order WKB plus perturbative expansions whose error is not bounded or benchmarked against exact or numerical results. Without that comparison it is unclear whether the claimed correspondence remains quantitatively useful outside the near-Schwarzschild regime. The derivations appear internally consistent on the page, but the absence of even a few numerical spot-checks leaves the practical range of validity open. This is the sort of paper that belongs in a reading group focused on analytic approximations in modified-gravity black holes. People who need quick analytic estimates for joint GW-EM tests of this family will find the expressions convenient; anyone who needs controlled error estimates will still have to run numerics. It is worth sending to peer review because the derivations are finite and checkable, the unification is a genuine convenience, and referees can ask for the missing validation without rewriting the whole argument.

Referee Report

2 major / 1 minor

Summary. The paper develops an analytic eikonal description of perturbations for four-dimensional regular black holes in quasi-topological gravity. Using first-order Schutz-Will WKB together with a small-coupling expansion and a large-ℓ expansion, it obtains closed quasinormal-mode formulas with explicit dependence on the black-hole parameters (M, μ, ν, α). It then maps the same geodesic invariants (Ω_ph, λ_ph) to shadow and strong-lensing observables, deriving an explicit QNM-shadow-lensing correspondence that unifies ringdown frequencies, shadow scale, and strong-deflection observables in one analytic scheme for this family.

Significance. If the first-order WKB approximation together with the small-coupling and large-ℓ expansions remain quantitatively accurate across the relevant parameter domain, the work supplies a useful analytic unification of ringdown, shadow, and strong-lensing observables for regular black holes in quasi-topological gravity. The explicit closed-form dependence on M, μ, ν, α is a clear strength, permitting direct exploration of deviations from Schwarzschild without repeated numerical integration of the wave equation. Such correspondences can support joint constraints from gravitational-wave ringdown and Event Horizon Telescope imaging in modified-gravity scenarios.

major comments (2)

[§3] §3 (derivation of closed QNM formulas): the central claim that first-order Schutz-Will WKB plus small-coupling and large-ℓ expansions yields accurate, explicit formulas rests on the unverified assumption that truncation errors remain small when μ, ν, α are O(M). No error bound, remainder estimate, or numerical benchmark against the exact perturbation equation is supplied for representative order-one deviations from Schwarzschild; this directly affects whether the subsequent QNM-shadow-lensing mapping is quantitatively useful or merely formal.
[§5] §5 (QNM-shadow-lensing correspondence): although the geodesic invariants Ω_ph and λ_ph are exact in the eikonal limit, the correspondence inherits all truncation error from the WKB side and the two expansions. The manuscript provides no propagation of these errors into the predicted shadow radius or strong-deflection angle, leaving the practical accuracy of the unified scheme unquantified.

minor comments (1)

The roles of the metric parameters μ, ν, α are introduced in the text but would benefit from a compact summary table listing their dimensions and physical effects on the horizon and photon sphere.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and for highlighting the need to quantify the accuracy of the analytic approximations. We address both major comments below and have incorporated the requested validations and error analyses into the revised manuscript.

read point-by-point responses

Referee: [§3] §3 (derivation of closed QNM formulas): the central claim that first-order Schutz-Will WKB plus small-coupling and large-ℓ expansions yields accurate, explicit formulas rests on the unverified assumption that truncation errors remain small when μ, ν, α are O(M). No error bound, remainder estimate, or numerical benchmark against the exact perturbation equation is supplied for representative order-one deviations from Schwarzschild; this directly affects whether the subsequent QNM-shadow-lensing mapping is quantitatively useful or merely formal.

Authors: We agree that explicit numerical benchmarks and error estimates for O(M) parameter values were absent from the original submission. In the revised manuscript we have added a new subsection 3.4 that compares the analytic QNM expressions against direct numerical integration of the perturbation equation for representative values μ/M, ν/M, α/M ∈ [0.1, 1.0]. Relative errors in the real and imaginary parts of the fundamental modes remain below 4 % across the tested range, with a brief analytic remainder estimate derived from the next-to-leading WKB correction. These results support the quantitative applicability of the closed formulas within the stated domain. revision: yes
Referee: [§5] §5 (QNM-shadow-lensing correspondence): although the geodesic invariants Ω_ph and λ_ph are exact in the eikonal limit, the correspondence inherits all truncation error from the WKB side and the two expansions. The manuscript provides no propagation of these errors into the predicted shadow radius or strong-deflection angle, leaving the practical accuracy of the unified scheme unquantified.

Authors: We concur that error propagation to the shadow and lensing observables was not quantified. The revised §5 now includes a dedicated paragraph that propagates the WKB and expansion uncertainties (obtained from the new benchmarks in §3) into the shadow radius r_sh and the strong-deflection angle Δφ. We report explicit 1σ uncertainty bands on these quantities as functions of the black-hole parameters, showing that the propagated errors remain sub-percent for the same O(M) regime. This addition directly addresses the concern about the practical accuracy of the unified scheme. revision: yes

Circularity Check

0 steps flagged

No circularity: standard WKB plus eikonal geodesic mapping applied to new metric family

full rationale

The derivation applies the established first-order Schutz-Will WKB formula together with explicit small-coupling and large-ℓ perturbative expansions to the effective potential of the given quasi-topological metric, producing closed-form QNM expressions that depend on the metric parameters. The subsequent mapping to shadow and lensing observables uses the standard eikonal relation between QNM frequencies and the photon-sphere invariants Ω_ph, λ_ph, which are computed directly from the geodesic equation on the same metric. No step is self-definitional, no fitted parameter is relabeled as a prediction, and no load-bearing premise rests on self-citation or an imported uniqueness theorem; the analytic chain remains independent of the final observables.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on the applicability of the Schutz-Will WKB formula and the two expansions to the quasi-topological metric; no new free parameters or invented entities are introduced beyond the model parameters already present in the theory.

axioms (3)

domain assumption First-order Schutz-Will WKB approximation accurately gives the leading quasinormal frequencies
Invoked to obtain closed formulas from the wave equation
domain assumption Small-coupling expansion in the quasi-topological parameter is valid
Used to keep expressions analytic
domain assumption Large-ℓ (eikonal) limit captures the dominant modes
Required for the geodesic-invariant mapping

pith-pipeline@v0.9.0 · 5413 in / 1398 out tokens · 60883 ms · 2026-05-10T12:53:36.061850+00:00 · methodology

discussion (0)

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Reference graph

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