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REVIEW 5 major objections 6 minor 59 references

Black hole shadow parameters and quasi-normal modes for Weyl-incorporated gravity

T0 review · 5 major / 6 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Weyl-matter coupling yields mass-dependent black-hole shadows and long-lived scalar ringdowns under a constant-density halo.

desk verdict First WIG shadows and scalar QNMs, but the numbers sit outside the controlled expansion the authors themselves define. read the letter →

arxiv 2607.23552 v1 pith:7ILB6FKH submitted 2026-07-26 gr-qc

Noraiz Tahir , Mubasher Jamil , Kaynat Fatima , Tajammal Hussain Khokhar This is my paper · ORCID
classification gr-qc MSC 83C5783D0583C56
keywords blackholeshadowsmodifiedgravityquasi-normalmodesdarkmatterhalosWeyl-incorporatedplasmatest-fieldapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper explores Weyl-incorporated gravity, a minimal extension of general relativity that adds a single coupling λ between matter and the free gravitational field (the Weyl tensor). The authors place static spherical black holes inside a constant-density baryonic halo, solve for the corrected lapse function, and extract photon-sphere radii and shadow sizes for masses spanning many orders of magnitude. They find unique λ values near 3.5–3.9 that keep a horizon near the Schwarzschild radius; the resulting shadows grow with mass and shrink when a homogeneous plasma is added. Under the test-field approximation they also compute massive-scalar quasi-normal frequencies and time-domain waveforms, all of which decay (ω_i < 0) yet can ring for thousands of cycles. A sympathetic reader cares because the same coupling already claimed to explain galactic rotation curves is here shown to leave measurable imprints on strong-field observables that future shadow and gravitational-wave data could test.

What carries the argument

The first-order corrected lapse function f^(1)(r) obtained by expanding the Weyl-incorporated field equations for constant density ρ_c; photon-sphere and shadow radii are read from its effective potential, and the same f^(1) supplies the potential barrier for WKB/AIM quasi-normal frequencies and characteristic time-domain integration.

What would settle it

Recompute the horizon condition and shadow radius with a realistic, radially falling halo density that vanishes at the horizon and a fully consistent two-function metric; if no unique λ near 3.5–3.9 survives, or if the resulting shadows become indistinguishable from Schwarzschild, the reported phenomenology collapses.

Watch

Extended reading notes

Core claim

For spherically symmetric static black holes embedded in a constant-density baryonic halo within Weyl-incorporated gravity, there exist unique values of the coupling λ ≈ 3.5–3.9 that preserve a horizon near r = 2m; the associated photon-sphere impact parameters produce shadow radii that increase with black-hole mass, decrease under homogeneous plasma, and support test-field massive scalar modes that all decay while reaching quality factors of order 10^3–10^4.

Load-bearing premise

The calculation assumes a single lapse function even though pressureless dust plus the Weyl interaction make the two metric potentials unequal, and the authors’ own error estimates show the expansion is not under control at the working densities.

Editorial extensions

If this is right

  • Shadow size in this framework is predicted to grow with black-hole mass once λ is fixed by the halo model.
  • A homogeneous plasma background systematically shrinks the apparent shadow relative to the pure baryonic-halo case.
  • Massive scalar test-field ringdowns on these backgrounds are long-lived (quality factors 10^3–10^4), offering a potential gravitational-wave signature.
  • Time-domain waveforms show slower late-time decay than Schwarzschild for low multipoles, consistent with enhanced wave trapping.
  • The same λ that was previously used for galactic rotation curves is here shown to leave strong-field imprints that can be compared with EHT and LIGO/Virgo data once realistic halos are included.

Reading between the lines

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

  • Because the reported λ values sit outside the controlled weak-coupling regime, any observational claim will first require a non-perturbative or fully consistent two-metric-function solution.
  • If the long-lived massive-scalar modes survive a full gravitational perturbation analysis, they could appear as slowly damped echoes or prolonged ringdowns distinguishable from Kerr templates.
  • Joint EHT-plus-galaxy-rotation constraints on a single λ would constitute a sharp, mass-scale-spanning test of whether the Weyl-matter coupling is universal.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 6 minor

Summary. The manuscript studies spherically symmetric static black holes in "Weyl-incorporated gravity" (WIG/MORD), which adds a coupling λ C_{αμβν}T^{αβ}T^{μν} to the Einstein–Hilbert action. Assuming a constant-density baryonic halo (ρ_c = 0.5) and a single-lapse ansatz ds² = −fdt² + dr²/f + r²dΩ², the authors solve the field equations iteratively to first order in λρ_c, determine λ for each black hole mass by imposing that the lapse vanish at r_s = 2m, and compute photon-sphere radii, shadow impact parameters (Table 1), energy emission using the Schwarzschild Hawking temperature, shadow modifications from a "homogeneous plasma" (Table 2), and massive-scalar QNM frequencies via 6th-order WKB and AIM, supplemented by a characteristic time-domain integration. The authors state all computed QNMs have ω_i < 0 with quality factors up to ~10⁴. The paper is explicitly framed as a theoretical exploration, and it is unusually candid about its own approximations — eqs. (32)–(33) quantify ε ≈ 0.5 and δ ≈ 1.9 — but the central numerical results are nonetheless presented at face value.

Significance. If the computations were controlled, the paper would provide the first black-hole shadow and QNM phenomenology for WIG/MORD, a framework whose galactic-scale fit (λ = 4.3125, Ref. [23]) makes a strong-field cross-check genuinely interesting. The manuscript has real strengths: it derives and reports its own consistency diagnostics (eqs. 29–33) rather than hiding the ansatz error; the QNM sector cross-checks two semi-analytic methods (6th-order WKB and AIM) against an explicit time-domain integration, a reasonable robustness standard for test-field claims; and the long-lived massive-scalar modes (Q ~ 10³–10⁴) are a falsifiable phenomenological prediction. However, the central numerical outputs currently rest on a first-order expansion at δ ≈ 1.9 and an internally inconsistent metric ansatz, so the results as tabulated cannot yet support the paper's conclusions.

major comments (5)
  1. [§2.1] §2.1, eqs. (29)–(33): The single-lapse ansatz (4) is inconsistent with the field equations for the matter model used: eq. (31) gives G^t_t − G^r_r = −κρ_c + λO(ρ_c²) ≠ 0 for pressureless dust, and the authors' own error measures at the working point are ε ≡ ρ_c r² ≈ 0.5 and δ ≡ λρ_c ≈ 1.9. Every quantitative output of the paper (Tables 1–2, Figs. 1–7) is built on f^(1), a first-order truncation whose neglected terms are the same order as those retained. The authors acknowledge this and call the exercise 'pedagogical', but the text then presents Table 1/2 numbers to three significant figures and draws trend conclusions ('positive correlation between mass and shadow size', '~7% barrier reduction'). Either the two-lapse system (A ≠ B) must be solved, or the claims must be explicitly re-scoped so that no quantitative precision is asserted beyond the uncontrolled regime.
  2. [Table 1] §2.1, Table 1: λ is not predicted; it is solved per-mass from the condition f^(1)(r_s) = 0. This has two load-bearing consequences. (a) The reported deviations of r_p and R_s from Schwarzschild are partially by construction, since λ is tuned to hold the horizon at r_s. (b) The text states 'for the larger λ the horizon lies slightly inside r = 2m', which contradicts the construction f^(1)(r_s)=0 with r_s = 2m — the horizon location should be re-solved and reported, not asserted. (c) The near-constancy λ ≈ 3.864–3.869 over nine decades of m followed by a jump to 3.568 at m = 10³ km is unexplained; it suggests either numerical conditioning of the root-finding or a branch change, and must be diagnosed.
  3. [Table 1] Table 1, columns 3–5: For m = 1 km the table gives r_p = 1.504 km while r_s = 2m = 2 km; for m = 10³ km it gives r_p = 1.873 km while 2m = 2000 km. The 'photon sphere' thus lies inside the nominal horizon for the two largest masses, and r_p is nearly mass-independent (~0.36–1.87 km) while m spans 10⁻⁶–10³ km. If r_p < r_h, the maximum of V_eff is not causally accessible to distant observers and R_s = r_p/√f(r_p) (eq. 42) is not an observable shadow radius. Either the units/normalization of Table 1 are mislabeled, the horizon is not at ~2m (in which case r_h must be tabulated), or the reported configurations do not describe shadows. This must be resolved before any conclusion from Table 1 stands.
  4. [§4] §4, eq. (46): the 'homogeneous plasma' refractive index n(r) = √(1 − ρ_p/r) is (i) explicitly r-dependent and therefore not homogeneous; (ii) dimensionally inconsistent if ρ_p is a density, since ρ_p/r must be dimensionless; and (iii) not the standard cold non-magnetized plasma dispersion n² = 1 − ω_p²/ω² with constant ω_p used in the cited formalism (Synge; Perlick/Tsupko). No derivation or reference is given. Since all of Table 2 and Fig. 4 follow from eq. (46), the plasma section must either derive this index from a stated microphysical model or be reworked with the standard plasma dispersion, including a clear definition of ρ_p and its units.
  5. [§3] §3, eq. (45): the energy-emission estimate uses the Schwarzschild temperature T = 1/(8πm) while the background lapse is explicitly non-Schwarzschild. The surface gravity T = f^(1)′(r_h)/(4π) is directly computable from the f^(1) the authors already construct, so the approximation is not forced by complexity. As it stands, §3 combines a WIG-corrected σ_lim with an uncorrected T, which is internally inconsistent and removes precisely the WIG effect the section is meant to estimate. At minimum the exact T should be computed or the induced error quantified.
minor comments (6)
  1. [§2.1] Units of the densities are never stated: ρ_c = 0.5 and ρ_p = 0.4/0.2 are given as pure numbers, but in geometric units with m in km a density has units km⁻². Please state units for ρ_c, ρ_p, and λ, and note how far ρ_c = 0.5 km⁻² sits from realistic halo densities (~10⁻²⁴ kg/m³).
  2. [§2.1] Eq. (42) and Fig. 2: the Carter constant is set to K = 1 without justification. For spherical symmetry K is fixed by the orbit, not chosen; please clarify the role of K and why K = 1 is consistent with the critical orbit used for R_s.
  3. [§2.1] Eq. (35): the conserved energy E is written with an overall factor 1/2 multiplying the g_tt ṫ term; as written this is not the standard normalization E = f(r) ṫ. Please check and correct, since eqs. (40)–(42) inherit this normalization.
  4. [§5.2] §5: the ~30% WKB–AIM discrepancy for (ℓ=0, µ=0) is reported only in words. A small table of representative ω values for both methods would make the claimed '<1% agreement for µ > 1' verifiable and properly delimit the WKB results.
  5. [Figures] Fig. 2 caption: '3600' should be '360°'; 'homogenous' in Fig. 4 caption should be 'homogeneous'; 'yeild' above eq. (43); author name spelled 'Khokhar' on p.1 but 'Khookhar' in running heads.
  6. [§6] The claimed galactic value λ = 4.3125 ± 0.0012 from Ref. [23] and the black-hole values λ ≈ 3.5–3.9 differ by ~20% in a theory whose motivating hypothesis (§1) is that λ is unique. Even granting the different halo models, this tension should be stated explicitly and its implications for the uniqueness hypothesis discussed, including whether the units of λ in the two fits coincide.

Circularity Check

2 steps flagged · score 4.0 of 10

λ in Table 1 is solved from the horizon condition, not an independent input; shadows/QNMs on that background are real computations, not forced equalities.

  1. fitted input called prediction [§2.1, eqs. (21)–(28), Table 1 and surrounding text]
    "Specifically, we determine the values of λ that satisfy the horizon condition f(1)(rs)=0 at the Schwarzschild radius rs. Table 1 presents the obtained λ values for black holes spanning several orders of magnitude in mass with a constant halo density ρc=0.5. For each mass configuration, we find a unique value of λ that preserves the horizon structure..."

    λ is not varied independently and then used to predict horizons or shadows. It is solved for so that the horizon sits at rs for the chosen m and ρc. Table 1’s λ column is therefore the fit residual of that condition. Reporting rp, η+ζ², and Rs next to those λ values as the theory’s shadow parameters for those masses presents a tuned background’s outputs as a λ-dependence analysis; the ‘unique λ’ is unique only as the root of f^(1)(rs)=0, not as an external prediction.

  2. self citation load bearing [§1 Introduction; cf. also Results §6 discussion of unique λ]
    "It was proposed that if WIG is truly a minimal extension to GR, the new coupling constant λ should be unique. ... Recently, for a sample of eight edge-on spiral galaxies in the local group, a single value of the coupling constant, λ=4.3125±0.0012, was shown to account for the asymptotic rotational velocities [23]."

    The narrative that λ is a single universal constant is supported inside this paper only by citation [23] (overlapping authorship with the present work). That galactic fit is not used for the BH calculation (Table 1 uses λ∼3.5–3.9 from the horizon condition instead), so it does not force the shadow/QNM numbers, but it is the sole load-bearing external warrant for the ‘unique λ’ framing that motivates treating the horizon-tuned values as physically distinguished rather than model knobs.

full rationale

The load-bearing procedural step is that, for each mass m at fixed ρ_c=0.5, λ is chosen so that the first-order lapse satisfies f^(1)(r_s)=0. The paper then reports photon-sphere radii, impact parameters, shadow radii, plasma corrections, and test-field QNMs on exactly those backgrounds, and phrases the exercise as analyzing dependence on λ and finding unique λ per mass. That is a fitted-parameter study, not a first-principles prediction of λ, and the galactic uniqueness claim is only motivational self-citation ([23]) unused in the numerics. Once f^(1) is fixed, however, R_s=r_p/√f(r_p) and the scalar potential/QNMs are standard derived quantities and do not reduce by construction to the horizon fit (unlike fitting λ to shadow data and re-predicting shadows). The GR limit and the authors’ own caveats (conditional λ, uncontrolled δ∼1.9) are stated. Score 4 reflects partial circular framing of λ without collapse of the central observables.

Assumptions & free parameters 4 free parameters · 7 assumptions · 1 invented entities

The central numerical claims rest on the WIG action (prior literature), a pressureless constant-density halo with hand-chosen ρ_c=0.5, an inconsistent spherical ansatz treated perturbatively outside its control parameter, and λ values solved from the horizon condition rather than measured. QNM claims further assume a frozen background and a massive minimally coupled scalar. No new particle is introduced, but the Weyl-matter coupling λ is an invented interaction whose only external handle cited here is a separate galactic fit by overlapping authors.

free parameters (4)
  • λ (WIG coupling) = ≈3.57–3.87 (mass-dependent; ρ_c=0.5)
    Solved per mass so that the first-order lapse vanishes at r_s; Table 1 values 3.568–3.869 are outputs of that fit, not first-principles predictions.
  • ρ_c (constant halo density) = 0.5
    Set by hand to 0.5 to amplify Weyl-matter effects; authors state it is not a realistic astrophysical density.
  • ρ_p (homogeneous plasma density) = 0.2 and 0.4
    Two illustrative values used for Table 2 and Fig. 4; not fitted to data.
  • r_0 (integration reference radius)
    Lower limit in the integral expressions for F_c and f^(1); needed to define the definite integrals but not observationally fixed.
assumptions (7)
  • domain assumption WIG field equations from varying √−g(R−2Λ−κT+λ C_αμβν T^αβ T^μν), with I_μν as given in eq. (3).
    Taken from Lee & Qadir 2019; vacuum limit recovers Schwarzschild–de Sitter. Invoked as the starting point of §2.
  • ad hoc to paper Spherically symmetric static ansatz with a single lapse f(r)=A(r)=B(r), despite G^t_t≠G^r_r for the matter model.
    Authors explicitly call this a leading-order approximation and defer the A≠B case (§2.1).
  • domain assumption Pressureless dust T_μν=ρ u_μ u_ν with constant ρ→ρ_c and static four-velocity.
    Standard idealization for a halo; used to simplify I_μν to eqs. (12)–(14).
  • ad hoc to paper First-order iterative truncation: λρ_c small enough that f^(0)=1−2m/r and F^(1) built from E^(0) suffice.
    Contradicted by their own δ∼1.9 estimate; still used for all tables.
  • domain assumption Test-field approximation: scalar Φ on fixed g_μν, T_μν, I_μν; no backreaction or gravitational QNMs.
    Stated at the opening of §5; standard but limits stability claims.
  • ad hoc to paper Hawking temperature ≈1/(8πm) for energy-emission estimates.
    §3 replaces T=f'(r_h)/4π from the WIG metric with the Schwarzschild value.
  • standard math Geometric units G=c=1; Carter separation for null geodesics; 6th-order WKB and AIM quantization as standard.
    Textbook GR and QNM technology (Carter, Konoplya, Cho et al.).
invented entities (1)
  • Weyl-matter coupling interaction λ C_αμβν T^αβ T^μν (WIG/MORD)
    purpose: Provide a minimal non-minimal coupling between matter and the free gravitational field, intended to replace or reduce the need for particle dark matter and to modify strong-field observables.
    Introduced in prior Lee–Qadir work; this paper does not derive λ from a more fundamental principle. Independent handle claimed via galactic rotation curves in a companion paper by overlapping authors, not via a collider or laboratory prediction here.

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Cite this review

Pith. "Pith review of Black hole shadow parameters and quasi-normal modes for Weyl-incorporated gravity." pith.science (2026). https://pith.science/paper/7ILB6FKH

@misc{pith2026260723552,
  author       = {Pith},
  title        = {Pith review of: Black hole shadow parameters and quasi-normal modes for Weyl-incorporated gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ILB6FKH}},
  note         = {Machine review of arXiv:2607.23552}
}
abstract

An additional term of the form $\lambda \mathbf{T} \cdot \mathbf{C} \cdot \mathbf{T}$ in the Einstein-Hilbert Lagrangian was introduced to explain the interaction between matter and pure gravitational field [H. W. Lee and A. Qadir, Motion of test particle for Weyl-interaction gravity, {\it Int. Jour. Mod. Phys. D} {\bf 28}(16) (2019) 2040014], the modified relativistic dynamics (MORD). In this paper, we estimate the shadow parameters of black hole in a spherically symmetric static spacetime within the MORD framework. As a first approximation, we assume that the black hole is surrounded by a constant-density baryonic matter halo. The analysis is then extended by introducing a homogeneous plasma background. We compute the shadow radii for various black hole masses and analyze their dependence on the WIG coupling parameter $\lambda$. In addition, we compute the fundamental quasi-normal mode (QNM) frequencies under test-field approximation, and perform a time-domain integration analysis.

Figures

Figures reproduced from arXiv: 2607.23552 by the authors.

Figure 1
Figure 1. The lapse function for spherically symmetric static black holes with different masses [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Effective potential profiles (left panel) and corresponding photon trajectories (right panel) [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The energy emission rates of the spherically symmetric static black holes with WIG. The [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: The impact of m and corresponding λ on the impact parameter with a homogenous plasma background with ρp = 0.4 (upper panel) and ρp = 0.2 (lower panel) [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: Comparison of the real parts of quasi-normal mode frequencies ( [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]
Figure 6
Figure 6. Figure 6: Comparison of the damping rates (|ωi|) computed using WKB (blue lines) and AIM (red lines) methods for massive scalar fields in a WIG black hole with mass m = 1 km and Weyl coupling λ = 3.869. The panels show different angular momentum quantum numbers: (a) ℓ = 0, (b) ℓ…
Figure 7
Figure 7. Figure 7: Time-domain profiles of scalar field perturbations for multipoles [PITH_FULL_IMAGE:figures/full_fig_p019_7.png]

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