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REVIEW 3 major objections 5 minor 20 references

Simple Analytic Estimate of Black Hole Shadow Size in an Expanding Universe

T0 review · 3 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read The paper claims the shadow angle of any non-rotating black hole in an expanding universe is simply the inverse sine of its photon-sphere radius over the angular-diameter distance.

desk verdict A clean but essentially content-free substitution of the Schwarzschild impact parameter into the standard angular-diameter-distance formula; cites the literature that already does this properly. read the letter →

arxiv 2510.19857 v2 pith:F74DLLDQ submitted 2025-10-21 gr-qc

Debarshi Mukherjee This is my paper · ORCID
classification gr-qc MSC 83C5783C1083F05 PACS 04.70.-s98.80.-k04.20.-q
keywords blackholeshadowcosmologicalexpansionangulardiameterdistancephotonsphereMcVittiemetricSchwarzschildspacetimeFLRWcosmologyLambda-CDM
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

The paper tries to establish a one-line formula for how cosmic expansion changes the apparent shadow size of a non-rotating black hole: the angular radius is the inverse sine of the local photon-sphere impact parameter divided by the standard angular-diameter distance to the source. All cosmological dependence enters through that distance, so the shadow becomes a probe of the distance-redshift relation rather than a purely local quantity. The authors motivate the formula by embedding Schwarzschild in an expanding background and then evaluate it numerically for flat Lambda-CDM, including a small-redshift expansion that shows the shadow angle falls roughly as 1/z with corrections set by the deceleration parameter. A sympathetic reader would care because this reduces a would-be ray-tracing problem in a dynamic spacetime to a known cosmological integral, making the expansion's influence on shadows analytically tractable and pedagogically transparent.

What carries the argument

The central object is the critical impact parameter b_c = 3*sqrt(3)*G*M/c^2 of the Schwarzschild photon sphere, the boundary between photons that fall into the black hole and those that escape. The load-bearing step is to substitute this b_c for the generic proper radius R in the cosmological relation alpha = arcsin(R/D_A(z)), yielding Eq. (9). The McVittie metric provides the conceptual bridge: it describes a point mass embedded in an expanding FLRW background and reduces to Schwarzschild locally and FLRW at large distances, which the paper uses to justify replacing the static-observer distance with D_A(z). All the cosmological work in the formula is done by the distance integral D_A(z), wh

What would settle it

A full null-geodesic ray-tracing calculation in the McVittie metric (or the Kottler metric) that computes the exact apparent shadow boundary would settle whether Eq. (9) holds; if the true boundary differs from b_c/D_A(z) by a term of order H_0*b_c/c or larger, the formula is only an approximation. Observationally, a high-redshift black hole with independently measured mass and distance whose shadow angle disagrees with Eq. (9) beyond measurement uncertainty would falsify it.

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Extended reading notes

Core claim

The paper's central claim is Eq. (9): alpha_shadow(z) = arcsin(3*sqrt(3)*G*M/c^2 / D_A(z)), where D_A(z) is the standard angular-diameter distance in the background cosmology. It asserts that the apparent angular radius of a Schwarzschild black hole's shadow in an expanding universe is obtained by taking the static critical impact parameter b_c = 3*sqrt(3)*G*M/c^2, the radius of the photon-sphere boundary for captured versus escaping light, and treating it as a rigid proper size at the source location. In flat Lambda-CDM, D_A(z) is the usual integral over H_0, Omega_m, and Omega_Lambda, so the formula carries all expansion effects through the distance-redshift relation. The paper then derive

Load-bearing premise

The paper assumes the shadow behaves as a rigid standard ruler of proper radius 3*sqrt(3)*G*M/c^2 placed at the black hole's location, so its observed angle is just that radius divided by the unmodified FLRW angular-diameter distance; this substitution is stated between Eqs. (8) and (9) rather than derived from the McVittie or Kottler null-geodesic equations.

Editorial extensions

If this is right

  • For the Milky Way's central black hole at redshift roughly 10^-6, the cosmological correction to the shadow radius is below one part in 10^10, so existing shadow measurements are unaffected by expansion.
  • For hypothetical black holes at redshift 1 to 5, the apparent shadow angle falls sharply; a 10^5 solar-mass black hole would subtend less than 10^-3 microarcseconds, below foreseeable interferometric resolution.
  • Because D_A(z) peaks near z about 1.6 in flat Lambda-CDM, a fixed-mass black hole appears slightly larger beyond that redshift; shadow size is non-monotonic in z.
  • Different background cosmologies, such as matter-dominated, de Sitter, and Lambda-CDM, yield systematically different shadow sizes at the same redshift, so shadow angles in principle encode H_0 and the matter and dark-energy densities.
  • In the static limit the formula reduces to the familiar Schwarzschild shadow result, making Eq. (9) an interpolation between local strong-field gravity and large-scale expansion.

Reading between the lines

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

  • If Eq. (9) is taken as a quantitative prediction rather than an estimate, its main unstated risk is the rigid-ruler substitution; a natural next step would be a full null-geodesic ray tracing in the McVittie or Kottler metric to compute the true shadow boundary and measure the error in b_c/D_A(z).
  • The same construction should extend to rotating black holes: replacing b_c with the appropriate spin-dependent critical impact parameter would give a closed-form, cosmology-dependent shadow size for Kerr-de Sitter black holes, a direction the paper only gestures at.
  • The non-monotonic behaviour of alpha(z) means a measured shadow size does not uniquely determine a black hole's redshift on its own; using shadows as cosmological probes would require independent distance or mass information to resolve the degeneracy.
  • Since D_A(z) depends on H_0, Eq. (9) could be inverted to estimate H_0 from a shadow angle plus a mass measurement, but the microarcsecond scales involved make this a long-term prospect rather than an immediate test.
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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

3 major / 5 minor

Summary. The paper proposes a simple analytic formula, Eq. (9), for the apparent angular radius of a Schwarzschild black hole shadow in an expanding universe: α_shadow(z) = arcsin[3√3 GM/c^2 / D_A(z)], where D_A(z) is the FLRW angular-diameter distance. The derivation is based on the McVittie and Kottler metrics, but the actual calculation only substitutes the Schwarzschild impact parameter b_c into the standard angular-size relation for an object of proper radius R. The paper then expands D_A(z) at small z, numerically evaluates the formula for various masses and cosmologies, and concludes that cosmological expansion has a negligible effect for local sources but a conceptually interesting effect at high redshift.

Significance. If Eq. (9) were established, the paper would provide a simple pedagogical bridge between strong-field black hole physics and cosmology. The manuscript is clearly organized and the numerical integration is straightforward and reproducible. However, the central relation is not derived from the null-geodesic structure of the McVittie or Kottler spacetimes; it is an ansatz that reduces the problem to the background distance-redshift relation. The cited exact calculations by Perlick, Tsupko, and Bisnovatyi-Kogan (Refs. [6,7]) show that the shadow impact parameter is modified in these spacetimes, and the present work neither uses nor compares against them. Consequently, the main claim that cosmic expansion influences the shadow in the manner described is unverified, and the significance is therefore limited to a heuristic, pedagogically framed estimate.

major comments (3)
  1. [II.B, Eq. (9)] The central formula is introduced by substituting R = b_c into the FLRW angular-size relation Eq. (8). No null-geodesic computation in the McVittie metric is performed. In the static Schwarzschild case, the shadow angle follows from the bending of null geodesics and the critical impact parameter; in McVittie/Kottler, the photon-sphere impact parameter is modified (see Refs. [6,7]). Identifying the shadow's 'proper radius' with the flat-space b_c is an unproven ansatz, not a derivation. The paper should either derive Eq. (9) from the McVittie geodesic equation or state explicitly that it is an approximation and quantify its error.
  2. [II.C, Eq. (12) and III.B] The small-z expansion and the numerical 'verification' check the same assumed relation. Eq. (12) is just the Taylor expansion of D_A(z) inserted into Eq. (9), and the numerical evaluation in Eq. (15) is the same expression. The agreement between the two does not verify the physics; it merely confirms the numerical integration is internally consistent. The redshift dependence of α is inherited from the background D_A(z), which is an input, not a new prediction about shadow optics.
  3. [II.B and III.D] The manuscript cites Refs. [6,7] but does not use or compare with their exact results. For Kottler, the shadow depends on Λ and on the observer's position relative to the cosmological horizon; for McVittie, Tsupko and Bisnovatyi-Kogan derived an analytic expression with additional dependence on the Hubble rate. These corrections are precisely the 'influence of cosmic expansion' the paper claims to estimate. Without showing that those corrections are negligible in the regime considered, Eq. (9) cannot be validated even as an approximation.
minor comments (5)
  1. [II.A, Eq. (5)] Typo: 'for larger o' should presumably be 'for large r_o'.
  2. [III.C, Fig. 1] The dashed portions labeled 'R > D_A(z)' correspond to physically unrealizable cases for the masses discussed; for any astrophysical black hole, b_c << D_A(z) at all redshifts. The figure caption should state which masses produce the dashed curves and why this regime is included.
  3. [III.A, Eq. (15)] The factor '206265×10^6 rad arcsin(...)' is dimensionally unclear; it should read '206265×10^6 μas/rad × arcsin(...)' or similar.
  4. [References] Reference [14] is incomplete: it lists 'Perlick, V., Oleg Yu. Tsupko' without a title, journal volume, or page numbers in the reference list.
  5. [General notation] The paper uses α, θ_sh, and α_shadow interchangeably for the shadow angular radius; please define one symbol and use it consistently.

Circularity Check

2 steps flagged · score 6.0 of 10

Eq. (9) is constructed by substituting the Schwarzschild impact parameter b_c into the definition of angular-diameter distance; the claimed cosmological dependence is inherited from D_A(z), not derived from McVittie/Kottler geodesics.

  1. self definitional [§II.B, Eqs. (8)–(9)]
    "The apparent angular size of any object of proper radius R is then α(z) = arcsin(R/D_A(z)). (8) For the case of a black hole shadow, we substitute R = b_c = 3√3GM/c^2, giving α_shadow(z) = arcsin[3√3GM/c^2 / D_A(z)]. (9)"

    Eq. (9) is exactly Eq. (8) with R replaced by b_c; Eq. (8) is the definitional angular-diameter-distance relation for a rigid proper radius. The McVittie and Kottler metrics written in §II.B are not used to obtain this relation or to compute b_c in a time-dependent background. Thus the central 'derived' formula is the input distance-redshift relation multiplied by a constant, so its redshift dependence is an input by construction.

  2. fitted input called prediction [§III.A–C, Eqs. (13)–(15)]
    "We adopt the cosmological parameters consistent with the Planck 2018 results: H0 = 67.4 km/s/Mpc, Ωm = 0.315, ΩΛ = 0.685. ... The angular size is computed from Eq. (9) in microarcseconds: α_shadow(z) = 206265×10^6 arcsin[3√3GM/c^2 / D_A(z)]. (15)"

    The numerical angular-size curves are b_c/D_A(z) with D_A(z) obtained by integrating the fitted Planck distance-redshift relation (13). All H0, Ωm, ΩΛ dependence and the non-monotonic behavior near z≈1.6 are properties of the input D_A(z); the shadow calculation contributes only the constant numerator. The claimed 'redshift dependence' is therefore the fitted distance-redshift relation renamed as a shadow prediction.

full rationale

The central step of the paper is Eq. (9), and the paper itself tells the reader how it was obtained: Eq. (8) states the standard angular-diameter-distance relation for any object of proper radius R, and then R is set equal to the Schwarzschild critical impact parameter b_c = 3√3 GM/c². That is not a geodesic computation in McVittie or Kottler spacetime; it is the definitional relation α = R/D_A with a constant chosen from the flat-space photon-sphere impact parameter. Consequently, the shadow size as a function of redshift is, by construction, b_c divided by the input cosmological angular-diameter distance. The numerical section then adopts Planck-fitted H0, Ωm, ΩΛ and integrates the same D_A(z), so the figures and 'non-monotonic trend' merely reproduce the input distance-redshift curve. This is a partial circularity: the paper is transparent that 'the only cosmological dependence enters through D_A(z)', but its central formula and numerical results reduce by construction to the input distance-redshift relation. There is no evidence of self-citation circularity here; the issue is the definitional/fitted-input structure of Eq. (9).

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central formula is a composition of two established results: the Schwarzschild critical impact parameter and the FLRW angular-diameter distance. The paper's only contribution is the substitution R=b_c into α=R/D_A, which is an unproven modeling step (axiom 3). All cosmological parameter dependence is fed in from Planck; no independent prediction is produced.

assumptions (6)
  • standard math Schwarzschild shadow: for a photon sphere at r_ph=3M, critical impact parameter b_c=3√3GM/c².
    Textbook geodesic result restated in §II.A; not supplied by this paper.
  • standard math An object of proper radius R at redshift z in FLRW subtends angle α = arcsin(R/D_A(z)), with D_A(z) from Eq. (7).
    Definition of angular-diameter distance; standard cosmology.
  • ad hoc to paper A black hole shadow can be treated as such an object with R = b_c.
    This is the unproven bridge in §II.B (Eqs. 8-9); no calculation shows the shadow boundary maps to a proper ruler at the source.
  • domain assumption The universe is flat ΛCDM with Planck 2018 parameters (H0=67.4, Ωm=0.315, ΩΛ=0.685) and radiation Ωr=9.2e-5.
    Adopted from Planck [10] and used for all numerical evaluations; standard but external.
  • standard math Small-z expansion of D_A with q0=½Ωm−ΩΛ.
    Textbook Taylor expansion used in Eq. (11)-(12).
  • domain assumption McVittie/Kottler metrics correctly embed a Schwarzschild black hole in an expanding universe, and a comoving observer's distance is the FLRW D_A.
    Invoked in the introduction but never used; the analysis does not actually integrate geodesics in these metrics.

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Pith. "Pith review of Simple Analytic Estimate of Black Hole Shadow Size in an Expanding Universe." pith.science (2026). https://pith.science/paper/F74DLLDQ

@misc{pith2026251019857,
  author       = {Pith},
  title        = {Pith review of: Simple Analytic Estimate of Black Hole Shadow Size in an Expanding Universe},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/F74DLLDQ}},
  note         = {Machine review of arXiv:2510.19857}
}
abstract

The apparent shadow of a black hole provides one of the most direct probes of strong-field general relativity. While the shadow size in asymptotically flat spacetimes is well understood, the influence of cosmic expansion on its apparent angular diameter remains less explored. In this work, we present a simple analytic framework to estimate the shadow size of a non-rotating black hole embedded in an expanding universe. By combining the local Schwarzschild geometry with large-scale cosmological dynamics through the McVittie and Kottler metrics, we derive a compact relation between the shadow angular size and the angular diameter distance $D_A(z)$. This approach captures the essential dependence on cosmological parameters such as the Hubble constant $H_0$ and the cosmological constant $\Lambda$, while remaining analytically tractable. We further perform numerical estimates to quantify the redshift dependence of the apparent shadow size, showing that the effect of cosmic expansion is negligible for nearby sources but becomes relevant for high-redshift black holes. Our results demonstrate a clear conceptual connection between strong-gravity optics and cosmological expansion, providing a pedagogically transparent and physically motivated extension of black hole shadow theory to a cosmological context.

Figures

Figures reproduced from arXiv: 2510.19857 by the authors.

Figure 1
Figure 1. FIG. 1. Numerical computation of the apparent shadow an [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

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Reference graph

Works this paper leans on

20 extracted references · 2 canonical work pages

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    A hypothetical intermediate-mass black hole of M= 10 5 M⊙ placed at cosmological distances (z∼1–5). The angular size is computed from Eq. (9) in microarc- seconds (µas): αshadow(z) = 206265×10 6 rad arcsin " 3 √ 3GM/c 2 DA(z) # .(15) For all practical cases,α≪1 rad, so the small-angle approximation sinα≈αis used in the numerical evalu- ation. B. V erifica...

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