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REVIEW 3 major objections 6 minor 259 references

Shadow in the Galactic Center: Theoretical Concept -- Prediction -- Realization

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Pith's one-line read The paper argues that black-hole shadow observations cannot, even in principle, distinguish the Schwarzschild metric from a modified metric differing only inside r=3, so EHT images constitute model preference rather than proof.

desk verdict A well-sourced historical review that overreaches in Section 8: the 'even in principle' degeneracy claim rests on a toy metric that is not a field-equation solution. read the letter →

arxiv 2506.16927 v1 pith:24VDUOXB submitted 2025-06-20 physics.hist-ph astro-ph.COastro-ph.GAgr-qc

Alexander F. Zakharov This is my paper · ORCID
classification physics.hist-phastro-ph.COastro-ph.GAgr-qc
keywords blackholeshadowGalacticCenterSgrA*M87*EventHorizonTelescopeSchwarzschildmetricdegeneracyVLBIobservations
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 historical review's scientific core is a caveat about what black-hole shadows can prove. The paper argues that a Schwarzschild black hole and a modified metric that coincides with it for r≥3 but has A(r)=$r^{2}$/27 for r<3 produce identical photon scattering and shadow, so an observer cannot, even in principle, tell them apart. It therefore claims the Event Horizon Telescope reconstructions of M87* and Sgr A* shadows should be read as showing that a black-hole model is preferable among alternatives, not as proof that a black hole is present. Along the way it also asserts that its 2005 prediction of a roughly 50 microarcsecond shadow for the Galactic Center was confirmed by the EHT Sgr A* result in 2022. A sympathetic reader would care because the claim sets a limit on what shadow imaging alone can establish about the nature of compact objects.

What carries the argument

The load-bearing object is the modified spherically symmetric metric function $A(r)=1-2/r$ for $r\ge 3$ and $A(r)=r^2/27$ for $r<3$, inside the line element $ds^2=A(r)dt^2-A^{-1}(r)dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$. The argument works through the comparison function $B(r)=A(r)/r^2$: photon motion is possible only where $B(r)\le 1/b^2$, and the capture-scatter boundary $b=3\sqrt{3}$ is the same for Schwarzschild and the modified metric because $B(r)$ is unchanged wherever it matters. This single construction carries the paper's claim that shadows cannot uniquely certify black holes.

What would settle it

Construct a non-black-hole metric that satisfies the Einstein equations with a specified matter source, differs from Schwarzschild only in $2<r<3$, and reproduces the Schwarzschild photon-scattering cross section; if no such source exists, the in-principle degeneracy fails for physical metrics. The converse test is to compute an observable that samples $r<3$, such as gravitational time delay or lensing of background sources, and show that it differs between the two metrics.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is a degeneracy in shadow observations. For a static spherically symmetric metric $ds^2=A(r)dt^2-A^{-1}(r)dr^2+r^2(d\theta^2+\sin^2\theta\,d\phi^2)$, photon motion is governed by $B(r)=A(r)/r^2$, and the critical impact parameter separating capture from scattering is $b=3\sqrt{3}$. Replacing $A(r)=1-2/r$ by $A(r)=r^2/27$ in the interval $2<r<3$ leaves the regions where $B(r)\le 1/b^2$ unchanged, hence photon scattering from infinity and the shadow edge are the same, while bound orbits test only $r\ge 3$ where the two metrics coincide. The author concludes that neither scattering experiments nor bound-orbit tests can distinguish the Schwarzschild black hole from this modified metric, even in principle, so shadow and stellar-orbit observations choose between models rather than prove the existence of black holes.

Load-bearing premise

The modified metric $A(r)=r^2/27$ for $r<3$ is introduced by hand, without being derived from Einstein's equations or assigned a matter source, so the claimed indistinguishability covers toy metrics unless every physical alternative must obey the field equations.

Editorial extensions

If this is right

  • If the degeneracy argument holds, no shadow image can by itself certify that M87* or Sgr A* is a black hole; it establishes that a black-hole model fits the image better than the alternatives considered.
  • Public statements that the EHT imaged a black hole should be rephrased as imaging a shadow consistent with a black-hole metric.
  • The same limitation likely applies to any test confined to photon scattering and bound stellar orbits, the two observables the paper examines.
  • The confirmed roughly 50 microarcsecond shadow diameter for Sgr A* remains a valid prediction, but its confirmatory force is model preference rather than uniqueness.

Reading between the lines

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

  • A reader could push further: if the modified metric must solve the Einstein equations, the degeneracy may disappear, because no known matter source is shown to produce $A(r)=r^2/27$ with the required matching.
  • The paper's reasoning suggests searching for observables that probe the $r<3$ region directly, such as gravitational-wave ringdown, photon-ring autocorrelation, or gravitational time delays, which could break the shadow degeneracy.
  • Historically, the argument implies that future event-horizon-scale images of other galactic centers should be reported as model comparisons, and public communication about black-hole imaging may need the same caveat.
  • One could test the construction's robustness by checking whether including spin, which the paper sets aside by restricting to spherical symmetry, reopens a distinction between a rotating Kerr black hole and a modified rotating metric.
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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 / 6 minor

Summary. This manuscript is a historical review of general relativity, cosmology, and black-hole shadow observations, with particular attention to Russian contributions (Friedman, Gamow, Shklovsky, Matveenko, and others). It traces the shadow concept from Bardeen's thought experiment to the EHT images of M87* and Sgr A*, argues that the author's 2005 prediction of a roughly 50 microarcsecond shadow for the Galactic Center was confirmed by the EHT in 2022, and closes with a philosophical Section 8 in which a modified static spherically symmetric metric is constructed that has the same photon capture impact parameter as Schwarzschild and the same bound orbits with periapsis r >= 3. The paper concludes that shadow observations alone cannot uniquely prove the existence of a black hole, and that the EHT results should be interpreted as model preference rather than proof.

Significance. The historical material is extensive and mostly reliable, and the review usefully documents the pre-EHT prediction literature. The non-circularity of the 2005 shadow-size prediction is correctly handled: the estimate was based on independently measured masses and general relativity, not on the EHT image itself. Section 8's geodesic calculation is analytically correct, and the critical impact parameter b = 3*sqrt(3) is reproduced for the modified metric, making the paper a clear illustration of observational degeneracy within a deliberately broad class of metrics. However, the central 'even in principle' claim is not established for physically meaningful alternatives, because the toy metric is not a solution of the Einstein equations with standard matter and has a naked singularity at r = 0. With appropriate qualification, the paper would be a valuable contribution to the history and philosophy of black-hole observations.

major comments (3)
  1. [Section 8, Eq. (1) and the definition A(r) = r^2/27 for r < 3] The modified metric is introduced without being derived from field equations or assigned a matter source. In the region 0 < r < 3 this spacetime is not a vacuum solution (its Einstein tensor is nonzero) and it has a curvature singularity at r = 0; it is also not a solution of the Einstein equations with standard energy conditions. Therefore the statement that 'we will not be able, even in principle, to distinguish the metric of a black hole from a metric where the function A(r) is replaced by a modified one' is established only for the class of arbitrary static, spherically symmetric metrics that are not required to satisfy any field equations. Please state this restriction explicitly, and either provide a physically sourced example or soften the 'even in principle' formulation to 'within the class of metrics allowed by the stated assumptions'.
  2. [Section 8, sentences after Eq. (7)] The equivalence argument covers only (i) null geodesics arriving from infinity and (ii) bound timelike orbits with periapsis r >= 3. It does not cover photons emitted inside r < 3 (for example, from the accretion flow or a hot spot in a high-spin model), nor other observational channels such as gravitational-wave ringdown, tidal heating, or direct horizon signatures, which can in principle probe the region r < 3. The phrase 'even in principle' is therefore too strong and is also inconsistent with the paper's own later statement that 'we have considered only two ways to test the metric'. The sentence should be revised to say that the two metrics are indistinguishable with respect to the specific observables considered.
  3. [Section 8, bound-orbit argument] The claim that bound orbits cannot distinguish the two metrics is correct but should be stated more precisely: for the S2 star, the periapsis is approximately 1400 Schwarzschild radii, so the entire orbit lies far outside r = 3 and provides no constraint whatsoever on the region r < 3. The bound-orbit part of the argument is thus a consistency check rather than an independent probe of the modified region, and the paper should say so explicitly.
minor comments (6)
  1. [Throughout] The text contains numerous repeated words and typographical errors, for example 'branches of of science' in the abstract, 'and and energy release' in Section 3, and 'he was the first who the first to make the attempt' in Section 2; a careful copyedit is needed.
  2. [Section 8] The cross-reference 'done in Eq. ()' should be 'Eq. (1)'.
  3. [References] Reference [12] is empty; please complete or remove it.
  4. [Section 8, sentence after Eq. (7)] The phrase 'in the interval 2 < r < 3' is inaccurate: the modified A(r) differs from Schwarzschild for all r < 3, and the Schwarzschild A(r) is negative for r < 2. This should read 'for r < 3'.
  5. [Section 4] The sentence 'whose shadow size is about a million times smaller than that of the shadow itself, the size in the center of the Galaxy' is garbled and should be rephrased.
  6. [Section 7] The statement that the 2005 prediction 'became true in May 2022' should be softened to 'is consistent with the EHT measurement', given the uncertainties in the 2005 mass estimates and in the measured shadow diameter.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the 2005 shadow estimate is an ex ante GR-based prediction confirmed by independent EHT data, and Section 8's toy metric is an explicitly constructed underdetermination example, not a fitted prediction.

full rationale

The paper's central confirmation claim (Sections 4 and 7) is not circular: the ~50 microarcsecond shadow estimate for Sgr A* in Zakharov et al. (2005) was derived from then-known black hole mass and distance together with general-relativistic photon deflection, before the EHT observations, and the EHT 2022 measurement (51.8 ± 2.3 microarcseconds) is an independent external result. Section 8's indistinguishability argument is an explicitly constructed counterexample to uniqueness, not a prediction: the metric (1) is defined piecewise with A(r)=1-2/r for r>=3 and A(r)=r^2/27 for r<3, so photons scattered from infinity with impact parameter b>3√3 and bound orbits with periapsis >=3 sample only the region where the two metrics coincide by definition. The paper itself limits the conclusion: 'we have considered only two ways to test the metric of a compact object' and concludes that the black-hole model 'describes the observational data best among the alternatives considered.' That is a model-preference statement, not a derivation whose output is secretly one of its inputs. The absence of a field-equation source for the modified metric in r<3 is a physical-reasonableness caveat (the toy metric is not shown to satisfy the Einstein equations and has a naked singularity), but it is a correctness concern, not circularity. Heavy self-citation is present but not load-bearing: the key confirmations are external (EHT, GRAVITY, historical literature), and Section 8's mathematics is self-contained. No circular step can be exhibited by quoting a reduction of the conclusion to its own input, so the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no fitted parameters and no new physical entities. Its analytical claim relies on one ad hoc metric choice for the interior region r<3, which is not a solution of any stated field equations, and on the standard geodesic equations of general relativity.

assumptions (3)
  • standard math Photon geodesics in static spherically symmetric metrics are governed by Eq. (4) and Eq. (5), derived from ds^2=0.
    Used in Section 8 to compute the critical impact parameter and shadow size.
  • ad hoc to paper The metric function A(r) may be arbitrarily specified for r<3 as long as it is continuous and differentiable at r=3.
    Section 8 constructs A(r)=r^2/27 inside r=3 to illustrate shadow degeneracy without deriving it from a gravitational field equation.
  • domain assumption The shadow size and capture cross-section are the relevant observables for comparing black hole and non-black-hole metrics.
    The paper restricts its non-uniqueness claim to these observables; it does not consider gravitational wave ringdown or accretion disk variability.

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Cite this review

Pith. "Pith review of Shadow in the Galactic Center: Theoretical Concept -- Prediction -- Realization." pith.science (2026). https://pith.science/paper/24VDUOXB

@misc{pith2026250616927,
  author       = {Pith},
  title        = {Pith review of: Shadow in the Galactic Center: Theoretical Concept -- Prediction -- Realization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/24VDUOXB}},
  note         = {Machine review of arXiv:2506.16927}
}
read the original abstract

General Relativity (GR) was created in November 1915 and since its creation and up to now this theory has undergone many tests. The first realistic cosmological models were proposed in the works of Friedman, written in the 1920s. For a long time Friedman's cosmological works were actually banned in Soviet Union due to philosophical reasons, since the models where the birth and evolution of the Universe occurs were considered ideologically unacceptable. Due to great achievements in relativity and cosmology and due to increasing interest to these branches of of science in last decades we recall a development of relativistic astrophysics and contribution of Russian researchers in these studies. Since one of the world leaders in physical cosmology A. A. Friedman passed away in September 1925, it is reasonable to outline the main achievements of physical cosmology over the past 100 years. We discuss also observational and theoretical achievements in confirmations of relativistic observational predictions for black holes, including the closest supermassive black hole in our Galactic Center. We outline an evolution of black hole shadow from the purely theoretical concept to observable quantities for supermassive black holes in Sgr A* and M87*.

Figures

Figures reproduced from arXiv: 2506.16927 by the authors.

Figure 1
Figure 1. Blue curve represents (1 − 2/r)/r2 for all r. Function B(r) corresponds to the blue curve for r ≥ 3 and corresponds to the black horizontal straight line B(r) = 1/27 for r < 3. Red horizontal straight line b = 5.063 (B(r) = 0.039) corresponds to capture of photon, the green horizontal straight line b = 6 (B(r) = 0.028 for r > 4.4) corresponds to the case when photon moves from infinity toward the until r ≈ 4.4, at t… view at source ↗
Figure 2
Figure 2. Blue curve presents function A(r) for the Schwarzschild, while red curve corresponds to our modified A(r). These red and blue curves are different only for r < 3. we usually underestimated our own and overestimated foreign...Many of the organizational shortcomings still exist today, and one of the main ones is the underestimation of one’s own and overestimation of foreign forces....After all, excessive modesty is an… view at source ↗

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