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REVIEW 3 major objections 4 minor 80 references

Causal concept for black hole shadows

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Black hole shadows are produced by the accumulation of light rays around wandering null geodesics, not by stationary photon orbits.

desk verdict A genuinely new causal redefinition of photon-sphere concepts, but the central implication from wandering null geodesics to observable shadows is asserted rather than proven. read the letter →

arxiv 1908.02921 v2 pith:W4D3ZJO6 submitted 2019-08-08 gr-qc astro-ph.HE

Masaru Siino This is my paper · ORCID
classification gr-qcastro-ph.HE MSC 83C5783C75 PACS 04.20.-q04.70.-s
keywords blackholeshadowsphotonspherewanderingnullgeodesicsconjugatepointsWeylfocusingcausalstructureasymptoticallyflatspacetimesdynamicalholes
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 argues that the black hole shadow—the dark region silhouetted against light behind a black hole—does not need an eternal photon sphere or stationary circular photon orbits. Its proposal is to define the shadow-producing structure by causality alone: a 'wandering null geodesic,' a complete light ray that repeatedly develops conjugate points, and the 'wandering set' of such rays that wander in both time directions. In Schwarzschild spacetime these rays are exactly the unstable circular orbits on the photon sphere, so the new concept reduces to the old one where the old one exists. If the argument is right, shadows become well defined for dynamical, merging, and newly formed black holes, where the photon-sphere concept is unavailable, and asymptotic flatness alone forces at least one such wandering ray to exist.

What carries the argument

The load-bearing object is the wandering null geodesic, a complete null geodesic with an unlimited number of conjugate points; its totally wandering version, with repeated conjugate points toward both past and future, is what replaces the photon sphere in general spacetimes. Conjugate points are detected through the null geodesic deviation and Raychaudhuri equations, where the Weyl curvature acts as an effective focusing source; the paper isolates Weyl focusing by a conformal transformation and derives an affine-length scale, roughly $1/\sqrt{C_1}$ with $C_1$ a bound on Weyl curvature, below which no truncated wandering geodesic can form. The shadow-relevance argument is carried by a singular null congruence and a number function $N(p)$ counting conjugate points along each past-directed ray, with the oldest conjugate point $q_f$ defining the causal boundary $\partial I^-(q_f)$. The paper then uses the projection of this boundary along timelike source worldlines onto a spatial slice to conclude that complete past rays cross infinitely many light sources, producing the accumulation that appears as a shadow.

What would settle it

Take a past-directed light cone in Minkowski spacetime and compute the projection of $\partial I^-(q_f)$ onto a spatial slice: it is a sphere of finite radius, not the whole slice, so the open-and-closed premise already fails in the simplest globally hyperbolic spacetime; repeating this computation in a numerical collapse and finding a proper subset there would break the claim that asymptotic flatness forces an accumulation somewhere.

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Extended reading notes

Core claim

The paper's central claim is that the essence of a black hole shadow is not stationary cycling of photon orbits but the accumulation of null geodesics near a ray that repeatedly acquires conjugate points. It defines a wandering null geodesic as a complete, inextendible null geodesic with unlimited conjugate points, and a totally wandering null geodesic as one with unlimited conjugate points in both future and past directions; the set of these is the wandering set, the proposed general replacement for the photon sphere. Using the geodesic deviation equation and Weyl focusing, it argues that such wandering rays exist in general asymptotically flat black-hole spacetimes, and that a truncated version can be used to discuss when a shadow structure forms during collapse. The paper also contends that near a totally wandering ray the number of conjugate points on nearby rays becomes unbounded and rapidly varying, which is the causal mechanism for the brightness contrast seen as a shadow, and that this structure is invariant under conformal transformations.

Load-bearing premise

The argument leans on the premise that the boundary traced by the oldest conjugate points projects onto the whole spatial slice of spacetime rather than only part of it; if that projection is only partial, the conclusion that every past light ray crosses infinitely many light-source worldlines does not follow.

Editorial extensions

If this is right

  • Shadow theory no longer needs stationarity, so dynamical, merging, and collapsing black holes can have a well-defined shadow locus defined by conjugate-point accumulation.
  • In any asymptotically flat spacetime with enough light sources, the existence of at least one totally wandering null geodesic implies a brightness contrast—the shadow—will appear somewhere in the observer's sky.
  • Because conjugate points are conformally invariant, the wandering set survives conformal rescalings that can change event-horizon topology, giving a causal way to optically distinguish such spacetimes.
  • The absence of truncated wandering geodesics in globally conformally flat collapse means homogeneous spherical collapse suppresses shadow-structure formation until Weyl curvature grows.
  • The arguments apply to sufficiently old black holes and not to young ones, so the predicted contrast depends on the black hole's age and curvature scale rather than on stationarity.

Reading between the lines

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

  • Editorial inference: the conjugate-point count $N(p)$ along simulated past rays could be used as a numerical diagnostic for locating the wandering set in dynamical spacetimes without solving for photon spheres.
  • Editorial inference: observed or simulated shadow brightness profiles might be compared with $N(p)$ to estimate how many windings light rays make, offering an observational probe of the wandering structure.
  • Editorial inference: since the same accumulation underlies strong-deflection lensing, the wandering-set idea could connect shadow formation to deflection-angle divergences in time-dependent spacetimes.
  • Editorial inference: the conformal-invariance result suggests that shadow observations probe conformal structure, which could separate spacetimes that share horizon topology but differ by a conformal factor.
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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 / 4 minor

Summary. The paper proposes a causal, coordinate-free replacement for the notion of a photon sphere in general black hole spacetimes. It defines a 'wandering null geodesic' as a complete null geodesic with infinitely many conjugate points, a 'totally wandering null geodesic' as one that is wandering in both temporal directions, and a 'wandering set' as the set of such geodesics. A 'truncated wandering null geodesic' is introduced to discuss the formation of the structure in dynamical collapses. The central claim is that the physical essence of a black hole shadow is not the stationary cycling of photon orbits but the accumulation of null geodesics near a wandering null geodesic, and that the existence of a wandering null geodesic implies such accumulation somewhere in an asymptotically flat spacetime. Section III gives a theorem on the absence of truncated wandering geodesics in globally conformally flat regions and a perturbative Weyl-focusing estimate of the scale below which they do not form. Section IV attempts to show, via a congruence argument, that a totally wandering null geodesic produces an accumulation of light rays and hence a shadow.

Significance. The conceptual framework is interesting and could be valuable for discussing black hole shadows in non-stationary spacetimes, where the standard photon sphere is undefined. The paper is careful to reproduce the Schwarzschild photon-sphere limit, and Theorem 1 is a clean, checkable statement. The perturbative Weyl-focusing estimate provides a concrete scale, and the paper is transparent about several limitations. However, the central implication from a wandering null geodesic to an observable accumulation is not established: the topological step in Section IV is unsupported, the existence of wandering geodesics is assumed rather than proved, and the link from conjugate-point counts to brightness is purely qualitative. As it stands, the paper is best described as a research proposal rather than a proof of the claimed general phenomenon.

major comments (3)
  1. [Section IV] The assertion that A = ψ[∂I^-(q_f)] is "a closed and open subset in the topology of C, so that A is C" is not derived from the hypotheses and is not a general topological fact. The image of a closed achronal C^1 submanifold under a timelike flow is homeomorphic to the submanifold, but homeomorphism does not make the image either open or closed in the Cauchy surface. In Minkowski spacetime with a vertical timelike flow the image happens to be all of C because the light cone is a graph over the Cauchy surface, but that is a special case, not a general theorem. The subsequent conclusion that every timelike source worldline intersects ∂I^-(q_f) precisely once, and hence that every complete past null geodesic crosses infinitely many source worldlines, rests on this unsupported clopen conclusion and therefore collapses.
  2. [Sections IV and V; Ref. [58]] The paper's central implication is conditional on the existence of a totally wandering null geodesic, but no existence theorem is proved. Section IV begins "We suppose a situation that there is at least one totally wandering null geodesic through pw," and Section V explicitly defers the existence theorem to Ref. [58], which is listed as "our forthcoming task" and is not available to the reader. The perturbative analysis in Section III.A only bounds a scale below which truncated wandering geodesics are absent; it does not show that any wandering geodesic appears. Moreover, the final paragraph of Section IV argues that "not all null geodesics... can be totally wandering" implies "there must be a totally wandering null geodesic λrw," which is a non-sequitur: non-emptiness of the wandering set cannot be inferred from the fact that a large congruence contains some non-wandering geodesics. Thus the claimed relevance to general black hole shadows is not established.
  3. [Section IV] The step from the number function N(p) to observable brightness is asserted rather than derived. The text says "the closer O' approaches pw, the wider the range of N(p) becomes and the brighter the light rays are," but no map from N(p) to surface brightness or angular density is supplied. Similarly, the claim that each null geodesic of the sub-congruence c[O'] "will cross an infinite (numerous) number of orbits for the luminous sources" is not justified by the topological condition that C\ψ[c[O']] is not dense; a complete null geodesic can intersect a non-dense set only in a limited portion of its affine parameter, for instance a single worldline. The paper's own caveat that these considerations "may be far from astrophysical generality" does not repair the logical gap between conjugate-point counts and image brightness.
minor comments (4)
  1. [References] Reference [58] is given as "our forthcoming task"; this is not a usable citation and should be replaced by a real reference or removed along with the passages that rely on it.
  2. [Section III.A] There is a typo in the text: "vanishing Wyle curvature" should read "vanishing Weyl curvature."
  3. [Section II] The statement that "the conjugate point is sufficient for a null geodesic to stay a chronal region" is imprecise; a conjugate point causes a geodesic to fail to be a maximum of Lorentzian length, and the relation to remaining in a chronal region should be stated more carefully.
  4. [Section IV] The regularity of the achronal boundary is referred to inconsistently as C^1 and C^1-; the notation should be unified, and the smoothness assumptions on the number function N(p) near the totally wandering geodesic should be stated explicitly.

Circularity Check

0 steps flagged · score 2.0 of 10

No load-bearing step reduces to its inputs by construction; the deferred existence theorem in Ref. [58] is a completeness gap, not a circular derivation.

full rationale

The paper's central claim is conditional: if a totally wandering null geodesic (a complete null geodesic with unbounded conjugate points) exists, then it implies accumulation and brightness contrast in an asymptotically flat spacetime. That conditional claim is not equivalent to the definitions by construction: the wandering geodesic is defined through conjugate points, while the shadow conclusion is argued via achronal boundaries and luminous-source crossings in Section IV, and the definitions are checked against the Schwarzschild photon sphere as an external benchmark. No parameter is fitted and no known result is merely renamed. The main self-referential element is Ref. [58], listed in the bibliography as 'our forthcoming task'; Section IV says 'the other theorem for the existence of wandering null geodesics will be introduced in our forthcoming research...' This is an explicit deferral of the general existence proof, so it is a completeness gap rather than a circular step: the paper does not cite the self-reference as if it were an established external theorem. The Section IV assertion that the image A of partial I^-(q_f) under a timelike flow is 'a closed and open subset in the topology of C, so that A is C' is asserted without proof and is a mathematical gap, but it is not a definitional reduction. Overall, no load-bearing step reduces to its own input, so the circularity score is low.

Assumptions & free parameters 1 free parameters · 7 assumptions · 3 invented entities

The central claim rests on several unproven assumptions, the most consequential being the assumed existence of a totally wandering geodesic and the topological claim about A. The curvature bound C1 is a hand-chosen quantity in the perturbative estimate. The wandering-set concepts are new definitions rather than fitted parameters.

free parameters (1)
  • curvature bound C1 = unspecified, assumed C1 < 1
    Introduced in Section IIIA to bound Weyl curvature components and justify the perturbative expansion; the derived no-conjugate-point length scale 1/sqrt(C1) depends on this hand-chosen quantity.
assumptions (7)
  • domain assumption The spacetime is globally hyperbolic.
    Stated in the introduction: the spacetime is considered to be globally hyperbolic without any caution; the conjugate-point and chronological-future arguments use this.
  • domain assumption The null generic condition k[e Ca]bc[dk f] ke kc != 0 holds along wandering geodesics.
    Used in Section II to argue curvature terms eventually dominate the expansion and shear equations; this fails in special spacetimes such as conformally flat regions.
  • ad hoc to paper At least one totally wandering null geodesic exists in the spacetime under study.
    Section IV begins by supposing that there is at least one totally wandering null geodesic through pw; the general existence theorem is deferred to Ref [58].
  • domain assumption Past null geodesic completeness of the spacetime.
    Footnote 9 assumes geodesic completeness in the past direction for the crossing-count argument, and the author notes this is not true for cosmological spacetimes with an initial singularity.
  • ad hoc to paper The image of partial I^-(q_f) under the timelike-flow homeomorphism is both closed and open in the Cauchy surface.
    Section IV asserts that A is a closed and open subset of C, hence equal to C; this premise is load-bearing for the brightness argument and is not proven. It is false for a past light cone in Minkowski spacetime.
  • ad hoc to paper Weyl curvature is small and bounded by C1, with Ricci terms negligible, near the formation region.
    Section IIIA adopts the assumption as only a superficial one to justify the perturbative expansion that excludes truncated wandering geodesics on scales below 1/sqrt(C1).
  • standard math Conjugate points along null geodesics are invariant under global conformal transformations.
    Used for Theorem 1; conformal transformations preserve null geodesics and their conjugate-point structure in the globally conformally flat setting.
invented entities (3)
  • Wandering null geodesic and totally wandering variant
    purpose: Causal generalization of the photon sphere for general black hole spacetimes, defined by unlimited repetitive conjugate points.
    Introduced by definition in Section II; no independent observable prediction is derived, and the promised existence theorem is deferred to Ref [58].
  • Wandering set W
    purpose: Set of totally wandering null geodesics, proposed as the counterpart of the photon sphere.
    Defined in Section II as W = {gamma_wT}; it is a mathematical construct whose observational relevance is argued heuristically in Section IV.
  • Truncated wandering null geodesic
    purpose: Symbolic device to discuss formation of the wandering structure during black hole collapse.
    Defined in Section III by cutting a future wandering geodesic at its first conjugate point; the author states it carries only symbolic meaning.

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Pith. "Pith review of Causal concept for black hole shadows." pith.science (2026). https://pith.science/paper/W4D3ZJO6

@misc{pith2026190802921,
  author       = {Pith},
  title        = {Pith review of: Causal concept for black hole shadows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W4D3ZJO6}},
  note         = {Machine review of arXiv:1908.02921}
}
read the original abstract

Causal concept for the general black hole shadow is investigated, instead of the photon sphere. We define several `wandering null geodesics' as complete null geodesics accompanied by repetitive conjugate points, which would correspond to null geodesics on the photon sphere in Schwarzschild spacetime. We also define a `wandering set', that is, a set of totally wandering null geodesics as a counterpart of the photon sphere, and moreover, a truncated wandering null geodesic to symbolically discuss its formation. Then we examine the existence of a wandering null geodesic in general black hole spacetimes mainly in terms of Weyl focusing. We will see the essence of the black hole shadow is not the stationary cycling of the photon orbits which is the concept only available in a stationary spacetime, but their accumulation. A wandering null geodesic implies that this accumulation will be occur somewhere in an asymptotically flat spacetime.

Figures

Figures reproduced from arXiv: 1908.02921 by the authors.

Figure 1
Figure 1. FIG. 1. A figure of the null congruence near the region [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. An image of a null congruence emanating from the south pole on [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Through the conjugate point [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Singular congruences are considered around a totally wandering null geodesic. While there are an infinite number of conjugate points [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

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