Pith. sign in

REVIEW 4 major objections 5 minor 26 references

Shadow Formation Conditions Beyond the Kerr Black Hole Paradigm

T0 review · 4 major / 5 minor · reviewed 2026-08-04 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A compact-object shadow can form without a photon sphere whenever the null-geodesic effective potential has a positive finite upper bound that traps or scatters light.

desk verdict The paper's 'necessary and sufficient' shadow condition is a definition dressed as a theorem, and its main-text finiteness condition is falsified by Schwarzschild. read the letter →

arxiv 2509.06244 v1 pith:KKJIGLB5 submitted 2025-09-07 gr-qc astro-ph.GAastro-ph.HE

Parth Bambhaniya , Saurabh , Elisabete M. de Gouveia Dal Pino This is my paper
classification gr-qcastro-ph.GAastro-ph.HE
keywords blackholeshadowphotonsphereeffectivepotentialnullgeodesicsnakedsingularitiescompactobjectsEventHorizonTelescope
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 usual requirement for a black hole shadow—a photon sphere, a shell of unstable photon orbits—is too narrow. The true condition, it claims, is that the effective potential for radial null geodesics stays finite with a positive upper bound and contains a region that either traps or scatters photons. That condition holds for Kerr black holes as a special case, but it also covers naked singularities, regular black holes, wormholes, and other horizonless compact objects, including cases with no photon sphere at all. The authors give this as a necessary and sufficient condition for axisymmetric, stationary, rotating spacetimes, and they distinguish the shadow's dark edge from the bright photon ring that often accompanies it.

What carries the argument

The effective potential Veff(r) for null geodesics, Eq. (7), reduced to a function of radius on the equatorial plane via conservation of energy E and angular momentum Lz. The paper's shadow criterion is that this potential must have a positive finite upper bound and include a trapping or scattering region; the photon sphere appears only as the special case where Veff' = 0 and Veff'' < 0. The apparent shadow boundary is then given by the critical impact parameter bcrit = Lz/E, through Eq. (11).

What would settle it

Ray-trace null geodesics in a rotating spacetime that satisfies the finite-positive-upper-bound condition but has no photon sphere, and observe the screen at a large inclination angle; if a clear shadow exists and its edge deviates from Eq. (11) computed from equatorial orbits, the equatorial-reduction claim is falsified.

Watch

Extended reading notes

Core claim

For a general axisymmetric, stationary, rotating spacetime whose null geodesic motion separates at least in the equatorial plane, the paper claims that a distant observer sees a shadow if and only if the radial effective potential Veff(r) is finite and continuous, has a positive finite upper bound, and includes a region where photons are trapped or scattered. The Kerr photon shell is only a particular realization of this condition: it corresponds to unstable circular orbits, Veff' = 0 and Veff'' < 0, sitting at the potential's maximum. When that maximum exists without unstable circular orbits, shadows still form, though the brightness depression may lack a sharp ring. The shadow boundary is

Load-bearing premise

The result assumes that a rotating spacetime's shadow edge is set by circular photon orbits in the equatorial plane, so any photon orbits that leave that plane would have to behave equivalently or be ignored for the condition to hold exactly.

Editorial extensions

If this is right

  • If the condition is correct, the absence of a detected photon ring does not imply the absence of a shadow: a finite potential barrier that scatters photons can still create a dark silhouette.
  • Shadows of horizonless compact objects, including naked singularities without photon spheres, become viable targets for EHT and next-generation imaging tests of cosmic censorship.
  • The distinction between the shadow edge and the bright photon ring gives observers two independent diagnostics: one tells whether light is captured or scattered, the other whether unstable photon orbits exist.
  • In rotating spacetimes, spin-induced asymmetry in the shadow boundary is governed by the same finite-barrier condition, so high-spin horizonless objects could cast shadows mimicking Kerr black holes without an event horizon.

Reading between the lines

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

  • Inference: the 'sufficient' direction of the claim likely depends on how the shadow edge is defined; a gradual brightness depression without a sharp intensity drop may not give a unique boundary even when the effective-potential condition holds.
  • Inference: the equatorial-plane reduction could be tested numerically by ray-tracing null geodesics in a rotating naked-singularity metric with no photon sphere at finite observer inclination; any mismatch between the true screen boundary and Eq. (11) would mark the separability approximation's limit.
  • Inference: the same finite-barrier logic should apply to gravitational-wave lensing or X-ray reverberation mapping, where the trapping/scattering region would imprint a characteristic dimming independent of a photon ring.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper claims to establish necessary and sufficient conditions for the formation of a black-hole shadow in general axisymmetric, stationary, rotating spacetimes. The proposed condition is that the effective potential of null geodesics has a positive finite upper bound and contains a region where photons are trapped or scattered; the authors argue that this generalizes the standard photon-sphere/photon-shell criterion and applies to naked singularities and other horizonless compact objects. The main text gives a generic metric, an effective-potential definition, a critical-impact-parameter formula, and a qualitative discussion of potential shapes, but it does not present a formal derivation or a worked non-Kerr rotating example.

Significance. If correct and properly formalized, a model-independent shadow-formation criterion would be useful for interpreting EHT images and for organizing known results on exotic compact objects. The paper does identify a legitimate conceptual point: a photon sphere is not synonymous with a shadow, and cases exist where a shadow is produced without one. However, as written the contribution is qualitative and incomplete: the central claim is not proved, the main condition is partly circular, and the axisymmetric extension is asserted rather than demonstrated. The paper's value is primarily pedagogical, not a new rigorous result.

major comments (4)
  1. [Sec. II and Sec. III] The central 'necessary and sufficient' claim is not proved. No mathematical definition of the shadow or of the apparent boundary is given, and the only quantitative boundary criterion, Eq. (11), is stated for circular photon orbits. The bullet condition 'includes a region where photons are either trapped or scattered' essentially restates the existence of a capture region, which is close to the definition of a shadow; without a derivation of the threshold impact parameter and the observer-screen projection, the sufficiency claim is circular. The necessity claim in Sec. III is likewise unsupported.
  2. [Eq. (11) and surrounding text] For a stationary axisymmetric spacetime, the shadow boundary on the observer's sky at finite inclination is determined by non-equatorial spherical photon orbits, not only by equatorial circular orbits. The paper explicitly restricts to separability, at least in the equatorial plane, and then uses Eq. (11) to define the critical boundary. This does not provide a derivation of the shadow boundary for the general axisymmetric rotating case claimed in the title and Sec. II. The equatorial-plane reduction must be stated as a limitation of the result, not as the basis for a general necessary-and-sufficient condition.
  3. [Sec. II, bullet list and Eq. (9)] The main-text condition differs from the abstract condition in a way that matters. The abstract requires a 'positive finite upper bound', while Sec. II and Eq. (9) require Veff(r)<∞ for all r<r_ph and say the potential 'must remain finite and continuous throughout the region'. These are not equivalent: a potential can be finite at every regular radius while having a negatively divergent limit at the singularity, as in Schwarzschild. The paper invokes naked-singularity shadows from refs. [24,25] without showing that they satisfy Eq. (9). The necessary condition is therefore ambiguously and inconsistently stated.
  4. [Sec. II, general framework] No worked axisymmetric non-Kerr example is presented. The text claims the framework 'extends beyond conventional solutions', but the only concrete examples mentioned are spherically symmetric static geometries from refs. [3,24,25]. A claim of necessary and sufficient conditions for rotating spacetimes requires either a proof from the geodesic equation or at least one nontrivial rotating example in which the condition is verified and the shadow boundary is computed. In the present form, the demonstration is a qualitative discussion rather than a derivation.
minor comments (5)
  1. [Eq. (7)] Eq. (7) defines Veff without the gθθ(dθ/dλ)^2 term that appears in Eq. (5). If this is meant as the equatorial-plane reduction, that should be stated explicitly before Eqs. (8)-(9) are used.
  2. [After Eq. (3)] The phrase 'were P µ is the four-momentum' should read 'where P µ is the four-momentum'.
  3. [PACS/references] The PACS numbers line is blank, and reference [23] appears incomplete; the general bibliography formatting is inconsistent.
  4. [Fig. 1] The axes of Fig. 1 are unlabeled, and panel (c), labeled 'anti-photon sphere', is not defined in the text and no explicit example of such a potential is given.
  5. [Sec. III] The closing caveat that the results 'may not generalize to all non-Kerr scenarios' is in tension with the abstract's general claim. The assumptions and limitations should be stated at the outset and reflected in the conclusions.

Circularity Check

1 steps flagged · score 7.0 of 10

The proposed sufficient condition is a restatement of the definition of a shadow; the necessary condition (Eq. 9) is an assumed, non-derived restriction inconsistent with Schwarzschild.

  1. self definitional [Abstract; Sec. III (Conclusions); cf. Sec. II bullet conditions]
    "We demonstrate that a shadow forms if the effective potential of null geodesics has a positive finite upper bound and includes a region where photons are either trapped or scattered."

    Bardeen's shadow (the apparent boundary) is defined as the dark silhouette produced by photon trajectories that are captured by the compact object or scattered away from the observer. The operative clause 'includes a region where photons are either trapped or scattered' is therefore not an independent physical condition but a restatement of the existence of a shadow. The added 'positive finite upper bound' is never connected to the shadow boundary by a derived equation; Eq. (11) merely restates the circular-orbit critical impact parameter bcrit = (-g_tφ + sqrt(g_tφ^2 - g_tt g_φφ))/g_tt. Hence the claimed necessary-and-sufficient criterion reduces, at its core, to: a shadow exists iff there are photon trajectories that do not reach the observer, which is true by definition.

full rationale

The paper contains no fitted parameters and does not rely on a uniqueness theorem. Self-citations [3,24] appear as examples of shadows without photon spheres, but the central derivation does not reduce to those citations, so no self-citation circularity is charged. The main circularity is definitional: the sufficient condition's 'trapped or scattered' clause is the shadow phenomenon itself. Separately, and not as circularity, the asserted necessary condition Eq. (9) is an unproved assumption ('we assume ... that the effective potential for null geodesics remains finite') and is violated by Schwarzschild, for which Veff → -∞ as r→0 < r_ph while a shadow exists; this is a correctness/falsification concern that further weakens the 'necessary and sufficient' claim. The score of 7 reflects that the central claim reduces to a definition rather than to a fit or a self-citation chain.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted. The framework relies on standard geodesic equations and explicit assumptions about separability and smoothness. The central condition itself is treated as an axiom-like definition rather than derived.

assumptions (4)
  • domain assumption The spacetime admits two Killing vectors for time translation and axial rotation, giving conserved energy and angular momentum.
    Stated in Section II, used to define E and L_z.
  • domain assumption Null geodesic equations are separable, at least in the equatorial plane.
    Stated in Section II, required for Eq. (11) to define the critical impact parameter.
  • domain assumption The metric functions are smooth and differentiable and the effective potential for null geodesics remains finite.
    Stated as a limitation in Section II, but also used as part of the claimed condition.
  • standard math The radial null geodesic motion is governed by the effective potential in Eq. (6) with the sign convention of Eq. (7).
    Derived from the null condition, but the sign convention is inconsistent between Eqs. (5)-(7).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Shadow Formation Conditions Beyond the Kerr Black Hole Paradigm." pith.science (2026). https://pith.science/paper/KKJIGLB5

@misc{pith2026250906244,
  author       = {Pith},
  title        = {Pith review of: Shadow Formation Conditions Beyond the Kerr Black Hole Paradigm},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KKJIGLB5}},
  note         = {Machine review of arXiv:2509.06244}
}
read the original abstract

A compact object illuminated by background radiation produces a dark silhouette. The edge of the silhouette or shadow (alternatively, the apparent boundary or the critical curve) is commonly determined by the presence of the photon sphere (or photon shell in the case of rotating spacetime), corresponding to the maximum of the effective potential for null geodesics. While this statement stands true for Kerr black holes, here we remark that the apparent boundary (as defined by Bardeen) forms under a more general condition. We demonstrate that a shadow forms if the effective potential of null geodesics has a positive finite upper bound and includes a region where photons are trapped or scattered. Our framework extends beyond conventional solutions, including but not limited to naked singularities. Furthermore, we clarify the difference between the apparent boundary of a dark shadow and the bright ring on the screen of a distant observer. These results provide a unified theoretical basis for interpreting observations from the Event Horizon Telescope (EHT) and guiding future efforts towards extreme-resolution observations of compact objects.

Figures

Figures reproduced from arXiv: 2509.06244 by the authors.

Figure 1
Figure 1. FIG. 1: Representation of various (but not limited to) effective potentials. Here (a) A finite upper bound with a photon sphere, [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

26 extracted references · 25 canonical work pages

  1. [1]

    J. M. Bardeen, Proceedings, Ecole d’Et´ e de Physique Th´ eorique: Les Astres Occlus : Les Houches, France, August, 1972, 215-240, 215-240 (1973)

    work page 1972

  2. [2]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L17 (2022)

    work page 2022

  3. [3]

    Vagnozzi, R

    S. Vagnozzi, R. Roy, Y. D. Tsai, L. Visinelli, M. Afrin, A. Allahyari, P. Bambhaniya, D. Dey, S. G. Ghosh and P. S. Joshi, et al. Class. Quant. Grav. 40, no.16, 165007 (2023)

    work page 2023

  4. [4]

    P. S. Joshi and I. H. Dwivedi, Phys. Rev. D 47, 5357 (1993)

    work page 1993

  5. [5]

    Goswami and P

    R. Goswami and P. S. Joshi, Classical Quantum Gravity 21, 3645 (2004)

    work page 2004

  6. [6]

    Mosani, D

    K. Mosani, D. Dey, and P. S. Joshi, Phys. Rev. D 101, 044052 (2020)

    work page 2020

  7. [7]

    Mosani, D

    K. Mosani, D. Dey and P. S. Joshi, Phys. Rev. D 102, no.4, 044037 (2020)

    work page 2020

  8. [8]

    Mosani, D

    K. Mosani, D. Dey and P. S. Joshi, Monthly Notices of the Royal Astronomical Society, 504, 4, 4743–4750 (2021)

    work page 2021

Show all 26 references
  1. [9]

    D. Dey, P. S. Joshi, K. Mosani and V. Vertogradov, Eur. Phys. J. C 82 (2022) no.5, 431

    2022

  2. [10]

    S. S. Deshingkar, S. Jhingan and P. S. Joshi, Gen. Rel. Grav. 30 (1998), 1477-1499

    1998

  3. [11]

    Jhingan and S

    S. Jhingan and S. Kaushik, Phys. Rev. D 90 (2014) no.2, 024009

    2014

  4. [12]

    P. S. Joshi, D. Malafarina and R. Narayan, Class. Quant. Grav. 28 (2011), 235018. 6

    2011

  5. [13]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, L1 (2019)

    2019

  6. [14]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 875, no.1, L2 (2019)

    2019

  7. [15]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 930, no.2, L12 (2022)

    2022

  8. [16]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 964, no.2, L25 (2024)

    2024

  9. [17]

    Akiyama et al

    K. Akiyama et al. [Event Horizon Telescope], Astrophys. J. Lett. 964, no.2, L26 (2024)

    2024

  10. [18]

    Ayzenberg, L

    D. Ayzenberg, L. Blackburn, R. Brito, S. Britzen, A. Broderick, R. Carballo-Rubio, V. Cardoso, A. Chael, K. Chatterjee and Y. Chen, et al. arXiv:2312.02130 [astro-ph.HE]

    arXiv

  11. [19]

    Psaltis et al

    D. Psaltis et al. [Event Horizon Telescope], Phys. Rev. Lett., 125, no.14, 141104 (2020)

    2020

  12. [20]

    Shaikh, P

    R. Shaikh, P. Kocherlakota, R. Narayan and P. S. Joshi, Mon. Not. Roy. Astron. Soc. 482, no.1, 52-64 (2019)

    2019

  13. [21]

    Bambhaniya and P

    Saurabh, P. Bambhaniya and P. S. Joshi, arXiv:2202.00588 [gr-qc]

    arXiv

  14. [22]

    A. E. Broderick and K. Salehi, arXiv:2406.05181 [astro- ph.HE]

    arXiv

  15. [23]

    Bambhaniya and P

    Saurabh, P. Bambhaniya and P. S. Joshi, A & A 628 A113, 8, (2024)

    2024

  16. [24]

    A. B. Joshi, D. Dey, P. S. Joshi and P. Bambhaniya, Phys. Rev. D 102, no.2, 024022 (2020)

    2020

  17. [25]

    D. Dey, R. Shaikh and P. S. Joshi, Phys. Rev. D 103, no.2, 024015 (2021)

    2021

  18. [26]

    Zhi-Fu Gao et al., Eur. Phys. J. C, 855, 433 (2025)

    2025

Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.