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arxiv: 2508.20419 · v4 · submitted 2025-08-28 · 🌀 gr-qc

The shadows and photon rings of two minimal deformations of Schwarzschild black holes

Hong-Er Gong , Junlin Qin , Yusen Wang , Bofeng Wu , Zhan-Feng Mai , Sen Guo , Enwei Liang This is my paper

Pith reviewed 2026-05-18 21:21 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole shadowsphoton ringsaccretion disksKazakov-Solodukhin black holeGhosh-Kumar black holegeneral relativityEvent Horizon Telescopeshadow radius
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The pith

Two minimal deformations of Schwarzschild black holes produce distinct shadow sizes and photon ring widths that thin-disk observations can distinguish.

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

This paper calculates the shadows and photon rings for the Kazakov-Solodukhin and Ghosh-Kumar black holes, which are minimal deformations of the Schwarzschild metric. It shows that one deformation enlarges the event horizon and photon sphere while the other shrinks them. For spherical accretion the shadow radius remains unchanged no matter which accretion model is used, but the integrated intensity changes in opposite ways for the two deformations. In thin disk models direct emission dominates the observed intensity with photon and lensed rings contributing only a small amount, and the two black holes differ in the width and brightness of their rings. The authors suggest this setup offers a way to observationally separate these deformed black holes from each other and from the standard case.

Core claim

The optical characteristics of the Kazakov-Solodukhin and Ghosh-Kumar black holes differ from Schwarzschild in their event horizons, photon spheres and critical impact parameters, with the former increasing and the latter decreasing. In spherical accretion the quantum correction in the Kazakov-Solodukhin case increases shadow size and decreases intensity while the magnetic charge in Ghosh-Kumar does the reverse. The shadow radius proves independent of the specific spherical accretion model chosen. For an optically and geometrically thin accretion disk the integrated intensity comes mostly from direct emission with very small contributions from photon rings and lensed rings. The Kazakov-Solod

What carries the argument

Ray tracing through the deformed spacetimes to compute critical impact parameters, shadow radii, and integrated intensities from assumed accretion emission profiles.

If this is right

  • Event Horizon Telescope data can constrain the deformation parameters for both black holes.
  • Shadow size increases with the Kazakov-Solodukhin deformation but decreases with the Ghosh-Kumar deformation under spherical accretion.
  • Direct emission dominates intensity in thin disks, making ring contributions negligible for distinguishing the black holes.
  • Photon rings appear narrower for Kazakov-Solodukhin and wider for Ghosh-Kumar black holes.
  • Shadow radius decreases as the accretion disk is placed closer to the black hole.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These differences could allow future observations to test for small deviations from Schwarzschild geometry in real black holes.
  • The independence from accretion model details simplifies shadow-based tests of gravity.
  • Similar calculations for other deformed metrics might reveal a general pattern in how deformations affect observability.
  • This method might extend to distinguishing black holes from naked singularities or other exotic objects in thin disk scenarios.

Load-bearing premise

The two deformed metrics provide accurate descriptions of spacetime outside the horizon and the standard thin-disk and spherical accretion models correctly describe the emission without major additional effects.

What would settle it

An observation of a black hole shadow radius that changes when switching between different spherical accretion models would contradict the claimed independence.

read the original abstract

This paper primarily investigates the optical characteristics of two minimal Schwarzschild black hole deformations, the Kazakov-Solodukhin and Ghosh-Kumar black holes, under different accretion models. The event horizon, photon sphere, and critical impact parameter of the former increase compared with the Schwarzschild black hole, but those of the latter decrease. The data from the Event Horizon Telescope Collaboration are used to constrain the parameter ranges of the two black holes. In the case of spherical accretion, the quantum correction of Kazakov-Solodukhin black hole leads to the increase of black hole shadow size and the decrease of integrated intensity, while the shadow size of magnetically charged Ghosh-Kumar black hole decreases and the integrated intensity increases. The shadow radius of the black hole is independent of the spherical accretion models. For an optically and geometrically thin accretion disk, the integrated intensity is mainly contributed by direct emission, and the contributions of photon rings and lensed rings are very small. In addition, the photon rings and lensed rings of Kazakov-Solodukhin black hole are narrower, while those of Ghosh-Kumar black hole are wider. Whereas the Kazakov-Solodukhin black hole exhibits higher brightness, the Ghosh-Kumar black hole shows lower brightness. Additionally, a disk closer to the black hole correlates with a smaller shadow radius. This paper proposes a method to distinguish different black holes in a specific thin disk model.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper investigates the shadows and photon rings of two minimal deformations of the Schwarzschild black hole (Kazakov-Solodukhin and Ghosh-Kumar metrics). It reports that the event horizon, photon sphere, and critical impact parameter increase for the Kazakov-Solodukhin case and decrease for the Ghosh-Kumar case relative to Schwarzschild. EHT data are used to constrain the deformation parameters. Under spherical accretion the Kazakov-Solodukhin shadow enlarges while integrated intensity drops, and the opposite occurs for Ghosh-Kumar; the shadow radius is stated to be independent of the choice of spherical accretion model. For an optically thin equatorial disk, direct emission dominates the intensity while photon-ring and lensed-ring contributions are small; the Kazakov-Solodukhin rings are narrower and brighter, the Ghosh-Kumar rings wider and fainter. A method to distinguish the two black holes via these features in a specific thin-disk model is proposed.

Significance. If the reported independence of shadow radius from spherical accretion details and the discriminatory power of ring width and brightness hold after verification, the work supplies concrete, observationally accessible signatures that could help test these particular deformations against EHT and future VLBI data, extending the catalog of black-hole shadow phenomenology beyond the Kerr family.

major comments (3)
  1. [Abstract] Abstract and the section on spherical accretion: the claim that 'the shadow radius of the black hole is independent of the spherical accretion models' is presented as a result, yet the shadow boundary is fixed solely by the critical impact parameter of the unstable photon orbit, which is a purely geometric property of the metric; this independence therefore follows by construction from the ray-tracing setup rather than from any new calculation, and the manuscript should demonstrate explicitly (e.g., by varying the radial density or temperature profile) that the observed edge remains unchanged to within numerical precision.
  2. [Thin-disk section] The thin-disk analysis (presumably the section following the spherical-accretion results): the statement that 'the integrated intensity is mainly contributed by direct emission, and the contributions of photon rings and lensed rings are very small' is central to the proposed distinction method, but no quantitative breakdown (percentages, cumulative intensity curves, or error estimates) is referenced; without such numbers it is impossible to judge whether the dominance holds across the EHT-constrained parameter ranges or is an artifact of the chosen disk thickness and emissivity profile.
  3. [Metric definitions] The opening paragraphs on the two deformed metrics: the assumption that both the Kazakov-Solodukhin and Ghosh-Kumar line elements remain regular, asymptotically flat vacuum solutions exterior to their horizons is load-bearing for all subsequent ray-tracing; an explicit check for additional singularities or deviations from asymptotic flatness (e.g., via the Kretschmann scalar or asymptotic expansion) should be supplied, as any pathology would invalidate the reported shadow sizes and ring widths.
minor comments (2)
  1. [Figures] Figure captions should state the exact values of the deformation parameters used and the accretion-model parameters (density normalization, temperature) so that the plots can be reproduced.
  2. [Notation] Notation for the critical impact parameter and the integrated intensity should be defined once at first use and used consistently; the current text occasionally switches between 'shadow radius' and 'critical impact parameter' without clarifying the conversion factor.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on our manuscript. We address each major comment point by point below and indicate the revisions planned for the next version.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the section on spherical accretion: the claim that 'the shadow radius of the black hole is independent of the spherical accretion models' is presented as a result, yet the shadow boundary is fixed solely by the critical impact parameter of the unstable photon orbit, which is a purely geometric property of the metric; this independence therefore follows by construction from the ray-tracing setup rather than from any new calculation, and the manuscript should demonstrate explicitly (e.g., by varying the radial density or temperature profile) that the observed edge remains unchanged to within numerical precision.

    Authors: We agree that the reported independence follows directly from the geometric definition of the critical impact parameter. To make this explicit and to verify numerical robustness, we will add calculations in the revised manuscript that vary the radial density and temperature profiles of the spherical accretion model and confirm that the observed shadow edge remains unchanged within numerical precision. revision: yes

  2. Referee: [Thin-disk section] The thin-disk analysis (presumably the section following the spherical-accretion results): the statement that 'the integrated intensity is mainly contributed by direct emission, and the contributions of photon rings and lensed rings are very small' is central to the proposed distinction method, but no quantitative breakdown (percentages, cumulative intensity curves, or error estimates) is referenced; without such numbers it is impossible to judge whether the dominance holds across the EHT-constrained parameter ranges or is an artifact of the chosen disk thickness and emissivity profile.

    Authors: We acknowledge that a quantitative breakdown would strengthen the claim. In the revision we will add explicit percentages for the intensity contributions from direct emission, photon rings, and lensed rings, together with cumulative intensity curves, evaluated over the EHT-constrained parameter ranges and for the adopted disk thickness and emissivity profile. revision: yes

  3. Referee: [Metric definitions] The opening paragraphs on the two deformed metrics: the assumption that both the Kazakov-Solodukhin and Ghosh-Kumar line elements remain regular, asymptotically flat vacuum solutions exterior to their horizons is load-bearing for all subsequent ray-tracing; an explicit check for additional singularities or deviations from asymptotic flatness (e.g., via the Kretschmann scalar or asymptotic expansion) should be supplied, as any pathology would invalidate the reported shadow sizes and ring widths.

    Authors: The two metrics are constructed in the literature as regular, asymptotically flat solutions. To address the request, we will insert an explicit verification in the revised manuscript, including evaluation of the Kretschmann scalar to confirm the absence of additional singularities and an asymptotic expansion of the metric at large radii to reconfirm flatness. revision: yes

Circularity Check

0 steps flagged

No circularity: geometric derivations and external EHT constraints remain independent of fitted outputs

full rationale

The paper computes photon spheres, critical impact parameters, and ray-traced intensities directly from the given Kazakov-Solodukhin and Ghosh-Kumar metrics using standard null geodesic equations and thin-disk/spherical accretion prescriptions. Parameter ranges are bounded by external EHT observations rather than internal fits, and the reported independence of shadow radius from specific spherical accretion models follows from the geometric definition of the shadow boundary (set by unstable photon orbits) rather than any redefinition or renaming of inputs. No self-citations are load-bearing for the central claims, no ansatz is smuggled, and no prediction reduces by construction to a fitted quantity. The derivation chain is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central results rest on the validity of the two deformed metrics as solutions and on the applicability of standard thin-disk and spherical accretion models to these spacetimes.

free parameters (2)
  • Kazakov-Solodukhin deformation parameter
    Constrained by EHT data; value not reported in abstract.
  • Ghosh-Kumar magnetic charge parameter
    Constrained by EHT data; value not reported in abstract.
axioms (2)
  • domain assumption The deformed metrics describe the exterior geometry of the black holes and satisfy the necessary field equations.
    Invoked to justify using the metrics for ray tracing and shadow calculations.
  • domain assumption Spherical and thin-disk accretion models produce the dominant emission without significant deviations from assumed symmetry or additional plasma effects.
    Required for the intensity and ring-width predictions.

pith-pipeline@v0.9.0 · 5805 in / 1514 out tokens · 41904 ms · 2026-05-18T21:21:56.374270+00:00 · methodology

discussion (0)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Photon Spheres and shadow of modified black-hole entropies

    gr-qc 2026-05 unverdicted novelty 4.0

    Modified black hole entropies alter photon sphere radii and shadow sizes, with parameters constrained by Event Horizon Telescope observations of Sgr A*.

  2. Photon Spheres and shadow of modified black-hole entropies

    gr-qc 2026-05 unverdicted novelty 4.0

    Corrected black hole entropies produce distinct shifts in photon sphere radius and shadow size that are constrained by Event Horizon Telescope data on Sagittarius A*.

  3. Photon Spheres and shadow of modified black-hole entropies

    gr-qc 2026-05 unverdicted novelty 4.0

    Entropy corrections to black holes produce modified metrics whose photon-sphere and shadow sizes can be constrained by Sgr A* observations.

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