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arxiv: 2604.18628 · v1 · submitted 2026-04-18 · 🌌 astro-ph.HE · gr-qc

Visual Characteristics of a Rotating Black Hole in 4D Einstein-Gauss-Bonnet Gravity with Thin Accretion Disk Under EHT Constraints

Muhammad Israr Aslam , Manahil Ali , Abdul Malik Sultan , Xiao-Xiong Zeng , Sultan Hussain This is my paper

Pith reviewed 2026-05-10 06:41 UTC · model grok-4.3

classification 🌌 astro-ph.HE gr-qc
keywords black hole shadowEinstein-Gauss-Bonnet gravityaccretion diskray tracingEvent Horizon TelescopeM87*Sgr A*shadow deviation
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The pith

In 4D Einstein-Gauss-Bonnet gravity, the shadow radius of a rotating black hole decreases and its deviation from circularity increases as the coupling parameter α grows, while EHT data from M87* and Sgr A* constrain α to ranges that keep a

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

The paper examines how the Gauss-Bonnet coupling parameter α and black hole spin a shape the appearance of shadows and thin accretion disks around rotating black holes in four-dimensional Einstein-Gauss-Bonnet gravity. Using ray-tracing calculations and a fisheye camera model, it shows that larger α reduces shadow size but increases asymmetry, while higher spin a slightly enlarges the shadow and strengthens space-dragging effects. For a thin accretion disk whose inner edge reaches the event horizon, the inner shadow shrinks with α and asymmetry rises with a, accompanied by distinct redshift patterns in direct and lensed images. Parameter constraints derived from Event Horizon Telescope images of M87* and Sagittarius A* show that the model remains viable within certain ranges of α.

Core claim

The central claim is that the visual features of rotating black holes in 4D Einstein-Gauss-Bonnet gravity, including shadow radius, deviation, photon rings, and disk redshifts, vary systematically with α and a, with shadow size decreasing and asymmetry increasing with α, and that parameter constraints derived from EHT data for M87* and Sgr A* demonstrate the model's consistency with observations.

What carries the argument

The rotating black hole metric in 4D Einstein-Gauss-Bonnet gravity, combined with ray-tracing through a thin accretion disk whose inner edge lies at the event horizon and a fisheye camera model for image formation.

If this is right

  • Shadow radius decreases while deviation increases with growing α.
  • Space-dragging becomes more prominent as spin a rises.
  • Inner shadow size reduces with α and asymmetry grows with a for the thin disk.
  • Redshift configurations differ between direct and lensed disk images.
  • EHT data from M87* and Sgr A* constrain α to values supporting the model's consistency.

Where Pith is reading between the lines

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

  • Future higher-precision shadow measurements could distinguish 4D EGB gravity from general relativity by detecting deviations in shadow size or asymmetry.
  • The inner shadow predictions may change if emission from inside the ISCO is included in the disk model.
  • The ray-tracing and fisheye approach can be applied to test other modified-gravity black hole metrics against the same EHT observations.

Load-bearing premise

The 4D Einstein-Gauss-Bonnet rotating metric is assumed valid and the thin accretion disk inner edge is set exactly at the event horizon with standard particle motion inside and outside the ISCO.

What would settle it

A measurement of the shadow radius or deviation parameter in M87* or Sgr A* that falls outside the range allowed by the constrained values of α would indicate inconsistency with the model.

Figures

Figures reproduced from arXiv: 2604.18628 by Abdul Malik Sultan, Manahil Ali, Muhammad Israr Aslam, Sultan Hussain, Xiao-Xiong Zeng.

Figure 1
Figure 1. Figure 1: FIG. 1: The physical behavior of horizons with varying BH parameters [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The shadow contours are shown in the left panel for a fixed spin parameter [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: illustrates the behavior of the shadow observables Rd and δd. In the upper panels, it is evident that the shadow radius Rd gradually decreases, while the deviation parameter δd increases as the GB coupling α becomes larger. These findings are compatible with [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The BH shadow profiles corresponding to celestial light source are examined for different values of [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Optical images of rotating BHs in 4 [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: The redshift distribution of direct images for rotating BHs in 4 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: The redshift profiles of lensed images for rotating BHs in 4 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: The lensing bands of rotating BHs in 4 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Optical images of rotating BHs in 4 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: The red-shift distribution for the direct images of rotating BHs in 4 [PITH_FULL_IMAGE:figures/full_fig_p023_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: The red-shift distribution for the lensed images of rotating BHs in 4 [PITH_FULL_IMAGE:figures/full_fig_p023_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: The lensing band of rotating BHs in 4 [PITH_FULL_IMAGE:figures/full_fig_p024_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: The profiles represent the variation of the approximated shadow angular diameter of Sgr A [PITH_FULL_IMAGE:figures/full_fig_p025_13.png] view at source ↗
read the original abstract

This study investigates the visual characteristics of a rotating black hole (BH) within the fabric of $4$D Einstein-Gauss-Bonnet gravity illuminated with two illumination models, such as a celestial light sphere and a thin accretion disk. To visualize the BH shadow images, we use a recent fisheye camera model and ray-tracing method. And then, we focus on investigating the impact of the coupling parameter $\alpha$ and the spin parameter $a$ on the shadow images. The results exhibit that the shadow radius decreases, while the shadow deviation increases with the aid of $\alpha$. However, with respect to $a$, the shadow radius is slightly increased compared to the corresponding shadow deviation. For a celestial light sphere, the increasing values of $\alpha$, lead to a decrease in the corresponding photon ring, while the space-dragging effect becomes more prominent with increasing $a$. For a thin accretion disk, we enhance its inner edge to the BH event horizon, and the particle motion is different in the regions inside and outside the innermost stable circular orbit. The result demonstrates that the shadow becomes progressively asymmetric with $a$, while the overall size of the inner shadow gradually decreases with the variations of $\alpha$. Subsequently, we also investigated the distinct features of red-shift configurations on the disk for both direct and lensed images. Additionally, we used the latest observational data from M87* and Sgr A* to impose certain parameter constraints on $\alpha$; the results depict the consistency of our considering the BH 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 manuscript numerically investigates the shadow and visual appearance of a rotating black hole in 4D Einstein-Gauss-Bonnet gravity, illuminated by a celestial light sphere or thin accretion disk. Using ray-tracing and a fisheye camera model, it examines the dependence of shadow radius, asymmetry, photon ring, and disk redshifts on the Gauss-Bonnet coupling α and spin a. The inner disk edge is placed at the event horizon with piecewise geodesic motion inside/outside the ISCO. Finally, EHT data on M87* and Sgr A* are used to constrain α, with the claim that the model is consistent with observations.

Significance. If the rotating 4D EGB metric is an exact solution and the disk truncation is physically justified, the work would supply new bounds on α from strong-field imaging, extending tests of modified gravity beyond Kerr. The numerical ray-tracing approach for both illumination models is a standard strength, but the absence of error analysis or code release limits immediate reproducibility.

major comments (3)
  1. [§2] §2 (Metric): The rotating 4D EGB line element is introduced without derivation or explicit validation that it satisfies the field equations exactly. The D→4 limit for rotating solutions is known to require additional regularization or ansatz assumptions; without addressing this, all subsequent shadow sizes and α constraints rest on an unverified foundation.
  2. [§4] §4 (Thin-disk model): The accretion disk inner edge is fixed exactly at the event horizon rather than the ISCO, with distinct geodesic prescriptions inside and outside the ISCO. This choice changes the inner-shadow boundary and emissivity profile relative to standard thin-disk calculations; a sensitivity test against ISCO truncation is required to support the reported α dependence.
  3. [§5] §5 (Observational constraints): α is constrained by direct fitting to M87* and Sgr A* shadow diameters, yet the abstract asserts 'consistency' without reporting the fitting procedure, χ² values, error propagation, or an independent validation set. This leaves open the possibility that the consistency is tautological with the fitting step itself.
minor comments (2)
  1. [Abstract] Abstract: Replace imprecise phrasing ('with the aid of α', 'enhance its inner edge', 'depict the consistency') with precise language (e.g., 'as α increases', 'set the inner edge at', 'are consistent within the fitting uncertainties').
  2. [Introduction and §2] Throughout: Add citations to prior 4D EGB shadow studies and EHT constraint papers to place the present results in context; several standard references on the D→4 regularization are missing.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their thorough review and valuable suggestions. We have addressed all the major comments in our point-by-point response below and revised the manuscript accordingly to improve its clarity and rigor.

read point-by-point responses

  1. Referee: §2 (Metric): The rotating 4D EGB line element is introduced without derivation or explicit validation that it satisfies the field equations exactly. The D→4 limit for rotating solutions is known to require additional regularization or ansatz assumptions; without addressing this, all subsequent shadow sizes and α constraints rest on an unverified foundation.

    Authors: We are grateful for this comment, which points to a crucial aspect of the theoretical foundation. The metric we employ is the rotating black hole solution in 4D Einstein-Gauss-Bonnet gravity as obtained through the regularization procedure detailed in the works of Ghosh et al. (2020) and subsequent papers on rotating solutions. In the original manuscript, we referenced these derivations implicitly through citations. To address the referee's concern explicitly, we have revised Section 2 to include a short paragraph summarizing the key steps of the D→4 limit regularization and confirming that the metric satisfies the field equations under the adopted ansatz. This addition provides the necessary validation without expanding into a full re-derivation, as the focus remains on the visual characteristics and observational implications. revision: yes

  2. Referee: §4 (Thin-disk model): The accretion disk inner edge is fixed exactly at the event horizon rather than the ISCO, with distinct geodesic prescriptions inside and outside the ISCO. This choice changes the inner-shadow boundary and emissivity profile relative to standard thin-disk calculations; a sensitivity test against ISCO truncation is required to support the reported α dependence.

    Authors: We acknowledge the referee's point regarding the disk truncation. Our choice to place the inner edge at the event horizon allows us to explore the photon ring and shadow features in the near-horizon region, with the piecewise geodesic motion (Keplerian outside ISCO and free-fall inside) to model the emissivity appropriately. This is a deliberate extension beyond standard thin-disk models to probe the strong gravity effects in modified gravity. We agree that a sensitivity test would strengthen the results. In the revised manuscript, we have included an additional discussion in Section 4 explaining the physical motivation for this setup and have performed a limited comparison by noting the differences when truncating at the ISCO for a few representative values of α and a. The primary trends in shadow size and asymmetry with α remain consistent, supporting our conclusions. A more comprehensive sensitivity analysis is planned for future work. revision: partial

  3. Referee: §5 (Observational constraints): α is constrained by direct fitting to M87* and Sgr A* shadow diameters, yet the abstract asserts 'consistency' without reporting the fitting procedure, χ² values, error propagation, or an independent validation set. This leaves open the possibility that the consistency is tautological with the fitting step itself.

    Authors: We thank the referee for this important feedback on the observational section. The constraints on α were derived by requiring that the calculated shadow diameters match the EHT-measured values for M87* (approximately 42 μas) and Sgr A* within their respective uncertainties, using the reported masses and distances. To make this transparent, we have revised Section 5 to explicitly describe the comparison procedure, list the specific bounds on α, and discuss the propagation of errors from observational parameters. Although a full χ² fitting with multiple parameters was not necessary given the direct nature of the shadow size constraint, we have clarified that the consistency is not tautological but based on independent EHT measurements. No separate validation set was used, but the two sources (M87* and Sgr A*) provide cross-checks. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation uses external data for constraints

full rationale

The paper assumes the 4D EGB rotating metric and thin-disk model (inner edge at horizon, geodesic motion inside/outside ISCO), computes shadow images via ray-tracing and fisheye camera for celestial sphere and disk illuminations, then applies independent EHT angular-diameter data from M87* and Sgr A* to bound the coupling α. The consistency statement is the direct outcome of this external fitting step and does not reduce any derived shadow size, asymmetry, or redshift map to the inputs by construction. No self-citation chain, ansatz smuggling, or relabeling of fitted quantities as predictions appears in the load-bearing steps; the visual-characteristics calculations remain independent of the final α bounds.

Axiom & Free-Parameter Ledger

1 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit list; the coupling parameter α functions as the central free parameter whose value is adjusted to match observations.

free parameters (1)
  • coupling parameter α
    Introduced by the 4D Einstein-Gauss-Bonnet action and constrained against EHT shadow data; no numerical value given in abstract.

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