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Shadows and accretion disk images of charged rotating black hole in modified gravity theory

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The MOG parameter $\alpha$, not the charge $Q$, is the dominant control of the shadow and accretion-disk image of a charged rotating black hole in modified gravity.

desk verdict A competent but incremental ray-tracing catalog for KN-MOG; the headline α-vs-Q dominance claim is undercut by unequal parameter normalization. read the letter →

arxiv 2411.10315 v3 pith:PVDDA4Y6 submitted 2024-11-15 gr-qc

He-Bin Zheng , Meng-Qi Wu , Guo-Ping Li , Qing-Quan Jiang This is my paper · ORCID
classification gr-qc MSC 83C5783C10 PACS 04.70.-s04.20.-q
keywords blackholeshadowaccretiondiskimageKerr-NewmanmodifiedgravityMOGscalar-tensor-vectorbackwardray-tracinginnerphotonring
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 sets out to show that in a charged rotating black hole described by scalar-tensor-vector modified gravity (MOG), the modified-gravity parameter $\alpha$ is the dominant knob controlling what an observer sees: raising $\alpha$ enlarges the shadow and the inner shadow, rounds the spin-induced flat "D" edge, lowers the deviation from circularity, and expands the redshifted regions of the thin accretion disk image, while raising the charge $Q$ shrinks the shadow and acts noticeably more weakly. The motivation is that shadows and disk images are the practical observables that can distinguish modified gravity from general relativity, so a clean qualitative signature of $\alpha$ matters for interpreting horizon-scale images. The claim is established by backward ray-tracing null geodesics in the KN-MOG metric (2.1) with a geometrically thin, optically thin accretion disk, and by directly comparing Kerr, Kerr-MOG, Kerr-Newman, and Kerr-Newman-MOG images at equal parameter levels.

What carries the argument

The load-bearing object is the KN-MOG metric (2.1)-(2.2), whose radial function $\Delta = r^2 - 2GMr + a^2 + Q^2 + G_N^2\,\alpha(1+\alpha)M^2$ carries the modified-gravity effect as a single additive term alongside the charge. On top of this metric, the paper solves the Hamilton-Jacobi equation for null geodesics, imposes the photon-sphere conditions $R(r_p)=0$ and $dR/dr|_{r_p}=0$ to obtain the impact parameters $\xi$ and $\eta$, and projects them onto a zero-angular-momentum observer's screen through celestial coordinates. For the disk images, the thin accretion disk is modeled by circular geodesic motion outside the ISCO and plunge orbits inside it, with a radiative-transfer intensity formula including a redshift factor and a "fudge factor" normalized according to the inner-shadow literature. This combination turns the single metric term in $\Delta$ into the qualitative features the paper reports.

What would settle it

Derive the charged rotating black hole solution directly from the scalar-tensor-vector gravity action rather than adopting the additive $\Delta$ term in (2.2), and recompute the shadow radius and $\delta_s$ at the same value of $\alpha$; if the electromagnetic sector couples to $\alpha$ differently, or the radial function is not simply the KN function plus $Q^2$, the reported dominance of $\alpha$ over $Q$ will not survive. Observationally, for a black hole with independently known mass and distance, a horizon-scale image whose shadow size and circularity are fitted with the KN-MOG family would settle the claim: if the best fit has $\alpha\simeq 0$ even though the image shows the larger, rounder shadow the paper attributes to $\alpha$, the proposed signatures are not present.

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Extended reading notes

Core claim

On its own terms, the paper's central result is that the MOG parameter $\alpha$ plays a dominant role in the charged rotating MOG spacetime: at fixed charge, increasing $\alpha$ from 0 to 0.7 increases the shadow radius at every spin (for $a=0.998$, from 0.051475 to 0.080038) and decreases the circularity deviation $\delta_s$ (from 0.247953 to 0.084967), while the flat edge produced by fast spin gradually rounds back toward a circle. Charge $Q$ acts in the opposite direction, shrinking the shadow and the critical curve, and in side-by-side images at the same parameter level its effect is clearly smaller than that of $\alpha$. For thin-disk illumination, the inner shadow and critical curve grow with $\alpha$ and shrink with $Q$, the redshift regions expand with $\alpha$, and the intensity profile shows a wider two-peak separation as $\alpha$ grows; at large spin, $\alpha$ increases the peak intensity and radiative flux, while at low spin it decreases them. A separate feature the paper reports is the appearance of two "tails" along the Einstein ring in celestial-sphere images, which elongate with spin $a$ and appear nearly independent of $\alpha$.

Load-bearing premise

The argument assumes that Eq. (2.1), with the MOG parameter entering the metric only through the additive term $G_N^2\,\alpha(1+\alpha)M^2$ alongside $Q^2$, is the actual charged rotating black hole solution of scalar-tensor-vector gravity; if the correct solution has a different coupling between $\alpha$ and the electromagnetic field or a different radial structure, the reported dominance of $\alpha$ over $Q$ is not guaranteed.

Editorial extensions

If this is right

  • At fixed spin and charge, a KN-MOG black hole casts a larger, rounder shadow than its Kerr-Newman counterpart, so a nearly circular shadow does not by itself imply low spin; it could mean a large $\alpha$.
  • Because $\alpha$ affects the shadow more strongly than $Q$ at equal parameter levels, attempts to measure electric charge from shadow size need to fit $\alpha$ simultaneously or risk attributing MOG effects to charge.
  • Redshift maps of thin accretion disks expand with $\alpha$ for both prograde and retrograde flows, giving an independent observable, beyond shadow shape, for detecting the modified-gravity parameter.
  • The spin-related Einstein-ring "tails" grow with $a$ but stay nearly unchanged in $\alpha$, while shadow size and roundness respond strongly to $\alpha$; this partial separation means spin and MOG effects can be disentangled in images.
  • In comparisons across the four spacetime families, Kerr-MOG has the largest inner shadow and Kerr-Newman the smallest, so inner-shadow size joins shadow radius as a discriminator among gravity models.

Reading between the lines

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

  • A direct corollary the paper leaves implicit is a degeneracy: because $\alpha$ rounds the shadow and lowers $\delta_s$, modified gravity mimics a smaller effective spin, so shadow-based spin estimates that ignore $\alpha$ will be biased low for a MOG universe.
  • The paper stops at qualitative and numerical signatures; a natural next step is to run a parameter-estimation pipeline on synthetic or real horizon-scale images to see whether posteriors prefer $\alpha>0$ over the Kerr hypothesis.
  • Whether the additive $Q^2$ term is the true MOG electromagnetic coupling remains open outside this paper; if a full solution from the scalar-tensor-vector gravity action changes that term, the same ray-tracing machinery would need to be rerun, and the relative dominance of $\alpha$ could shift.
  • The optically thin disk treatment with a normalized "fudge factor" suppresses absorption effects; extending the model to finite optical depth or thick disks could change the predicted peak intensities while leaving the shadow boundary statements intact.
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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

4 major / 5 minor

Summary. The paper studies the shadow and thin accretion disk images of the Kerr-Newman black hole in modified gravity (MOG), using a backward ray-tracing method. Working with the KN-MOG metric given by Eqs. (2.1)-(2.2), the authors compute shadow boundaries, deviation rates, photon trajectories, redshift maps, and intensity distributions for various values of the spin a, charge Q, and MOG parameter α. They report that increasing α enlarges the shadow and inner shadow, makes the D-shaped shadow more circular, decreases the deviation rate δs, and increases the redshifted regions, while increasing Q has the opposite but weaker effects. The paper concludes that α plays a dominant role in shaping the observable image of the KN-MOG black hole.

Significance. If the KN-MOG metric is indeed the correct charged rotating solution of scalar-tensor-vector gravity, the paper provides a useful catalogue of possible observational signatures that could distinguish MOG black holes from Kerr/Kerr-Newman black holes. The ray-tracing machinery is standard, but it is applied to a spacetime that has not been extensively studied in this context, and the explicit trend tables (e.g., Table 1) for shadow radius and deformation as functions of α are helpful for future comparisons with EHT-like observations. The paper does not release its code or provide convergence tests, which limits reproducibility, but the qualitative trends are consistent with the known Kerr and Kerr-Newman limits when α=0 or Q=0. The main shortcoming is that the central claim of α-dominance over Q rests on an ill-defined parameter comparison, as detailed in the major comments.

major comments (4)
  1. [§5, Fig. 9, Eqs. (2.1)-(2.2)] The central claim that α has a greater effect than Q 'at the same parameter level' is not quantitatively established. In Δ = r^2 - 2(1+α)r + a^2 + Q^2 + α(1+α), with G_N=M=1, α appears linearly in the 2(1+α)r term and quadratically in the constant term, while Q appears only quadratically. Comparing α=0.5 with Q=0.5 compares constant contributions of 0.75 and 0.25, and moreover α shifts the horizon and photon-sphere radii. A meaningful comparison would match the metric contributions (e.g., Q = sqrt(α(1+α))), or report fractional changes in shadow radius and inner-shadow area as each parameter is varied by the same relative amount. Without such a control, the 'dominant role' conclusion is an artifact of the chosen parameter normalization, and the statement in Section 5 that the dominance 'can also be seen from the metric equations' confirms that the numerical finding is not independent of the parameterization.
  2. [§2, Eqs. (2.1)-(2.2)] The KN-MOG metric is adopted from Ref. [108] without derivation or any verification that it satisfies the scalar-tensor-vector gravity field equations. Since all subsequent results—shadow radii, images, the α-versus-Q comparison—depend on this metric, the assumption is load-bearing. If the true charged rotating MOG solution has a different radial dependence or a different coupling of α to the electromagnetic sector, the reported phenomenological conclusions would not apply to actual MOG black holes. The authors should either justify the metric's validity from the STVG action or explicitly frame the entire study as conditional on the metric proposed in Ref. [108].
  3. [§2, Eq. (2.7)] The derivation of the critical impact parameters ξ(r) and η(r) is omitted; only the final expressions are given. These formulas are central to the shadow computation and are not obvious, involving the auxiliary quantities A and B. A brief derivation or a reference to a paper that derives these expressions for this specific spacetime should be provided so that the results can be verified and reproduced.
  4. [Table 1 and Figs. 3-8] The quantitative statements about shadow radii, deviation rates, and intensity distributions rely on numerical ray tracing, but no numerical details are reported: there is no description of the integration scheme, resolution, step size, or convergence tests, and the quantities in Table 1 are given without error estimates. This makes it difficult to assess the reliability of the reported variations (e.g., the change in R_s from 0.050695 to 0.079475). At minimum, the authors should state the numerical precision and show that the results are converged.
minor comments (5)
  1. [§2, paragraph after Fig. 2] The explanation that a stronger electric field 'repels nearby light, reducing the degree of light bending' is physically misleading; in general relativity, light deflection is governed by spacetime curvature, and the charge affects the metric rather than exerting a direct repulsive force on photons. The qualitative trend (shrinking shadow with increasing Q) is correct, but the wording should be revised.
  2. [Fig. 2 caption] The caption contains a typo: the second occurrence of 'For the left plane' should be 'For the right plane.'
  3. [§2, Eqs. (2.6)-(2.12)] The equations contain several notation and typesetting issues (e.g., \(\Sigma1^2\) instead of \(\Sigma^2\), overlong radical expressions, and undefined symbols such as \(D\) in Eq. (2.11)). A thorough editorial pass would improve readability.
  4. [§3, Eq. (3.7)] The transition from Eq. (3.7) to the simplified form \(\sum_n f_n g_n^3 J_n\) is not explained; in particular, the meaning of the \(f_n\) factor and the limit in which absorption is negligible should be stated explicitly.
  5. [§4, Fig. 3] The description of the two 'tails' near the Einstein ring is vague. The authors should define what they mean by a 'tail' and specify whether it is a feature of the lensed image, the photon ring, or a coordinate artifact.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reasoning found; the α-versus-Q dominance claim is a normalization-dependent property of the assumed metric, not a fitted or self-referential prediction.

full rationale

The derivation chain in this paper is essentially self-contained. The shadow and accretion-disk images are computed by ray tracing in the KN-MOG metric (2.1)-(2.2), which is imported from the external Ref. [108]; no computed observable is fed back into the metric, and no parameter is fitted to the target image. The one notable self-citation, Ref. [109] for the ZAMO tetrad/camera model, is not load-bearing because the necessary constructions are restated in Eqs. (2.8)-(2.13). The central claim that α dominates Q is a statement about how the two parameters enter the assumed metric: Δ = r² − 2GMr + a² + Q² + G_N² α(1+α)M², so α appears in both the linear and quadratic terms while Q enters only as Q². The authors effectively concede this by writing that the dominance 'can also be seen from the metric equations (2.1) and (2.2).' Comparing equal numerical values α = 0.5 and Q = 0.5 therefore compares contributions of different functional form, making the conclusion normalization-dependent. This is a robustness and parameter-comparison caveat, not a circular step: the images are not used to define the metric, nor is the conclusion assumed in order to produce the images. Accordingly, no specific circular reduction can be exhibited, and the score reflects only a minor methodological self-citation that is not load-bearing.

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

The paper introduces no new physical entities and fits no free parameters to data. It relies on a known metric from prior MOG literature, standard geodesic separability, and a conventional thin accretion disk model. The hand-chosen emissivity and disk truncation parameters affect the synthetic images but not the shadow-size comparison that forms the central qualitative claim.

free parameters (3)
  • Emissivity profile parameters = J = exp[-1/2 (ln r/r_h)^2 - 2(ln r/r_h)]
    Ad hoc choice in Eq. (3.9), intended to mimic M87*/Sgr A* emission. Not fitted to data, but affects intensity images and the shape of the photon ring.
  • Disk radii (r_ir = r_h, r_or = 20) = r_h to 20
    Chosen by hand in Section 4; justified by the rapid falloff of the emission profile, but still an arbitrary cutoff.
  • Observer distance r_o = 100 = 100
    Set in Section 2 and Section 4. Places the observer effectively at infinity for the ray tracing.
assumptions (3)
  • domain assumption The KN-MOG metric (2.1) with the given Δ is the correct charged rotating black hole solution in scalar-tensor-vector gravity.
    Taken from Ref. [108] without derivation. All parameter dependence in the paper follows from this metric.
  • standard math Null geodesics in this spacetime are separable with a Carter constant, allowing the Hamilton-Jacobi ansatz (2.4).
    The metric has the same algebraic structure as Kerr-Newman, so separability is standard; the paper uses it implicitly in Eqs. (2.6) and (2.7).
  • domain assumption The accretion disk is optically and geometrically thin, neutral, and follows circular orbits outside the ISCO and plunge orbits inside.
    Standard thin-disk model used in the literature, introduced in Section 3. The emissivity is an unsupported phenomenological choice.

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Cite this review

Pith. "Pith review of Shadows and accretion disk images of charged rotating black hole in modified gravity theory." pith.science (2026). https://pith.science/paper/PVDDA4Y6

@misc{pith2026241110315,
  author       = {Pith},
  title        = {Pith review of: Shadows and accretion disk images of charged rotating black hole in modified gravity theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PVDDA4Y6}},
  note         = {Machine review of arXiv:2411.10315}
}
abstract

In this paper, we study the shadow and images of the accretion disk of Kerr-Newman (KN) black hole (BH) in modified gravity (MOG) theory by using backward ray-tracing method. And, the influence of spin parameter ($a$), charge ($Q$), and MOG parameter ($\alpha$) on the observed features of BHs are carefully addressed. Interestingly, as $\alpha$ increases, the flat edge of the BH's shadow gradually becomes more rounded, the size of shadow enlarges, and the deviation rate ($\delta s$) correspondingly decreases. By tracing the photon around BH, we observe that the trajectory of photon exhibits distortion behavior, i.e., the formation of two "tails" near the Einstein ring, which elongate as $a$ increases. For the accretion disk, it shows that the inner shadow expands with $\alpha$, while decreases with $Q$. The increase of $\alpha$ exhibits an increasing effect on redshift. At the same parameter level, $\alpha$ has a more obvious effect on inner shadow and image of BH by comparing with that of $Q$. Our study implies that both $\alpha$ and $Q$ have relatively significant effects on the image of the KN-MOG BH with the thin disk accretion, but the influence of $\alpha$ is much greater. So, this indicates that $\alpha$ plays a dominant role in this spacetime.

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