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REVIEW 3 major objections 6 minor 66 references

Spherical photon orbits around Kerr-MOG black hole

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Photon orbits around Kerr-MOG black holes are governed by a sextic polynomial whose real root count is set by a critical inclination angle, with four orbits below it, two above, and three at it.

desk verdict A promising extension of Tavlayan-Tekin to Kerr-MOG, but the key factorization behind the critical inclination angle is unverified and the paper has too many internal typos to be reliable as written. read the letter →

arxiv 2412.17520 v1 pith:K5MHY3K4 submitted 2024-12-23 gr-qc

Shi-Yu Li , Song-Shan Luo , Zhong-Wen Feng This is my paper · ORCID
classification gr-qc MSC 83C5783C1083D05 PACS 04.70.-s04.20.Jb04.50.Kd
keywords Kerr-MOGblackholesphericalphotonorbitsmodifiedgravityHamilton-Jacobiequationcriticalinclinationangleshadowsexticpolynomialsphere
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 derives the sextic polynomial that governs spherical photon orbits around a Kerr-MOG black hole, a rotating black hole in scalar-tensor-vector modified gravity. It claims that the number of such photon orbits is determined by an effective inclination angle $v$: in the non-extremal case a critical angle $v_{cr}$ separates four real photon orbits (below) from two (above), with three at the critical angle. In the polar plane there are two effective orbits, one outside and one inside the event horizon; in the equatorial plane there are four, two outside and two inside. The paper also finds that the MOG deformation parameter $\alpha$ shrinks the allowed range of the rotation parameter $u$ and lowers the critical impact parameter, changing the predicted shadow brightness. All orbits it finds are radially unstable.

What carries the argument

The central object is the sixth-order polynomial $f(x)=0$ in Eq. (21), obtained by combining the separability of the Hamilton-Jacobi equation (with Carter constant) with the two conditions $R(r)=0$ and $dR/dr=0$ for constant-radius photon orbits. The polynomial encodes the orbit radius $x=r/M$ as a function of the rotation parameter $u=a^2/M^2$, the effective inclination angle $v=\sin^2 i$, and the MOG deformation parameter $\alpha$. The critical-angle analysis rests on the claimed factorization $f(x)=P_4(x)(x+A_5)^2$ at $v_{cr}$, which turns the sextic into solvable pieces whose roots are identified as the photon orbits.

What would settle it

For a concrete parameter pair such as $\alpha=0.3$ and $u=0.5$, compute $v_{cr}$ from Eqs. (30)-(33) and check numerically whether the sextic (21) equals $P_4(x)(x+A_5)^2$ with the coefficients of Eq. (35); any nonzero residual refutes the factorization, and a direct root-count scan in $v$ would confirm or refute the claimed four-to-two transition.

Watch

Extended reading notes

Core claim

For Kerr-MOG black holes, the radial photon motion reduces to the sextic polynomial (21), whose real roots are the radii of spherical photon orbits. In the extremal case the polynomial factors as $(x-1)^2P_4(x)$; a slowly rotating extremal hole has two photon orbits outside the horizon, while a rapidly rotating one has only one. In the non-extremal case the paper claims there is a critical inclination angle $v_{cr}=v_{cr}(u,\alpha)$, with four orbits below it, two above it, and three at $v_{cr}$; at the critical angle the sextic is claimed to factor into a quartic times a squared linear term. All of these orbits are radially unstable, and $\alpha$ constrains the spin to $u<1/(1+\alpha)$.

Load-bearing premise

The four-to-two transition at the critical inclination angle rests on the unverified factorization of the sextic into a quartic times a squared linear term at $v_{cr}$; if that factorization is not exact, the formula for $v_{cr}$ and the claimed orbit counts would not follow.

Editorial extensions

If this is right

  • The Kerr critical-inclination result of Tavlayan and Tekin extends to MOG, with $v_{cr}$ now a function of the deformation parameter $\alpha$ and the rotation parameter $u$.
  • The allowed spin range shrinks to $u<1/(1+\alpha)$, so the extremal hierarchy of orbits is different from Kerr for every nonzero $\alpha$.
  • The critical impact parameters decrease as $\alpha$ grows, which would change the size and brightness of the predicted black hole shadow and its photon ring.
  • Because all spherical photon orbits are radially unstable, their observable signatures are transient lensing features rather than stable light rings.

Reading between the lines

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

  • The same factorization-plus-root-count method could be reapplied to other rotating modified-gravity metrics, such as Kerr-Newman in scalar-tensor theories, to test whether a critical inclination angle is a generic feature of photon orbits.
  • If the factorization at $v_{cr}$ is exact, it implies the discriminant of the sextic vanishes along the $v_{cr}(u,\alpha)$ surface; locating that discriminant surface directly would provide an independent check of the paper's central claim.
  • The near-horizon location of the prograde orbit at large $\alpha$ suggests that very-high-spin MOG black holes could be distinguished from Kerr by future near-horizon imaging, if the MOG parameter is not too small.
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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 / 6 minor

Summary. This paper studies spherical photon orbits in the Kerr-MOG spacetime of scalar-tensor-vector gravity. Starting from the Hamilton-Jacobi equations, the authors derive a sixth-order polynomial f(x) in the dimensionless radius x, involving the rotation parameter u, the effective inclination angle v, and the MOG deformation parameter α (Eq. 21). They analyze the polar (v=1), equatorial (v=0), and intermediate (0<v<1) cases, for both extremal and non-extremal black holes. The central new claim is that in the non-extremal case there is a critical inclination angle vcr(u,α) such that the sextic has four real roots below vcr, two above vcr, and three at vcr, with two roots outside and one inside the event horizon at the critical point. The paper also reports radial instability of all photon orbits and studies how the critical impact parameter and hence the black hole shadow depend on α.

Significance. If correct, the result would generalize the spherical photon orbit analysis of Tavlayan and Tekin for Kerr black holes to Kerr-MOG black holes and would give a concrete α-dependent shadow prediction that could, in principle, distinguish MOG from GR. The manuscript is self-contained: the polynomial is derived from the Hamilton-Jacobi equation rather than imported, and the paper checks several familiar limits such as Schwarzschild and Kerr. The numerical survey of orbits and stability is extensive. However, at present the central critical-angle result rests on an unverified factorization, and the displayed sextic has apparent algebraic and dimensional inconsistencies that prevent the stated limits from being reproduced. Because the main object of the paper, Eq. (21), and the main new result, the vcr transition, are both affected, the significance cannot be evaluated until the derivation is corrected and the factorization is verified.

major comments (3)
  1. [III, Eq. (21)] As printed, the sextic polynomial is not consistent with the equations of motion from which it is supposed to follow. In the Kerr limit α=0 and with M=1, the x^4 coefficient is 9+2uva=9+2u^{3/2}v, because u=a^2, whereas the polar condition v=1 (L_z=0) obtained from R(r)=0 and dR/dr=0 is (x^3-3x^2+ux+u)^2=0, whose x^4 coefficient is 9+2u. The x^3 coefficient in Eq. (21) is -4u for α=0, but the correct coefficient at general v is -4uv, so at the equatorial value v=0 the polynomial retains a cubic term instead of reducing to the condition η=0. Thus Eq. (21) does not reduce to Eq. (22) or to Eq. (27), and the root counts shown in the figures are not those of the displayed polynomial. If the intended terms are 2uv and -4uv, the derivation and all subsequent numerical results must be redone with the corrected polynomial.
  2. [V, Eqs. (30)-(35)] The critical inclination angle is the central new result, but the formulas supporting it cannot be checked. Eqs. (30)-(33) are stated without derivation, and Eq. (31) contains an unbalanced bracket and an unclear division by w. More seriously, at vcr the sextic is asserted to factor as f(x)=P4(x)(x+A5)^2, Eq. (34), but no substitution or verification is supplied; the factorization is not demonstrated. The coefficient list (35) is unusable as printed: in Eq. (35b) the numerator and denominator are identical, forcing A2=1; Eq. (35c) contains an undefined symbol d; and Eq. (35d) contains a bare a instead of α. Since the four-to-two transition and the existence of three orbits at vcr rest on this factorization, this is a load-bearing gap rather than a presentation issue.
  3. [IV.B, Eq. (27)] The equatorial-plane section contains a contradiction in the definition of v. Eqs. (18)-(19) define v=sin^2 i, so for equatorial orbits (K=0, i=0,π) one has v=0, while the text of Section IV.B states that 'i=0 or π, which leads to v=1'. The abstract and Section IV.A use v=1 for polar orbits, so the two conventions are incompatible. In addition, Eq. (27) does not follow from Eq. (21) for either v=0 or v=1: substituting v=0 into Eq. (21) and factoring out x^2 yields a quartic whose x coefficient differs from that of Eq. (27) by a term involving u and α. The derivation of Eq. (27) must be shown explicitly and the v convention fixed.
minor comments (6)
  1. [IV.A] In the sentence defining polar orbits, the text writes 'ν=1' where v=1 is meant; the symbol should be made consistent with Eq. (19).
  2. [IV.B] In the description of the extremal equatorial case, the paper says the black hole has one photon orbit 'outside the event horizon and the other (x1) inside'; the inside orbit should be labeled x3 or x4, not x1.
  3. [IV.A and IV.B, impact parameter discussion] The critical impact parameter is defined by Eq. (25) and evaluated in Eq. (26), but the text and Fig. 4 refer to 'Eq. (29)' when discussing β; the cross-reference should be corrected.
  4. [Section VI, Conclusion] The conclusion states that as α increases the radii of the prograde and retrograde equatorial orbits expand, which contradicts the description in Section IV.B and the behavior shown in Fig. 6; this discrepancy should be resolved.
  5. [References] Reference [55] duplicates reference [32], and reference [57] duplicates reference [53]; the bibliography should be consolidated to avoid duplicate entries.
  6. [Eq. (24)] The notation ddR(2)_i in Eq. (24) is nonstandard and is not defined in the text; it should be written explicitly as the second derivative of R(x) with respect to x evaluated at the orbit x_i.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the photon-orbit polynomial and critical-angle analysis are derived from the external Kerr-MOG metric via the Hamilton-Jacobi equation, with no fitted parameter renamed as a prediction and no load-bearing self-citation.

full rationale

The paper's derivation chain is self-contained once the Kerr-MOG metric (Eq. 9-10) is adopted from the cited MOG literature. The sextic polynomial (Eq. 21) is obtained by combining the radial conditions R(r)=0 and dR/dr=0 with the Hamilton-Jacobi equations, and the polar and equatorial reductions follow algebraically from that polynomial. The critical inclination angle in the non-extremal case is obtained from an algebraic factorization ansatz (Eq. 34) with coefficients determined by comparison with the sextic, not by fitting data or by importing a result from the authors' own prior work. The extremal critical angle v_cr is also derived from the quartic P4(x), again algebraically. No parameter is fitted to the quantity it is later said to predict, and no central claim reduces by definition to an input. The paper contains no self-citations by the authors at all; references to prior work by other groups supply the spacetime metric, the stability criterion, and the Kerr-case motivation, which are standard external inputs rather than circular load-bearing assumptions. The concerns raised about Eq. (35) being internally inconsistent or the factorization (34) being unverified are mathematical correctness issues, not circularity: an unproven or erroneous algebraic step is not the same as a derivation that presupposes its conclusion. Therefore, on the circularity axis the paper receives score 0.

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

The paper introduces no new entities or fitted parameters. It relies on the MOG metric, the separability ansatz, a stability criterion from the literature, and an unproven factorization ansatz for the sextic at the critical angle.

assumptions (4)
  • domain assumption The Kerr-MOG metric (9) is a valid vacuum solution of STVG.
    The paper imports the metric from Refs. [35,36] and does not derive or test it.
  • domain assumption The Hamilton-Jacobi equation is separable in the Kerr-MOG spacetime, allowing a Carter constant.
    Used implicitly in Section III when writing the action (13) and equations (14)-(15).
  • domain assumption The radial stability criterion of Ref. [54] applies, namely an orbit is unstable if d²R/dx² > 0.
    Used in Section IV.A and throughout to classify all orbits as unstable.
  • ad hoc to paper At the critical inclination angle vcr, the sextic f(x) factorizes as P4(x)P2(x) with P2 = (x + A5)^2.
    Equation (34) is asserted without proof; it is the basis for computing vcr and the roots in Section V.

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Pith. "Pith review of Spherical photon orbits around Kerr-MOG black hole." pith.science (2026). https://pith.science/paper/K5MHY3K4

@misc{pith2026241217520,
  author       = {Pith},
  title        = {Pith review of: Spherical photon orbits around Kerr-MOG black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5MHY3K4}},
  note         = {Machine review of arXiv:2412.17520}
}
abstract

This study investigates photon orbits around Kerr-MOG black holes. The equation of photon of motion around the Kerr-MOG black hole is derived by solving the Hamilton-Jacobi equation, expressed as a sixth-order polynomial involving the inclination angle $v$, the rotation parameter $u$, and the deformation parameter $\alpha$ that characterizes modified gravity. We find that $\alpha$ constrains the rotation of the black hole, modifying its gravitational field and leading to distinct photon orbital characteristics. Numerical analysis reveals that the polar plane ($v=1$) has two effective orbits: one outside and one inside the event horizon, while the equatorial plane ($v=0$) has four effective orbits: two outside and two inside the event horizon. Moreover, we derive the exact formula for general photon orbits between the polar and equatorial planes ($0<v<1$). In the extremal case, the rotation speed significantly impacts general photon orbits. A slowly rotating extremal black hole has two general photon orbits outside the event horizon, whereas a rapidly rotating extremal black hole has only one such orbit. In the non-extremal case, a critical inclination angle $v_{cr}$ exists in the parameter space $\left(v, u, \alpha \right)$. Below $v_{cr}$, there are four general photon orbits, while above $v_{cr}$, there are two orbits. At the critical inclination angle, three solutions are found: two photon orbits outside and one inside the event horizon. Additionally, the results indicate that all orbits are radially unstable. Furthermore, by analyzing photon impact parameter, we argue that $\alpha$ influences observational properties of the black hole.

Figures

Figures reproduced from arXiv: 2412.17520 by the authors.

Figure 1
Figure 1. FIG. 1. The polar photon orbits [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The behavior of positive polar photon orbits ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The ddR [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: illustrates the critical impact parameter for photons on the polar orbit x1 of a Kerr-MOG black hole, as derived from Eq. (29). The critical impact parameter delineates three possible outcomes for photons near the black hole: (i) Photons with impact parameters smaller …
Figure 5
Figure 5. Figure 5: FIG. 5. The equatorial photon orbits [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The behavior of equatorial photon orbits [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The ddR [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The critical impact parameters [PITH_FULL_IMAGE:figures/full_fig_p015_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The general photon general orbits and event horizon o [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The critical inclination angle as the functions of t [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The general photon orbits [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The behavior of general photon orbits [PITH_FULL_IMAGE:figures/full_fig_p020_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. The ddR [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The critical impact parameters [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.