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Geodesics, accretion disk, gravitational lensing, time delay, and effects on neutrinos induced by a non-commutative black hole

T0 review · 4 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that a non-commutative Schwarzschild black hole is equivalent, for all computed observables, to a Schwarzschild hole with mass $M_\Theta = M - \Theta^2/(64M)$, yielding definite non-commutative corrections to geodesics…

desk verdict A wide-ranging but internally inconsistent application of a mass-deformed Schwarzschild model; the paper's own equations contradict its headline shadow and deflection trends, and the phenomenology is far below observability. read the letter →

arxiv 2412.08369 v2 pith:OK6UYFRR submitted 2024-12-11 gr-qc hep-th

A. A. Araújo Filho , N. Heidari , Ali Övgün This is my paper · ORCID
classification gr-qchep-th PACS 04.70.-s04.20.-q98.62.Sb
keywords non-commutativeblackholeSeiberg-WittenmapmassdeformationshadowgravitationallensingtimedelayneutrinooscillationSagittariusA*
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 spacetime non-commutativity, introduced through the Seiberg–Witten map, leaves the Schwarzschild black hole unchanged in shape and only shifts its mass parameter by a second-order correction in the non-commutativity scale $\Theta$. Working from the deformed metric, the authors define $M_\Theta = M - \Theta^2/(64M)$ and then use the standard Schwarzschild metric with $f_\Theta(r) = 1 - 2M_\Theta/r$ to derive photon geodesics, a thin-accretion-disk shadow image, weak- and strong-field deflection angles, time delay, neutrino pair-annihilation energy deposition, and neutrino oscillation phases and lensing probabilities. If this mass-deformation picture is correct, every one of those observables carries a definite, small correction controlled by $\Theta^2/M$, and the shadow size can be compared with Event Horizon Telescope data on Sagittarius A* to bound the non-commutativity scale. The paper's value would be to turn an abstract Planck-scale hypothesis into a concrete menu of testable predictions for black hole imaging and neutrino observations.

What carries the argument

The load-bearing object is the mass-deformed Schwarzschild metric function $f_\Theta(r) = 1 - 2M_\Theta/r$ with $M_\Theta = M - \Theta^2/(64M)$, obtained from the deformed tetrad metric of Eq. (1) by reading off the horizon radius $r_s^\Theta = 2M - \Theta^2/(32M)$. This single function carries the whole calculation: it enters the null-geodesic system, the optical metric for the Gauss–Bonnet deflection angle, the Tsukamoto strong-field integrals through $A(r)=B(r)^{-1}=f_\Theta(r)$ and $C(r)=r^2$, the time-delay integrals, the redshift factors in the accretion-disk intensity, and the neutrino phase integrals. The other named machinery is standard: backward ray tracing for the disk image, the Gauss–Bonnet theorem for the weak deflection, Tsukamoto's logarithmic regularization for the strong deflection, Bozza's observables for relativistic images, and the Salmonson–Wilson angular integration for neutrino annihilation.

What would settle it

A shadow image of Sagittarius A* with sub-percent precision on the ring radius would test the mass-shift picture: the paper's Table I predicts a $\Theta/M$-dependent deviation of the shadow radius from the Schwarzschild value $3\sqrt{3}M$, at the level of about 0.16% for the quoted EHT bound $\Theta/M \leq 0.3166$. The full metric of Eq. (1) also predicts angle-dependent distortions of the ring that a purely mass-shifted Schwarzschild metric cannot produce, so detection of any quadrupolar warping of the shadow, or a measured shift of the opposite sign to the paper's Table I, would falsify the shortcut.

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Extended reading notes

Core claim

The central claim is that the non-commutative Schwarzschild black hole obtained from the Seiberg–Witten map is faithfully represented by the ordinary Schwarzschild metric with the mass replaced by $M_\Theta = M - \Theta^2/(64M)$, so the event horizon sits at $r_s^\Theta = 2M_\Theta$. All subsequent results follow from this replacement: geodesic equations, the backward-ray-traced shadow of an optically thin infalling flow, the Gauss–Bonnet weak deflection angle $\alpha(b,\Theta)$, the Tsukamoto strong-field deflection, the gravitational time delay, the neutrino pair-annihilation energy deposition, and the neutrino oscillation phase and lensing probability. In the strong-field limit the critical impact parameter is $b_c = 3\sqrt{3}\,(M - \Theta^2/(64M))$, the photon sphere is shifted accordingly, and the shadow radius is computed as a function of $\Theta/M$ in Table I. The paper reports that the shadow radius grows with $\Theta$, that the deflection angles increase with $\Theta$ in the strong-field section, and that the $\Theta = 0$ limit of the neutrino oscillation results reduces to the known Schwarzschild case.

Load-bearing premise

The load-bearing premise is that the full non-commutative deformed metric of Eq. (1), including the corrections to $g_{\theta\theta}$ and $g_{\phi\phi}$ and their angle dependence, can be replaced by a purely radial Schwarzschild metric with mass $M_\Theta = M - \Theta^2/(64M)$, and that this replacement preserves the physics of photons and neutrinos at the orders computed here.

Editorial extensions

If this is right

  • For Sagittarius A*, the relativistic image positions shift at second order in $\Theta$: the critical impact parameter is $b_c = 3\sqrt{3}(M - \Theta^2/(64M))$ and $\theta_\infty \approx 25.24\,\mu$as $+\,O(\Theta^2)$.
  • The Event Horizon Telescope observation of Sgr A* at 68% confidence is translated into an upper bound $\Theta/M \leq 0.3166$, giving a quantitative limit on the non-commutative scale.
  • Gravitational time delay picks up explicit $\Theta^2$ terms, so timing observations of lensed signals offer an independent channel to constrain non-commutativity.
  • Neutrino oscillation phases and the lensed flavor-transition probability acquire $\Theta$-dependent corrections, and both reduce to the known Schwarzschild results in the limit $\Theta \to 0$.
  • Non-commutativity modifies the neutrino pair-annihilation energy deposition rate, with the ratio $\dot{Q}/\dot{Q}_{\rm Newt}$ growing as $\Theta$ increases in the paper's numerical plots.

Reading between the lines

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

  • If the mass-deformation shortcut holds, any existing Schwarzschild prediction that the paper does not recompute — quasi-normal mode frequencies, ISCO radius, Hawking temperature, ring-down waveforms — can be converted to non-commutative predictions by the same replacement $M \to M - \Theta^2/(64M)$, giving an instant catalogue of signatures.
  • For the astrophysically motivated $\Theta \sim 10^{-35}$ m cited in the paper, $\Theta^2/M$ is utterly negligible at Sgr A* scales, so the computed effects are best read as a proof of principle; a detectable signal would require $\Theta$ many orders of magnitude larger.
  • The neutrino-lensing phase differences depend on $\Delta m^2_{ij}$ and on path-dependent impact parameters, so a future high-statistics neutrino source could in principle separate the non-commutative correction from the mass-hierarchy term, a measurement decoupled from photon imaging.
  • The angle-dependent terms dropped in the mass-shift approximation are the cleanest target for falsifying the shortcut: if the full metric of Eq. (1) is the true spacetime, the shadow should show a small $\Theta^2$ quadrupolar distortion that the mass-shifted Schwarzschild model cannot reproduce.
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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 a non-commutative Schwarzschild black hole by replacing the mass M in the standard Schwarzschild metric with MΘ = M − Θ²/(64M), and then computes geodesics, thin-accretion-disk images and shadows, weak- and strong-field deflection angles, lensing observables for Sagittarius A*, time delay, neutrino annihilation energy deposition, and neutrino oscillation/lensing effects. The central quantitative claims are that the shadow radius grows with the non-commutative parameter Θ, that both weak and strong deflection angles increase with Θ, and that EHT observations place an upper bound Θ/M ≤ 0.316632.

Significance. If the calculations were correct, the paper would provide a systematic set of observable signatures for a plausible non-commutative deformation of Schwarzschild, including a concrete EHT-based constraint on Θ. The manuscript is broad in scope and contains a large number of analytic derivations, which is a useful feature. However, the central quantitative claims are not consistent with the model defined in Eq. (3): the reported shadow growth and deflection increase have the opposite sign to what the mass-deformed metric predicts. Because the headline observational predictions and the EHT bound are built on this inconsistent sign, the paper does not currently deliver a reliable prediction for any of its main observables.

major comments (4)
  1. [Section II, Eq. (3); Section IV, Table I and Fig. 4] For the mass-deformed metric fΘ = 1 − 2MΘ/r used throughout the paper, the photon sphere radius is r_ph = 3MΘ = 3M − 3Θ²/(64M) and the shadow radius is r_sh = 3√3 MΘ, so both should decrease as Θ increases. Table I and Fig. 4 instead report r_ph and r_sh growing with Θ; for example, the Θ/M = 2 row gives r_ph = 3.1875, which equals 3M + 3Θ²/(64M), i.e., the opposite sign. The model defined by Eq. (3) is therefore not the model used for the shadow calculation, and the claimed EHT bound Θ/M ≤ 0.316632 in Section IV is read off the wrong-sign branch.
  2. [Section V, Eq. (22); Section XII, Conclusion] Equation (22) contains only negative Θ² corrections, namely −Θ²/(4b³), −Θ²/(16bM), −Θ²M/(2b³), and −17Θ²M²/(10b⁵), so for fixed M and b the weak-field deflection angle decreases as Θ increases. The Conclusion states that the weak deflection angle increases with Θ, contradicting Eq. (22). This is not a minor wording issue because the direction of the Θ dependence is the paper's main physical claim.
  3. [Section VI A, Eq. (39); Section XII, Conclusion] The strong-field deflection angle in Eq. (39) depends on Θ only through the critical impact parameter b_c = 3√3(M − Θ²/(64M)). Since b_c decreases with Θ, the logarithm in Eq. (39) is evaluated at a larger argument for larger Θ, making a(b) more negative (or, for fixed b, smaller). This again contradicts the Conclusion's statement that the strong deflection angle increases with Θ, and is inconsistent with the weak-field behavior reported in Eq. (22).
  4. [Section II, Eq. (1) and the paragraph following Eq. (3)] The starting point is the full Seiberg-Witten deformed metric of Eq. (1), which contains angle-dependent corrections to gθθ and gφφ. The paper then states that it uses the standard Schwarzschild metric with only the deformed mass parameter MΘ, discarding the angular and other metric corrections without any estimate of their size or relevance. Every subsequent geodesic, shadow, lensing, and neutrino calculation is therefore performed for a different spacetime than the one derived from Eq. (1). The paper needs to justify this truncation or show that the dropped terms are negligible for the computed observables; without that, the interpretation of all results as 'non-commutative black hole effects' is not established.
minor comments (5)
  1. [Section III, after Eq. (8)] The text contains an unrendered placeholder 'Fig. ??' before the actual Figure 1; the figure reference needs to be fixed.
  2. [Section V, Fig. 5 caption and Section VI A, Fig. 6 caption] The captions say that 'higher charge values' increase the deflection angle, but the model has no charge parameter; the intended variable is the non-commutative parameter Θ. This wording should be corrected.
  3. [Section VI A, Eq. (38)] The expression for IR(rm) contains a factor sgn(Θ² − 64M²) whose derivation is not explained; since Θ is normally treated as a small parameter, this sign function is puzzling and should be clarified or removed.
  4. [Section II, first paragraph] The deformed metric of Eq. (1) is attributed to Ref. [97], but Ref. [97] is a paper on Gödel-type universes in bumblebee gravity and is unrelated to the Seiberg-Witten deformed Schwarzschild metric; the correct attribution appears to be Ref. [31] (Chaichian, Tureanu, and Zet). This reference error should be corrected.
  5. [Section XI, Eq. (95) and the unnumbered formula after it] Several display equations in the neutrino lensing section are unnumbered, and the derivation of the final probability expression would be easier to follow if each step were labeled consistently.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper's observables follow from an explicitly stated mass-deformed Schwarzschild ansatz, and no fitted parameter is relabeled as a prediction.

full rationale

The derivation chain is self-contained in the sense relevant to circularity. Section II defines the input through the Seiberg-Witten-deformed metric of Eq. (1), obtains the horizon shift in Eq. (2), and then explicitly declares the working model: "This study utilizes the standard Schwarzschild metric in conjunction with the deformed non-commutative mass parameter outlined in Eq. (3)." All later shadow, lensing, time-delay, and neutrino results are then derived from fΘ = 1 - 2MΘ/r, with MΘ = M - Θ²/(64M). This makes the Θ-dependence a reparameterization of ordinary Schwarzschild formulas, but that is the paper's open modeling assumption rather than a hidden equivalence between inputs and outputs. The mass shift is not fitted to any quantity later advertised as a prediction; the EHT/Sgr A* comparison in Section IV and VII.A is an external constraint, not a calibration of MΘ against the same observable. The self-citations [98,99] for the mass-deformation formula are not load-bearing because Eq. (3) is re-derived in Section II from the horizon condition of Eq. (1). A separate internal consistency problem does exist: Eq. (3) gives r_ph = 3MΘ = 3M - 3Θ²/(64M), which decreases with Θ, while Table I and Figure 4 report r_ph increasing with Θ (e.g., the Θ/M = 2 row gives 3.1875, corresponding to 3M + 3Θ²/(64M), the opposite sign); likewise Eq. (22) makes the weak deflection angle decrease with Θ while the Conclusion states that deflection angles increase with Θ. This is a correctness and consistency flaw, but it is not circularity: the outputs are not fitted inputs, and no load-bearing conclusion rests on a self-citation alone.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The ledger records one free parameter (Θ), the metric-simplification assumption that is ad hoc to this paper, and standard domain assumptions from the lensing and neutrino literature. No new particles, forces, or entities are introduced.

free parameters (1)
  • Θ (non-commutative parameter) = varied over Θ/M ∈ [0.001, 2] in tables/figures; Θ ≈ 1.235e-35 m for Sgr A*
    All observables depend on Θ through MΘ. The paper explores a range and adopts a literature value, but Θ is not measured or fit here, and the EHT constraint is not derived transparently.
assumptions (5)
  • ad hoc to paper The full NC deformed metric of Eq. (1) can be replaced by Schwarzschild with mass MΘ = M - Θ²/(64M).
    Stated explicitly after Eq. (3); the angle-dependent g_θθ and g_φφ corrections in Eq. (1) are dropped without justification, changing the spacetime geometry used for all subsequent calculations.
  • standard math Tsukamoto's strong deflection framework for static, spherically symmetric, asymptotically flat spacetimes is applicable.
    Used in Section VI with metric functions A=fΘ, B=1/fΘ, C=r²; the framework is standard and cited, but its applicability depends on the metric being exactly of the Schwarzschild-deformed form.
  • domain assumption Plane-wave approximation gives the correct neutrino oscillation phase in curved spacetime.
    Section X integrates g_μν p^μ dx^ν to get Φ_k following Cardall-Fuller style treatments; this is standard in the neutrino-lensing literature but not derived from first principles here.
  • domain assumption The EHT shadow-radius measurement maps to the stated 68% C.L. limit on Θ/M.
    Section IV and Fig. 4 cite Ref. [101] but do not show the conversion from the measured shadow deviation to Θ/M. The implied fractional shadow change is about 0.16%, which is far smaller than the EHT precision quoted elsewhere in the paper.
  • standard math Weak deflection angle can be computed via the Gauss-Bonnet theorem on the optical metric.
    Section V integrates the Gaussian curvature over the equatorial plane; this is a standard method from Gibbons-Werner and is not in question, but the final expression inherits the sign ambiguity of the mass deformation.

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Cite this review

Pith. "Pith review of Geodesics, accretion disk, gravitational lensing, time delay, and effects on neutrinos induced by a non-commutative black hole." pith.science (2026). https://pith.science/paper/OK6UYFRR

@misc{pith2026241208369,
  author       = {Pith},
  title        = {Pith review of: Geodesics, accretion disk, gravitational lensing, time delay, and effects on neutrinos induced by a non-commutative black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OK6UYFRR}},
  note         = {Machine review of arXiv:2412.08369}
}
read the original abstract

This paper explores gravitational phenomena associated with a non-commutative black hole. Geodesic equations are derived, and a thin accretion disk is analyzed to model the black hole shadow image, considering an optically thin, radiating, and infalling gas. Retrolensing effects are examined to trace photon emission configurations, while gravitational lensing is investigated through weak and strong deflection limits, with lensing equations and observables applied to Sagittarius A*. The study also includes calculations of time delay, energy deposition rate from neutrino annihilation, phase and probability of neutrino oscillation, and neutrino gravitational lensing.

Figures

Figures reproduced from arXiv: 2412.08369 by the authors.

Figure 1
Figure 1. Geodesic trajectories are computed for different values of Θ and [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The shadows are shown for different values of the non–commutative parameter Θ. [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗
Figure 3
Figure 3. The regions linked to the black hole’s direct emission, lensing rings, and photon [PITH_FULL_IMAGE:figures/full_fig_p012_3.png] view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: Analyze the variation of the normalized black hole shadow radius, [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: The deflection angle as a function of b for different values of M and Θ. Accordingly, after performing the integration, Eq. (20) becomes α(b, Θ) ≃ 4M b − Θ2 4b 3 + 8M2 b 3 + 32M3 3b 3 − Θ2 16bM + 136M4 5b 5 − Θ2M 2b 3 − 17Θ2M2 10b 5 . (22) The derivation of the above e…
Figure 6
Figure 6. Figure 6: The deflection angle as a function of b for different values of M and Θ. VII. LENSING EQUATIONS AND OBSERVABLES This section focuses on analyzing the parameters that govern the bending of light in the strong gravitational field surrounding the black hole. Light emitted…
Figure 7
Figure 7. Figure 7: Representation of the gravitational lensing. The light emitted from the source [PITH_FULL_IMAGE:figures/full_fig_p021_7.png]
Figure 8
Figure 8. Figure 8: The time delay ∆T is evaluated as a function of Θ (left panel) and rS (right panel) for the parameters Θ = 0.1 (only for the right panel), M = 0.1, r0 = 3, rO = 10, b = 0.1 and rS = 4 (only for the left panel). IX. THE ENERGY DEPOSITION RATE BY THE NEUTRINO ANNIHILATIO…
Figure 9
Figure 9. Figure 9: The solid, dotted and dashed curves of the ratio [PITH_FULL_IMAGE:figures/full_fig_p028_9.png]

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Pith tools

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