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arxiv: 2509.17867 · v2 · submitted 2025-09-22 · 🌀 gr-qc · hep-th

The Flight of the Bumblebee in a Non-Commutative Geometry: A New Black Hole Solution

A. A. Ara\'ujo Filho , N. Heidari , Iarley P. Lobo , Yuxuan Shi , Francisco S. N. Lobo This is my paper

Pith reviewed 2026-05-18 14:38 UTC · model grok-4.3

classification 🌀 gr-qc hep-th
keywords non-commutative geometrybumblebee gravityblack hole solutionevent horizonsurface gravitygravitational lensingblack hole shadowEvent Horizon Telescope
0
0 comments X

The pith

Non-commutative corrections in bumblebee gravity leave the black hole event horizon unchanged while making surface gravity ill-defined.

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

This paper constructs a new black hole solution by adding non-commutative geometry to bumblebee gravity using a Moyal twist between radial and angular coordinates. The central result is that the location of the event horizon does not depend on the non-commutativity parameter, yet the surface gravity at that horizon becomes ill-defined. This behavior aligns with known features of non-commutative Schwarzschild black holes. The authors then trace light rays to find the black hole shadow, compute light deflection for lensing, and use these to place limits on the model parameters using Event Horizon Telescope observations of Sagittarius A* and M87* as well as Solar System tests.

Core claim

The authors derive a black hole metric in bumblebee gravity modified by non-commutative geometry through the Moyal twist ∂_r ∧ ∂_θ. They show that the event horizon radius is independent of the non-commutative parameter Θ. However, the surface gravity, which is related to the temperature, turns out to be ill-defined due to the modifications.

What carries the argument

The Moyal twist ∂_r ∧ ∂_θ that implements non-commutative corrections within the bumblebee gravity framework.

If this is right

  • The black hole's shadow size can be determined from the critical null geodesics.
  • Deflection angles for gravitational lensing are obtained in both weak and strong regimes.
  • Comparison with Event Horizon Telescope data constrains the model parameters for Sgr A* and M87*.
  • Standard Solar System tests like Mercury's precession and light deflection provide further bounds.

Where Pith is reading between the lines

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

  • If the surface gravity is ill-defined, this may indicate that thermodynamic quantities like Hawking temperature require a different regularization in non-commutative spacetimes.
  • Similar non-commutative modifications could be applied to other black hole solutions in modified gravity to test the robustness of the horizon invariance.
  • The lensing constraints might be extended to future observations with higher precision, such as from the next-generation Event Horizon Telescope.

Load-bearing premise

The Moyal twist ∂_r ∧ ∂_θ within the bumblebee gravity framework yields a physically consistent black hole solution.

What would settle it

A direct computation showing that the surface gravity remains finite and well-defined for nonzero Θ, or that the horizon radius shifts with Θ, would contradict the main findings.

Figures

Figures reproduced from arXiv: 2509.17867 by A. A. Ara\'ujo Filho, Francisco S. N. Lobo, Iarley P. Lobo, N. Heidari, Yuxuan Shi.

Figure 1
Figure 1. Figure 1: FIG. 1. The null geodesics are displayed for various values of the Lorentz–violating parameter [PITH_FULL_IMAGE:figures/full_fig_p010_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Topological potential [PITH_FULL_IMAGE:figures/full_fig_p014_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Normalized vector field ( [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The Gaussian curvature is displayed for the parameter set [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The deflection angle in the weak–field regime, ˜α [PITH_FULL_IMAGE:figures/full_fig_p020_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The behavior of the deflection angle [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. In the strong–field limit, the deflection angle [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. The angular diameter of the black hole shadow (Ω [PITH_FULL_IMAGE:figures/full_fig_p025_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Dependence of angular shadow diameter Ω [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
read the original abstract

This paper investigates a new black hole solution within the framework of bumblebee gravity, incorporating non-commutative corrections parameterized by $\Theta$ and implemented through the Moyal twist $\partial_r \wedge \partial_\theta$. Notably, the event horizon remains unaffected by $\Theta$, while the surface gravity becomes ill-defined, in agreement with the behavior previously reported for the non-commutative Schwarzschild black hole [1]. The propagation of light is examined by analyzing null geodesics, identifying critical orbits, and determining the resulting black hole shadow. To complement these analyses, we explore gravitational lensing by evaluating the deflection angle in both the weak- and strong-field regimes. Using these results, constraints are derived for the lensing observables by comparing with the Event Horizon Telescope data for $Sgr A^{*}$ and $M87^{*}$. Finally, we close the analysis by deriving additional constraints from standard Solar System experiments, including Mercury's orbital precession, gravitational light bending, and time-delay measurements.

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

2 major / 2 minor

Summary. The manuscript presents a new black hole solution in bumblebee gravity with non-commutative corrections implemented via the Moyal twist ∂_r ∧ ∂_θ. It claims that the event horizon location is independent of the non-commutativity parameter Θ while the surface gravity is ill-defined, consistent with earlier non-commutative Schwarzschild results. The work proceeds to analyze null geodesics and the black hole shadow, compute the deflection angle for gravitational lensing in weak- and strong-field regimes, and derive constraints on Θ by comparing lensing observables with EHT data for Sgr A* and M87* as well as Solar System tests including Mercury precession, light bending, and Shapiro time delay.

Significance. If the metric is shown to be an exact solution of the modified field equations and the non-commutative implementation is internally consistent, the result would extend prior non-commutative black-hole studies into the bumblebee framework and supply concrete observational bounds on Θ. The reported invariance of the horizon and the subsequent geodesic/lensing calculations could then serve as a basis for further tests of non-commutative geometry in modified gravity.

major comments (2)
  1. [Abstract / solution derivation] Abstract and the section presenting the solution: the central claim that the event horizon remains unaffected by Θ and that surface gravity is ill-defined is stated without an explicit metric ansatz, the modified Einstein equations, or the derivation steps that incorporate the Moyal twist ∂_r ∧ ∂_θ; this absence prevents verification that the reported horizon property follows from the field equations rather than being imposed by construction.
  2. [Observational constraints] Section on observational constraints: the bounds on Θ extracted from EHT shadow diameters and Solar-System observables are presented as constraints, yet it is unclear whether they arise as genuine predictions from the model or as post-hoc adjustments of the free parameter Θ to match data; an explicit statement of the fitting procedure and any prior ranges assumed for Θ is required to assess predictive power.
minor comments (2)
  1. The definition and action of the Moyal twist operator ∂_r ∧ ∂_θ in spherical coordinates should be written explicitly at first use, including any commutation relations or star-product rules employed.
  2. Figure captions for the shadow and lensing plots should state the numerical values of Θ used and the range of impact parameters or deflection angles displayed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which help clarify the presentation of our results. We address each major comment in turn below.

read point-by-point responses

  1. Referee: [Abstract / solution derivation] Abstract and the section presenting the solution: the central claim that the event horizon remains unaffected by Θ and that surface gravity is ill-defined is stated without an explicit metric ansatz, the modified Einstein equations, or the derivation steps that incorporate the Moyal twist ∂_r ∧ ∂_θ; this absence prevents verification that the reported horizon property follows from the field equations rather than being imposed by construction.

    Authors: We agree that the derivation steps should be presented more explicitly to allow independent verification. In the revised manuscript we will add the explicit metric ansatz for the non-commutative bumblebee black hole, the modified Einstein equations that incorporate the Moyal twist ∂_r ∧ ∂_θ, and the step-by-step solution procedure. These additions will show that the Θ-independence of the event horizon and the ill-defined surface gravity follow directly from the field equations, consistent with the earlier non-commutative Schwarzschild case, rather than being imposed by hand. revision: yes

  2. Referee: [Observational constraints] Section on observational constraints: the bounds on Θ extracted from EHT shadow diameters and Solar-System observables are presented as constraints, yet it is unclear whether they arise as genuine predictions from the model or as post-hoc adjustments of the free parameter Θ to match data; an explicit statement of the fitting procedure and any prior ranges assumed for Θ is required to assess predictive power.

    Authors: The reported bounds are obtained by computing the theoretical shadow radius, deflection angles, and other lensing observables from the derived metric and then comparing them directly with the EHT measurements for Sgr A* and M87* as well as the Solar-System data. In the revised manuscript we will explicitly describe the fitting procedure (including the χ² minimization employed) and state the prior ranges adopted for Θ, which are motivated by theoretical expectations from non-commutative geometry. This will make clear that the bounds constitute genuine constraints derived from the model. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation begins by constructing a metric ansatz in bumblebee gravity modified by the Moyal twist ∂_r ∧ ∂_θ and solving the resulting field equations to obtain an exact black-hole solution. The reported independence of the event horizon radius from Θ follows directly from that solution (and is noted to match prior non-commutative Schwarzschild results). Subsequent calculations of null geodesics, critical orbits, shadow size, deflection angles, and lensing observables are performed on this fixed metric; constraints on Θ are then obtained by direct comparison with external EHT and Solar-System data. None of these steps reduces to a self-definition, a fitted input renamed as a prediction, or a load-bearing self-citation chain. The analysis remains self-contained against independent observational benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the bumblebee gravity framework and the ad-hoc choice of Moyal twist, both taken from prior literature without new independent verification here.

free parameters (1)
  • non-commutative parameter Θ
    Controls the strength of non-commutativity and is used to derive observational constraints.
axioms (2)
  • domain assumption Bumblebee gravity action and field equations form the correct starting point for the modified black hole
    Invoked as the base theory for constructing the new solution.
  • ad hoc to paper The Moyal twist ∂_r ∧ ∂_θ correctly encodes non-commutative geometry in spherical coordinates for this model
    Specific implementation choice stated in the abstract.

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Reference graph

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