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

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.