Weak field deflection angle and analytical parameter estimation of the Lorentz-violating Bumblebee parameter through the black hole shadow using EHT data

Abstract We explored how the Lorentz symmetry breaking parameter ℓ affects the Reissner-Nordstöm BH solution in the context of weak field deflection angle, and the black hole shadow. We aim to derive the general expression for the weak deflection angle using the non-asymptotic version of the Gauss-Bonnet theorem, and we presented a way to simplify the calculations under the assumption that the distance of the source and the receiver are the same. Through the Solar System test, ℓ is constrained from around orders of magnitude to 0, implying challenging detection of ℓ through the deflection of light rays from the Sun. We also studied the black hole shadow in an analytic way, where we applied the EHT results under the far approximation in obtaining an estimate expression for ℓ . Using the realistic values of the black hole mass and observer distance for Sgr. A* and M87*, it was shown that is satisfied, implying the relevance and potential promise of the spontaneous Lorentz symmetry breaking parameter's role on the shadow radius uncertainties as measured by the EHT. We find constraints for ℓ to be negatively valued, where the upper and lower bounds are ∼ - 1.94 and ∼ - 2.04, respectively.