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

TL;DR

This paper investigates how Lorentz symmetry breaking affects black hole properties, deriving expressions for light deflection and shadow size, and constrains the Lorentz-violating parameter using Solar System tests and EHT data.

Contribution

It provides a novel analytical framework for estimating the Lorentz-violating parameter from black hole shadow observations and weak field deflection angles.

Findings

Lorentz-violating parameter constrained between -2.04 and -1.94.

Weak deflection angle affected by Lorentz symmetry breaking.

Black hole shadow size sensitive to Lorentz-violating parameter.

Abstract

We explored how the Lorentz symmetry breaking parameter $\ell$ affects the Reissner-Nordst\"{o}m 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. \textcolor{black}{Through the Solar System test, $\ell$ is constrained from around $-10^{-9}$ orders of magnitude to $0$, implying challenging detection of $\ell$ 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 $\ell$. Using the realistic values of the black hole mass and observer distance for Sgr. A* and M87*, it was shown that $M/r_{\rm o} \neq 1$ 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. \textcolor{black}{We find constraints for $\ell$ to be negatively valued, where the upper and lower bounds are $\sim -1.94$ and $\sim -2.04$, respectively.