# Probing Lorentz Symmetry Violation through Lensing Observables of Rotating Black Holes

## TL;DR

This paper investigates how Lorentz symmetry violation affects gravitational lensing signatures of rotating black holes, proposing observational methods to detect such violations using strong lensing data from supermassive black holes like Sgr A* and M87*.

## Contribution

It introduces a Kerr-like black hole solution with Lorentz-violating parameter and analyzes its lensing observables, providing a novel way to test Lorentz symmetry in strong gravity regimes.

## Key findings

- Lorentz-violating parameter significantly alters lensing observables.
- Strong lensing signatures can distinguish Lorentz-violating black holes from Kerr black holes.
- Constraints on Lorentz violation parameter are derived from observational data.

## Abstract

We find a Kerr-like black hole solution-a rotating Bumblebee black hole (RBBH) with a Lorentz-violating parameter $\ell$ and examine the strong lensing by it. The parameter $\ell$ changes the event horizon radius and photon sphere, resulting in a different lensing signature compared to the Kerr black hole of general relativity. Using the strong deflection limit formalism, we compute key observables such as the angular positions of relativistic images, their separation, magnification, and time delays for supermassive black holes Sgr A* and M87*. Our results show that the parameter $\ell$ has a profound influence on these observables, with $\ell > 0$ suppressing and $\ell < 0$ increasing the deflection angle compared to the Kerr case. We compare RBBH observables with those of Kerr black holes, using Sgr A* and M87* as lenses to observe the effect of the Lorentz symmetry-breaking parameter $\ell$. For Sgr A*, the angular position $\theta_\infty$ in $\in~(18.25-33.3)~\mu as$, while for M87* $\in~(13.71-25.02)~\mu as$. The angular separation $s$, for supermassive black holes (SMBHs) Sgr A* and M87*, differs significantly, with values ranging $\in~(0.005-0.81)~\mu as$ for Sgr A* and $\in~(0.003-0.6)~\mu as$ for M87*. The relative magnitude $r_{\text{mag}}$ $\in~(3.04-8.15)~\mu as$. We also compared the time delays between the relativistic images in the SMBHs and found that RBBH can be quantitatively distinguished from Kerr black holes. Our analysis concludes that, within the 1$\sigma$ region, a significant portion of the parameter space agrees with the EHT results of M87* and Sgr A*. This demonstrates the feasibility of utilizing strong gravitational lensing to identify Lorentz symmetry violations in extreme gravity regimes. Weak lensing analysis and Einstein ring observations provide further constraints, producing an upper bound of $\ell \lesssim \mathcal{O}(10^{-6})$.