Gravitational lens effect of a holonomy corrected Schwarzschild black hole

TL;DR

This paper investigates the gravitational lensing effects of a holonomy corrected Schwarzschild black hole, analyzing deflection angles and observables using weak and strong field approximations, and assessing the impact of quantum gravity parameter  on lensing phenomena.

Contribution

It introduces a detailed analysis of holonomy corrections in Schwarzschild black holes and computes lensing observables incorporating quantum gravity effects using both approximation methods.

Findings

 increases the deflection angle and image separation.

 decreases the brightness of the primary lensed image.

The study provides explicit formulas for lensing observables in quantum-corrected black holes.

Abstract

In this paper we study the gravitational lensing effect for the Schwarzschild solution with holonomy corrections. We use two types of approximation methods to calculate the deflection angle, namely the weak and strong field limits. For the first method, we calculate the deflection angle up to the fifth order of approximation and show the influence of the parameter $\lambda$ (in terms of loop quantum gravity) on it. In addition, we construct expressions for the magnification, the position of the lensed images and the time delay as functions of the coefficients from the deflection angle expansion. We find that $\lambda$ increases the deflection angle. In the strong field limit, we use a logarithmic approximation to compute the deflection angle. We then write four observables, in terms of the coefficients $b_1$, $b_2$ and $u_m$, namely: the asymptotic position approached by a set of images $\theta_{\infty}$, the distance between the first image and the others $s$, the ratio between the flux of the first image and the flux of all other images $r_m$, and the time delay between two photons $\Delta T_{2,1}$. We then use the experimental data of the black hole Sagittarius $A^{\star}$ and calculate the observables and the coefficients of the logarithmic expansion. We find that the parameter $\lambda$ increases the deflection angle, the separation between the lensed images and the delay time between them. In contrast, it decreases the brightness of the first image compared to the others.