Strong Gravitational Lensing by Static Black Holes in Effective Quantum Gravity

Abstract

We investigate strong gravitational lensing by two static black hole models (Model-1 and Model-2) within the Effective Quantum Gravity (EQG) framework, characterized by mass $M$ and parameter $\zeta$. For $\zeta = 0$, they reduce to the Schwarzschild solution, and depending on the parameters, they describe black holes with an event and Cauchy horizon (Model-1), a single horizon (Model-2), or no horizons. Using SMBHs Sgr A* and M87* as lenses and integrating theoretical predictions with recent EHT data, we identify significant differences in lensing signatures due to quantum corrections. For Model-1, the deviations of the lensing observables: $|\delta\theta_{\infty}|$ of black holes in EQG from Schwarzschild black hole, for SMBHs Sgr A* and M87, can reach as much as $1.75~\mu$as and $1.32~\mu$as, while $|\delta s|$ is about $30.12$~nas for Sgr A* and $22.63$~nas for M87*. The flux ratio of the first image to all subsequent packed images indicates that EQG black hole images are brighter than their Schwarzschild counterparts, with a deviation in the brightness ratio $|\delta r_{mag}|$ reaching up to 2.02. The time delays between the second and first images, denoted $|\delta T_{2,1}|$, exhibit substantial deviations from the GR counterpart, reaching up to 1.53 min for Sgr A* and 1159.9 min for M87*. The EHT constraints on $\theta_{sh}$ of Sgr A* and M87* within the $1\sigma$ region limit the parameters $\zeta$. Our analysis concludes that EQG black holes are consistent with the EHT observations within this finite space.