Gravitational lensing by black holes in the $4D$ Einstein-Gauss-Bonnet gravity

Abstract

Recently, a non-trivial $4D$ Einstein-Gauss-Bonnet (EGB) theory of gravity, by rescaling the GB coupling parameter as $\alpha/(D-4)$, was formulated in \cite{Glavan:2019inb}, which bypasses Lovelock's theorem and avoids Ostrogradsky instability. The theory admits a static spherically symmetric black hole, unlike $5D$ EGB or general relativity counterpart, which can have both Cauchy and event horizons. We generalize previous work, on gravitational lensing by a Schwarzschild black hole, in the strong and weak deflection limits to the $4D$ EGB black holes to calculate the deflection coefficients $\bar{a}$ and $\bar{b}$, while former increases and later decrease with increasing $\alpha$. We also find that the deflection angle $\alpha_D$, angular position $\theta_{\infty}$ and $u_{m}$ decreases, but angular separation $s$ increases with $\alpha$. The effect of the GB coupling parameter $\alpha$ on positions and magnification of the source relativistic images is discussed in the context of SgrA* and M87* black holes. A brief description of the weak gravitational lensing using the Gauss-Bonnet theorem is presented.