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 [D. Glavan and C. Lin, Phys. Rev. Lett. 124, 081301 (2020)], 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 and also free from the singularity. 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.

中文翻译：

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4D 爱因斯坦-高斯-博内引力中黑洞的引力透镜效应

最近，通过将 GB 耦合参数重新缩放为 $\alpha/(D-4)$，在 [D. Glavan 和 C. Lin，物理。牧师莱特。124, 081301 (2020)]，它绕过了洛夫洛克定理并避免了 Ostrogradsky 不稳定性。该理论承认一个静态球对称黑洞，不同于 $5D$ EGB 或广义相对论对应物，它可以同时具有柯西和事件视界，并且没有奇点。我们将先前关于施瓦西黑洞引力透镜的工作推广到 $4D$ EGB 黑洞的强和弱偏转极限，以计算偏转系数 $\bar{a}$ 和 $\bar{b}$，而前者随着$\alpha$的增加而增加，后来减少。我们还发现偏转角$\alpha_D$，角位置 $\theta_{\infty}$ 和 $u_{m}$ 减小，但角间距 $s$ 随 $\alpha$ 增加。在 SgrA* 和 M87* 黑洞的背景下讨论了 GB 耦合参数 $\alpha$ 对源相对论图像的位置和放大率的影响。使用 Gauss-Bonnet 定理简要描述了弱引力透镜。