Gravitational Lensing Under the Effect of Weyl and Bumblebee Gravities: Applications of Gauss-Bonnet Theorem.

In this paper, we use the Gauss Bonnet theorem to obtain the deflection angle by the photons coupled to Weyl tensor in a Schwarzschild black hole and Schwarzschild-like black hole in bumblebee gravity in the weak limit approximation. To do so, we first calculate the corresponding optical metrics, and then we find the Gaussian curvature to use it in Gauss-Bonnet theorem, which is first done by Gibbons and Werner. Hence, in the leading order terms we compute the deflection angle, which is affected by the coupling between the photon and Weyl tensor. It is shown that there is a difference in the deviations of light passing near to the Schwarzschild black hole and to the Schwarzschild-like black hole in the bumblebee gravity. Moreover, we compare the deflection angles formed by Einstein-Rosen type wormhole in Weyl gravity and in bumblebee gravity. Remarkably, the deflection angle by Einstein-Rosen type wormhole in bumblebee gravity is found as being to be larger than the deflection angle by Einstein-Rosen type wormhole in Weyl gravity.