Geodesics of both lightrays and timelike particles with nonzero mass are deflected in a gravitational field. In this work we apply the perturbative method developed in Ref.[1] to compute the deflection angle of both null and timelike rays in the weak field limit for four spacetimes. We obtained the deflection angles for the Bardeen spacetime to the eleventh order of m/b where m is the ADM mass and b is the impact parameter, and for the Hayward, Janis-Newman-Winicour and Einstein-Born-Infeld spacetimes to the ninth, seventh and eleventh order respectively. The effect of the impact parameter b, velocity v and spacetime parameters on the deflection angle are analyzed in each of the four spacetimes. It is found that in general, the perturbative deflection angle depends on and only on the asymptotic behavior of the metric functions, and in an order-correlated way. Moreover, it is shown that although these deflection angles are calculated in the large b/m limit, their minimal valid b can be as small as a few m's as long as the order is high enough. At these impact parameters, the deflection angle itself is also found large. As velocity decreases, the deflection angle in all spacetime studied increases. For a given b, if the spacetime parameters allows a critical velocity vc, then the perturbative deflection angle will deviate from its true value as v decreases to vc. It is also found that if the variation of spacetime parameters can only change the spacetime qualitatively at small but not large radius, then these spacetime parameter will not cause a qualitative change of the deflection angle, although its value is still quantitatively affected. The application and possible extension of the work are discussed.

Based on the Jacobi metric method, this paper studies the deflection of a charged massive particle by a novel four-dimensional charged Einstein–Gauss–Bonnet black hole. We focus on the weak field approximation and consider the deflection angle with finite distance effects. To this end, we use a geometric and topological method, which is to apply the Gauss–Bonnet theorem to the Jacobi space to calculate the deflection angle. We find that the deflection angle contains a pure gravitational contribution δ g, a pure electrostatic δ c and a gravitational–electrostatic coupling term δ gc. We find that the deflection angle increases (decreases) if the Gauss–Bonnet coupling constant α is negative (positive). Furthermore, the effects of the BH charge, the particle charge-to-mass ratio and the particle velocity on the deflection angle are analyzed.

We study the deflection and gravitational lensing (GL) of both timelike and null signals in the equatorial plane of arbitrary stationary and axisymmetric spacetimes in the strong deflection limit. Our approach employs a perturbative method to show that both the deflection angle and the total travel time take quasi-series forms ∑ n=0 [Cn ln (1-bc/b) + Dn ] (1-bc/b) n , with the coefficients Cn and Dn incorporating the signal velocity and finite distance effect of the source and detector. This new deflection angle allows us to establish an accurate GL equation from which the apparent angles of the relativistic images and their time delays are found. These results are applied to the Kerr and the rotating Kalb-Ramond (KR) spacetimes to investigate the effect of the spacetime spin in both spacetimes, and the effective charge parameter and a transition parameter in the rotating KR spacetime on various observables. Moreover, using our approach, the effect of the signal velocity and the source angular position on these variables is also studied.