The gravitational deflection of test particles including light, due to a radially moving Kerr–Newman (KN) black hole with an arbitrary constant velocity that is perpendicular to its angular momentum, is investigated. In harmonic coordinates, we derive the second post-Minkowskian (2PM) equations of motion for test particles, and solve them by high-accuracy numerical calculations. We then concentrate on discussing the kinematical corrections caused by the motion of the gravitational source to second-order deflection. The analytical formula of the light-deflection angle up to the second order by a moving lens is obtained. For a massive particle moving with a relativistic velocity, there are two different analytical results for the Schwarzschild deflection angle up to the second order reported in the previous works, i.e. and , where and w are the mass of the lens, the impact parameter, and the particle’s initial velocity, respectively. Our numerical result is in perfect agreement with the former result. Furthermore, the analytical formula for massive particle deflection up to the second order in the Kerr geometry is achieved. Finally, the possibilities of detecting the motion effects on the second-order deflection are also analyzed.

In this paper, the gravitational deflection of relativistic massive particles up to the second post-Minkowskian order by static and spherically symmetric wormholes is investigated in the weak-field limit. These wormholes include the Janis-Newman-Winicour wormhole, a class of zero Ricci scalar scalar-tensor wormholes, and a class of charged Einstein-Maxwell-dilaton wormholes. With the Jacobi metric approach, the Gauss-Bonnet theorem is employed to study the gravitational deflection. In this scheme, the deflection angle as a topological effect is considered. Moreover, we analyze the influence of the spacetime parameters on the results. PACS numbers: 98.62.Sb, 95.30.Sf

In this work, we base on the second-order post-Minkowskian equations of motion, and apply an iterative technique to analytically derive the gravitational deflection of the relativistic particles in the equatorial plane of Kerr-Newman black hole with a radial (or longitudinal) and constant velocity. We find that the kinematically correctional effects on the second-order contributions to the deflection can be expressed into a very compact form, which are valid for both the massive particle and photon. Our result reduces to the previous formulation for the first-order deflection caused by a moving Schwarzschild black hole when the second-order contributions are dropped.

We derive the analytical time delay of light propagating in the equatorial plane and parallel to the velocity of a moving Kerr-Newman black hole up to the second post-Minkowskian order via integrating the null geodesic equations. The velocity effects are expressed by a very compact form. We then concentrate on analyzing the magnitudes of the correctional effects on the second-order contributions to the delay and discuss their possible detection. Our result in the first post-Minkowskian approximation is in agreement with Kopeikin and Schäfer's formulation which is based on the retarded Liénard-Wiechert potential.

The gravitational deflection of test particles including light, due to a radially moving Kerr-Newman black hole with an arbitrary constant velocity being perpendicular to its angular momentum, is investigated. In harmonic coordinates, we derive the second post-Minkowskian equations of motion for test particles, and solve them by high-accuracy numerical calculations. We then concentrate on discussing the kinematical corrections caused by the motion of the gravitational source to the second-order deflection. The analytical formula of light deflection angle up to second order by the moving lens is obtained. For a massive particle moving with a relativistic velocity, there are two different analytical results for Schwarzschild deflection angle up to second order reported in the previous works, i.e., α(w) = 2 1 + 1 w 2 M b + 3π 1 4 + 1 w 2 M 2 b 2 and α(w) = 2 1 + 1 w 2 M b + 3π 1 4 + 1 w 2 + 2 1 − 1 w 4 M 2 b 2 , where M, b, and w are the mass of the lens, impact parameter, and the particle's initial velocity, respectively. Our numerical result is in perfect agreement with the former. Furthermore, the analytical formula for massive particle deflection up to second order in the Kerr geometry is achieved. Finally, the possibilities of detecting the motion effects on the second-order deflection are also analyzed.