Gravitational deflection of light and massive particles by a moving Kerr–Newman black hole

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. α ( w ) = 2 1 + 1 w 2 M b + 3 π 1 4 + 1 w 2 M 2 b 2 and α ( w ) = 2 1 + 1 w 2 M b + 3 π 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, 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.

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