Addendum to ‘Gravitational deflection of light and massive particle by a moving Kerr–Newman black hole’

Abstract

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.

Analytical derivation of second-order deflection in the equatorial plane of a radially moving Kerr–Newman black hole

“…For the case of no translational motion (v = 0) of the gravitational source, the four kinematical coefficients reduce to N 1 = N 2 = N 3 = N 4 = 1, and Eq. (20) becomes the second-order Kerr-Newman deflection angle of a massive particle (w < 1) [19,24] α(0, w) = 2 1 + 1…”

Weak gravitational lensing by stringy black holes

“…According to GR, spacetime singularities rise various issues, both scientific and physical [46,47], by applying the nonlinear electrodynamics it is reasonable to solve these singularities by fabricating a regular BH solution [48][49][50]. He and Lin [51] have investigated the deflection of light for test particles, due to a axially moving Kerr-Newman BH with an arbitrary constant velocity that is perpendicular to its angular momentum. Additionally, it is demonstrated that here is no peculiarity of the electric field quality at the cause for the point-like particles and it has an attractive charge.…”

Equivalence of Gibbons-Werner method to geodesics method in the study of gravitational lensing

“…Finally, with the first-order parameter transformation dp = dx [50] and integrating y, one can get second-order light ray as fallows…”