Gravitational Deflection of Light and Massive Particle by a Moving Kerr-Newman Black Hole

TL;DR

This paper derives analytical formulas for the second-order gravitational deflection of light and massive particles by a moving Kerr-Newman black hole, incorporating source motion effects and confirming previous results for relativistic particles.

Contribution

It provides new analytical expressions for second-order deflection angles in Kerr geometry with a moving source, including numerical validation and discussion of detection possibilities.

Findings

Analytical formula for light deflection up to second order by a moving Kerr-Newman black hole.

Numerical results confirm previous analytical results for relativistic massive particles.

Discussion on the detectability of motion effects on gravitational deflection.

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., $\alpha(w)=2\left(1+\frac{1}{w^2}\right)\frac{M}{b}+3\pi\left(\frac{1}{4}+\frac{1}{w^2}\right)\frac{M^2}{b^2}$ and $\alpha(w)=2\left(1+\frac{1}{w^2}\right)\frac{M}{b}+\left[3\pi\left(\frac{1}{4}+\frac{1}{w^2}\right)+2\left(1-\frac{1}{w^4}\right)\right]\frac{M^2}{b^2}$, 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.