Non-Equatorial Deflection of Light due to Kerr-Newman Black Hole: A Material Medium Approach

Abstract

We explored the effect of space-time geometry on the trajectory of light rays in the context of a charged, rotating black hole. We derived an analytical expression for the deflection of light rays in Kerr-Newman space-time geometry, using a material medium approach, on non-equatorial plane. From this deflection angle expression it is evident that the charge and rotation of the black hole can affect the light rays' paths. Additionally, we calculated the refractive index of light rays by treating space-time as a medium. We demonstrated how the rotation parameter and charge influence the refractive index and hence the deflection angle of light rays. For Kerr-Newman geometry, the deflection angle decreases with increasing charge when the rotation parameter is held constant. Conversely, for a constant charge, the deflection angle increases with the rotation parameter for prograde and decreases for retrograde trajectories. Applying both factors results in the deflection angle being lower than that of the Schwarzschild geometry. Non-equatorial study of the deflection angle reveals that it is maximum in the equatorial plane than in the pole. The frame-dragging effects in the Kerr-Newman field were taken into account to calculate the velocity of light rays, leading to the determination of the refractive index in this field geometry. This study concludes that depending on the values of the rotation parameter and charge parameter both prograde and retrograde trajectories coincide, resulting in the conclusion that at some point the frame dragging effect is the same for prograde and retrograde motion. Also, the frame dragging increases towards poles for retrograde trajectories while decreasing for prograde trajectories, and these nontrivial nature results because of the interplay between charge and rotation.

Figures (14)

• Figure 1: (a) Schematic view for the deflection of light due to a graded refractive index near a BH. Here, light bends because it follows the path of least optical distance in the refractive medium and vacuum acts as an effective medium influenced by the gravitational field. (b) The sample refractive index profile indicating the decline of refractive index, $n(r)$ with the distance roy2025deflection.
• Figure 2: The variation of horizon radii with respect to the rotation parameter ($\alpha$) for KNBH as compared to other BHs solution in GR. Here we consider $M=1$, $u=0.4$ and $q=0.5$.
• Figure 3: The graphical representation of photon sphere radius at polar plane. Subfigure (a) corresponds to as a function of charge parameter with different values of rotation parameter, while subfigure (b) corresponds to as a function of rotation parameter with different values of charge parameter. Here we consider $M=1$ and $\theta=0$.
• Figure 4: The graphical representation of photon sphere radius at equatorial plane. Subfigure (a) corresponds to as a function of charge parameter with different values of rotation parameter, while subfigure (b) corresponds to as a function of rotation parameter with different values of charge parameter. Here we consider $M=1$ and $\theta=\pi/2$.
• Figure 5: The variation of frame dragging as a function of the rotation parameter, charge parameter, radial distance, and polar angle, along with a comparison to other BHs solution in GR. Subfigure (f) is the zoom in view of subfigure (e). Here we consider $M=1$ and $L=\mp 0.5 (Pro/Retro)$. We used $q=0.3$, $u=0.4$, $\theta=\pi/2$ and $r$ near the horizon, where frame dragging not plotted as a function of these parameters.