General Relativistic Aberration Equation and Measurable Angle of Light Ray in Kerr Spacetime.

In this paper, we will mainly focus on the measurable angle of the light ray $\psi$ at the position of the observer instead of the total deflection angle $\alpha$ in Kerr spacetime. We will investigate not only the effect of the gravito-magnetic field or frame dragging characterized by the spin parameter $a$ but also the contribution of the motion of the observer with a coordinate radial velocity $v^r$ and a coordinate transverse velocity $bv^{\phi}$ ($b$ is the impact parameter) or coordinate angular velocity $v^{\phi}$ which are converted from the components of the 4-velocity of the observer $u^r$ and $u^{\phi}$, respectively. In order to take into account the influence of the motion of the observer on the measurable angle $\psi$, we will employ the general relativistic aberration equation because the motion of the observer changes the direction of the light ray approaching the observer. The general relativistic aberration equation makes it possible to calculate the measurable angle $\psi$ more easily and straightforwardly even if the effect of the velocity is included. The measurable angle $\psi$ obtained in this paper can be applied not only to the case of the observer located in an asymptotically flat region but also to the case of the observer placed within the curved and finite-distance region in spacetime. Moreover, when the observer is in radial motion, the total deflection angle $\alpha_{\rm radial}$ can be expressed by $\alpha_{\rm radial} = (1 + v^r)\alpha_{\rm static}$ which is consistent with the overall scaling factor $1 - v$ instead of $1 - 2v$ where $v$ is the velocity of the lens object. On the other hand, when the observer is in transverse motion, the total deflection angle is given by the form $\alpha_{\rm transverse} = (1 + bv^{\phi}/2)\alpha_{\rm static}$ if we define the transverse velocity as having the form $bv^{\phi}$.