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

Abstract

We will mainly discuss the measurable angle (local angle) of the light ray $\psi_P$ at the position of the observer $P$ instead of the total deflection angle (global angle) $\alpha$ in Kerr spacetime. We will investigate not only the effect of the gravito-magnetic field or frame dragging 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 and $v^{\phi}$ is a coordinate angular velocity) which are converted from the components of the 4-velocity of the observer $u^r$ and $u^{\phi}$, respectively. Because the motion of observer causes an aberration, we will employ the general relativistic aberration equation to obtain the measurable angle $\psi$. The measurable angle $\psi$ given 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. 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 $1 - v$ with respect to the total deflection angle $\alpha{\rm static}$ in the static case. 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 $bv^{\phi}$.