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

Abstract

As an extension of our previous paper, instead of the total deflection angle $\alpha$, we will mainly focus on discussing the measurable angle of the light ray $\psi$ at the position of the observer in Kerr--de Sitter spacetime which includes the cosmological constant $\Lambda$. We will investigate the contributions of the radial and transverse motions of the observer which are connected with the radial velocity $v^r$ and transverse velocity $bv^{\phi}$ ($b$ is the impact parameter) as well as the spin parameter $a$ of the central object which induces the gravitomagnetic field or frame dragging and the cosmological constant $\Lambda$. The general relativistic aberration equation is employed to take into account the influence of the motion of the observer on the measurable angle $\psi$. The measurable angle $\psi$ derived in this paper can be applied to the observer placed within the curved and finite-distance region in the spacetime. The equation of the light trajectory will be obtained in such a way that the background is de Sitter spacetime instead of Minkowski spacetime. If we assume that the lens object is the typical galaxy, the static terms ${\cal O}(\Lambda bm, \Lambda ba)$ are basically comparable with the second order deflection term ${\cal O}(m^2)$, and they are almost one order smaller that of the Kerr deflection $-4ma/b^2$. The velocity-dependent terms ${\cal O}(\Lambda bm v^r, \Lambda bav^r)$ for radial motion and ${\cal O}(\Lambda b^2mv^{\phi}, \Lambda b^2av^{\phi})$ for transverse motion are at most two orders of magnitude smaller than the second order deflection ${\cal O}(m^2)$. We also find that even when the radial and transverse velocities have the same sign, their asymptotic behavior as $\phi$ approaches $0$ is differs, and each diverges to the opposite infinity.