Frequency shift and viewing direction variations in gravitational lensing

In a gravitational lensing system, the relative transverse velocities of the lens, source, and observer induce a frequency shift in the observed radiation. While this shift is typically negligible in most astrophysical contexts, strategies for its detection have been proposed for both electromagnetic and gravitational waves. This paper provides a rigorous theoretical treatment of the effect, deriving general expressions for the frequency shift within a lensing system embedded in a cosmological spacetime. Our formulation remains valid for arbitrary distances and velocities, including highly relativistic regimes, under any Friedmann-Lemaître-Robertson-Walker metric. Expanding upon previous papers on moving lenses, we provide a detailed analysis of frequency effects induced by lenses moving at relativistic speeds. Furthermore, we extend standard lensing theory by deriving an exact formula for the variation in the source's viewing direction. This result is of interest for strongly anisotropic emitters, such as compact binary systems emitting gravitational waves. Finally, we quantify the apparent misalignment between the lens and the source's two images produced by time-delay effects in lens systems moving with high velocity.