Frequency shift in gravitational lensing by Kerr metric

Abstract

In this study, we calculate the relative frequency shift ($Δν/ ν$) caused by gravitational lensing resulting from the transverse motion of a spinning lens described by the Kerr metric \cite{Kerr:1963ud}. While the effect of transverse velocity on the frequency shift has previously been investigated using the Schwarzschild metric \cite{Rahvar_2020,10.1093/mnras/stae1938}, extending this to the Kerr metric provides a new window for measuring the transverse velocity of a lens or its spin parameter, a parameter often subject to degeneracy in gravitational microlensing observations. Recent microlensing observations have revealed black holes as viable lensing candidates \cite{2022ApJ...933...83S}. Since spinning black holes form naturally from stellar evolution, astrophysical black hole lenses are best described by the Kerr metric. Furthermore, recent observations of supermassive black holes at the centers of galaxies provide an ideal laboratory for testing the observational effects of the Kerr metric by measuring the frequency shifts of lensed sources.