On the Doppler effect for light from orbiting sources in Kerr-type metrics

A formula is derived for the combined motional and gravitational Doppler effect in general stationary axisymmetric metrics for a photon emitted parallel or antiparallel to the assumed circular orbital motion of its source. The same formula is derived by both the eikonal approximation and Killing vector approaches to elucidate connections between observational astronomy and modern relativity. The formula yields expected results in the limits of a moving or stationary source in the exterior Kerr and Schwarzschild metrics and is useful for broad range astrophysical analyses.