Doppler effect in Schwarzschild and Kerr geometries

Calculation of the Doppler shift in general relativity involves contributions of gravitational and kinematical origins and for most metrics or trajectories these contributions are coupled. The exact expression for this Doppler shift may simplify for particular symmetries. Here the specific case for a light signal emitted by a distant inertial observer and received by an in-falling observer in a Schwarzschild geometry is discussed. The resulting expression the Doppler shift is composed of simple factors that can be clearly identified with contributions arising from classical kinematical, special relativistic and general relativistic origins. This result turns out to be more general and it holds for a case of an arbitrary radial in-fall in Schwarzschild geometry and for a particular type of in-fall in the case of a Kerr metric