Doppler effect in Schwarzschild geometry

TL;DR

This paper analyzes the Doppler effect in Schwarzschild spacetime, revealing simplified expressions that separate classical, special relativistic, and general relativistic contributions, with implications for observers in radial fall and certain Kerr metric scenarios.

Contribution

It derives a simplified, factorized form of the Doppler shift in Schwarzschild spacetime, applicable to a broad class of observers and specific in-fall conditions, enhancing understanding of relativistic Doppler effects.

Findings

Doppler shift factorizes for radial fall in Schwarzschild spacetime.

The expression separates into classical, special relativistic, and general relativistic parts.

The result extends to observers moving parallel to null geodesics and certain Kerr in-fall cases.

Abstract

The Doppler shift considered in general relativity involves mixed contributions of distinct, gravitational and kinematical origins and for most metrics or trajectories it takes a complex form. The expression for the Doppler shift may simplify due to particular symmetries. In Schwarzschild spacetime it factorizes in the case of radial fall for an observer and radial null geodesic. The resulting expression is composed of factors that can be identified with contributions arising from classical, special relativistic and general relativistic origins. This result turns out to be more general: it holds for the whole class of observers travelling parallel to the spatial path of null geodesics when receiving the signal. It also holds for a particular type of in-fall in the case of a Kerr metric.