Relativity, Doppler shifts, and retarded times in deriving the correction for the finite speed of light: a comment on 'Second-order Doppler-shift corrections in free-fall absolute gravimeters'

TL;DR

This paper critiques the relativistic derivation of the finite speed of light correction in absolute gravimeters, highlighting issues with the beat frequency and retarded times approaches and clarifying their equivalence.

Contribution

It provides a critical analysis of the relativistic methods used to derive the correction, identifying problems and clarifying the relationship between different approaches.

Findings

The beat frequency and retarded times methods are equivalent in deriving the correction.

The analysis reveals issues with the treatment of the correction in the original article.

The correction can be understood as a single Doppler shift, not two separate shifts.

Abstract

In the article (Rothleitner and Francis 2011 Metrologia 48 187-195) the correction due to the finite speed of light in absolute gravimeters is analyzed from the viewpoint of special relativity. The relativistic concepts eventually lead to the two classical approaches to the problem: analysis of the beat frequency, and introduction of the retarded times. In the first approach, an additional time delay has to be assumed, because the frequency of the beam bounced from the accelerated reflector differs at the point of reflection from that at the point of interference. The retarded times formalism is equivalent to a single Doppler shift, but results in the same correction as the beat frequency approach, even though the latter is explicitly combines two Doppler shifts. In our comments we discuss these and other problems we found with the suggested treatment of the correction.