While teaching a course in modern physics, several years ago, I noticed that the special relativity velocity addition formula could be obtained from the special relativistic radial Doppler shift equation after only a few easy steps of algebra. This calculation is simpler than explicitly using the group property of Lorentz transformations, although a derivation of the Doppler shift may require it. Also, the algebra described here is a suitable alternative to straightforward differentiation, in non-calculus presentations. So far as I know, this derivation has not been mentioned elsewhere.

Consider the composition of two radial Doppler shifts along the same line of sight. Let frames of reference S, S′, and S″ all be in standard configuration,1 where S and S′ have relative velocity β′, S′ and S″ β″, and S and S″ have relative velocity β. (As usual, β = v/c etc.) Then the Doppler...
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