Note: Axiomatic Derivation of the Doppler Factor and Related Relativistic Laws

TL;DR

This paper derives the relativistic Doppler effect and related laws from invariance axioms using an abstract velocity addition operation, revealing conditions under which these laws are consistent or equivalent.

Contribution

It provides an axiomatic derivation of the Doppler effect and related relativistic laws, clarifying their logical relationships and conditions for consistency.

Findings

Axioms are consistent with Doppler effect and Lorentz contraction without standard operation assumptions.

Under standard operation assumptions, axioms are equivalent to the Doppler effect formula.

Different mathematical representations emerge when assuming Lorentz contraction, especially for perpendicular motions.

Abstract

The formula for the relativistic Doppler effect is investigated in the context of two compelling invariance axioms. The axioms are expressed in terms of an abstract operation generalizing the relativistic addition of velocities. We prove the following results. (1) If the standard representation for the operation is not assumed a priori, then each of the two axioms is consistent with both the relativistic Doppler effect formula and the Lorentz-Fitzgerald Contraction. (2) If the standard representation for the operation is assumed, then the two axioms are equivalent to each other and to the relativistic Doppler effect formula. Thus, the axioms are inconsistent with the Lorentz-FitzGerald Contraction in this case. (3) If the Lorentz-FitzGerald Contraction is assumed, then the two axioms are equivalent to each other and to a different mathematical representation for the operation which applies in the case of perpendicular motions. The relativistic Doppler effect is derived up to one positive exponent parameter (replacing the square root). We prove these facts under regularity and other reasonable background conditions.