UC Irvine

In the context of the relativistic Doppler effect [DE] and the Lorentz–Fitzgerald contraction [LF], we investigate the consequences of two abstract axioms [R] and [M] expressed in terms of an operation $${\oplus}$$ generalizing the addition of velocities and a function $${L:(\lambda, v)\mapsto L(\lambda, v)}$$ . The latter can represent either the Doppler effect or the Lorentz– Fitzgerald Contraction. In words, these axioms state the following: [R] iterating the function L has the same effect as adding velocities; [M] adding a velocity via the operation $${\oplus}$$ preserves the order of the function L. We prove that these axioms are equivalent to each other, and also to generalized forms of the Doppler effect and the standard expression [AV] for the relativistic addition of velocities, taken jointly. We also show that if [AV] holds, then the axioms [R] and [M] are inconsistent with [LF].