A note on the Lorentz transformations linking initial and final four-vectors

Abstract

The set of all (homogeneous, proper, orthochronous) Lorentz transformations that link two given forward-timelike four-vectors with equal norms, as well as the condition under which a unique Lorentz transformation is singled out, is completely determined, and presented in a form suitable for immediate reduction to Galilean transformations by letting c→∞,c being the speed of light in empty space. Analogies to the intuitively well-understood Galilean transformation group are obvious. Thus, for instance, the ordinary velocity addition operator ‘‘+’’ involved in the determination of a Galilean transformation link becomes ⊕, the relativistic velocity addition operator involved in the determination of a Lorentz transformation link. The analogies shared by Galilean and Lorentz transformation links were overlooked by explorers since, as opposed to the associative–commutative binary operation + in the Euclidean three-space R3, the binary operation ⊕ in R3c = {v∈R3:‖v‖<c} is neither associative nor commutative.