Reconsidering Velocity Addition/Subtraction in Special Relativity

TL;DR

This paper revisits velocity addition and subtraction in Special Relativity, providing new derivations, explicit formulas, and a geometric perspective that clarifies the invariant nature of relative velocity and compares it to classical mechanics.

Contribution

It offers a novel geometric approach to defining relative velocity in Special Relativity, along with explicit derivations of velocity algebra properties using the polar-decomposition theorem.

Findings

Explicit expressions for the Thomas angle derived

A new invariant definition of relative velocity proposed

Comparison with Galilei-Newton spacetime highlights key differences

Abstract

We reconsider velocity addition/subtraction in Special Relativity and re-derive its well-known non-commutative and non-associative algebraic properties in a self contained way, including various explicit expressions for the Thomas angle, the derivation of which will be seen to be not as challenging as often suggested. All this is based on the polar-decomposition theorem in the traditional component language, in which Lorentz transformations are ordinary matrices. In the second part of this paper we offer a less familiar alternative geometric view, that leads to an invariant definition of the concept of relative velocity between two states of motion, which is based on the boost-link-theorem, of which we also offer an elementary proof that does not seem to be widely known in the relativity literature. Finally we compare this to the corresponding geometric definitions in Galilei-Newton spacetime, emphasising similarities and differences.