Velocity Addition/Subtraction in Special Relativity

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.