The Thomas rotation formalism underlying a nonassociative group structure for relativistic velocities

Abstract

Physics is a major external source of mathematical theories. It is shown here that the Thomas rotation effect of special theory of relativity (STR) gives rise to a “group” structure for the set of relativistically admissible velocities. This group structure turns out to be noncommutative and nonassociative. The term nonassociative group is justified by two illustrative examples demonstrating that the new concept of the nonassociative group is forced on us by the study of the laws of relativistic velocities. Thomas rotation is studied in STR as an isolated notion; and the bizarre and counterintuitive noncommutativity and nonassociativity of the relativistic composition of nonparallel admissible velocities is sometimes interpreted as a peculiarity of STR. However, it turns out that the Thomas rotation plays a central role in STR, giving rise to an elegant formalism underlying the noncommutative, nonassociative “group” of relativistically admissible velocities. To demonstrate the Thomas rotation formalism and the group structure to which it gives rise let E?R9,