Abstract
The bizarre and counterintuitive noncommutativity and nonassociativity of the relativistic composition of nonparallel admissible velocities is sometimes interpreted as a peculiarity of special theory of relativity. It is related to the fact that Lorentz acceleration transformations in three space dimensions do not form a group due to the presence of the so called Thomas rotation. The Thomas rotation turns out to be the effect that provides a means to the presentation of the set of relativistically admissible velocities as an interesting noncommutative, nonassociative group with a group operation given by the relativistic velocity composition. Interestingly, the algebraic structure induced by the Thomas rotation is not an isolated result in special relativity. It was earlier discovered by Karzel, and studied by Kerby and by Wefelscheid in a totally different context.
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Ungar, A.A. The Relativistic Noncommutative Nonassociative Group of Velocities and the Thomas Rotation. Results. Math. 16, 168–179 (1989). https://doi.org/10.1007/BF03322653
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DOI: https://doi.org/10.1007/BF03322653