, Volume 27, Issue 6, pp 881951
Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics
 Abraham A. UngarAffiliated withDepartment of Mathematics, North Dakota State University
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Gyrogroup theory and its applications is introduced and explored, exposing the fascinating interplay between Thomas precession of special relativity theory and hyperbolic geometry. The abstract Thomas precession, called Thomas gyration, gives rise to grouplike objects called gyrogroups [A, A. Ungar, Am. J. Phys.59, 824 (1991)] the underlying axions of which are presented. The prefix gyro extensively used in terms like gyrogroups, gyroassociative and gyrocommutative laws, gyroautomorphisms, and gyrosemidirect products, stems from their underlying abstract Thomas gyration. Thomas gyration is tailor made for hyperbolic geometry. In a similar way that commutative groups underlie vector spaces, gyrocommutative gyrogroups underlie gyrovector spaces. Gyrovector spaces, in turn, provide a most natural setting for hyperbolic geometry in full analogy with vector spaces that provide the setting for Euclidean geometry. As such, their applicability to relativistic physics and its spacetime geometry is obvious.
 Title
 Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics
 Journal

Foundations of Physics
Volume 27, Issue 6 , pp 881951
 Cover Date
 199706
 DOI
 10.1007/BF02550347
 Print ISSN
 00159018
 Online ISSN
 15729516
 Publisher
 Kluwer Academic PublishersPlenum Publishers
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 Authors

 Abraham A. Ungar ^{(1)}
 Author Affiliations

 1. Department of Mathematics, North Dakota State University, 58105, Fargo, North Dakota