Foundations of Physics

, Volume 27, Issue 6, pp 881–951

Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics

  • Abraham A. Ungar

DOI: 10.1007/BF02550347

Cite this article as:
Ungar, A.A. Found Phys (1997) 27: 881. doi:10.1007/BF02550347


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.

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • Abraham A. Ungar
    • 1
  1. 1.Department of MathematicsNorth Dakota State UniversityFargo

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