Thomas precession: Its underlying gyrogroup axioms and their use in hyperbolic geometry and relativistic physics
 Abraham A. Ungar
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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.
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 During a seminar on spacetime geometry that the author delivered at The University of Sydney, School of Mathematics and Statistics, April 18, 1996, Dr. Hugh Luckock stated that the splitting of spacetime into time and space, offered in gyrogroup theory by means of Thomas precession, may provide an answer to the desire to split the general relativistic notion of spacetime into space and time, expressed in: Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler,Gravitation (W. H. Freeman, San Francisco, 1973), Section 24.1, p. 505.
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 An observation made by Michael K. Kinyon.
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 “The possible existence of extra dimensions to spacetime can be tested astrophysically” perhaps by Gravity Probe B^{(68)}; see D. Kalligas, P. S. Wesson, and C. F. W. Everitt, “The Classical tests in KaluzaKlein gravity,” preprint. A few references from the literature on sixdimensional relativity are: H. C. Chandola and B. S. Rajput, “Maxwell's equations in sixdimensional spacetime,”Indian J. Pure Appl. Phys. 24, 58–64 (1986); E. A. B. Cole, “Centreofmass frame in sixdimensional special relativity,”Lett. Nuovo Cimento,28, 171–174 (1980); E. A. B. Cole, “New electromagnetic fields in sixdimensional special relativity,”Nuovo Cimento,60A, 1–11 (1980); E. A. B. Cole and S. A. Buchanan, “Spacetime transformations in sixdimensional special relativity,”J. Phys. A. Math. Gen. 15, L255–L257 (1982); I. Merches, and C. Dariescu, “The use of sixvectors in the theory of special relativity,”Acta Phys. Hung. 70, 63–70 (1991); P. T. Papas, “The threedimensional time equation,”Lett. Nuovo Cimento 25, 429–434 J. Strnad, “Sixdimensional spacetime and the Thomas precession,”Lett Nuovo Cimento 26, 535–536 (1970); and M. T. Teli, “General Lorentz transformations in sixdimensional spacetime,”Phys. Lett. A 128, 447–450 (1987).
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 “Thumbs partly up for Gravity Probe B.”Science News 147, No. 23, p. 367 (1995). Gravity Probe B is a dragfree satellite carrying gyroscopes around Earth. For details see C. W. Francis Everitt, William M. Fairbank, and L. I. Schiff, “Theoretical background and present status of the Stanford relativitygyroscope experiment,” inThe Significance of Space Research for Fundamental Physics, Proc. Colloq. of the European Space Research Org. at Interlaken, Swizerland, 4 Sept. 1969; R. Vassar, J. V. Breakwell, C. W. F. Everitt, and R. A. VanPatten, “Orbit selection for the Stanford relativity gyroscope experiment,”J. Spacecraft Rockets 19, 66–71 (1986). The generalrelativistic Thomas precession involves several terms one of which is the specialrelativistic Thomas precession studied in this article. The NASA program to perform a Thomas precession test of Einstein's theory of general relativity by measuring the precession of gyroscopes in Earth orbit was initiated by William M. Fairbank; see C. W. F. Everitt “Gravity Probe B: I. The scientific implications,” The Sixth Marcel Grossmann Meeting on Relativity, Kyoto, Japan, June 23–29, 1991 (World Scientific Publ.); J. D. Fairbank, B. S. Deaver, Jr., C. W. F. Everitt, and P. F. Michelson,Near Zero: New Frontiers of Physics (Freeman, New York, 1988); “William Martin Fairbank (1917–1989)”,Nature 342, 125 (1989).
 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
 19970601
 DOI
 10.1007/BF02550347
 Print ISSN
 00159018
 Online ISSN
 15729516
 Publisher
 Kluwer Academic PublishersPlenum Publishers
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 Abraham A. Ungar ^{(1)}
 Author Affiliations

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