Weakly Associative Groups
- 12 Downloads
The space ℝ c 3 of 3-dimensional relativistically admissible velocities possesses (i) a binary operation which represents the relativistic velocity composition law; and (ii) a mapping from the cartesian product ℝ c 3 ×ℝ c 3 into a subgroup of its automorphism group, Aut(ℝ c 3 ), representing the Thomas precession of special relativity. These binary operation and mapping are studied in special relativity as two isolated phenomena. It was recently discovered, however, that they are linked by an algebraic structure which gives rise to a theory of weakly associative and weakly associative-commutative groups. The axioms of these groups are presented in this paper and employed to obtain various interesting results. The algebraic structure underlying these nonstandard groups has been discovered and studied in a totally different context by Karzel (1965), Kerby and Wefelscheid.
KeywordsSpecial Relativity Binary Operation Semidirect Product Identity Automorphism Thomas Precession
Unable to display preview. Download preview PDF.
- R.H. Brück, A Survey of Binary Systems, 2nd ed., Springer-Verlag, New York 1966.Google Scholar
- H. Karzel, Inzidenzgruppen I, lecture notes by I. Pieper and K. Sörensen, Univ. Hamburg (1965), 123–135.Google Scholar
- W. Kerby, Infinite Sharply Multiply Transitive Groups, Hamburger Mathematische Einelschriften, Neue Folge, Heft 6. Vandenhoek und Ruprecht, Göttingen 1974.Google Scholar
- I.N. Sneddon, ed., Encyclopedic Dictionary of Mathematics for Engineers and Applied Scientists, p. 320, Pergamon, New York, 1976.Google Scholar
- L.H. Thomas, Recollections of the discovery of the Thomas precessional frequency, AIP Conf Proc. No. 95, High Energy Spin Physics Brookhaven National Lab, ed. G.M. Bunce, (1982), 4–12.Google Scholar
- G.E. Uhlenbeck, Fifty years of spin: personal reminiscences, Phys. Today, June (1976), 43–48.Google Scholar
- A.A. Ungar, Quasidirect product groups and the Lorentz transformation group, in T.M. Rassias (ed.), Constantin Caratheodory: An International Tribute, World Scientific Pub., NJ, 1991.Google Scholar
- H. Wefelscheid, personal communication.Google Scholar
- H.E. Wolfe, Introduction to Non-Euclidean Geometry, p. vi, Dryden Press, New York, 1945.Google Scholar