Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces [Paperback]

I cannot define coincidence [in mathematics]. But 1 shall argue that coincidence can always be elevated or organized into a superstructure which perfonns a unification along the coincidental elements. The existence of a coincidence is strong evidence for the existence of a covering theory. -Philip 1. Davis [Dav81] Alluding to the Thomas gyration, this book presents the Theory of gy? rogroups and gyrovector spaces, taking the reader to the immensity of hyper? bolic geometry that lies beyond the Einstein special theory of relativity. Soon after its introduction by Einstein in 1905 [Ein05], special relativity theory (as named by Einstein ten years later) became overshadowed by the ap? pearance of general relativity. Subsequently, the exposition of special relativity followed the lines laid down by Minkowski, in which the role of hyperbolic ge? ometry is not emphasized. This can doubtlessly be explained by the strangeness and unfamiliarity of hyperbolic geometry [Bar98]. The aim of this book is to reverse the trend of neglecting the role of hy? perbolic geometry in the special theory of relativity, initiated by Minkowski, by emphasizing the central role that hyperbolic geometry plays in the theory.1. Thomas Precession: The Missing Link.- 2. Gyrogroups: Modeled on Einsteins Addition.- 3. The Einstein Gyrovector Space.- 4. Hyperbolic Geometry of Gyrovector Spaces.- 5. The Ungar Gyrovector Space.- 6. The M?bius Gyrovector Space.- 7. Gyrogeometry.- 8. Gyrooperations  The SL(2, C) Approach.- 9. The Cocycle Form.- 10.The Lorentz Group and Its Abstraction.- 11.The Lorentz Transformation Link.- 12.Other Lorentz Groups.- 13.References.- About the Author.- Topic Index.- Author Index.