Dr. Abraham A. Ungar (auth.)0792369092, 9780792369097, 9780306471346, 1402003536, 9781402003530
“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. |
Table of contents :
Front Matter….Pages i-xlii
Thomas Precession: The Missing Link….Pages 1-34
Gyrogroups: Modeled on Einstein’s Addition….Pages 35-71
The Einstein Gyrovector Space….Pages 73-94
Hyperbolic Geometry of Gyrovector Spaces….Pages 95-139
The Ungar Gyrovector Space….Pages 141-160
The Möbius Gyrovector Space….Pages 161-210
Gyrogeometry….Pages 211-252
Gyrooperations — The SL (2, C ) Approach….Pages 253-278
The Cocycle Form….Pages 279-311
The Lorentz Group and Its Abstraction….Pages 313-328
The Lorentz Transformation Link….Pages 329-370
Other Lorentz Groups….Pages 371-380
References….Pages 381-401
Back Matter….Pages 403-419 |
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