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Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession

The Theory of Gyrogroups and Gyrovector Spaces

  • Book
  • © 2001

Overview

Authors:
  1. Abraham A. Ungar

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

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Part of the book series: Fundamental Theories of Physics (FTPH, volume 117)

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Table of contents (13 chapters)

  1. Front Matter

    Pages i-xlii
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  2. Thomas Precession: The Missing Link

    • Abraham A. Ungar
    Pages 1-34

  3. Gyrogroups: Modeled on Einstein’s Addition

    • Abraham A. Ungar
    Pages 35-71

  4. The Einstein Gyrovector Space

    • Abraham A. Ungar
    Pages 73-94

  5. Hyperbolic Geometry of Gyrovector Spaces

    • Abraham A. Ungar
    Pages 95-139

  6. The Ungar Gyrovector Space

    • Abraham A. Ungar
    Pages 141-160

  7. The Möbius Gyrovector Space

    • Abraham A. Ungar
    Pages 161-210

  8. Gyrogeometry

    • Abraham A. Ungar
    Pages 211-252

  9. Gyrooperations — The SL(2, C) Approach

    • Abraham A. Ungar
    Pages 253-278

  10. The Cocycle Form

    • Abraham A. Ungar
    Pages 279-311

  11. The Lorentz Group and Its Abstraction

    • Abraham A. Ungar
    Pages 313-328

  12. The Lorentz Transformation Link

    • Abraham A. Ungar
    Pages 329-370

  13. Other Lorentz Groups

    • Abraham A. Ungar
    Pages 371-380

  14. References

    • Abraham A. Ungar
    Pages 381-401

  15. Back Matter

    Pages 403-419
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About this book

"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.

Authors and Affiliations

  • Department of Mathematics, North Dakota State University, Fargo, USA

    Abraham A. Ungar

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