Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession

1. Front Matter

   Pages i-xlii

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2. Thomas Precession: The Missing Link

   Pages 1-34

3. Gyrogroups: Modeled on Einstein’S Addition

   Pages 35-71

4. The Einstein Gyrovector Space

   Pages 73-94

5. Hyperbolic Geometry of Gyrovector Spaces

   Pages 95-139

6. The Ungar Gyrovector Space

   Pages 141-160

7. The MÖbius Gyrovector Space

   Pages 161-210

8. Gyrogeometry

   Pages 211-252

9. Gyrooprations — the SL(2, c) Approach

   Pages 253-278

10. The Cocycle Form

    Pages 279-311

11. The Lorentz Group and its Abstraction

    Pages 313-328

12. The Lorentz Transformation Link

    Pages 329-370

13. Other Lorentz Groups

    Pages 371-380

14. Back Matter

    Pages 381-418

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Evidence that Einstein's addition is regulated by the Thomas precession has come to light, turning the notorious Thomas precession, previously considered the ugly duckling of special relativity theory, into the beautiful swan of gyrogroup and gyrovector space theory, where it has been extended by abstraction into an automorphism generator, called the Thomas gyration. The Thomas gyration, in turn, allows the introduction of vectors into hyperbolic geometry, where they are called gyrovectors, in such a way that Einstein's velocity additions turns out to be a gyrovector addition. Einstein's addition thus becomes a gyrocommutative, gyroassociative gyrogroup operation in the same way that ordinary vector addition is a commutative, associative group operation. Some gyrogroups of gyrovectors admit scalar multiplication, giving rise to gyrovector spaces in the same way that some groups of vectors that admit scalar multiplication give rise to vector spaces. Furthermore, gyrovector spaces form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. In particular, the gyrovector space with gyrovector addition given by Einstein's (Möbius') addition forms the setting for the Beltrami (Poincaré) ball model of hyperbolic geometry.
The gyrogroup-theoretic techniques developed in this book for use in relativity physics and in hyperbolic geometry allow one to solve old and new important problems in relativity physics. A case in point is Einstein's 1905 view of the Lorentz length contraction, which was contradicted in 1959 by Penrose, Terrell and others. The application of gyrogroup-theoretic techniques clearly tilt the balance in favor of Einstein.

• Vector space
• automorphism
• geometry
• relativity
• special relativity
• theory of relativity

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

  Abraham A. Ungar

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