[...]... special relativity by the Thomas precession The book is divided into twelve chapters (1) Chapter 1 presents the history and the physical background of the introduction of the Thomas precession The remainder of the book is independent of this chapter (2) Chapter 2 presents the gyrogroup notion, modeled on Einstein s addition of relativistically admissible velocities and their Thomas precession Elementary... Algebra: The ultimate unity of mathematics and physics makes a strong case for the inclusion of the generalized groups and vector spaces, that is, gyrogroups and gyrovector spaces, (i) since they provide the mathematical model which underlies Preface xxix Einstein s addition, Möbius addition, and their respective scalar multiplication, and (ii) since, moreover, they provide a theory which unifies these additions... reigns over the section A fi S for E, which represents the experiments that revealed the spectral multiplicity and the anomalous Zeeman effect that led to the idea of the electron spin Thomas discovery of the significance of the relativistic precession of the electron spin on Christmas 1925 led to the understanding of the relativistic effect which became known as the Thomas precession Thomas precession. .. GYROGROUPS AND GYROVECTOR SPACES Exploring the special theory of relativity and its structure beyond the Einstein addition law and its gyroscopic Thomas precession, we take advantage, in this book, of the idea that hyperbolic geometry governs velocities in relativity physics in the same way that Euclidean geometry governs velocities in prerelativity physics, allowing us to partially restore the classical... Earth Program Rather than studying the Thomas precession of electronic gyroscopes in atoms, NASA plans to study the Thomas precession of gyroscopes in space It is the sensitivity of the Thomas precession to the nonEuclidean nature of the geometry of our spacetime that attracts both NASA’s and our interest in measuring it, in understanding it, and in exploiting it Reading This Book The main merit of... evidence for the existence of a covering theory." —Philip J Davis [Dav81] Alluding to the Thomas gyration, this book presents the Theory of gyrogroups and gyrovector spaces, taking the reader to the immensity of hyperbolic 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... [Dra74b], [Dra74a] The majestic scientific achievement of twentieth century in mathematical beauty and experimental verifications has been the theory of relativity [Ein05] [LEMWed] [Ein98] [Ein89] with its Einstein s velocity addition and Thomas precession It has been of the utmost importance in the development of physics, and its concepts have penetrated several mathematical areas Following Einstein s 1905... effect known as the Thomas precession, or gyration, comes to the rescue This book is, accordingly, the fascinating story of the role that the Thomas gyration plays in physics, in nonassociative algebra, and in non-Euclidean geometry, and of the theory of gyrogroups and gyrovector spaces to which it gives rise The story is of fairly recent origin, dating back to the author’s work starting in the late 1980’s...x GYROGROUPS AND GYROVECTOR SPACES 7 8 9 9 THE 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Gyrovector Space Isomorphisms Conclusion Exercises 275 277 277 COCYCLE FORM 279 The Real Einstein Gyrogroup and its Cocycle Form 279 The Complex Einstein Gyrogroup and its Cocycle Form 281 The Möbius Gyrogroup and its Cocycle Form 283 284 The Ungar Gyrogroup and its Cocycle Form Abstract Gyrocommutative... in the Möbius addition we associate it with the gyration (or, rotation) generated by a, b ˛ Preface xxi giving rise to the gyrocommutative law of Möbius addition, Gyrocommutative law Following the gyration definition the gyrocommutative law is not terribly surprising, but we are not finished Coincidentally, the gyration that repairs the breakdown of commutativity in the Möbius addition repairs the . Zentralinstitut für Astrophysik der Akademie der Volume 117 Addition Law Beyond the Einstein and its Gyroscopic Thomas Precession The Theory of Gyrogroups and Gyrovector Spaces by Abraham A. Ungar Department. 0-306-47134-5 To my Mother Chayah Sarah and to the memory of my Father Chayim Yehudah and to my Son Ofer for their love and support, and to Rabbi Yom-Tov Lipman-Heler ben Nathan Halevy, Ba’al Hatosafot Yom-Tov,. Surface, II The Thomas Precession Cosine of The Thomas Rotation Angle Sine of The Thomas Rotation Angle The Minimum Points of the Cosine of the Thomas Angle Thomas Rotation Animation by its Generating