Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession Fundamental Theories of Physics

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Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession Fundamental Theories of Physics

An International Book Series on The Fundamental Theories of Physics: Their Clarification, Development and Application

Editor: ALWYN VAN DER MERWE, University of Denver, U.S.A.

Editorial Advisory Board: JAMES T. CUSHING, University of Notre Dame, U.S.A. GIANCARLO GHIRARDI, University of Trieste, Italy LAWRENCE P. HORWITZ, Tel-Aviv University, Israel BRIAN D. JOSEPHSON, University of Cambridge, U.K. CLIVE KILMISTER, University of London, U.K. PEKKA J. LAHTI, University of Turku, Finland ASHER PERES, Israel Institute of Technology, Israel EDUARD PRUGOVECKI, University of Toronto, Canada TONY SUDBURY, University of York, U.K. HANS-JÜRGEN TREDER, Zentralinstitut für Astrophysik der Akademie der Wissenschaften, Germany

Volume 117 Beyond the Einstein Addition Law and its Gyroscopic Thomas Precession The Theory of Gyrogroups and Gyrovector Spaces

Abraham A. Ungar Department of Mathematics, North Dakota State University, Fargo, North Dakota, U.S.A.

KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON , DORDRECHT, LONDON , MOSCOW eBook ISBN 0-306-47134-5 Print ISBN 0-792-36909-2

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Created in the United States of America

Visit Kluwer Online at: http://www.kluweronline.com and Kluwer's eBookstore at: http://www.ebooks.kluweronline.com 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, born in 1579 in the city of Valershtein, Bavaria, and died in 1654 in the city of krakow, Galitzia, who was the first known mathematician in the author's family tree. This book is dedicated to: (i) Llewellyn H. Thomas (1902-1992); and (ii) the development of greater understanding of the central role that the Thomas gyration plays in relativity physics, in nonassociative algebra, in non-Euclidean geometry and, particularly, in the theory of gyrogroups and gyrovector spaces. Contents

List of Figures xiii List of Tables xvii Preface xix Acknowledgments xxxi Introduction xxxv Abraham A. Ungar 1. THOMAS PRECESSION: THE MISSING LINK 1 1 A Brief History of the Thomas Precession 1 2 The Einstein Velocity Addition 3 3 Thomas Precession and Gyrogroups 6 4 The Relativistic Composite Velocity Reciprocity Principle 8 5 From Thomas Precession to Thomas Gyration 11 6 Solving Equations in Einstein’s Addition, and the Einstein Coaddition 13 7 The Abstract Einstein Addition 16 8 Verifying Algebraic Identities of Einstein’s Addition 18 9 Matrix Representation of the Thomas Precession 24 10 Graphical Presentation of the Thomas Precession 27 1 1 The Thomas Rotation Angle 29 1 2 The Circular Functions of the Thomas Rotation Angle 31 1 3 Exercises 34 2. GYROGROUPS: MODELED ON EINSTEIN’S ADDITION 35 1 Definition of a Gyrogroup 36 2 Examples of Gyrogroups 39 3 First Theorems of Gyrogroup Theory 43 4 Solving Gyrogroup Equations 47 5 The Gyrosemidirect Product Group 49 6 Understanding Gyrogroups by Gyrosemidirect Product Groups 52 7 Some Basic Gyrogroup Identities 57 vii viii GYROGROUPS AND GYROVECTOR SPACES

8 Exercises 71 3. THE EINSTEIN GYROVECTOR SPACE 73 1 Einstein Scalar Multiplication 73 2 Einstein’s Half 76 3 Einstein’s Metric 77 4 Metric Geometry of Einstein Gyrovector Spaces 80 5 The Einstein Geodesics 84 6 Gyrovector Spaces 86 7 Solving a Simple System of Two Equations in a Gyrovector Space 89 8 Einstein’s Addition and The Beltrami Model of Hyperbolic Geometry 90 9 The Riemannian Line Element of Einstein’s Metric 93 1 0 Exercises 94 4. HYPERBOLIC GEOMETRY OF GYROVECTOR SPACES 95 1 Rooted Gyrovectors 95 2 Equivalence Classes of Gyrovectors 98 3 The Hyperbolic Angle 104 4 Hyperbolic Trigonometry in Einstein’s Gyrovector Spaces 107 5 From Pythagoras to Einstein: The Hyperbolic Pythagorean Theorem 110 6 The Relativistic Dual Uniform Accelerations 112 7 Einstein’s Dual Geodesics 114 8 The Riemannian Line Element of Einstein’s Cometric 119 9 Moving Cogyrovectors in Einstein Gyrovector Spaces 122 10 Einstein’s Hyperbolic Coangles 123 11 The Gyrogroup Duality Symmetry 126 12 Parallelism in Cohyperbolic Geometry 127 13 Duality, And The Dual Gyrotransitive Laws of Mutually Dual Geodesics 128 14 The Bifurcation Approach to Hyperbolic Geometry 130 15 The Gyroparallelogram Addition Rule 132 16 Gyroterminology 137 17 Exercises 139 5 . THE UNGAR GYROVECTOR SPACE 141 1 The Ungar Gyrovector Space of Relativistic Proper Velocities 141 2 Some Identities for Ungar’s Addition 145 3 The Gyrovector Space Isomorphism Between Einstein’s and Ungar’s Gyrovector Spaces 146 4 The Riemannian Line Elements of The Ungar Dual Metrics 148 5 The Ungar Model of Hyperbolic Geometry 153 Contents ix

6 Angles in The Ungar Model of Hyperbolic Geometry 154 7 The Angle Measure in Einstein’s and in Ungar’s Gyrovector Spaces 156 8 The Hyperbolic Law of Cosines and Sines in the Ungar Model of Hyperbolic Geometry 158 9 Exercises 160 6. THE MöBIUS GYROVECTOR SPACE 161 1 The Gyrovector Space Isomorphism 161 2 Möbius Gyrovector Spaces 163 3 Gyrotranslations – Left and Right 168 4 The Hyperbolic Pythagorean Theorem in the Poincaré Disc Model of Hyperbolic Geometry 170 5 Gyrolines and the Cancellation Laws 174 6 The Riemannian Line Elements of the Möbius Dual Metrics 176 7 Rudiments of Riemannian Geometry 183 8 The Möbius Geodesics and Angles 184 9 Hyperbolic Trigonometry in Möbius Gyrovector Spaces 186 1 0 Numerical Demonstration 193 1 1 The Equilateral Gyrotriangle 201 1 2 Exercises 210 7. GYROGEOMETRY 211 1 The Möbius Gyroparallelogram 211 2 The Triangle Angular Defect in Gyrovector Spaces 213 3 Parallel Transport Along Geodesics in Gyrovector Spaces 216 4 The Triangular Angular Defect And Gyrophase Shift 222 5 Polygonal And Circular Gyrophase Shift 224 6 Gyrovector Translation in Möbius Gyrovector Spaces 226 7 Triangular Gyrovector Translation of Rooted Gyrovectors 232 8 The Hyperbolic Angle and Gyrovector Translation 234 9 Triangular Parallel Translation of Rooted Gyrovectors 236 10 The Nonclosed Circular Path Angular Defect 240 11 Gyroderivative: The Hyperbolic Derivative 245 1 2 Parallelism in Cohyperbolic Geometry 249 13 Exercises 252 8. GYROOPRATIONS – THE SL(2, C ) APPROACH 253 1 The Algebra Of The SL(2, C ) Group 253 2 The SL(2, C ) General Vector Addition 259 3 Case I –The Einstein Gyrovector Spaces 264 4 Case II – The Möbius Gyrovector Spaces 266 5 Case III – The Ungar Gyrovector Spaces 269 6 Case IV – The Chen Gyrovector Spaces 272 x GYROGROUPS AND GYROVECTOR SPACES

7 Gyrovector Space Isomorphisms 275 8 Conclusion 277 9 Exercises 277 9. THE COCYCLE FORM 279 1 The Real Einstein Gyrogroup and its Cocycle Form 279 2 The Complex Einstein Gyrogroup and its Cocycle Form 281 3 The Möbius Gyrogroup and its Cocycle Form 283 4 The Ungar Gyrogroup and its Cocycle Form 284 5 Abstract Gyrocommutative Gyrogroups with Cocycle Forms 285 6 Cocycle Forms, By Examples 287 7 Basic Properties of Cocycle Forms 290 8 Applications of the Real Even Cocycle Form Representation 293 9 The Secondary Gyration of a Gyrocommutative Gyrogroup with a Complex Cocycle Form 294 10 The Gyrogroup Extension of a Gyrogroup with a Cocycle Form295 11 Cocyclic Gyrocommutative Gyrogroups 304 1 2 Applications of Gyrogroups to Cocycle Forms 309 13 Gyrocommutative Gyrogroup Extension by Cocyclic Maps 310 1 4 Exercises 311 10.THE LORENTZ GROUP AND ITS ABSTRACTION 313 1 Inner Product and the Abstract Lorentz Boost 314 2 Extended Automorphisms of Extended Gyrogroups 316 3 The Lorentz Boost of Relativity Theory 321 4 The Parametrized Lorentz Group and its Composition Law 323 5 The Parametrized Lorentz Group of Special Relativity 325 11.THE LORENTZ TRANSFORMATION LINK 329 1 Group Action on Sets 330 2 The Galilei Transformation of Structured Spacetime Points 332 3 The Galilean Link 335 4 The Galilean Link By a Rotation 335 5 The Lorentz Transformation of Structured Spacetime Points 338 6 The Lorentz Link By a Rotation 343 7 The Lorentz Boost Link 347 8 The Little Lorentz Groups 348 9 The Relativistic Shape of Moving Objects 349 10 The Shape of Moving Circles 352 11 The Shape of Moving Spheres 354 12 The Shape of Moving Straight Lines 358 1 3 The Shape of Moving Curves 359 14 The Shape of Moving Harmonic Waves 360 1 5 The Relativistic Doppler Shift 362 Contents xi

16 Simultaneity: Is Length Contraction Real? 367 17 Einstein’s Length Contraction: An Idea Whose Time Has Come Back 369 18 Exercises 370 12.OTHER LORENTZ GROUPS 371 1 The Proper Velocity Ungar–Lorentz Boost 371 2 The Proper Velocity Ungar–Lorentz Transformation Group 373 3 The Unique Ungar–Lorentz Boost that Links Two Points 374 4 The Möbius-Lorentz Boost 375 5 The Unique Möbius–Lorentz Boost that Links Two Points 376 6 The Möbius–Lorentz Transformation Group 377 13.REFERENCES 381

About the Author 403

Topic Index 405

Author Index 411 List of Figures

0.1. Artful Application of the Möbius Transformation, I xxii 0.2 Artful Application of the Möbius Transformation, II xxiii 0.3 The Shape of a Relativistically Moving Surface, I xxix 0.4 The Shape of a Relativistically Moving Surface, II xxix 1.1 The Thomas Precession 6 1.2 Cosine of The Thomas Rotation Angle 28 1.3 Sine of The Thomas Rotation Angle 28 1.4 The Minimum Points of the Cosine of the Thomas Angle 33 1.5 Thomas Rotation Animation by its Generating Angle 33 2.1 Multiplication Table of a Finite Gyrogroup 41 2.2 Gyration Table of the Finite Gyrogroup 41 3.1 A Gyroline Segment 84 3.2 The Effect of Gyrotranslation 84 3.3 Hyperbolic Triangle Medians are Concurrent 85 4.1 Successive gyrovector Translations 103 4.2 The Hyperbolic Angle 104 4.3 The Hyperbolic Triangle and its Angles 109 4.4 The Effects of Left Gyrotranslating Right Triangles 111 4.5 The Hyperbolic Pythagorean Theorem in the Beltrami Model 111 4.6 The gyroline 113 4.7 The cogyroline 113 4.8 A 3-dimensional Einstein gyroline 114 4.9 A 3-dimensional Einstein cogyroline 114 4.10 The Cogyroline and its Supporting Diameter 116 4.11 Hyperbolic Dual Triangle Medians are Not Concurrent 118 4.12 Coangle – The Hyperbolic Dual Angle 124 4.13 Cotriangle – The Hyperbolic Dual Triangle 124 4.14 Hyperbolic Alternate Interior Coangles—Einstein 127 4.15 The Hyperbolic p Theorem in the Beltrami Disc Model 127 xiii xiv GYROGROUPS AND GYROVECTOR SPACES

4.16 The Hyperbolic Bifurcation Diagram 130 4.17 Gyroparallelogram—the Hyperbolic Parallelogram 132 4.18 Gyrosquare—the Hyperbolic square 132 4.19 A relocated gyroparallelogram 134 4.20 A relocated gyrosquare 134 51. A Gyroline in the Ungar Gyrovector Plane 153 5.2 A Cogyroline in the Ungar Gyrovector Plane 153 5.3 Hyperbolic Triangle Medians are Concurrent 155 5.4 Hyperbolic Dual Triangle Medians are Not Concurrent 155 5.5 The Hyperbolic Pythagorean Theorem in Ungar’s Model 160 6.1 The Möbius gyroline 165 6.2 The Möbius dual gyroline (cogyroline) 165 6.3 Successive Gyrotranslations – Left and Right, I 168 6.4 Successive Gyrotranslations – Left and Right, II 168 6.5 Successive Right Gyrotranslations In 3–D. 169 6.6 The Möbius Hyperbolic Pythagorean Theorem, I 172 6.7 The third kind gyroline 175 6.8 The Möbius gyroline 176 6.9 The Möbius cogyroline 176 6.10 Tangential Transport in the Poincaré Disc, I 177 6.11 The Möbius Gyrocircle 178 6.12 The 2-dimensional Möbius Geodesic 184 6.13 The 3-dimensional Möbius Geodesic 184 6.14 The Möbius angle 185 6.15 A Möbius Triangle 187 6.16 The Möbius Hyperbolic Pythagorean Theorem, II 191 6.17 A Möbius Triangle and its Height 193 6.18 A Möbius Triangle and its Three Heights 197 6.19 Equilateral Gyrotriangles 201 6.20 Equilateral Gyrotriangles, I 205 6.21 Equilateral Gyrotriangles, II 205 6.22 Isosceles Gyrotriangles, I 207 6.23 Isosceles Gyrotriangles, II 207 7.1 A Möbius Gyroparallelogram 212 7.2 A Möbius Gyrosquare 212 7.3 Equidefect Hyperbolic Triangles 214 7.4 Parallel Transport Along Geodesics 216 7.5 Parallel Gyrovector Fields Along Geodesics 217 7.6 Parallel Transport Along Closed Gyropolygonal Contour 220 7.7 Triangular Parallel Transport 222 7.8 Triangular Parallel Transport from the Origin 223 7.9 The accrued polygonal gyrophase shift 224 List of Figures xv

7.10 The accrued circular gyrophase shift 225 7.11 Parallel Transport and Gyrovector Translation 226 7.12 The Gyrovector Angular Defect in Plane Hyperbolic Geometry I 227 7.13 The Gyrovector Angular Defect in Plane Hyperbolic Geometry II 227 7.14 Parallel and Gyrovector Translations, I 229 7.15 Parallel and Gyrovector Translations, II 231 7.16 Hyperbolic Angles by Gyrovector Translation 235 7.17 The Triangle Defect, I 239 7.18 The Triangle Defect, II 239 7.19 Evolution of the Nonclosed Circular Path Defect, I 240 7.20 Evolution of the Nonclosed Circular Path Defect, II 242 7.21 Evolution of the Nonclosed Circular Path Defect, III 242 7.22 Tangential Transport in the Poincaré Disc, II 247 7.23 Hyperbolic Alternate Interior Coangles—Möbius 249 7.24 The Hyperbolic p Theorem in the Poincaré Disc Model 249 8.1 A Gyroline in a Chen’s Gyrovector Plane 274 8.2 A Cogyroline in a Chen’s Gyrovector Plane 274 List of Tables

0.1 Analogies for Möbius Addition xxiv 4.1 Duality symmetries 126 4.2 Gyroterminology 138 6.1 Riemannian line elements of gyrovector spaces 183 6.2 Euclidean and Hyperbolic Geometry 192 7.1 Parallel and Gyrovector Translation 230 11.1 Galilei and Lorentz Transformation Analogies 344

xvii Preface

"I cannot define coincidence [in mathematics]. But I shall argue that coincidence can always be elevated or organized into a superstructure which performs a unification along the coincidental elements. The existence of a coincidence is strong 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 later) became overshadowed by the appearance of general relativity. Subsequently, the exposition of special relativity followed the lines laid down by Minkowski, in which the role of hyperbolic geometry 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 hyperbolic geometry in the special theory of relativity, initiated by Minkowski, by emphasizing the central role that hyperbolic geometry plays in the theory. We will find in this book that the special theory of relativity and hyperbolic geometry cross-pollinate to produce important new results: (1) The relativistic gyroscopic precession, known as the Thomas precession, turns out to play an important role in the foundations of hyperbo geometry, uncovering hitherto unnoticed analogies with Euclidean geometry; and (2) guided by the analogies that hyperbolic geometry shares with Euclidean geometry, uncovered by means of the Thomas precession, we discover related analogies that the Lorentz transformation shares with the Galilei transformation. The emerging analogies reverse a trend initiated by Minkowski: The relativistic spacetime emerges as a derived concept rather than a primitive (that is, unexplained) notion. xix XX GYROGROUPS AND GYROVECTOR SPACES

Exploring the special theory of relativity and its structure beyond the Ein- stein 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 picture. In particular, we take advantage of the result that the Einstein velocity addition of relativistically admissible velocities is a gyrocommutative gyrogroup operation in the same way that the Galilei velocity addition (that is, ordinary vector addition) is a commutative group operation. Furthermore, Einstein’s addition is a gyrocommutative gyrogroup operation that admits scalar multiplication, thus giving rise to a gyrovector space. Gyrovector spaces, in turn, form the setting for hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry. The primary purpose of this book is, accordingly, to provide readers with a self-contained account of the two topics in the subtitle. A gyrogroup is a grouplike algebraic structure that provides a most natural generalization of the group notion. Historically, the first gyrogroup was discovered by the author in the algebraic structure that underlies Einstein’s velocity addition [Ung88a]. However, the best way to introduce the gyrogroup notion by example is offered by the Möbius transformation of the disc. Ahlfors’ book [Ahl73], Conformal Invariants: Topics in Geometric Func- tion Theory, begins with the presentation of the most general Möbius transformation of the complex open unit disc

which we write as

z0 ˛q ˛Suggestively, we define the Möbius addition ¯ in the disc, allowing the generic Möbius transformation of the disc to be viewed as a Möbius left gyrotranslation

followed by a rotation. The prefix “gyro” which we use to emphasize analogies with classical notions, stems from the Thomas gyration, which will soon become clear. The resulting Möbius addition in the disc is neither commutative nor associative. To ‘repair’ the breakdown of commutativity 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 breakdown of associativity as well, giving rise to identities that capture analogies, Left Gyroassociative Law Right Gyroassociative Law Left Loop Property Right Loop Property

The Möbius addition is thus regulated by its associated gyration so that, in fact, the Möbius addition and its associated gyration are inextricably linked. Where there are coincidences there is significance. The emerging coincidences to which the gyration gives rise uncover an interesting algebraic structure that merits extension by abstraction, leading to the grouplike structure called a gyrogroup. Gyrogroups are generalized groups that share remarkable analogies with groups. In full analogy with groups:

(1) Gyrogroups are classified into gyrocommutative gyrogroups and non- gyrocommuntative gyrogroups. (2) Some gyrocommutative gyrogroups admit scalar multiplication, turning them into gyrovector spaces. (3) Gyrovector spaces, in turn, provide the setting for hyperbolic geometry in the same way that vector spaces provide the setting for Euclidean geometry, thus enabling the two geometries to be unified. (4) Moreover, the resulting analogies shared by the motions of Euclidean geometry (that is, a commutative group of translations and a group of rotations) and the motions of hyperbolic geometry (that is, a gyrocommutative gyrogroup of left gyrotranslations and a group of rotations) induce analogies shared by the Galilei transformation and the Lorentz transformation. These analogies, in turn, enable Lorentz transformation problems, hitherto poorly understood, to be straightforwardly solved in full analogy with the respective solutions of their Galilean counterparts. A point in case is, for instance, the determination of the visible shape of relativistically moving objects in Chapter 11. xxii GYROGROUPS AND GYROVECTOR SPACES

Figure 0.1. Artful Application of the Möbius Transformation in Hyperbolic Geometry, I. The hyperbolic Pythagorean theorem for Möbius right angled hyperbolic triangles in the complex unit disc in a form fully analogous to its Euclidean counterpart [Ung99].

The flavor of this book is illustrated by Figs. 0.1 and 0.2. To demonstrate the ability of gyro-formalism to capture analogies:

(1) We present graphically in Fig. 0.1 the hyperbolic Pythagorean Theorem in the Möbius gyrovector plane ¯,˜), which we will study in Chapters 4 and 6; and (2) We present graphically in Fig. 0.2 the algebra of the hyperbolic parallel transport along geodesics in the Möbius gyrovector plane ¯,˜), which we will study in Chapter 7.

Typically to the study of hyperbolic geometry as the geometry of gyrovector spaces, the hyperbolic geometry concepts shown in Figs. 0.1 and 0.2 turn out to be fully analogous to their Euclidean counterparts. In Contrast, prior to the emergence of gyrogroup and gyrovector space theory:

(1) The Hyperbolic Pythagorean Theorem appeared in the literature in a form which shares no obvious analogies with its Euclidean counterpart. Preface xxiii

Figure 0.2. Artful Application of the Möbius Transformation in Hyperbolic Geometry, II. The Poincaré n-dimensional ball model of hyperbolic geometry turns out to be the n-dimensional Möbius gyrovector space. In Chapter 7 we will find that in gyro-formalism the nonassociative algebra of the hyperbolic parallel transport of a gyrovector (– a0 ¯ b0 ) rooted at a0 to the gyrovector (– a1 ¯ b1 ) rooted at a 1 along the Möbius geodesicMobius, geodesic which links a0 and a1 in a Möbius gyrovector space ( , ¯, ˜) is fully analogous to the algebra of its Euclidean counterpart. The special case of n = 2 is shown here graphically.

(2) Parallel transport in classical hyperbolic geometry is achieved by methods of differential geometry rather than by methods of nonassociative algebra.

One of the attractive features of this book is that the prerequisites are minimal, encouraging readers to perform their own research at an early stage once they master the manipulation of analogies to which the Thomas gyration gives rise. The theory of gyrogroups and gyrovector spaces recasts hyperbolic geometry and aspires to further conquests in the immense domain of mathematical physics and hyperbolic geometry. Some gyrogroup identities are presented without proof. At the end of chapters exercises are provided to enhance practical experience. This book can be used by teachers and researchers as a source for research programs, classroom projects, and master theses (e.g., [Big94]), and as a self-study book, perhaps as part of a directed reading course. For that xxiv GYROGROUPS AND GYROVECTOR SPACES

Table 0.1. Euclidean-Hyperbolic Analogies for the Möbius Addition ¯ in gyrovector spaces

reason the exposition is as self-contained as possible, consisting of new results that are linked to well known ones by elegant novel analogies such as those we have already seen. These and some other remarkable analogies that we will study are presented in Table 0.1. We are particularly interested, in this book, with Einstein’s velocity addition. Like Möbius addition, Einstein’s addition of relativistically admissible velocities is neither commutative nor associative. The breakdown of commutativity and associativity in Einstein’s addition is not well known, since most books on relativity physics, with only a few outstanding exceptions known to the author, for instance [Foc64], [Bac77], and [SU92], present Einstein’s addition for only parallel velocities, in which case it is both commutative and associative. However, like Möbius addition, Einstein’s addition turns out to be a gyrocommutative gyrogroup operation. Following the breakdown of com- Preface xxv mutativity and associativity in Einstein’s addition, the relativistic 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 [Ung88a]. The book is written for mathematical physicists, geometers, and algebraists, and for readers who enjoy the harmonious interplay between algebra, geometry, and physics. These include experienced researchers as well as third or fourth year students who have encountered some elements of calculus with analytic geometry, Newtonian and Einsteinian mechanics, linear algebra, and elementary group theory. The Thomas precession is a relativistic rotation that does not exist classically. The writer has long believed that the study of topics in relativity physics, non-Euclidean geometry, and nonassociative algebra, to which the Thomas precession gives rise, deserves early place in the curriculum. The concept of Thomas gyration, the abstract Thomas precession, is simple and without equal as a means of giving a unifying approach to Euclidean and non- Euclidean geometry, the geometries governing velocities in Newtonian and in Einsteinian mechanics. Physics is a major external source of mathematical inspiration, and mathematics is the tool for understanding physics. This book provides a leisurely and elementary introduction to (i) gyrogroup theory and (ii) gyrovector space theory. The prerequisites should be met by a third year undergraduate student of physics or mathematics. It may serve as a text for physics and mathematics students and researchers, thereby highlighting Gravity Probe B, a NASA–Stanford University project led by C.W. Francis Everitt aimed at the measurement of the gyroscopic precession of gyroscopes of unprecedented accuracy in Earth orbit. NASA’s Gravity Probe B (GP-B) [EFS69], [Tau97], initiated by William M. Fairbank (1917–1989) [Edi89], is a drag-free satellite carrying gyroscopes around 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 non- Euclidean 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 this book is that it is written in an accessible language and does not require of the reader additional deep knowledge of related fields of mathematics. The detailed description of the Thomas precession in Chapter 1 should not discourage the geometer and the algebraist from reading the remainder of the xxvi GYROGROUPS AND GYROVECTOR SPACES book about: (i) gyrogroups (which are generalized groups); (ii) gyrovector spaces (which are generalized vector spaces); and (iii) hyperbolic geometry, for which gyrogroups and gyrovector spaces set the stage. The remainder of the book is independent of Chapter 1. It is, however, likely that once the reader becomes familiar with the fundamental importance and elegance of the ability of gyrations to capture analogies, he or she will be willing to read in Chapter 1 how the abstract gyration is realized in 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 properties of gyrogroups are presented and analogies that gyrogroups share with groups are emphasized. (3) Chapter 3 extends the notion of the gyrogroup to the notion of a gyrovector space, and, in particular, presents a study of Einstein’s gyrovector spaces. Elementary properties of gyrovector spaces are presented, and analogies they share with vector spaces are emphasized. It is shown that Einstein’s vector spaces form the setting for the Beltrami (also known as the Klein) ball model of hyperbolic geometry in the same way that vector spaces form the setting for Euclidean geometry in any dimension. (4) Chapter 4 presents the study of hyperbolic geometry in terms of its underlying gyrovector space structure. Some known results are obtained in a new way, as well as some novel results, such as the new hyperbolic Pythagorean theorem. (5) Chapter 5 presents the Ungar gyrovector spaces, which are studied in a way similar to the study of the Einstein gyrovector spaces by replacing relativistically admissible velocities by proper relativistic admissible velocities. The resulting model of hyperbolic geometry is a whole space model, rather than a ball model (as in the case of the models of Beltrami and of Poincaré). (6) Chapter 6 presents the study of the Poincaré ball model of hyperbolic geometry in terms of its underlying Möbius gyrovector space structure. Some known results are obtained in a new way and some novel results, such as the hyperbolic Pythagorean theorem, emerge. (7) Chapter 7 presents the gyroparallelogram and the gyroderivative in hyperbolic geometry, and the parallelism in the so called cohyperbolic Preface xxvii

geometry. The conformality of the Poincaré ball model of hyperbolic geometry makes it an attractive model for the study of parallel transport by the gyrovector space formalism. (8) Chapter 8 The remainder of the book is independent of this chapter, the sole purpose of which is to provide a motivational approach to readers who appreciate the importance of the SL(2, C)) algebra in relativity physics. Readers who wish to concentrate on the study of gyrogroups and gyrovector spaces may, therefore, skip reading this chapter and, perhaps, return to read it later on when their curiosity about links between old and new arises. (9) Chapter 9 presents the theory of gyrogroup extension of gyrocommutative gyrogroups, which in Chapter 10 will allow us to present and study the abstract Lorentz group. (10) Chapter 10 employs the results of Chapter 9 to study the abstract Lorentz group and its composition law. As opposed to the four- dimensionally covariant formalism commonly used in the literature to deal with the applications of the Lorentz group, gyrogroup theory allows in this chapter a formalism that is fully analogous to the formalism commonly used with the Galilean group and its applications. (11) Chapter 11 employs the gyrogroup formalism which has been developed in this book to determine the Lorentz transformations that link any two given spacetime events. As a useful tool in quantum mechanics, the so called little Lorentz groups emerge in an elegant way in terms of the Lorentz link. Furthermore, the Lorentz link theorem allows in this chapter an extraordinarily simple and transparent derivation of the shape of relativistically moving objects. The story of the moving rigid sphere that this chapter presents is particularly fascinating. Einstein believed in 1905 that a moving sphere appears to be contracted in the direction of its motion. Following Penrose and Terrell it became widely accepted in relativity physics, since 1959, that the moving sphere does not at all appear to be contracted. The gyrogroup-theoretic techniques that are developed in this book, however, clearly tilt the balance in favor of Einstein, giving a way to calculate the visible shape of moving objects, Figs. 0.3 and 0.4, that is superior to those employed by Penrose, by Terrell, and by others. (12) Chapter 12 presents briefly (i) a non-standard Lorentz transformation group which is parametrized by proper, rather than coordinate, velocities; and (ii) a Möbius–Lorentz transformation group based on the Möbius addition. xxviii GYROGROUPS AND GYROVECTOR SPACES

Teaching From This Book This book presents the remarkable ability of gyrogroups and gyrovector spaces to capture analogies that allow the teacher to tie the modern and unknown with the classical and familiar. As such, this book can accompany any conventional undergraduate and graduate course in (i) modern physics, (ii) non-Euclidean geometry, and (iii) abstract algebra, as part of the process of introducing novel ideas that link the old and known with the new and unfamiliar. Gyrogroup and gyrovector space theory is still in its infancy. It is too early to write a book for teaching a course totally devoted to this area. However, the study of gyrogroups and gyrovector spaces of Einstein and of Möbius must be included: (1) In any improved course on Modern Physics: In the same way that physics students must know that vector addition is commutative and associative, they will have to know that Einstein’s vector addition of relativistically admissible velocities is, in full analogy, gyrocommutative and gyroassociative. The paradoxical breakdown of commutativity and associativity in Einstein’s addition will become clear. The lost commutativity and associativity reappear as gyrocommutativity and gyroassociativity, allowing the Lorentz group to be treated in full analogy with its simpler and better understood counterpart, the Galilean group. (2) In any improved course on Non-Euclidean Geometry: Geometry students will have to know that hyperbolic geometry shares remarkable analogies with Euclidean geometry, allowing the unification of the two geometries into a single super geometry, called gyro-geometry. (i) Vector addition with its scalar multiplication is a vector space operation, known to provide the setting for Euclidean geometry. In the same way, students should know that (ii) Möbius addition with its scalar multiplication is a gyrovector space operation, which provides the setting for the Poincaré ball (or, disc, in two dimensions) model of hyperbolic geometry; and (iii) Einstein’s addition with its scalar multiplication is a gyrovector space operation, which provides the setting for the Beltrami ball (or, disc, in two dimensions) model of hyperbolic geometry; and similarly for other infinitely many isomorphic gyrovector space operations, which provide the setting for infinitely many other models of hyperbolic geometry. (3) In improved courses on Abstract 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 with the common vector addition. It is recommended to include in graduate abstract algebra courses the two articles [FU00] and [FU01], which are beyond the scope of this book. These articles exhibit the natural emergence of gyrogroups, both gyrocommutative and non-gyrocommutative, in group theory. This book may serve as a preparation for reading the related [SSS98] [Iss99] [KU98], but more abstract book of Sabinin [Sab99], on smooth quasigroups and loops in nonassociative algebra and differential geometry. Guided by the analogies this book presents, and other analogies readers may discover, readers are likely to find new results in hyperbolic geometry and in relativity physics, including new gyrogroup identities. These, as well as corrections and terminology suggestions, will be gratefully received by the author, [email protected], for possible inclusion in the next edition of this book, with credit to their discoverers. The author hopes that the distinctive aspects of this book, reflected in Figs. 0.1 and 0.2 and in Table 0.1, will make it of interest to students, to instructors, and to researchers of both physics and mathematics, and that readers will approach to share the fascination that led him to write this book.

ABRAHAM A. UNGAR

Figure 0.3. Theorem 11.21 of Chapter 11, Figure 0.4. viewed from a frame relative to p. 369, established by gyrogroup-theoretic which it moves, is flattened in the direction techniques that are developed in this book, as- of its motion, as shown here, Fig. 0.4, for the serts that a stationary surface, Fig. 0.3, surface x 2 y 2 + y2 z 2 + z2 x2 = 1. Acknowledgments

Much of the insight presented in this book was obtained by useful dis- cussions with colleagues, collaborators, students and friends. These include Graciela Birman, Jingling Chen, Peter Eby, Flynn J. Dustrud, Tuval Foguel, Brian K. Hagelstrom, Oliver Jones, Michael K. Kinyon, Dan Loewenthal, Hala O. Pflugfelder, Alireza Ranjbar-Motlagh, Krzysztof Rozga, Lev V. Sabinin, Larissa V. Sbitneva, Jonathan D.H. Smith Janos Szenthe, Holger Teismann, and Helmuth K. Urbantke. The author is pleased to thank Haya Falk for inspiration, and Alwyn van der Merwe for his cooperation and encouragement since the 1988 publication of [Ung88a] that signaled the birth of gyrogroup and gyrovector space theory which, in turn, led to the appearance of this book. The author is particularly indebted to Helmuth K. Urbantke for help in tracing the prehistory of gyrogroup and gyrovector space theory, as presented in [SU00a], for discussing Penrose’s way of understanding the Lorentz contraction, and for drawing attention to important references about the role of hyperbolic geometry in special relativity theory. The author shares his wish to incorporate gyroscopic precession into science and mathematics education with the GP-B group of the NASA/Stanford Pro- gram to measure the precession of gyroscopes of unprecedented accuracy in Earth orbit. The hospitality during several visits with the GP-B group, led by Francis C.W Everitt, and the support of NASA’s Joint Venture (JOVE) Program, Grant NAG8–1007, are acknowledged. NSF support is discussed in [Ung00b]. The numerical calculations and the graphs presented in this book were obtained by using MATLAB.

xxxi Foreword

The obscured Thomas precession of the special theory of relativity soared in 1988 [Ung88a] into prominence by deciphering the mathematical structure, called a gyrocommutative gyrogroup, that the Thomas precession encodes. The notorious Thomas precession, seemingly the ugly duckling of special relativity theory, thus became the beautiful swan of the theory of gyrogroups and gyrovector spaces.

Abraham A. Ungar

xxxiii Introduction Abraham A. Ungar

Einstein modeled physics with the epistemological scheme EASE

E A S E

E being a variety of Experiments, leading to a system of Axioms A from which Statements S are deduced which, in turn, suggest new Experiments E [Gor93]. This book 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 is studied in relativity physics as an isolated phenomenon. However, the algebraic structure that it stores, discovered by the author in 1988 [Ung88a], suggested the axiomatic approach [Ung97] the consequences of which led to the appearance of this book. Accordingly, this book places the Thomas precession centrally in the foundations of special relativity and hyperbolic geometry. The book of nature is written in mathematical characters, wrote Galilei [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 paper that founded the special theory of relativity, the revolution of relativity physics erupted on November 7, 1919, when a British expedition led by Eddington reported that it had found dramatic confirmation of Einstein’s general theory of relativity. Six years later, on Christmas 1925, Thomas discovered the importance of a relativistic rotation that now bears his name. xxxv xxxvi GYROGROUPS AND GYROVECTOR SPACES

The Thomas precession enters into relativity physics, according to the current literature, merely as an isolated phenomenon, and so is deprived of its true role as the regulator of the Einstein velocity addition law. The latter fares no better: the Einstein velocity addition seems still to be, after almost a century from its discovery, an enigmatic mathematical object. Most texts on special and general relativity present the Einstein velocity addition only for collinear relativistically admissible velocities; among several outstanding exceptions are [Foc64], [Bac77] and [SU00b]. The reason is obvious: Restricted to parallel velocities, Einstein’s addition is both commutative and associative. In the general case, however, Einstein’s velocity addition presents algebraic difficulties [Moc86] since it is neither commutative nor associative. Einstein’s addition is a binary operation in the space of all relativistically admissible velocities,

c being the vacuum speed of light. The space of all relativistically admissible velocities is thus the open c-ball (a ball with radius c) of the Euclidean 3-space

Being nonassociative, Einstein’s velocity addition is not a group operation. Since groups measure symmetry and exhibit mathematical regularity, it seems that these have been lost in the transition to Einstein’s relativity theory. Is the progress from Newtonian to relativity physics associated with a loss of symmetry and mathematical regularity? The seemingly lost mathematical regularity in the transition from vector addition, which is a commutative and associative operation, to the Einstein velocity addition, which is neither commutative nor associative, is counterintuitive and paradoxical. If the relative velocity between two inertial frames is the composition of two relativistically admissible non-parallel velocities u and v, and if Einstein’s velocity addition is denoted by ¯,one may wonder as to which one of the two distinct composite velocities u¯ v and v ¯ u is the ‘right’ relative velocity between the two inertial frames. Similarly, if the relative velocity between two inertial frames is the composition of three relativistically admissible non- parallel velocities u, v and w, one may wonder as to which one of the two distinct composite velocities (u¯ v )¯ w and u ¯ ( v¯ w) is the ‘right’ relative velocity between the two inertial frames. Fortunately, the Thomas precession comes to the rescue, as this book will show. Since physics and mathematics go hand in hand, the paradoxical breakdown of commutativity and associativity in Einstein’s addition provides an irresistible challenge to decipher the mathematical structure Einstein’s addition must en- code if it is to restore mathematical regularity. We will see in this book that if not restricted to parallel velocities, Einstein’s addition possesses rich structure. There is a relativistic effect which, as we will see in this book, is specially Introduction xxxvii

‘designed’ to repair the breakdown of commutativity and associativity in Ein- stein’s addition. This effect, the Thomas precession, is currently studied in the literature as an isolated phenomenon, unheard of in most texts on relativity. Among outstanding exceptions is Jackson’s Classical Electrodynamics [Jac75].

Accordingly, this book deals with the symmetry and with the mathematical regularity which the Thomas precession encodes, presenting these as the very bedrock of relativity physics. At first sight the study of the Thomas precession seems to be a rash exercise with no chance of real success, better left to the experimental physicist. After all, the Einstein velocity addition in its full generality is a complicated operation with apparently poor mathematical regularity, let alone its associated Thomas precession. The seemingly bad behavior of the “notorious Thomas precession formula” (in the words of Rindler and Robinson, p. 431 in [RR99]) is well described by Herbert Goldstein in his book Classical Mechanics, pp. 285–286 [Go180]. Discouraging any attempt to simplify the Thomas rotation matrix to the point where its rotation-matrix behavior can be actually demonstrated and applied to related problems, Goldstein notes that The decomposition process [describing successive pure Lorentz transformations as a pure Lorentz transformation preceded, or followed, by a Thomas rotation] can be carried through on the product of two pure Lorentz transformations to obtain explicitly the rotation of the coordinate axes resulting from the two successive boosts [that is, the Thomas rotation]. In general, the algebra involved is quite forbidding, more than enough, usually, to discourage any actual demonstration of the rotation matrix [italics added]. Herbert Goldstein, Classical Mechanics The Einstein addition and its associated Thomas precession form an integral part of the greatest intellectual achievement of the twentieth century, that is, the understanding of spacetime geometry. However, it seems that the presence of relativistic velocities with their Einstein’s addition in spacetime geometry results in a loss of mathematical regularity since Einstein’s addition is not a group operation. Indeed, one of the goals this book is to show that this is not the case. Deciphering the mathematical regularity the Thomas precession encodes, an elegant theory of gyrogroups and gyrovector spaces emerges. It (i) places Einstein’s addition in a vectorlike algebraic context and (ii) generalizes the notions of the group and the vector space into that of the gyrogroup and the gyrovector space. Evidence that Einstein’s addition is regulated by the Thomas precession came to light in 1988 [Ung88a], turning the notorious Thomas precession, described by Goldstein as the ugly duckling of special relativity theory, into the beautiful swan of gyrogroup and gyrovector space theory. It is known that while the two composite relativistically admissible velocities u¯ v and v ¯ u are distinct when u and v are non-parallel, they have equal xxxviii GYROGROUPS AND GYROVECTOR SPACES magnitudes. Hence these two distinct velocities are linked by a rotation. The unique rotation that takes v ¯ u to u¯ v about a rotation axis that is perpendic- ular to the plane which u and v span, through an angle smaller than p, is the Thomas precession gyr[u, v] generated by the two velocities u and v. This well known property of the Thomas precession [Sil 14] is viewed in this book as a relaxed commutative law called the gyrocommutative law of Einstein’s velocity addition ¯.The Thomas precession, when viewed as an abstract mathematical object, is called the Thomas gyration. The latter suggests the prefix gyro, that we extensively use to emphasize analogies with classical notions and results, and to develop far reaching extensions of classical concepts. The Thomas gyration thus ‘repairs’ the breakdown of commutativity in Einstein’s velocity addition, giving rise to the gyrocommutative law that Einstein’s velocity addition possesses. Coincidentally, the same Thomas gyration that repairs the breakdown of commutativity, repairs the breakdown of associativity in Einstein’s velocity addition as well, giving rise to the gyroassociative law (left and right) [Ung88a]

of Einstein’s velocity addition. The Einstein addition and the Thomas precession are inextricably linked. The resulting grouplike object ,¯ ) formed by the set of all relativistically admissible velocities with their Einstein’s addition ¯ regulated by the Thomas gyration is a gyrocommutative gyrogroup called the Einstein gyrogroup or the relativity gyrogroup. Can we use the gyroassociative law of Einstein’s velocity addition to solve gyrogroup problems in the same way we commonly use the associative law to solve group problems? Luckily, this can be done since one more ‘coincidence’ comes to the rescue. To demonstrate that the gyroassociative law which Einstein’s addition possesses is as useful as the associative law which group operations possess, let us solve the gyrogroup equation for the unknown x in the Einstein gyrogroup ,¯). If a solution x exists, then by the left gyroassociative law we have the following chain of equations Introduction xxxix where u = –u, and where we abbreviate ab = a ¯ (–b) in We thus see the Thomas precession effect in operation: The classical picture of velocity addition is restored by employing the gyrogroup formalism to which the Thomas precession gives rise. By applying the left gyroassociative law in the above chain of equations we have been able to group an unknown x on the right hand sides of the chain of equations with u, (x¯ u), enabling this x to be eliminated by replacing (x ¯ u ) by v. But the application of the left gyroassociative law that allowed grouping x with u, leading to the elimination of an x, introduces a new x that too, must be eliminated. Even worse, the unknown x that the left gyroassociative law introduces into the chain of equations is buried inside the “notorious Thomas precession formula” gyr[·, ·]. Seemingly, we thus encounter the ‘law of con- servation of difficulty’: If one overcomes a difficult point, another will emerge. It seems that in order to be able to eliminate the second unknown x that sits inside the Thomas gyration gyr on the extreme right hand side of the chain of equations, in an elegant way, we need a miracle. Miraculously, indeed, the Thomas precession gyr is sensitive to our needs, possessing the loop property (left and right) which comes to the rescue:

The left loop property of the Thomas precession enables the chain of equations to be further manipulated towards the annihilation of x from the extreme right hand side, obtaining

Hence, if a solution x of the equation x¯ u = v exists, it must be given uniquely by To show that this x is indeed a solution we must substitute it in its equation and apply some of the gyrogroup identities developed in this book. We have thus seen that the relativistic rotation named after Llewellyn Thomas is sensitive to our need for mathematical regularity: (i) it repairs the breakdown of commutativity and associativity in Einstein’s addition and (ii) it possesses the loop property to render the resulting gyroassociative law effective. The sensitivity of the Thomas precession to the needs of the mathematician goes, in fact, beyond that. Being a one-to-one self-map of the space of all relativistically admissible velocities, the Thomas precession gyr[u, v] is bijective. The inverse of gyr[u, v] is gyr[v, u] for all u, v ˛ . Moreover, gyr[u, v] is xl GYROGROUPS AND GYROVECTOR SPACES an automorphism of , satisfying

for all a, b, u, v ˛ and r ˛ Here ˜ denotes the scalar multiplication that Einstein’s addition admits, which specializes to

when r = n is an integer. Since coincidences in mathematics do not emerge accidentally, the ability of the Thomas gyration to repair the breakdown of both commutativity and associativity in Einstein’s addition and to capture analogies must be the mani- festation of a super-theory that unifies Newtonian and Einsteinian mechanics. The discovery that the gyrocommutative law of Einstein’s addition accompa- nies a gyroassociative law (left and right) in 1988 [Ung88a] was a significant development, signaling the birth of the theory of gyrogroups and gyrovector spaces presented in this book. One of the goals of this book is, accordingly, to expound the mathematical theory to which the Thomas gyration gives rise and to employ it to unify the study of the Galilean and the Lorentz transformations between inertial frames with relative velocities and orientations. In the unified theory, the Einstein composition of relativistic velocities emerges as a gyrocommutative gyrogroup operation in the same way that the composition of Newtonian velocities is a commutative group operation. Furthermore, it will be shown in this book that 1. The Lorentz transformation group appears in the unified theory as the gyrosemidirect product of the gyrogroup of boosts (a boost, in the jargon, is a Lorentz transformation without rotation) and a group of rotations, in full analogy with 2. The Galilean transformation group that appears as the semidirect product of the group of Galilean boosts (that is, Galilean transformations without rotation) and a group of rotations. The unified theory enables the study of the Lorentz group to be guided by analogies it shares with the Galilean group. For instance, the algebraic determination of all the Galilean transformations that link any two given spacetime events is obvious and well known. In contrast, an analogous determination in the context of the Lorentz group is not obvious [vW86] [vW91] unless the gyrogroup formalism we develop in this book is employed [Ung92b]. In the years 1910–1914, the period which experienced a dramatic flower- ing of creativity in the special theory of relativity, the Croatian physicist and Introduction xli mathematician Vladimir Vari ak (1865–1942), professor and rector of Zagreb University, showed that this theory has a natural interpretation in hyperbolic geometry [Bar98]. Accordingly, following Thomas’ discovery of the importance of the Thomas precession in relativity theory, this book improves the understanding of that theory by exposing the hitherto unnoticed central role the Thomas precession plays in hyperbolic geometry. It will be discovered that the incorporation of the Thomas precession in geometry allows hyperbolic and Euclidean geometry to be unified, encouraging the search for a corresponding unification of some aspects of classical and relativistic mechanics, as well. Vectors are important in Euclidean geometry [Mey99]. It therefore seems that the unification of Euclidean and hyperbolic geometry is impossible since, in Yaglom’s words [Yag73], “a geometry very distant from Euclidean geometry is hyperbolic geometry, which does not use the notion of vector at all.” Indeed, Walter [Wal99b] points out that Vari ak [Var24] had to admit in 1924 that the adaption of ordinary vector algebra for use in hyperbolic space was just not possible. However, we will see in this book that the Thomas precession is tailor made for the introduction of vectors into hyperbolic geometry, where they are called gyrovectors. The resulting hyperbolic gyrovector algebra shares remarkable analogies with Euclidean vector algebra, that are expressed in terms of the Thomas gyration. The study of hyperbolic geometry flowered in the late nineteenth century as mathematicians increasingly questioned Euclid’s parallel postulate. By relaxing it they derived a wealth of new results, giving rise to a type of non-Euclidean geometry which later became known as hyperbolic geometry. By the 1820s János Bolyai in Hungary, Carl Friedrich Gauss in Germany, and Nikolai Ivanovich Lobachevski in Russia realized that a self-consistent geometry need not satisfy the parallel postulate. Gauss was the first, but typically, he chose not to publish his results. Bolyai received no recognition until long after his death. Hence, the resulting non-Euclidean geometry became known as Lobachevskian geometry, and is still sometimes called this. The term “hyperbolic geometry” was introduced by Felix Klein at the turn of the 20th century. Owing mainly to the work of Tibor Toró, cited in [Kis99], it is now known that János Bolyai was the forerunner of geometrizing physics. According to Kiss [Kis99], Lajos Dávid drew attention in a 1924 series of articles in Italian journals to the precursory role which János Bolyai played in the constructions of Einstein’s relativity theory. The seemingly abstract results of hyperbolic geometry found applications in physics upon Einstein’s introduction of the special theory of relativity in 1905, as was pointed out later by Vari ak, [Var08] [Var12], whose work has been cited by Pauli [Pau58]. The physical significance of a peculiar rotation in special relativity emerged in 1925 when Thomas relativistically re-computed xlii GYROGROUPS AND GYROVECTOR SPACES the precessional frequency of the doublet separation in the fine structure of the atom, and thus rectified a missing factor of 1/2. This correction has come to be known as the Thomas half, and one result of his computation was that the rotation, which now bears Thomas’ name, emerged as the missing link in the understanding of spin in the early development of quantum mechanics. It thus provides a link between Newtonian and relativistic mechanics, as well as between their respective underlying Euclidean and hyperbolic geometry. Hyperbolic geometry underlies velocities in relativistic mechanics in the same way that Euclidean geometry underlies velocities in Newtonian mechanics [Kar77][Sen88][FL97]. Accordingly, the Thomas precession, which plays a role in relativistic mechanics, is expected to play a role in hyperbolic geometry as well [RR95]. Indeed, Thomas precession is presented in this book as the missing link that unifies Euclidean and hyperbolic geometry. The strong links between Euclidean geometry and hyperbolic geometry that the Thomas precession provides are expounded by deciphering the mathematical regularity it stores. The Thomas precession will be found to constitute the missing link needed to establish remarkable analogies shared by Newtonian mechanics and relativistic mechanics and, similarly, analogies shared by hyperbolic geometry and Euclidean geometry. The discovery of a link between two theories allows the theories to be unified, a process of great advantage in both physics and mathematics. The story of this book is thus about the unification of hitherto separate physical and geometrical theories that the Thomas precession, which can no longer be dispensed with, allows. The sensitivity of Thomas precession to non-Euclideaness attracts our attention in relativity physics and in non-Euclidean geometry. For the same reason it attracts NASA’s interest as well. In 1960 Willam Fairbank, L.I. Schiff, and a Stanford engineer, Robert Cannon, initiated the Stanford Gyroscope Exper- iment to measure the gyroscopic precession of gyroscopes in space. Since 1963 it has become the NASA–Stanford GP-B Program. Led by the Stanford physicist Francis Everitt since 1971, it has grown into NASA’s largest running astrophysical program that will perform the most accurate confirmation ever of Einstein’s theory of relativity. Presently, in 2001, the NASA–Stanford GP-B Program is believed to be in the last stages of building the flight apparatus for testing relativistic gravity in space [Eea00]. The common task that the NASA–Stanford GP-B Program and this book face is to highlight the central role that the gyroscopic precession plays in the understanding of relativity physics and its underlying geometry.