Analytic Hyperbolic Geometry: Mathematical Foundations and Applications

Abraham A. Ungar9812564578, 9789812564573, 9789812703279

This is the first book on analytic hyperbolic geometry, fully analogous to analytic Euclidean geometry. Analytic hyperbolic geometry regulates relativistic mechanics just as analytic Euclidean geometry regulates classical mechanics. The book presents a novel gyrovector space approach to analytic hyperbolic geometry, fully analogous to the well-known vector space approach to Euclidean geometry. A gyrovector is a hyperbolic vector. Gyrovectors are equivalence classes of directed gyrosegments that add according to the gyroparallelogram law just as vectors are equivalence classes of directed segments that add according to the parallelogram law. In the resulting “gyrolanguage” of the book one attaches the prefix “gyro” to a classical term to mean the analogous term in hyperbolic geometry. The prefix stems from Thomas gyration, which is the mathematical abstraction of the relativistic effect known as Thomas precession. Gyrolanguage turns out to be the language one needs to articulate novel analogies that the classical and the modern in this book share.The scope of analytic hyperbolic geometry that the book presents is cross-disciplinary, involving nonassociative algebra, geometry and physics. As such, it is naturally compatible with the special theory of relativity and, particularly, with the nonassociativity of Einstein velocity addition law. Along with analogies with classical results that the book emphasizes, there are remarkable disanalogies as well. Thus, for instance, unlike Euclidean triangles, the sides of a hyperbolic triangle are uniquely determined by its hyperbolic angles. Elegant formulas for calculating the hyperbolic side-lengths of a hyperbolic triangle in terms of its hyperbolic angles are presented in the book.The book begins with the definition of gyrogroups, which is fully analogous to the definition of groups. Gyrogroups, both gyrocommutative and nongyrocommutative, abound in group theory. Surprisingly, the seemingly structureless Einstein velocity addition of special relativity turns out to be a gyrocommutative gyrogroup operation. Introducing scalar multiplication, some gyrocommutative gyrogroups of gyrovectors become gyrovector spaces. The latter, in turn, form the setting for analytic hyperbolic geometry just as vector spaces form the setting for analytic Euclidean geometry. By hybrid techniques of differential geometry and gyrovector spaces, it is shown that Einstein (Möbius) gyrovector spaces form the setting for Beltrami-Klein (Poincaré) ball models of hyperbolic geometry. Finally, novel applications of Möbius gyrovector spaces in quantum computation, and of Einstein gyrovector spaces in special relativity, are presented.

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Table of contents :
Contents……Page 14
Preface……Page 8
Acknowledgements……Page 12
1. Introduction……Page 20
1.1 The Vector and Gyrovector Approach to Euclidean and Hyperbolic Geometry……Page 21
1.2 Gyrolanguage……Page 23
1.3 Analytic Hyperbolic Geometry……Page 27
1.4 The Three Models……Page 34
1.5 Applications in Quantum and Special Relativity Theory……Page 37
2. Gyrogroups……Page 40
2.1 Definitions……Page 41
2.2 First Gyrogroup Theorems……Page 43
2.3 The Associative Gyropolygonal Gyroaddition……Page 48
2.4 Two Basic Gyrogroup Equations and Cancellation Laws……Page 50
2.5 Commuting Automorphisms with Gyroautomorphisms……Page 54
2.6 The Gyrosemidirect Product Group……Page 55
2.7 Basic Gyration Properties……Page 59
3.1 Gyrocommutative Gyrogroups……Page 68
3.2 Nested Gyroautomorphism Identities……Page 84
3.3 Two-Divisible Two-Torsion Free Gyrocommutative Gyrogroups……Page 88
3.4 The Mobius Complex Disc Gyrogroup……Page 91
3.5 Mobius Gyrogroups……Page 93
3.6 Einstein Gyrogroups……Page 96
3.7 Einstein Coaddition……Page 100
3.8 PV Gyrogroups……Page 101
3.9 Points and Vectors in a Real Inner Product Space……Page 103
3.10 Exercises……Page 104
4.1 Gyrogroup Extension……Page 106
4.2 The Gyroinner Product, the Gyronorm, and the Gyroboost……Page 109
4.3 The Extended Automorphisms……Page 115
4.4 Gyrotransformation Groups……Page 118
4.6 PV (Proper Velocity) Gyrotransformation Groups……Page 121
4.7 Galilei Transformation Groups……Page 122
4.8 From Gyroboosts to Boosts……Page 123
4.9 The Lorentz Boost……Page 125
4.10 The ( p :q)-Gyromidpoint……Page 127
4.11 The (p1:p2 : . . . : pn)-Gyromidpoint……Page 131
5.1 Equivalence Classes……Page 136
5.2 Gyrovectors……Page 137
5.3 Gyrovector Translation……Page 138
5.4 Gyrovector Translation Composition……Page 141
5.5 Points and Gyrovectors……Page 144
5.6 Cogyrovectors……Page 145
5.7 Cogyrovector Translation……Page 146
5.8 Cogyrovector Translation Composition……Page 149
5.10 Exercises……Page 153
6.1 Definition and First Gyrovector Space Theorems……Page 156
6.2 Solving a System of Two Equations in a Gyrovector Space……Page 163
6.3 Gyrolines and Cogyrolines……Page 166
6.4 Gyrolines……Page 169
6.5 Gyromidpoints……Page 175
6.6 Gyrocovariance……Page 177
6.7 Gyroparallelograms……Page 179
6.8 Gyrogeodesics……Page 185
6.9 Cogyrolines……Page 188
6.10 Cogyromidpoints……Page 199
6.11 Cogyrogeodesics……Page 200
6.12 Mobius Gyrovector Spaces……Page 204
6.13 Mobius Cogyroline Parallelism……Page 207
6.14 Illustrating the Gyroline Gyration Transitive Law……Page 208
6.15 Turning the Mobius Gyrometric into the Poincare Metric……Page 211
6.16 Einstein Gyrovector Spaces……Page 213
6.17 Turning Einstein Gyrometric into a Metric……Page 216
6.18 PV (Proper Velocity) Gyrovector Spaces……Page 218
6.19 Gyrovector Space Isomorphism……Page 220
6.20 Gyrotriangle Gyromedians and Gyrocentroids……Page 222
6.20.1 In Einstein Gyrovector Spaces……Page 223
6.20.2 In Mobius Gyrovector Spaces……Page 227
6.20.3 In PV Gyrovector Spaces……Page 230
6.21 Exercises……Page 232
7. Rudiments of Differential Geometry……Page 234
7.1 The Riemannian Line Element of Euclidean Metric……Page 235
7.2 The Gyroline and the Cogyroline Element……Page 236
7.3 The Gyroline Element of Mobius Gyrovector Spaces……Page 239
7.4 The Cogyroline Element of Mobius Gyrovector Spaces……Page 242
7.5 The Gyroline Element of Einstein Gyrovector Spaces……Page 245
7.6 The Cogyroline Element of Einstein Gyrovector Spaces……Page 248
7.7 The Gyroline Element of PV Gyrovector Spaces……Page 250
7.8 The Cogyroline Element of PV Gyrovector Spaces……Page 252
7.9 Table of Riemannian Line Elements……Page 254
8.1 Gyroangles……Page 256
8.2 Gyrovector Translation of Gyrorays……Page 267
8.3 Gyrorays Parallelism and Perpendicularity……Page 274
8.4 Gyrotrigonometry in Mobius Gyrovector Spaces……Page 276
8.5 Gyrotriangle Gyroangles and Side Gyrolengths……Page 288
8.6 The Gyroangular Defect of Right Gyroangles Gyrotriangles……Page 291
8.7 Gyroangular Defect of the Gyrotriangle……Page 292
8.8 Gyroangular Defect of the Gyrotriangle – a Synthetic Proof……Page 296
8.9 The Gyrotriangle Side Gyrolengths in Terms of its Gyroangles……Page 299
8.10 The Semi-Gyrocircle Gyrotriangle……Page 303
8.11 Gyrotriangular Gyration and Defect……Page 305
8.12 The Equilateral Gyrotriangle……Page 306
8.13 The Mobius Gyroparallelogram……Page 309
8.14 Gyrotriangle Defect in the Mobius Gyroparallelogram……Page 312
8.15 Parallel Transport……Page 318
8.16 Parallel Transport vs. Gyrovector Translation……Page 325
8.17 Gyrocircle Gyrotrigonometry……Page 328
8.18 Cogyroangles……Page 331
8.19 The Cogyroangle in the Three Models……Page 337
8.20 Parallelism in Gyrovector Spaces……Page 338
8.21 Reflection, Gyroreflection, and Cogyroreflection……Page 340
8.22 Tessellation of the Poincare Disc……Page 342
8.23 The Bifurcation Approach to Non-Euclidean Geometry……Page 344
8.24 Exercises……Page 346
9.1 The Density Matrix for Mixed State Qubits……Page 350
9.2 The Bloch Gyrovector……Page 356
9.3 The Bures Fidelity……Page 365
10. Special Theory of Relativity: The Analytic Hyperbolic Geometric Viewpoint……Page 368
10.1 Introduction……Page 369
10.2 Einstein Velocity Addition……Page 371
10.3 Status of the General Einstein Addition……Page 373
10.4 Einstein Addition is an Indispensable Relativistic Tool……Page 376
10.5 From Thomas Gyration to Thomas Precession……Page 380
10.6 The Relativistic Gyrovector Space……Page 384
10.7 Gyrogeodesics, Gyromidpoints and Gyrocentroids……Page 386
10.8 The Midpoint and the Gyromidpoint – Newtonian and Einsteinian Mechanical Interpretation……Page 388
10.9 The Einstein Gyroparallelogram……Page 394
10.10 The Relativistic Gyroparallelogram Law……Page 401
10.11 The Parallelepiped……Page 402
10.12 The Pre-Gyroparallelepiped……Page 406
10.13 The Gyroparallelepiped……Page 408
10.14 The Relativistic Gyroparallelepiped Law……Page 413
10.15 The Lorentz Transformation and its Gyro-Algebra……Page 415
10.16 Galilei and Lorentz Transformation Links……Page 421
10.17 (t1:t2)-Gyromidpoints as CM Velocities……Page 423
10.18 The Hyperbolic Theorems of Ceva and Menelaus……Page 429
10.19 Relativistic Two-Particle Systems……Page 434
10.20 The Covariant Relativistic Center of Momentum (CM) Velocity……Page 440
10.21 Barycentric Coordinates……Page 443
10.22 Einsteinian Gyrobarycentric Coordinates……Page 446
10.23 Gyrobarycentric Coordinates for the Universe……Page 449
10.24 The Proper Velocity Lorentz Group……Page 451
10.25 Demystifying the Proper Velocity Lorentz Group……Page 457
10.26 Exercises……Page 459
Notation And Special Symbols……Page 462
Bibliography……Page 464
Index……Page 476