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Analytic Hyperbolic Geometry and Albert Einstein's Special Theory of Relativity

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This book presents a powerful way to study Einstein's special theory of relativity and its underlying hyperbolic geometry in which analogies with classical results form the right tool. It introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors.

Newtonian velocity addition is the common vector addition, which is both commutative and associative. The resulting vector spaces, in turn, form the algebraic setting for the standard model of Euclidean geometry. In full analogy, Einsteinian velocity addition is a gyrovector addition, which is both gyrocommutative and gyroassociative. The resulting gyrovector spaces, in turn, form the algebraic setting for the Beltrami Klein ball model of the hyperbolic geometry of Bolyai and Lobachevsky. Similarly, Möbius addition gives rise to gyrovector spaces that form the algebraic setting for the Poincaré ball model of hyperbolic geometry.

In full analogy with classical results, the book presents a novel relativistic interpretation of stellar aberration in terms of relativistic gyrotrigonometry and gyrovector addition. Furthermore, the book presents, for the first time, the relativistic center of mass of an isolated system of noninteracting particles that coincided at some initial time t = 0. The novel relativistic resultant mass of the system, concentrated at the relativistic center of mass, dictates the validity of the dark matter and the dark energy that were introduced by cosmologists as ad hoc postulates to explain cosmological observations about missing gravitational force and late-time cosmic accelerated expansion.

The discovery of the relativistic center of mass in this book thus demonstrates once again the usefulness of the study of Einstein's special theory of relativity in terms of its underlying analytic hyperbolic geometry.

Contents: Gyrogroups; Gyrocommutative Gyrogroups; Gyrogroup Extension; Gyrovectors and Cogyrovectors; Gyrovector Spaces; Rudiments of Differential Geometry; Gyrotrigonometry; Bloch Gyrovector of Quantum Information and Computation; Special Theory of Relativity: The Analytic Hyperbolic Geometric Viewpoint; Relativistic Gyrotrigonometry; Stellar and Particle Aberration.

Author(s): Abraham A. Ungar
Edition: illustrated edition
Publisher: World Scientific Publishing Company
Year: 2008

Language: English
Pages: 649

Contents......Page 14
Preface......Page 8
Acknowledgements......Page 12
1. Introduction......Page 22
1.1 A Vector Space Approach to Euclidean Geometry and A Gyrovector Space Approach to Hyperbolic Geometry......Page 23
1.2 Gyrolanguage......Page 26
1.3 Analytic Hyperbolic Geometry......Page 28
1.4 The Three Models......Page 30
1.5 Applications in Quantum and Special Relativity Theory......Page 33
2. Gyrogroups......Page 36
2.1 Definitions......Page 37
2.2 First Gyrogroup Theorems......Page 40
2.3 The Associative Gyropolygonal Gyroaddition......Page 44
2.4 Two Basic Gyrogroup Equations and Cancellation Laws......Page 46
2.5 Commuting Automorphisms with Gyroautomorphisms......Page 53
2.6 The Gyrosemidirect Product Group......Page 55
2.7 Basic Gyration Properties......Page 60
3.1 Gyrocommutative Gyrogroups......Page 72
3.2 Nested Gyroautomorphism Identities......Page 89
3.3 Two-Divisible Two-Torsion Free Gyrocommutative Gyrogroups......Page 93
3.4 From M obius to Gyrogroups......Page 96
3.5 Higher Dimensional M obius Gyrogroups......Page 98
3.6 M obius gyrations......Page 102
3.7 Three-Dimensional M obius gyrations......Page 106
3.8 Einstein Gyrogroups......Page 107
3.9 Einstein Coaddition......Page 113
3.10 PV Gyrogroups......Page 114
3.11 Points and Vectors in a Real Inner Product Space......Page 118
3.12 Exercises......Page 119
4.1 Gyrogroup Extension......Page 122
4.2 The Gyroinner Product, the Gyronorm, and the Gyroboost......Page 126
4.3 The Extended Automorphisms......Page 132
4.4 Gyrotransformation Groups......Page 135
4.6 PV (Proper Velocity) Gyrotransformation Groups......Page 138
4.7 Galilei Transformation Groups......Page 139
4.8 From Gyroboosts to Boosts......Page 140
4.9 The Lorentz Boost......Page 142
4.10 The (p :q)-Gyromidpoint......Page 144
4.11 The (p1 :p2 :...: pn)-Gyromidpoint......Page 148
5.1 Equivalence Classes......Page 152
5.2 Gyrovectors......Page 153
5.3 Gyrovector Translation......Page 154
5.4 Gyrovector Translation Composition......Page 158
5.5 Points and Gyrovectors......Page 161
5.6 The Gyroparallelogram Addition Law......Page 162
5.7 Cogyrovectors......Page 164
5.8 Cogyrovector Translation......Page 165
5.9 Cogyrovector Translation Composition......Page 169
5.10 Points and Cogyrovectors......Page 172
5.11 Exercises......Page 173
6.1 Definition and First Gyrovector Space Theorems......Page 174
6.2 Solving a System of Two Equations in a Gyrovector Space......Page 181
6.3 Gyrolines and Cogyrolines......Page 184
6.4 Gyrolines......Page 187
6.5 Gyromidpoints......Page 193
6.6 Gyrocovariance......Page 196
6.7 Gyroparallelograms......Page 198
6.8 Gyrogeodesics......Page 204
6.9 Cogyrolines......Page 207
6.10 Carrier Cogyrolines of Cogyrovectors......Page 218
6.11 Cogyromidpoints......Page 219
6.12 Cogyrogeodesics......Page 220
6.13 Various Gyrolines and Cancellation Laws......Page 224
6.14 M obius Gyrovector Spaces......Page 226
6.15 M obius Cogyroline Parallelism......Page 233
6.16 Illustrating the Gyroline Gyration Transitive Law......Page 234
6.17 Turning the M obius Gyrometric into the Poincar e Metric......Page 237
6.18 Einstein Gyrovector Spaces......Page 239
6.19 Turning Einstein Gyrometric into a Metric......Page 243
6.20 PV(ProperVelocity) Gyrovector Spaces......Page 244
6.21 Gyrovector Space Isomorphisms......Page 246
6.22 Gyrotriangle Gyromedians and Gyrocentroids......Page 249
6.22.1 In Einstein Gyrovector Spaces......Page 250
6.22.2 In M obius Gyrovector Spaces......Page 254
6.22.3 In PV Gyrovector Spaces......Page 257
6.23 Exercises......Page 259
7. Rudiments of Differential Geometry......Page 260
7.1 The Riemannian Line Element of Euclidean Metric......Page 261
7.2 The Gyroline and the Cogyroline Element......Page 262
7.3 The Gyroline Element of M obius Gyrovector Spaces......Page 266
7.4 The Cogyroline Element of M obius Gyrovector Spaces......Page 269
7.5 The Gyroline Element of Einstein Gyrovector Spaces......Page 271
7.6 The Cogyroline Element of Einstein Gyrovector Spaces......Page 274
7.7 The Gyroline Element of PV Gyrovector Spaces......Page 276
7.8 The Cogyroline Element of PV Gyrovector Spaces......Page 278
7.9 Table of Riemannian Line Elements......Page 280
8.1 Vectors and Gyrovectors are Equivalence Classes......Page 282
8.2 Gyroangles......Page 284
8.3 Gyrovector Translation of Gyrorays......Page 296
8.4 Gyrorays Parallelism and Perpendicularity......Page 303
8.5 Gyrotrigonometry in M obius Gyrovector Spaces......Page 305
8.6 Gyrotriangle Gyroangles and Side Gyrolengths......Page 317
8.7 The Gyroangular Defect of Right Gyroangle Gyrotriangles......Page 321
8.8 Gyroangular Defect of the Gyrotriangle......Page 322
8.9 Gyroangular Defect of the Gyrotriangle – a Synthetic Proof......Page 325
8.10 The Gyrotriangle Side Gyrolengths in Terms of its Gyroangles......Page 328
8.11 The Semi-Gyrocircle Gyrotriangle......Page 335
8.12 Gyrotriangular Gyration and Defect......Page 337
8.13 The Equilateral Gyrotriangle......Page 339
8.14 The M obius Gyroparallelogram......Page 342
8.15 Gyrotriangle Defect in the M obius Gyroparallelogram......Page 345
8.16 Gyroparallelograms Inscribed in a Gyroparallelogram......Page 351
8.17 M obius Gyroparallelogram Addition Law......Page 354
8.18 The Gyrosquare......Page 357
8.19 Equidefect Gyrotriangles......Page 363
8.20 Parallel Transport......Page 365
8.21 Parallel Transport vs. Gyrovector Translation......Page 371
8.22 Gyrocircle Gyrotrigonometry......Page 374
8.23 Cogyroangles......Page 377
8.24 The Cogyroangle in the Three Models......Page 383
8.25 Parallelism in Gyrovector Spaces......Page 384
8.26 Reection, Gyroreection, and Cogyroreection......Page 386
8.27 Tessellation of the Poincare Disc......Page 388
8.28 Bifurcation Approach to Non-Euclidean Geometry......Page 390
8.29 Exercises......Page 392
9.1 The Density Matrix for Mixed State Qubits......Page 396
9.2 Bloch Gyrovector......Page 402
9.3 Trace Distance and Bures Fidelity......Page 411
9.4 The Real Density Matrix for Mixed State Qubits......Page 413
9.5 Extending the Real Density Matrix......Page 416
9.6 Exercises......Page 417
10. Special Theory of Relativity: The Analytic Hyperbolic Geometric Viewpoint Part I: Einstein Velocity Addition and its Consequences......Page 418
10.1 Introduction......Page 420
10.2 Einstein Velocity Addition......Page 422
10.3 From Thomas Gyration to Thomas Precession......Page 424
10.4 The Relativistic Gyrovector Space......Page 428
10.5 Gyrogeodesics, Gyromidpoints and Gyrocentroids......Page 430
10.6 The Midpoint and the Gyromidpoint – Newtonian and Einsteinian Mechanical Interpretation......Page 432
10.7 Einstein Gyroparallelograms......Page 439
10.8 The Relativistic Gyroparallelogram Law......Page 445
10.9 The Parallelepiped......Page 448
10.10 The Pre-Gyroparallelepiped......Page 451
10.11 The Gyroparallelepiped......Page 454
10.12 The Relativistic Gyroparallelepiped Addition Law......Page 459
10.13 Exercises......Page 464
11.1 The Lorentz Transformation and its Gyro-Algebra......Page 466
11.2 Galilei and Lorentz Transformation Links......Page 473
11.3 (t1:t2)-Gyromidpoints as CMM Velocities......Page 475
11.4 The Hyperbolic Theorems of Ceva and Menelaus......Page 481
11.5 Relativistic Two-Particle Systems......Page 486
11.6 The Covariant Relativistic CMM Frame Velocity......Page 492
11.7 The Relativistic Invariant Mass of an Isolated Particle System......Page 498
11.8 Relativistic CMM and the Kinetic Energy Theorem......Page 506
11.9 Additivity of Relativistic Energy and Momentum......Page 509
11.10 Bright (Baryonic) and Dark Matter......Page 512
11.11.1 The Newtonian Kinetic Energy......Page 515
11.11.2 The Relativistic Kinetic Energy......Page 516
11.11.3 Consequences of Classical Kinetic Energy Conservation During Elastic Collisions......Page 517
11.11.4 Consequences of Relativistic Kinetic Energy Conservation During Elastic Collisions......Page 519
11.11.5 On the Analogies and a Seeming Disanalogy......Page 522
11.12 Barycentric Coordinates......Page 523
11.13 Einsteinian Gyrobarycentric Coordinates......Page 526
11.14 The Proper Velocity Lorentz Group......Page 529
11.15 Demystifying the Proper Velocity Lorentz Group......Page 534
11.16 The Standard Lorentz Transformation Revisited......Page 537
11.17 The Inhomogeneous Lorentz Transformation......Page 538
11.18 The Relativistic Center of Momentum and Mass......Page 541
11.19 Relativistic Center of Mass: Example 1......Page 548
11.20 Relativistic Center of Mass: Example 2......Page 550
11.21 Dark Matter and Dark Energy......Page 552
11.22 Exercises......Page 553
12.1 The Relativistic Gyrotriangle......Page 558
12.3 The AAA to SSS Conversion Law......Page 563
12.5 The Relativistic Equilateral Gyrotriangle......Page 565
12.6 The Relativistic Gyrosquare......Page 566
12.7 The Einstein Gyrosquare with = /3......Page 568
12.8 The ASA to SAS Conversion Law......Page 571
12.9 The Relativistic Gyrotriangle Defect......Page 572
12.10 The Right-Gyroangled Gyrotriangle......Page 573
12.11 The Einsteinian Gyrotrigonometry......Page 575
12.12 The Relativistic Gyrotriangle Gyroarea......Page 579
12.13 The Gyrosquare Gyroarea......Page 581
12.14 The Gyrotriangle Constant Principle......Page 582
12.15 Ceva and Menelaus, Revisited......Page 584
12.16 Saccheri Gyroquadrilaterals......Page 587
12.17 Lambert Gyroquadrilaterals......Page 591
12.18 Exercises......Page 596
13. Stellar and Particle Aberration......Page 598
13.1 Particle Aberration: The Classical Interpretation......Page 600
13.2 Particle Aberration: The Relativistic Interpretation......Page 604
13.3 Particle Aberration: The Geometric Interpretation......Page 614
13.4 Relativistic Stellar Aberration......Page 617
13.5 Exercises......Page 620
Notation And Special Symbols......Page 622
Bibliography......Page 626
Index......Page 642