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

by Ungar, Abraham Albert

• ISBN13: 9789812772299
• ISBN10: 9812772294
• Format: Hardcover
• Copyright: 2008-02-11
• Publisher: World Scientific Pub Co Inc

Table of Contents

Preface	p. vii
Acknowledgements	p. xi
Introduction	p. 1
A Vector Space Approach to Euclidean Geometry and A Gyrovector Space Approach to Hyperbolic Geometry	p. 2
Gyrolanguage	p. 5
Analytic Hyperbolic Geometry	p. 7
The Three Models	p. 9
Applications in Quantum and Special Relativity Theory	p. 12
Gyrogroups	p. 15
Definitions	p. 16
First Gyrogroup Theorems	p. 19
The Associative Gyropolygonal Gyroaddition	p. 23
Two Basic Gyrogroup Equations and Cancellation Laws	p. 25
Commuting Automorphisms with Gyroautomorphisms	p. 32
The Gyrosemidirect Product Group	p. 34
Basic Gyration Properties	p. 39
Gyrocommutative Gyrogroups	p. 51
Gyrocommutative Gyrogroups	p. 51
Nested Gyroautomorphism Identities	p. 68
Two-Divisible Two-Torsion Free Gyrocommutative Gyrogroups	p. 72
From Mobius to Gyrogroups	p. 75
Higher Dimensional Mobius Gyrogroups	p. 77
Mobius gyrations	p. 81
Three-Dimensional Mobius gyrations	p. 85
Einstein Gyrogroups	p. 86
Einstein Coaddition	p. 92
PV Gyrogroups	p. 93
Points and Vectors in a Real Inner Product Space	p. 97
Exercises	p. 98
Gyrogroup Extension	p. 101
Gyrogroup Extension	p. 101
The Gyroinner Product, the Gyronorm, and the Gyroboost	p. 105
The Extended Automorphisms	p. 111
Gyrotransformation Groups	p. 114
Einstein Gyrotransformation Groups	p. 117
PV (Proper Velocity) Gyrotransformation Groups	p. 117
Galilei Transformation Groups	p. 118
From Gyroboosts to Boosts	p. 119
The Lorentz Boost	p. 121
The (p:q)-Gyromidpoint	p. 123
The (p[subscript 1]:p[subscript 2]: ... : p[subscript n])-Gyromidpoint	p. 127
Gyrovectors and Cogyrovectors	p. 131
Equivalence Classes	p. 131
Gyrovectors	p. 132
Gyrovector Translation	p. 133
Gyrovector Translation Composition	p. 137
Points and Gyrovectors	p. 140
The Gyroparallelogram Addition Law	p. 141
Cogyrovectors	p. 143
Cogyrovector Translation	p. 144
Cogyrovector Translation Composition	p. 148
Points and Cogyrovectors	p. 151
Exercises	p. 152
Gyrovector Spaces	p. 153
Definition and First Gyrovector Space Theorems	p. 153
Solving a System of Two Equations in a Gyrovector Space	p. 160
Gyrolines and Cogyrolines	p. 163
Gyrolines	p. 166
Gyromidpoints	p. 172
Gyrocovariance	p. 175
Gyroparallelograms	p. 177
Gyrogeodesics	p. 183
Cogyrolines	p. 186
Carrier Cogyrolines of Cogyrovectors	p. 197
Cogyromidpoints	p. 198
Cogyrogeodesics	p. 199
Various Gyrolines and Cancellation Laws	p. 203
Mobius Gyrovector Spaces	p. 205
Mobius Cogyroline Parallelism	p. 212
Illustrating the Gyroline Gyration Transitive Law	p. 213
Turning the Mobius Gyrometric into the Poincare Metric	p. 216
Einstein Gyrovector Spaces	p. 218
Turning Einstein Gyrometric into a Metric	p. 222
PV (Proper Velocity) Gyrovector Spaces	p. 223
Gyrovector Space Isomorphisms	p. 225
Gyrotriangle Gyromedians and Gyrocentroids	p. 228
In Einstein Gyrovector Spaces	p. 229
In Mobius Gyrovector Spaces	p. 233
In PV Gyrovector Spaces	p. 236
Exercises	p. 238
Rudiments of Differential Geometry	p. 239
The Riemannian Line Element of Euclidean Metric	p. 240
The Gyroline and the Cogyroline Element	p. 241
The Gyroline Element of Mobius Gyrovector Spaces	p. 245
The Cogyroline Element of Mobius Gyrovector Spaces	p. 248
The Gyroline Element of Einstein Gyrovector Spaces	p. 250
The Cogyroline Element of Einstein Gyrovector Spaces	p. 253
The Gyroline Element of PV Gyrovector Spaces	p. 255
The Cogyroline Element of PV Gyrovector Spaces	p. 257
Table of Riemannian Line Elements	p. 259
Gyrotrigonometry	p. 261
Vectors and Gyrovectors are Equivalence Classes	p. 261
Gyroangles	p. 263
Gyrovector Translation of Gyrorays	p. 275
Gyrorays Parallelism and Perpendicularity	p. 282
Gyrotrigonometry in Mobius Gyrovector Spaces	p. 284
Gyrotriangle Gyroangles and Side Gyrolengths	p. 296
The Gyroangular Defect of Right Gyroangle Gyrotriangles	p. 300
Gyroangular Defect of the Gyrotriangle	p. 301
Gyroangular Defect of the Gyrotriangle - a Synthetic Proof	p. 304
The Gyrotriangle Side Gyrolengths in Terms of its Gyroangles	p. 307
The Semi-Gyrocircle Gyrotriangle	p. 314
Gyrotriangular Gyration and Defect	p. 316
The Equilateral Gyrotriangle	p. 318
The Mobius Gyroparallelogram	p. 321
Gyrotriangle Defect in the Mobius Gyroparallelogram	p. 324
Gyroparallelograms Inscribed in a Gyroparallelogram	p. 330
Mobius Gyroparallelogram Addition Law	p. 333
The Gyrosquare	p. 336
Equidefect Gyrotriangles	p. 342
Parallel Transport	p. 344
Parallel Transport vs. Gyrovector Translation	p. 350
Gyrocircle Gyrotrigonometry	p. 353
Cogyroangles	p. 356
The Cogyroangle in the Three Models	p. 362
Parallelism in Gyrovector Spaces	p. 363
Reflection, Gyroreflection, and Cogyroreflection	p. 365
Tessellation of the Poincare Disc	p. 367
Bifurcation Approach to Non-Euclidean Geometry	p. 369
Exercises	p. 371
Bloch Gyrovector of Quantum Information and Computation	p. 375
The Density Matrix for Mixed State Qubits	p. 375
Bloch Gyrovector	p. 381
Trace Distance and Bures Fidelity	p. 390
The Real Density Matrix for Mixed State Qubits	p. 392
Extending the Real Density Matrix	p. 395
Exercises	p. 396
Special Theory of Relativity: The Analytic Hyperbolic Geometric Viewpoint: Part I: Einstein Velocity Addition and its Consequences	p. 397
Introduction	p. 399
Einstein Velocity Addition	p. 401
From Thomas Gyration to Thomas Precession	p. 403
The Relativistic Gyrovector Space	p. 407
Gyrogeodesics, Gyromidpoints and Gyrocentroids	p. 409
The Midpoint and the Gyromidpoint - Newtonian and Einsteinian Mechanical Interpretation	p. 411
Einstein Gyroparallelograms	p. 418
The Relativistic Gyroparallelogram Law	p. 424
The Parallelepiped	p. 427
The Pre-Gyroparallelepiped	p. 430
The Gyroparallelepiped	p. 433
The Relativistic Gyroparallelepiped Addition Law	p. 438
Exercises	p. 443
Special Theory of Relativity: The Analytic Hyperbolic Geometric Viewpoint Part II: Lorentz Transformation and its Consequences	p. 445
The Lorentz Transformation and its Gyro-Algebra	p. 445
Galilei and Lorentz Transformation Links	p. 452
(t[subscript 1]:t[subscript 2])-Gyromidpoints as CMM Velocities	p. 454
The Hyperbolic Theorems of Ceva and Menelaus	p. 460
Relativistic Two-Particle Systems	p. 465
The Covariant Relativistic CMM Frame Velocity	p. 471
The Relativistic Invariant Mass of an Isolated Particle System	p. 477
Relativistic CMM and the Kinetic Energy Theorem	p. 485
Additivity of Relativistic Energy and Momentum	p. 488
Bright (Baryonic) and Dark Matter	p. 491
Newtonian and Relativistic Kinetic Energy	p. 494
The Newtonian Kinetic Energy	p. 494
The Relativistic Kinetic Energy	p. 495
Consequences of Classical Kinetic Energy Conservation During Elastic Collisions	p. 496
Consequences of Relativistic Kinetic Energy Conservation During Elastic Collisions	p. 498
On the Analogies and a Seeming Disanalogy	p. 501
Barycentric Coordinates	p. 502
Einsteinian Gyrobarycentric Coordinates	p. 505
The Proper Velocity Lorentz Group	p. 508
Demystifying the Proper Velocity Lorentz Group	p. 513
The Standard Lorentz Transformation Revisited	p. 516
The Inhomogeneous Lorentz Transformation	p. 517
The Relativistic Center of Momentum and Mass	p. 520
Relativistic Center of Mass: Example 1	p. 527
Relativistic Center of Mass: Example 2	p. 529
Dark Matter and Dark Energy	p. 531
Exercises	p. 532
Relativistic Gyrotrigonometry	p. 537
The Relativistic Gyrotriangle	p. 537
Law of Gyrocosines, SSS to AAA Conversion Law	p. 542
The AAA to SSS Conversion Law	p. 542
The Law of Gyrosines	p. 544
The Relativistic Equilateral Gyrotriangle	p. 544
The Relativistic Gyrosquare	p. 545
The Einstein Gyrosquare with [theta] = [pi]/3	p. 547
The ASA to SAS Conversion Law	p. 550
The Relativistic Gyrotriangle Defect	p. 551
The Right-Gyroangled Gyrotriangle	p. 552
The Einsteinian Gyrotrigonometry	p. 554
The Relativistic Gyrotriangle Gyroarea	p. 558
The Gyrosquare Gyroarea	p. 560
The Gyrotriangle Constant Principle	p. 561
Ceva and Menelaus, Revisited	p. 563
Saccheri Gyroquadrilaterals	p. 566
Lambert Gyroquadrilaterals	p. 570
Exercises	p. 575
Stellar and Particle Aberration	p. 577
Particle Aberration: The Classical Interpretation	p. 579
Particle Aberration: The Relativistic Interpretation	p. 583
Particle Aberration: The Geometric Interpretation	p. 593
Relativistic Stellar Aberration	p. 596
Exercises	p. 599
Notation And Special Symbols	p. 601
Bibliography	p. 605
Index	p. 621
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