Beyond the Einstein Addition Law and Its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces

ISBN-13: 9780792369103 / Angielski / Miękka / 2001 / 419 str.

Abraham A. Ungar; Abraham A. Ungar

Opis

"I cannot define coincidence in mathematics]. But 1 shall argue that coincidence can always be elevated or organized into a superstructure which perfonns a unification along the coincidental elements. The existence of a coincidence is strong evidence for the existence of a covering theory. " -Philip 1. Davis Dav81] Alluding to the Thomas gyration, this book presents the Theory of gy rogroups and gyrovector spaces, taking the reader to the immensity of hyper bolic 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 ap pearance of general relativity. Subsequently, the exposition of special relativity followed the lines laid down by Minkowski, in which the role of hyperbolic ge ometry 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 hy perbolic geometry in the special theory of relativity, initiated by Minkowski, by emphasizing the central role that hyperbolic geometry plays in the theory."

Spis treści

1. Thomas Precession: The Missing Link.- 1 A Brief History of the Thomas Precession.- 2 The Einstein Velocity Addition.- 3 Thomas Precession and Gyrogroups.- 4 The Relativistic Composite Velocity Reciprocity Principle.- 5 From Thomas Precession to Thomas Gyration.- 6 Solving Equations in Einstein’s Addition, and the Einstein Coaddition.- 7 The Abstract Einstein Addition.- 8 Verifying Algebraic Identities of Einstein’s Addition.- 9 Matrix Representation of the Thomas Precession.- 10 Graphical Presentation of the Thomas Precession.- 11 The Thomas Rotation Angle.- 12 The Circular Functions of the Thomas Rotation Angle.- 13 Exercises.- 2. Gyrogroups: Modeled on Einstein’s Addition.- 1 Definition of a Gyrogroup.- 2 Examples of Gyrogroups.- 3 First Theorems of Gyrogroup Theory.- 4 Solving Gyrogroup Equations.- 5 The Gyrosemidirect Product Group.- 6 Understanding Gyrogroups by Gyrosemidirect Product Groups.- 7 Some Basic Gyrogroup Identities.- 8 Exercises.- 3. The Einstein Gyrovector Space.- 1 Einstein Scalar Multiplication.- 2 Einstein’s Half.- 3 Einstein’s Metric.- 4 Metric Geometry of Einstein Gyrovector Spaces.- 5 The Einstein Geodesics.- 6 Gyrovector Spaces.- 7 Solving a Simple System of Two Equations in a Gyrovector Space.- 8 Einstein’s Addition and The Beltrami Model of Hyperbolic Geometry.- 9 The Riemannian Line Element of Einstein’s Metric.- 10 Exercises.- 4. Hyperbolic Geometry of Gyrovector Spaces.- 1 Rooted Gyrovectors.- 2 Equivalence Classes of Gyrovectors.- 3 The Hyperbolic Angle.- 4 Hyperbolic Trigonometry in Einstein’s Gyrovector Spaces.- 5 From Pythagoras to Einstein: The Hyperbolic Pythagorean Theorem.- 6 The Relativistic Dual Uniform Accelerations.- 7 Einstein’s Dual Geodesics.- 8 The Riemannian Line Element of Einstein’s Cometric.- 9 Moving Cogyrovectors in Einstein Gyrovector Spaces.- 10 Einstein’s Hyperbolic Coangles.- 11 The Gyrogroup Duality Symmetry.- 12 Parallelism in Cohyperbolic Geometry.- 13 Duality, And The Dual Gyrotransitive Laws of Mutually Dual Geodesics.- 14 The Bifurcation Approach to Hyperbolic Geometry.- 15 The Gyroparallelogram Addition Rule.- 16 Gyroterminology.- 17 Exercises.- 5. The Ungar Gyrovector Space.- 1 The Ungar Gyrovector Space of Relativistic Proper Velocities.- 2 Some Identities for Ungar’s Addition.- 3 The Gyrovector Space Isomorphism Between Einstein’s and Ungar’s Gyrovector Spaces.- 4 The Riemannian Line Elements of The Ungar Dual Metrics.- 5 The Ungar Model of Hyperbolic Geometry.- 6 Angles in The Ungar Model of Hyperbolic Geometry.- 7 The Angle Measure in Einstein’s and in Ungar’s Gyrovector Spaces.- 8 The Hyperbolic Law of Cosines and Sines in the Ungar Model of Hyperbolic Geometry.- 9 Exercises.- 6. The Möbius Gyrovector Space.- 1 The Gyrovector Space Isomorphism.- 2 Möbius Gyrovector Spaces.- 3 Gyrotranslations — Left and Right.- 4 The Hyperbolic Pythagorean Theorem in the Poincaré Disc Model of Hyperbolic Geometry.- 5 Gyrolines and the Cancellation Laws.- 6 The Riemannian Line Elements of the Möbius Dual Metrics.- 7 Rudiments of Riemannian Geometry.- 8 The Möbius Geodesics and Angles.- 9 Hyperbolic Trigonometry in Möbius Gyrovector Spaces.- 10 Numerical Demonstration.- 11 The Equilateral Gyrotriangle.- 12 Exercises.- 7. Gyrogeometry.- 1 The Möbius Gyroparallelogram.- 2 The Triangle Angular Defect in Gyrovector Spaces.- 3 Parallel Transport Along Geodesics in Gyrovector Spaces.- 4 The Triangular Angular Defect And Gyrophase Shift.- 5 Polygonal And Circular Gyrophase Shift.- 6 Gyrovector Translation in Möbius Gyrovector Spaces.- 7 Triangular Gyrovector Translation of Rooted Gyrovectors.- 8 The Hyperbolic Angle and Gyrovector Translation.- 9 Triangular Parallel Translation of Rooted Gyrovectors.- 10 The Nonclosed Circular Path Angular Defect.- 11 Gyroderivative: The Hyperbolic Derivative.- 12 Parallelism in Cohyperbolic Geometry.- 13 Exercises.- 8. Gyrooperations — The SL(2, C) Approach.- 1 The Algebra Of The SL(2, C) Group.- 2 The SL(2, C) General Vector Addition.- 3 Case I — The Einstein Gyrovector Spaces.- 4 Case II — The Möbius Gyrovector Spaces.- 5 Case III — The Ungar Gyrovector Spaces.- 6 Case IV — The Chen Gyrovector Spaces.- 7 Gyrovector Space Isomorphisms.- 8 Conclusion.- 9 Exercises.- 9. The Cocycle Form.- 1 The Real Einstein Gyrogroup and its Cocycle Form.- 2 The Complex Einstein Gyrogroup and its Cocycle Form.- 3 The Möbius Gyrogroup and its Cocycle Form.- 4 The Ungar Gyrogroup and its Cocycle Form.- 5 Abstract Gyrocommutative Gyrogroups with Cocycle Forms.- 6 Cocycle Forms, By Examples.- 7 Basic Properties of Cocycle Forms.- 8 Applications of the Real Even Cocycle Form Representation.- 9 The Secondary Gyration of a Gyrocommutative Gyrogroup with a Complex Cocycle Form.- 10 The Gyrogroup Extension of a Gyrogroup with a Cocycle Form.- 11 Cocyclic Gyrocommutative Gyrogroups.- 12 Applications of Gyrogroups to Cocycle Forms.- 13 Gyrocommutative Gyrogroup Extension by Cocyclic Maps.- 14 Exercises.- 10.The Lorentz Group and Its Abstraction.- 1 Inner Product and the Abstract Lorentz Boost.- 2 Extended Automorphisms of Extended Gyrogroups.- 3 The Lorentz Boost of Relativity Theory.- 4 The Parametrized Lorentz Group and its Composition Law.- 5 The Parametrized Lorentz Group of Special Relativity.- 11.The Lorentz Transformation Link.- 1 Group Action on Sets.- 2 The Galilei Transformation of Structured Spacetime Points.- 3 The Galilean Link.- 4 The Galilean Link By a Rotation.- 5 The Lorentz Transformation of Structured Spacetime Points.- 6 The Lorentz Link By a Rotation.- 7 The Lorentz Boost Link.- 8 The Little Lorentz Groups.- 9 The Relativistic Shape of Moving Objects.- 10 The Shape of Moving Circles.- 11 The Shape of Moving Spheres.- 12 The Shape of Moving Straight Lines.- 13 The Shape of Moving Curves.- 14 The Shape of Moving Harmonic Waves.- 15 The Relativistic Doppler Shift.- 16 Simultaneity: Is Length Contraction Real?.- 17 Einstein’s Length Contraction: An Idea Whose Time Has Come Back.- 18 Exercises.- 12.Other Lorentz Groups.- 1 The Proper Velocity Ungar-Lorentz Boost.- 2 The Proper Velocity Ungar-Lorentz Transformation Group.- 3 The Unique Ungar-Lorentz Boost that Links Two Points.- 4 The Möbius-Lorentz Boost.- 5 The Unique Möbius-Lorentz Boost that Links Two Points.- 6 The Möbius-Lorentz Transformation Group.- 13.References.- About the Author.- Topic Index.- Author Index.