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

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"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.
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Product details

  • Paperback | 419 pages
  • 155.96 x 233.93 x 26.67mm | 979.75g
  • Dordrecht, Netherlands
  • English
  • 25 Illustrations, black and white; XLII, 419 p. 25 illus.
  • 0792369106
  • 9780792369103

Table of contents

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 Moebius Gyrovector Space.- 1 The Gyrovector Space Isomorphism.- 2 Moebius Gyrovector Spaces.- 3 Gyrotranslations - Left and Right.- 4 The Hyperbolic Pythagorean Theorem in the Poincare Disc Model of Hyperbolic Geometry.- 5 Gyrolines and the Cancellation Laws.- 6 The Riemannian Line Elements of the Moebius Dual Metrics.- 7 Rudiments of Riemannian Geometry.- 8 The Moebius Geodesics and Angles.- 9 Hyperbolic Trigonometry in Moebius Gyrovector Spaces.- 10 Numerical Demonstration.- 11 The Equilateral Gyrotriangle.- 12 Exercises.- 7. Gyrogeometry.- 1 The Moebius 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 Moebius 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 Moebius 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 Moebius 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 Moebius-Lorentz Boost.- 5 The Unique Moebius-Lorentz Boost that Links Two Points.- 6 The Moebius-Lorentz Transformation Group.- 13.References.- About the Author.- Topic Index.- Author Index.
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