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. The premise of analogy as a study strategy is to make the unfamiliar familiar. Accordingly, this book introduces the notion of vectors into analytic hyperbolic geometry, where they are called gyrovectors. Gyrovectors turn out to be equivalence classes that add according to the gyroparallelogram law just as vectors are equivalence classes that add according to the parallelogram law. In the gyrolanguage of this book, accordingly, one prefixes a gyro to a classical term to mean the analogous term in hyperbolic geometry. As an example, the relativistic gyrotrigonometry of Einstein's special relativity is developed and employed to the study of the stellar aberration phenomenon in astronomy.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. It turns out that the invariant mass of the relativistic center of mass of an expanding system (like galaxies) exceeds the sum of the masses of its constituent particles. This excess of mass suggests a viable mechanism for the formation of dark matter in the universe, which has not been detected but is needed to gravitationally 'glue' each galaxy in the universe. 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 hyperbolic geometry.
Author(s): Abraham Albert Ungar
Edition: 2
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 774
City: Singapore
Contents
Preface
Acknowledgements
1. Introduction
1.1 A Vector Space Approach to Euclidean Geometry and A Gyrovector Space Approach to Hyperbolic
Geometry
1.2 Gyrolanguage
1.3 Analytic Hyperbolic Geometry
1.4 The Three Models
1.5 Applications in Quantum and Special Relativity Theory
2. Gyrogroups
2.1 Definitions
2.2 First Gyrogroup Theorems
2.3 The Associative Gyropolygonal Gyroaddition
2.4 Two Basic Gyrogroup Equations and Cancellation
Laws
2.5 Commuting Automorphisms with Gyroautomorphisms
2.6 The Gyrosemidirect Product Group
2.7 Basic Gyration Properties
3. Gyrocommutative Gyrogroups
3.1 Gyrocommutative Gyrogroups
3.2 Nested Gyroautomorphism Identities
3.3 Two-Divisible Two-Torsion Free Gyrocommutative Gyrogroups
3.4 From Möbius to Gyrogroups
3.5 Higher Dimensional Möbius Gyrogroups
3.6 Möbius Gyrations
3.7 Three-Dimensional Möbius
gyrations
3.8 Einstein Gyrogroups
3.9 Einstein Gyrations
3.10 Einstein Coaddition
3.11 PV Gyrogroups
3.12 Points and Vectors in a Real Inner Product Space
3.13 Exercises
4. Gyrogroup Extension
4.1 Gyrogroup Extension
4.2 The Gyroinner Product, the Gyronorm, and the Gyroboost
4.3 Extended Automorphisms
4.4 Gyrotransformation Groups
4.5 Einstein Gyrotransformation Groups
4.6 PV (Proper Velocity) Gyrotransformation Groups
4.7 Galilei Transformation Groups
4.8 From Gyroboosts to Boosts
4.9 Lorentz Boost
4.10 The (p :q)-Gyromidpoint
4.11 The (p1 :p2 : . . . : pn)-Gyromidpoint
5. Gyrovectors and Cogyrovectors
5.1 Equivalence Classes
5.2 Gyrovectors
5.3 Gyrovector Translation
5.4 Gyrovector Translation Composition
5.5 Points and Gyrovectors
5.6 The Gyroparallelogram Addition Law
5.7 Cogyrovectors
5.8 Cogyrovector Translation
5.9 Cogyrovector Translation Composition
5.10 Points and Cogyrovectors
5.11 Exercises
6. Gyrovector Spaces
6.1 Definition and First Gyrovector Space Theorems
6.2 Solving a System of Two Equations in a Gyrovector Space
6.3 Gyrolines and Cogyrolines
6.4 Gyrolines
6.5 Gyromidpoints
6.6 Gyrocovariance
6.7 Gyroparallelograms
6.8 Gyrogeodesics
6.9 Cogyrolines
6.10 Carrier Cogyrolines of Cogyrovectors
6.11 Cogyromidpoints
6.12 Cogyrogeodesics
6.13 Various Gyrolines That Correspond to Cancellation Laws
6.14 Möbius Gyrovector Spaces
6.15 Möbius Cogyroline Parallelism
6.16 Illustrating the Gyroline Gyration Transitive Law
6.17 Turning Möbius Gyrometric into Poincaré Metric
6.18 Einstein Gyrovector Spaces
6.19 Turning Einstein Gyrometric into a Metric
6.20 PV (Proper Velocity) Gyrovector Spaces
6.21 Gyrovector Space Isomorphisms
6.22 Converting the Einstein Half into the PV Half
6.23 Gyrotriangle Gyromedians and Gyrocentroids
6.23.1 In Einstein Gyrovector Spaces
6.23.2 In Möbius Gyrovector Spaces
6.23.3 In PV Gyrovector Spaces
6.24 Exercises
7. Rudiments of Differential Geometry
7.1 The Riemannian Line Element of Euclidean Metric
7.2 The Gyroline and the Cogyroline Element
7.3 The Gyroline Element of Möbius Gyrovector Spaces
7.4 The Cogyroline Element of Möbius Gyrovector Spaces
7.5 The Gyroline Element of Einstein Gyrovector Spaces
7.6 The Cogyroline Element of Einstein Gyrovector Spaces
7.7 The Gyroline Element of PV Gyrovector Spaces
7.8 The Cogyroline Element of PV Gyrovector Spaces
7.9 Table of Riemannian Line Elements
8. Gyrotrigonometry
8.1 Vectors and Gyrovectors are Equivalence Classes
8.2 Gyroangles
8.3 Gyrovector Translation of Gyrorays
8.4 Gyrorays Parallelism and Perpendicularity
8.5 Gyrotrigonometry in Möbius Gyrovector Spaces
8.6 Gyrotriangle Gyroangles and Side Gyrolengths
8.7 The Gyroangular Defect of Right-Gyroangled Gyrotriangles
8.8 Gyroangular Defect of the Gyrotriangle
8.9 Gyroangular Defect of the Gyrotriangle — a Synthetic Proof
8.10 The Gyrotriangle Side Gyrolengths in Terms of Its Gyroangles
8.11 The Semi-Gyrocircle Gyrotriangle
8.12 Gyrotriangular Gyration and Defect
8.13 The Equilateral Gyrotriangle
8.14 The Möbius Gyroparallelogram
8.15 Gyrotriangle Defect in the Möbius Gyroparallelogram
8.16 Gyroparallelograms Inscribed in a Gyroparallelogram
8.17 Möbius Gyroparallelogram Addition Law
8.18 The Gyrosquare
8.19 Equidefect Gyrotriangles
8.20 Parallel Transport
8.21 Parallel Transport vs. Gyrovector Translation
8.22 From Parallel Transports to Binary Operations
8.23 Gyrocircle Gyrotrigonometry
8.24 Cogyroangles
8.25 The Cogyroangle in the Three Models
8.26 Parallelism in Gyrovector Spaces
8.27 Reflection, Gyroreflection, and Cogyroreflection
8.28 Tessellation of the Poincaré Disc
8.29 Bifurcation Approach to Non-Euclidean Geometry
8.30 Exercises
9. Bloch Gyrovector of Quantum Information and Computation
9.1 Bloch Vector and the Density Matrix for Mixed State Qubits
9.2 Bloch Gyrovector
9.2.1 Example 1
9.2.2 Example 2
9.2.3 Example 3
9.2.4 Example 4
9.3 Structure of the Qubit Density Matrix Space
9.4 Trace Distance and Bures Fidelity
9.5 Real Density Matrix for Mixed State Qubits
9.6 Extending the Real Density Matrix
9.7 Exercises
10. Special Theory of Relativity: The Analytic Hyperbolic
Geometric Viewpoint
10.1 Introduction
10.2 Einstein Velocity Addition
10.3 From Thomas Gyration to Thomas Precession
10.4 The Relativistic Gyrovector Space
10.5 Gyrogeodesics, Gyromidpoints and Gyrocentroids
10.6 The Midpoint and the Gyromidpoint — Newtonian and Einsteinian Mechanical Interpretation
10.7 Einstein Gyroparallelograms
10.8 The Gyroparallelogram Addition Law Is Experimentally Significant
10.9 Extending the Relativistic Gyroparallelogram Law
10.10 The Parallelepiped
10.11 The Pre-Gyroparallelepiped
10.12 The Gyroparallelepiped
10.13 The Relativistic Gyroparallelepiped Addition Law
11. Special Theory of Relativity: The Analytic Hyperbolic
Geometric Viewpoint
11.1 The Lorentz Transformation and Its Gyro-Algebra
11.2 Galilei and Lorentz Transformation Links
11.3 (t1: t2)-Gyromidpoints as CMM Velocities
11.4 The Hyperbolic Theorems of Ceva and Menelaus
11.5 Relativistic Two-Particle Systems
11.6 The Covariant Relativistic CMM Frame Velocity
11.7 The Relativistic Invariant Mass of an Isolated Particle System
11.8 On the Relativistic Particle System Mass Theorem
11.9 The Relativistic Law of Momentum for Particle Systems
11.10 Relativistic CMM and the Kinetic Energy Theorem
11.11 Additivity of Relativistic Energy and Momentum
11.12 Newtonian and Relativistic Kinetic Energy
11.12.1 The Newtonian Kinetic Energy
11.12.2 The Relativistic Kinetic Energy
11.12.3 An Unexpected Analogy That Classical and Relativistic Kinetic Energy Share
11.12.4 Consequences of Classical Kinetic Energy Conservation During Elastic Collisions
11.12.5 Consequences of Relativistic Kinetic Energy Conservation During Elastic Collisions
11.12.6 On the Analogies and a Seeming Disanalogy
11.13 Barycentric Coordinates
11.14 Einsteinian Gyrobarycentric Coordinates
11.15 The Proper Velocity Lorentz Group
11.16 Demystifying the Proper Velocity Lorentz Group
11.17 The Standard Lorentz Transformation Revisited
11.18 The Inhomogeneous Lorentz Transformation
11.19 The Relativistic Center of Momentum and Mass
11.20 Relativistic Center of Mass: Example 1
11.21 Relativistic Center of Mass: Example 2
11.22 Dark Matter and Dark Energy
11.23 Exercises
12. Relativistic Gyrotrigonometry
12.1 The Relativistic Gyrotriangle
12.2 Law of Gyrocosines, SSS to AAA Conversion Law
12.3 The AAA to SSS Conversion Law
12.4 The Law of Gyrosines
12.5 The Relativistic Equilateral Gyrotriangle
12.6 The Relativistic Gyrosquare
12.7 The Relativistic Gyrosquare with θ = π/3
12.8 The ASA to SAS Conversion Law
12.9 The Relativistic Gyrotriangle Defect
12.10 The Right Gyrotriangle
12.11 Einsteinian Gyrotrigonometry
12.12 Gyrodistance Between a Point and a Gyroline
12.13 Gyrotriangle Gyroaltitudes
12.14 Einstein Gyrotriangle Gyroorthocenter
12.15 Möbius Gyrotriangle Gyroorthocenter
12.16 Relativistic Gyrotriangle Gyroarea
12.17 Gyrotriangle Gyroarea Addition
12.18 Gyrosquare Gyroarea
12.19 The Gyrotriangle Constant Principle
12.20 Ceva and Menelaus, Revisited
12.21 Saccheri Gyroquadrilaterals
12.22 Lambert Gyroquadrilaterals
12.23 Gyrotetrahedron Gyroaltitudes
12.24 Exercises
13. Stellar and Particle Aberration
13.1 Particle Aberration: The Classical Interpretation
13.2 Particle Aberration: The Relativistic Interpretation
13.3 Particle Aberration: The Geometric Interpretation
13.4 Relativistic Stellar Aberration
13.5 Exercises
14. Enriched Special Relativity Theory: Special Relativity of
Signature (m,n)
14.1 Introduction
14.2 Galilei and Lorentz Boosts and Multi-boosts
14.3 Pseudo-Euclidean Spaces and Lorentz Transformations of Any Signature
14.4 Matrix Balls of Radius c
14.5 Bi-gamma Factor
14.6 V-Parametric Realization of Lorentz Transformations of Signature (m,n)
14.7 Additive Decomposition of the Lorentz Bi-boost
14.8 Application of the Galilei Bi-boost of Signature (1,3)
14.9 Application of the Galilei Bi-boost of Any Signature
14.10 Application of the Lorentz Bi-boost of Any Signature
14.11 Lorentz Bi-boost Composition Law
14.12 A Supporting Conjecture
14.13 Epilogue
14.14 Exercises
Notation and Special Symbols
Bibliography
Index
