This volume deals with the fundamental concepts of space, time and matter. It presents a novel reformulation of both the special and general theory of relativity, in which time does not constitute the fourth dimension in a conventional 4-dimensional space-time. Instead, the role of time is played by the flow of a vector field on a 3-dimensional space. The standard models of de Sitter, Schwarzschild and Kerr space-times are reformulated in a purely 3-dimensional manifold.
The volume also presents a theory of matter in which the fundamental particles, such as baryons and leptons, appear as a result of an interaction between left-handed and right-handed 2-component Weyl neutrinos. The Appendices contain a comprehensive treatment of classical mechanics in terms of Hamiltonian vector fields on symplectic manifolds. Graduate students of mathematical physics or theoretical physics, as well as academics, will find this volume of interest.

PREFACE v

1. SPACE AND TIME 1

1.0. Introduction 1

1.1. The Hyperbolic Structure of the Space of Relative Velocities 4

1.2. Relativistic Kinematics on 3-Manifolds 8

1.3. Class of Inertial Observers and Generalized Lorentz Matrices 11

1.4. Classical Relativistic Field Dynamics 13

      1.4.1. One-dimensional Heat equation 13

      1.4.2. One-dimensional Wave equation 17

      1.4.3. Gauss–Einstein equations 18

1.5. Relativistic Field Dynamics on 3-Manifolds 21

      1.5.1. Three-dimensional field equations and relationship with Einstein equations 21

1.6. Flat-Space Solutions 24

1.7. The de Sitter Solution 26

1.8. A New Solution of the Vacuum Einstein Field Equations 27

1.9. A Solution of Mathematical Interest 34

1.10. The Schwarzschild Solution 38

      1.10.1. The Schwarzschild metric 39

1.11. From Space-Time 4-Metric to 3-Metric and 3-Vector Field 44

1.12. The Kerr Solution 49

1.13. The Maxwell Equations 56

References 60

2. MATTER 61

2.0. Introduction 61

2.1. The Photon and the Weyl Neutrinos 63

2.2. A Neutrino Theory of Matter 68

      2.2.1. Composite νL–νR system without interaction 68

      2.2.2. Composite νL–νR system with interaction 70

      2.2.3. A reduced 1-dimensional model 72

      2.2.4. Conclusion 77

References 78

APPENDIX A. VECTOR FIELDS ON MANIFOLDS 79

A.1 Vector Fields on Manifolds 79

A.2 Example of a Vector Field whose Integral Curves are Knots 83

References 86

APPENDIX B. DYNAMICAL VECTOR FIELDS OF CLASSICAL MECHANICS 87

B.1 Introduction 87

B.2 Configuration Spaces of Mechanical Systems as Manifolds 89

B.3 Dynamical Vector Fields in Classical Mechanics 99

References 126

APPENDIX C. MAPLE PROGRAM 1 129

APPENDIX D. MAPLE PROGRAM 2 131

APPENDIX E. MAPLE PROGRAM 3 133

AUTHOR INDEX 137

SUBJECT INDEX 139

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