Gravitation as a Plastic Distortion of the Lorentz Vacuum (eBook)

Gravitation as a Plastic Distortion of the Lorentz 
3 
Preface 5
Contents 7
1 Introduction 11
1.1 Geometrical Space Structures, Curvature, Torsion and Nonmetricity Tensors 11
1.2 Flat Spaces, Affine Spaces, Curvature and Bending 13
1.3 Killing Vector Fields, Symmetries and Conservation Laws 16
References 20
2 Multiforms, Extensors, Canonical and Metric Clifford Algebras 23
2.1 Multiforms 24
2.1.1 The k-Part Operator and Involutions 24
2.1.2 Exterior Product 25
2.1.3 The Canonical Scalar Product 25
2.1.4 Canonical Contractions 27
2.2 The Canonical Clifford Algebra 28
2.3 Extensors 29
2.3.1 The Space extV 29
2.3.2 The Space (p,q)-extV of the (p,q)-Extensors 29
2.3.3 The Adjoint Operator 29
2.3.4 (1,1)-Extensors, Properties and Associated Extensors 30
Symmetric and Antisymmetric parts of (1,1)-Extensors 30
The extension of (1,1)-Extensors 30
The Characteristic Scalars tr[t]and det[t] 31
The Characteristic Biform Mapping bif 33
The Generalization Operator of (1,1)-Extensors 33
Normal (1,1)-Extensors 34
2.4 The Metric Clifford Algebra C(V,g) 34
The Metric Scalar Product 34
The Metric Left and Right Contractions 35
The Metric Clifford Product 36
2.5 Pseudo-Euclidean Metric Extensors on V 37
2.5.1 The metric extensor 37
2.5.2 Metric Extensor g with the Same Signature of 38
2.5.3 Some Remarkable Results 40
Golden Rule 40
Hodge Star Operators 40
Relation Between the Hodge Star Operators of g and the Canonical Hodge Star Operator 41
Relation Between the Hodge Star Operators of g and 41
2.5.4 Useful Identities 41
References 42
3 Multiform Functions and Multiform Functionals 43
3.1 Multiform Functions of Real Variable 43
3.1.1 Limit and Continuity 44
3.1.2 Derivative 44
3.2 Multiform Functions of Multiform Variables 45
3.2.1 Limit and Continuity 45
3.2.2 Differentiability 45
3.2.3 The Directional Derivative AX 45
Chain Rules 46
3.2.4 The Derivative Mapping X 47
3.2.5 Examples 47
3.2.6 The Operators X and their t-distortions 50
3.3 Multiform Functionals F(X1,…,Xk)[t] 51
3.3.1 Derivatives of Induced Multiform Functionals 51
The A-Directional At Derivative of a Multiform Functional 51
The Operators t 52
Examples 53
3.3.2 The Variational Operator tw 54
References 56
4 Multiform and Extensor Calculus on Manifolds 57
4.1 Canonical Space 58
The Position 1-Form 59
4.1.1 Multiform Fields 59
Extensor Fields 59
4.2 Parallelism Structure (U0,) and Covariant Derivatives 60
4.2.1 The Connection 2-Extensor Field on Uo and AssociatedExtensor Fields 60
4.2.2 Covariant Derivative of Multiform Fields Associated with (U0,) 60
4.2.3 Covariant Derivative of Extensor Fields Associated with (U0,) 62
4.2.4 Notable Identities 63
4.2.5 The 2-Exform Torsion Field of the Structure (Uo,) 64
4.3 Curvature Operator and Curvature Extensor Fields of the Structure (Uo,) 64
4.4 Covariant Derivatives Associated with Metric Structures (Uo,g) 66
4.4.1 Metric Structures 66
4.4.2 Christoffel Operators for the Metric Structure (Uo,g) 66
4.4.3 The 2-Extensor field 67
4.4.4 (Riemann and Lorentz)-Cartan MGSS's (Uo,g,) 67
4.4.5 Existence Theorem of the g-gauge Rotation Extensorof the MCGSS (Uo,g,) 68
4.4.6 Some Important Properties of a Metric Compatible Connection 68
4.4.7 The Riemann 4-Extensor Field of a MCGSS (Uo,g,) 69
4.4.8 Existence Theorem for the on (Uo,g,) 70
4.4.9 The Einstein (1,1)-Extensor Field 70
4.5 Riemann and Lorentz MCGSS's (Uo,g,) 71
4.5.1 Levi-Civita Covariant Derivative 71
4.5.2 Properties of Da 71
4.5.3 Properties of R2(B) and R1(b) 73
4.5.4 Levi-Civita Differential Operators 73
4.6 Deformation of MCGSS Structures 74
4.6.1 Enter the Plastic Distortion Field h 74
Minkowski Metric on Uo 74
Lorentzian Metric 74
4.6.2 On Elastic and Plastic Deformations 75
Construction of a Lorentzian Metric Field on Uo 75
4.7 Deformation of a Minkowski-Cartan MCGSS into a Lorentz-Cartan MCGSS 76
4.7.1 h-Distortions of Covariant Derivatives 77
4.8 Coupling Between the Minkowski-Cartan and the Lorentz-Cartan MCGSS 78
4.8.1 The Gauge Riemann and Ricci Fields 79
4.8.2 Gauge Extensor Fields of a Lorentz-Cartan MCGSS (Uo,g,) 80
4.8.3 Lorentz MCGSS as h-Deformation of a Particular 
82 
References 84
5 Gravitation as Plastic Distortion of the Lorentz Vacuum 85
5.1 Notation for This Chapter 85
5.2 Lagrangian for the Free h Field 86
5.3 Equation of Motion for h 87
5.4 Lagrangian for the Gravitational Field Plus Matter Field Including a Cosmological Constant Term 91
6 Gravitation Described by the Potentials g=h() 92
6.1 Definition of the Gravitational Potentials 92
6.2 Lagrangian Density for the Massive Gravitational Field Plus the Matter Fields 95
6.3 Energy-Momentum Conservation Law 97
6.4 Angular Momentum Conservation Law 98
6.5 Wave Equations for the g 100
References 101
7 Hamiltonian Formalism 102
7.1 The Hamiltonian 3-form Density H 102
7.2 The Quasi Local Energy 106
7.3 Hamilton's Equations 107
7.4 The ADM Energy. 109
References 111
8 Conclusions 113
References 114
Appendix A May a Torus with Null Riemann Curvature Exist on E3? 116
Appendix B Levi-Civita and Nunes Connections on 2 118
References 122
Appendix C Gravitational Theory for Independent h and Fields 123
References 127
Appendix D Proof of Eq.(6.13) 128
References 130
Appendix E Derivation of the Field Equations from Leh 131
References 137
Appendix F Comment on the LDG Gauge Theory of Gravitation 138
References 142
Appendix G Gravitational Field as a Nonmetricity Tensor Field 143
References 145
Acronyms and Abbreviations 146
List of Symbols 147
Index 151