Jr Gravitation as a Plastic Distortion of the Lorentz Vacuum

1. Auflage 2010
ISBN: 978-3-642-13589-7
Verlag: Springer
Format: PDF
Kopierschutz: Wasserzeichen (»Systemvoraussetzungen)

E-Book, Englisch, 154 Seiten

Addressing graduate students and researchers in theoretical physics and mathematics, this book presents a new formulation of the theory of gravity. In the new approach the gravitational field has the same ontology as the electromagnetic, strong, and weak fields. In other words it is a physical field living in Minkowski spacetime. Some necessary new mathematical concepts are introduced and carefully explained. Then they are used to describe the deformation of geometries, the key to describing the gravitational field as a plastic deformation of the Lorentz vacuum. It emerges after further analysis that the theory provides trustworthy energy-momentum and angular momentum conservation laws, a feature that is normally lacking in General Relativity.

Autoren/Hrsg.

Jr,

Inhaltsverzeichnis

1;Gravitation as a Plastic Distortion of the Lorentz
Vacuum;3
1.1;Preface;5
1.2;Contents;7
1.3;1 Introduction;11
1.3.1;1.1 Geometrical Space Structures, Curvature, Torsion and Nonmetricity Tensors;11
1.3.2;1.2 Flat Spaces, Affine Spaces, Curvature and Bending;13
1.3.3;1.3 Killing Vector Fields, Symmetries and Conservation Laws;16
1.3.4;References;20
1.4;2 Multiforms, Extensors, Canonical and Metric Clifford Algebras;23
1.4.1;2.1 Multiforms;24
1.4.1.1;2.1.1 The k-Part Operator and Involutions;24
1.4.1.2;2.1.2 Exterior Product;25
1.4.1.3;2.1.3 The Canonical Scalar Product;25
1.4.1.4;2.1.4 Canonical Contractions;27
1.4.2;2.2 The Canonical Clifford Algebra;28
1.4.3;2.3 Extensors;29
1.4.3.1;2.3.1 The Space extV;29
1.4.3.2;2.3.2 The Space (p,q)-extV of the (p,q)-Extensors;29
1.4.3.3;2.3.3 The Adjoint Operator;29
1.4.3.4;2.3.4 (1,1)-Extensors, Properties and Associated Extensors;30
1.4.3.4.1;Symmetric and Antisymmetric parts of (1,1)-Extensors;30
1.4.3.4.2;The extension of (1,1)-Extensors;30
1.4.3.4.3;The Characteristic Scalars tr[t]and det[t] ;31
1.4.3.4.4;The Characteristic Biform Mapping bif;33
1.4.3.4.5;The Generalization Operator of (1,1)-Extensors;33
1.4.3.4.6;Normal (1,1)-Extensors;34
1.4.4;2.4 The Metric Clifford Algebra C(V,g);34
1.4.4.1;The Metric Scalar Product;34
1.4.4.2;The Metric Left and Right Contractions;35
1.4.4.3;The Metric Clifford Product;36
1.4.5;2.5 Pseudo-Euclidean Metric Extensors on V;37
1.4.5.1;2.5.1 The metric extensor ;37
1.4.5.2;2.5.2 Metric Extensor g with the Same Signature of ;38
1.4.5.3;2.5.3 Some Remarkable Results;40
1.4.5.3.1;Golden Rule;40
1.4.5.3.2;Hodge Star Operators;40
1.4.5.3.3;Relation Between the Hodge Star Operators of g and the Canonical Hodge Star Operator;41
1.4.5.3.4;Relation Between the Hodge Star Operators of g and ;41
1.4.5.4;2.5.4 Useful Identities;41
1.4.6;References;42
1.5;3 Multiform Functions and Multiform Functionals;43
1.5.1;3.1 Multiform Functions of Real Variable;43
1.5.1.1;3.1.1 Limit and Continuity;44
1.5.1.2;3.1.2 Derivative;44
1.5.2;3.2 Multiform Functions of Multiform Variables;45
1.5.2.1;3.2.1 Limit and Continuity;45
1.5.2.2;3.2.2 Differentiability;45
1.5.2.3;3.2.3 The Directional Derivative AX ;45
1.5.2.3.1;Chain Rules;46
1.5.2.4;3.2.4 The Derivative Mapping X ;47
1.5.2.5;3.2.5 Examples;47
1.5.2.6;3.2.6 The Operators X and their t-distortions ;50
1.5.3;3.3 Multiform Functionals F(X1,…,Xk)[t];51
1.5.3.1;3.3.1 Derivatives of Induced Multiform Functionals;51
1.5.3.1.1;The A-Directional At Derivative of a Multiform Functional;51
1.5.3.1.2;The Operators t ;52
1.5.3.1.3;Examples;53
1.5.3.2;3.3.2 The Variational Operator tw;54
1.5.4;References;56
1.6;4 Multiform and Extensor Calculus on Manifolds;57
1.6.1;4.1 Canonical Space;58
1.6.1.1;The Position 1-Form;59
1.6.1.1.1;4.1.1 Multiform Fields;59
1.6.1.1.1.1;Extensor Fields;59
1.6.2;4.2 Parallelism Structure (U0,) and Covariant Derivatives ;60
1.6.2.1;4.2.1 The Connection 2-Extensor Field on Uo and AssociatedExtensor Fields;60
1.6.2.2;4.2.2 Covariant Derivative of Multiform Fields Associated with (U0,);60
1.6.2.3;4.2.3 Covariant Derivative of Extensor Fields Associated with (U0,);62
1.6.2.4;4.2.4 Notable Identities;63
1.6.2.5;4.2.5 The 2-Exform Torsion Field of the Structure (Uo,);64
1.6.3;4.3 Curvature Operator and Curvature Extensor Fields of the Structure (Uo,);64
1.6.4;4.4 Covariant Derivatives Associated with Metric Structures (Uo,g);66
1.6.4.1;4.4.1 Metric Structures;66
1.6.4.2;4.4.2 Christoffel Operators for the Metric Structure (Uo,g);66
1.6.4.3;4.4.3 The 2-Extensor field ;67
1.6.4.4;4.4.4 (Riemann and Lorentz)-Cartan MGSS's (Uo,g,);67
1.6.4.5;4.4.5 Existence Theorem of the g-gauge Rotation Extensorof the MCGSS (Uo,g,);68
1.6.4.6;4.4.6 Some Important Properties of a Metric Compatible Connection;68
1.6.4.7;4.4.7 The Riemann 4-Extensor Field of a MCGSS (Uo,g,);69
1.6.4.8;4.4.8 Existence Theorem for the on (Uo,g,);70
1.6.4.9;4.4.9 The Einstein (1,1)-Extensor Field;70
1.6.5;4.5 Riemann and Lorentz MCGSS's (Uo,g,);71
1.6.5.1;4.5.1 Levi-Civita Covariant Derivative;71
1.6.5.2;4.5.2 Properties of Da;71
1.6.5.3;4.5.3 Properties of R2(B) and R1(b);73
1.6.5.4;4.5.4 Levi-Civita Differential Operators;73
1.6.6;4.6 Deformation of MCGSS Structures;74
1.6.6.1;4.6.1 Enter the Plastic Distortion Field h;74
1.6.6.1.1;Minkowski Metric on Uo;74
1.6.6.1.2;Lorentzian Metric;74
1.6.6.2;4.6.2 On Elastic and Plastic Deformations;75
1.6.6.2.1;Construction of a Lorentzian Metric Field on Uo;75
1.6.7;4.7 Deformation of a Minkowski-Cartan MCGSS into a Lorentz-Cartan MCGSS;76
1.6.7.1;4.7.1 h-Distortions of Covariant Derivatives;77
1.6.8;4.8 Coupling Between the Minkowski-Cartan and the Lorentz-Cartan MCGSS;78
1.6.8.1;4.8.1 The Gauge Riemann and Ricci Fields;79
1.6.8.2;4.8.2 Gauge Extensor Fields of a Lorentz-Cartan MCGSS (Uo,g,);80
1.6.8.3;4.8.3 Lorentz MCGSS as h-Deformation of a Particular
Minkowski-Cartan MCGSS ;82
1.6.9;References;84
1.7;5 Gravitation as Plastic Distortion of the Lorentz Vacuum;85
1.7.1;5.1 Notation for This Chapter;85
1.7.2;5.2 Lagrangian for the Free h Field;86
1.7.3;5.3 Equation of Motion for h;87
1.7.4;5.4 Lagrangian for the Gravitational Field Plus Matter Field Including a Cosmological Constant Term;91
1.8;6 Gravitation Described by the Potentials g=h();92
1.8.1;6.1 Definition of the Gravitational Potentials;92
1.8.2;6.2 Lagrangian Density for the Massive Gravitational Field Plus the Matter Fields;95
1.8.3;6.3 Energy-Momentum Conservation Law;97
1.8.4;6.4 Angular Momentum Conservation Law;98
1.8.5;6.5 Wave Equations for the g;100
1.8.6;References;101
1.9;7 Hamiltonian Formalism;102
1.9.1;7.1 The Hamiltonian 3-form Density H;102
1.9.2;7.2 The Quasi Local Energy;106
1.9.3;7.3 Hamilton's Equations;107
1.9.4;7.4 The ADM Energy.;109
1.9.5;References;111
1.10;8 Conclusions;113
1.10.1;References;114
1.11;Appendix A May a Torus with Null Riemann Curvature Exist on E3?;116
1.12;Appendix B Levi-Civita and Nunes Connections on 2;118
1.12.1;References;122
1.13;Appendix C Gravitational Theory for Independent h and Fields;123
1.13.1;References;127
1.14;Appendix D Proof of Eq.(6.13);128
1.14.1;References;130
1.15;Appendix E Derivation of the Field Equations from Leh;131
1.15.1;References;137
1.16;Appendix F Comment on the LDG Gauge Theory of Gravitation ;138
1.16.1;References;142
1.17;Appendix G Gravitational Field as a Nonmetricity Tensor Field;143
1.17.1;References;145
1.18; Acronyms and Abbreviations;146
1.19; List of Symbols;147
1.20; Index;151