Fernandes, Rodrigues: Gravitation as a Plastic Distortion of the Lorentz Vacuum


Gravitation as a Plastic Distortion of the Lorentz Vacuum

by Virginia Velma Fernández • Waldyr A. Rodrigues Jr.

  • Hardcover: 153 pages
  • Publisher: Springer; 1st Edition. edition (September 7, 2010)
  • Language: English
  • ISBN-10: 3642135889
  • ISBN-13: 978-3642135880
  • Price: 129 USD

From the preface
In this book we present a theory of the gravitational field where this field (a kind of ‘square root’ of g) is represented by a (1, 1)-extensor field h describing a plastic distortion of the Lorentz vacuum (a complex substance that lives in a Minkowski spacetime) due to the presence of matter. The field h distorts the Minkowski metric extensor η generating what may be interpreted as an effective Lorentzian metric extensor g = h†ηh. Besides that, h permits the introduction of different kinds of parallelism rules on the world manifold, which may be interpreted as distortions of the parallelism structure of Minkowski spacetime and which may also have non null curvature and/or torsion and/or nonmetricity tensors. Therefore, we have different possible effective geometries which may be associated with the gravitational field and thus its description by a Lorentzian geometry is only a possibility, not an imposition from Nature. Moreover, we present with enough details the theory of multiform functions and multiform functionals which is the main new ingredient permitting us to write a Lagrangian for h successfully and to obtain its equations of motion, that results in our theory being equivalent to Einstein field equations of General Relativity (for all those solutions where the manifold M is diffeomorphic to R4). However, in our theory, differently from the case of General Relativity a trustworth energy-momentum conservation law and an orbital plus spin angular momentum conservation law exist. We also express the results of our theory in terms of the gravitational potentials gμ = h†(ϑμ) where {ϑμ} is an orthonormal basis of  Minkowski spacetime (representing the ground state of the Lorentz vacuum), in order to have results which may be easily expressed with the theory of differential forms. The nice Hamiltonian formalism for our theory (formulated in terms of the potentials gμ) is also discussed with details. The book contains also several important Appendices that complement the material in the main text. …

Table of Contents
Preface v
1 Introduction 1
1.1 Geometrical Space Structures, Curvature, Torsion and Nonmetricity Tensors . . . . 1
1.2 Flat Spaces, Affine Spaces, Curvature and Bending . . . . . . . . . . . . . . . . . . . 3
1.3 Killing Vector Fields, Symmetries and Conservation Laws . . . . . . . . . . . . . . . 6
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Multiforms, Extensors, Canonical and Metric Clifford Algebras 13
2.1 Multiforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.1 The k-Part Operator and Involutions . . . . . . . . . . . . . . . . . . . . . . . 14
2.1.2 Exterior Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.3 The Canonical Scalar Product . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.4 Canonical Contractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 The Canonical Clifford Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Extensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.1 The Space extV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 The Space (p, q)-extV of the (p, q)-Extensors . . . . . . . . . . . . . . . . . . 19
2.3.3 The Adjoint Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.4 (1, 1)-Extensors, Properties and Associated Extensors . . . . . . . . . . . . . 20
2.4 The Metric Clifford Algebra C(V, g) . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 Pseudo-Euclidean Metric Extensors on V . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.1 The metric extensor η . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.2 Metric Extensor g with the Same Signature of η . . . . . . . . . . . . . . . . 28
2.5.3 Some Remarkable Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.4 Useful Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3 Multiform Functions and Multiform Functionals 33
3.1 Multiform Functions of Real Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1.1 Limit and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.2 Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2 Multiform Functions of Multiform Variables . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.1 Limit and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.2 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 The Directional Derivative A · ∂X . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.4 The Derivative Mapping ∂X . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2.6 The Operators ∂X∗ and their t-distortions . . . . . . . . . . . . . . . . . . . 40
3.3 Multiform Functionals F(X1,…,Xk)[t] . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.1 Derivatives of Induced Multiform Functionals . . . . . . . . . . . . . . . . . . 41
3.3.2 The Variational Operator δwt. . . . . . . . . . . . . . . . . . . . . . . . . . . 44
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4 Multiform and Extensor Calculus on Manifolds 47
4.1 Canonical Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.1.1 Multiform Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Parallelism Structure (U0, γ) and Covariant Derivatives . . . . . . . . . . . . . . . . 50
4.2.1 The Connection 2-Extensor Field γ on Uo and Associated
Extensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.2 Covariant Derivative of Multiform Fields Associated with (U0, γ) . . . . . . . 50
4.2.3 Covariant Derivative of Extensor Fields Associated with (U0, γ) . . . . . . . . 52
4.2.4 Notable Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.2.5 The 2-Exform Torsion Field of the Structure (Uo, γ) . . . . . . . . . . . . . . 54
4.3 Curvature Operator and Curvature Extensor Fields of the Structure (Uo, γ) . . . . . 54
4.4 Covariant Derivatives Associated with Metric Structures (Uo, g) . . . . . . . . . . . . 56
4.4.1 Metric Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.4.2 Christoffel Operators for the Metric Structure (Uo, g) . . . . . . . . . . . . . 56
4.4.3 The 2-Extensor field λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.4 ( Riemann and Lorentz)-Cartan MGSS’s (Uo, g, γ) . . . . . . . . . . . . . . . 57
4.4.5 Existence Theorem of the γg-gauge Rotation Extensor
of the MCGSS (Uo, g, γ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.4.6 Some Important Properties of a Metric Compatible Connection . . . . . . . . 58
4.4.7 The Riemann 4-Extensor Field of a MCGSS (Uo, g, γ) . . . . . . . . . . . . . 59
4.4.8 Existence Theorem for the on (Uo, g, γ) . . . . . . . . . . . . . . . . . . . . . 60
4.4.9 The Einstein (1, 1)-Extensor Field . . . . . . . . . . . . . . . . . . . . . . . . 60
4.5 Riemann and Lorentz MCGSS’s (Uo, g, λ) . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5.1 Levi-Civita Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5.2 Properties of Da . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.5.3 Properties of R2(B) and R1(b) . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5.4 Levi-Civita Differential Operators . . . . . . . . . . . . . . . . . . . . . . . . 63
4.6 Deformation of MCGSS Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.1 Enter the Plastic Distortion Field h . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.2 On Elastic and Plastic Deformations . . . . . . . . . . . . . . . . . . . . . . . 65
4.7 Deformation of a Minkowski-Cartan MCGSS into a Lorentz-Cartan MCGSS . . . . 66
4.7.1 h-Distortions of Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . 67
4.8 Coupling Between the Minkowski-Cartan and the Lorentz-Cartan MCGSS . . . . . . 68
4.8.1 The Gauge Riemann and Ricci Fields . . . . . . . . . . . . . . . . . . . . . . 69
4.8.2 Gauge Extensor Fields of a Lorentz-Cartan MCGSS (Uo, g, γ) . . . . . . . . . 70
4.8.3 Lorentz MCGSS as h-Deformation of a Particular Minkowski-Cartan MCGSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5 Gravitation as Plastic Distortion of the Lorentz Vacuum 75
5.1 Notation for This Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.2 Lagrangian for the Free h♣ Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.3 Equation of Motion for h♣ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.4 Lagrangian for the Gravitational Field Plus Matter Field Including a Cosmological
Constant Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6 Gravitation Described by the Potentials gα = h†(ϑα) 83
6.1 Definition of the Gravitational Potentials . . . . . . . . . . . . . . . . . . . . . . . . 83
6.2 Lagrangian Density for the Massive Gravitational Field Plus the Matter Fields . . . 86
6.3 Energy-Momentum Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.4 Angular Momentum Conservation Law . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.5 Wave Equations for the gκ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Hamiltonian Formalism 93
7.1 The Hamiltonian 3-form Density H . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
7.2 The Quasi Local Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
7.3 Hamilton’s Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
7.4 The ADM Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
8 Conclusions 105
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A May a Torus with Null Riemann Curvature Exist on E3? 109
B Levi-Civita and Nunes Connections on ˚S2 111
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
C Gravitational Theory for Independent h and Ω Fields 117
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
D Proof of Eq.(6.13) 123
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
E Derivation of the Field Equations from Leh 127
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
F Comment on the LDG Gauge Theory of Gravitation 135
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
G Gravitational Field as a Nonmetricity Tensor Field 141
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Acronyms and Abbreviations 145
List of Symbols 147
Index 151

Sources: Email by W.A. Rodrigues Jr. (25 Dec. 2010, walrodATmpc.com.br), http://www.amazon.com

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