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Gravitation as a Plastic Distortion of the Lorentz Vacuum written by Virginia and Waldyr

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Gravitation as a Plastic Distortion of the Lorentz Vacuum written by Virginia Velma Fernández and Waldyr A. Rodrigues Jr . 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.

Gravitation as a Plastic Distortion of the Lorentz Vacuum written by Virginia Velma Fernández and Waldyr A. Rodrigues Jr cover the following topics.

• 1. Introduction
1.1 Geometrical Space Structures, Curvature, Torsion and Nonmetricity Tensors
1.2 Flat Spaces, Affine Spaces, Curvature and Bending
1.3 Killing Vector Fields, Symmetries and Conservation Laws
References

• 2. Multiforms, Extensors, Canonical and Metric Clifford Algebras
2.1 Multiforms
2.1.1 The k-Part Operator and Involutions
2.1.2 Exterior Product
2.1.3 The Canonical Scalar Product
2.1.4 Canonical Contractions
2.2 The Canonical Clifford Algebra
2.3 Extensors
2.3.1 The Space extV
2.3.2 The Space (p, q)-extV of the (p, q)-Extensors
2.3.4 (1, 1)-Extensors, Properties and Associated Extensors
2.4 The Metric Clifford Algebra C(V, g)
2.5 Pseudo-Euclidean Metric Extensors on V
2.5.1 The metric extensor ?
2.5.2 Metric Extensor g with the Same Signature of ?
2.5.3 Some Remarkable Results
2.5.4 Useful Identities
References

• 3. Multiform Functions and Multiform Functionals
3.1 Multiform Functions of Real Variable
3.1.1 Limit and Continuity
3.1.2 Derivative
3.2 Multiform Functions of Multiform Variables
3.2.1 Limit and Continuity
3.2.2 Differentiability
3.2.3 The Directional Derivative A · ?X
3.2.4 The Derivative Mapping ?X
3.2.5 Examples
3.2.6 The Operators ?X* and their t-distortions
3.3 Multiform Functionals F
3.3.1 Derivatives of Induced Multiform Functionals
3.3.2 The Variational Operator dwt
References

• 4. Multiform and Extensor Calculus on Manifolds
4.1 Canonical Space
4.1.1 Multiform Fields
4.2 Parallelism Structure (U0, ?) and Covariant Derivatives
4.2.1 The Connection 2-Extensor Field ? on Uo and Associated Extensor Fields
4.2.2 Covariant Derivative of Multiform Fields Associated with (U0, ?)
4.2.3 Covariant Derivative of Extensor Fields Associated with (U0, ?)
4.2.4 Notable Identities
4.2.5 The 2-Exform Torsion Field of the Structure (Uo, ?)
4.3 Curvature Operator and Curvature Extensor Fields of the Structure (Uo, ?)
4.4 Covariant Derivatives Associated with Metric Structures (Uo, g)
4.4.1 Metric Structures
4.4.2 Christoffel Operators for the Metric Structure (Uo, g)
4.4.3 The 2-Extensor field ?
4.4.4 ( Riemann and Lorentz)-Cartan MGSS’s (Uo, g, ?)
4.4.5 Existence Theorem of the ?g-gauge Rotation Extensor of the MCGSS (Uo, g, ?)
4.4.6 Some Important Properties of a Metric Compatible Connection
4.4.7 The Riemann 4-Extensor Field of a MCGSS (Uo, g, ?)
4.4.8 Existence Theorem for the on (Uo, g, ?)
4.4.9 The Einstein (1, 1)-Extensor Field
4.5 Riemann and Lorentz MCGSS’s (Uo, g, ?)
4.5.1 Levi-Civita Covariant Derivative
4.5.2 Properties of Da
4.5.3 Properties of R2(B) and R1(b)
4.5.4 Levi-Civita Differential Operators
4.6 Deformation of MCGSS Structures
4.6.1 Enter the Plastic Distortion Field h
4.6.2 On Elastic and Plastic Deformations
4.7 Deformation of a Minkowski-Cartan MCGSS into a Lorentz-Cartan MCGSS
4.7.1 h-Distortions of Covariant Derivatives
4.8 Coupling Between the Minkowski-Cartan and the Lorentz-Cartan MCGSS
4.8.1 The Gauge Riemann and Ricci Fields
4.8.2 Gauge Extensor Fields of a Lorentz-Cartan MCGSS (Uo, g, ?)
4.8.3 Lorentz MCGSS as h-Deformation of a Particular Minkowski-Cartan MCGSS
References

• 5. Gravitation as Plastic Distortion of the Lorentz Vacuum
5.1 Notation for This Chapter
5.2 Lagrangian for the Free h Print Field;
5.3 Equation of Motion for h ?
5.4 Lagrangian for the Gravitational Field Plus Matter Field Including a Cosmological Constant Term

• 6. Gravitation Described by the Potentials ga = h †(?a)
6.1 Definition of the Gravitational Potentials
6.2 Lagrangian Density for the Massive Gravitational Field Plus the Matter Fields
6.3 Energy-Momentum Conservation Law
6.4 Angular Momentum Conservation Law
6.5 Wave Equations for the g?
References

• 7. Hamiltonian Formalism
7.1 The Hamiltonian 3-form Density H
7.2 The Quasi Local Energy
7.3 Hamilton’s Equations
References

• 8. Conclusions
References
A May a Torus with Null Riemann Curvature Exist on E3?
B Levi-Civita and Nunes Connections on °S2
References
C Gravitational Theory for Independent h and O Fields
References
D Proof of Eq.(6.13)
References
E Derivation of the Field Equations from Leh
References
X Contents
F Comment on the LDG Gauge Theory of Gravitation
References
G Gravitational Field as a Nonmetricity Tensor Field
References
Acronyms and Abbreviations
List of Symbols
Index

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