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E-raamat: Covariance and Gauge Invariance in Continuum Physics: Application to Mechanics, Gravitation, and Electromagnetism

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Covariance and Gauge Invariance in Continuum Physics: Application to Mechanics, Gravitation, and ElectromagnetismSuurem pilt
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This book presents a Lagrangian approach model to formulate various fields of continuum physics, ranging from gradient continuum elasticity to relativistic gravito-electromagnetism. It extends the classical theories based on Riemann geometry to Riemann-Cartan geometry, and then describes non-homogeneous continuum and spacetime with torsion in Einstein-Cartan relativistic gravitation.

It investigates two aspects of invariance of the Lagrangian: covariance of formulation following the method of Lovelock and Rund, and gauge invariance where the active diffeomorphism invariance is considered by using local Poincaré gauge theory according to the Utiyama method.

Further, it develops various extensions of strain gradient continuum elasticity, relativistic gravitation and electromagnetism when the torsion field of the Riemann-Cartan continuum is not equal to zero. Lastly, it derives heterogeneous wave propagation equations within twisted and curved manifolds and proposes a relation between electromagnetic potential and torsion tensor.


1 General Introduction
1(8)
1.1 Classical Physics, Lagrangian, and Invariance
1(3)
1.2 General Covariance, Gauge Invariance
4(2)
1.3 Objectives and Planning
6(3)
2 Basic Concepts on Manifolds, Spacetimes, and Calculus of Variations
9(64)
2.1 Introduction
9(1)
2.2 Space-Time Background
10(2)
2.2.1 Basics on Flat Minkowski Spacetime
10(2)
2.2.2 Twisted and Curved Spacetimes
12(1)
2.3 Manifolds, Tensor Fields, and Connections
12(43)
2.3.1 Coordinate System, and Group of Transformations
12(7)
2.3.2 Elements on Spacetime and Invariance for Relativity
19(10)
2.3.3 Volume-Form
29(2)
2.3.4 Affine Connection
31(14)
2.3.5 Tetrads and Affine Connection: Continuum Transformations
45(10)
2.4 Invariance for Lagrangian and Euler-Lagrange Equations
55(9)
2.4.1 Covariant Formulation of Classical Mechanics of a Particle
55(3)
2.4.2 Basic Elements for Calculus of Variations
58(2)
2.4.3 Extended Euler-Lagrange Equations
60(4)
2.5 Simple Examples in Continuum and Relativistic Mechanics
64(9)
2.5.1 Particles in a Minkowski Spacetime
64(1)
2.5.2 Some Continua Examples
65(8)
3 Covariance of Lagrangian Density Function
73(22)
3.1 Introduction
73(1)
3.2 Some Basic Theorems
74(4)
3.2.1 Theorem of Cartan
74(1)
3.2.2 Theorem of Lovelock (1969)
75(1)
3.2.3 Theorem of Quotient
76(2)
3.3 Invariance with Respect to the Metric
78(6)
3.3.1 Transformation Rules for the Metric and Its Derivatives
80(1)
3.3.2 Introduction of Dual Variables
81(2)
3.3.3 Theorem
83(1)
3.4 Invariance with Respect to the Connection
84(11)
3.4.1 Preliminary
85(3)
3.4.2 Application: Covariance of L
88(1)
3.4.3 Summary for Lagrangian Covariance
89(1)
3.4.4 Covariance of Nonlinear Elastic Continuum
90(5)
4 Gauge Invariance for Gravitation and Gradient Continuum
95(82)
4.1 Introduction to Gauge Invariance
95(7)
4.1.1 Transition from Covariance to Gauge Invariance
95(3)
4.1.2 Mechanical Coupling of Matter and Spacetime
98(4)
4.2 Gravitation, Fields, and Matter
102(52)
4.2.1 Preliminaries
103(4)
4.2.2 Newton--Cartan Formalism for Classical Gravitation
107(8)
4.2.3 Matter Within Torsionless Gravitation
115(12)
4.2.4 Matter Within Curved Spacetime with Torsion
127(11)
4.2.5 Lagrangian for Coupled Spacetime and Matter
138(16)
4.3 Gauge Invariance on a Riemann--Cartan Continuum
154(23)
4.3.1 Lie Derivative and Gauge Invariance
155(4)
4.3.2 Poincare's Group of Transformations
159(2)
4.3.3 Poincare's Gauge Invariance and Conservation Laws
161(7)
4.3.4 Conservation Laws in a Curved Spacetime with Torsion
168(9)
5 Topics in Continuum Mechanics and Gravitation
177(62)
5.1 Introduction
177(1)
5.2 Continuum Mechanics in a Newton Spacetime
178(28)
5.2.1 Classical Continuum in Newtonian Spacetime
178(6)
5.2.2 Continuum with Torsion in a Newtonian Spacetime
184(8)
5.2.3 Curved Continuum in a Newtonian Spacetime
192(8)
5.2.4 Causal Model of Curved Continuum
200(6)
5.3 Gravitational Waves
206(17)
5.3.1 Basic Equations
206(2)
5.3.2 Equations of Linearized Gravitation Waves
208(2)
5.3.3 Limit Case of Newton Gravitation
210(1)
5.3.4 Gravitational Waves
211(4)
5.3.5 Elementary Bases for Measurement of Gravitational Waves
215(1)
5.3.6 Vacuum Spacetime with Torsion
216(7)
5.4 Geodesic and Autoparallel Deviation for Gravitational Waves
223(16)
5.4.1 Geodesic Equation for Newtonian Mechanics
223(2)
5.4.2 Geodesic Deviation Equation in Riemannian Manifold
225(5)
5.4.3 Autoparallel Deviation in Riemann--Cartan Spacetime
230(9)
6 Topics in Gravitation and Electromagnetism
239(62)
6.1 Introduction
239(1)
6.2 Electromagnetism in Minkowskian Vacuum
240(18)
6.2.1 Maxwell's 3D Equations in Vacuum
241(9)
6.2.2 Covariant Formulation of Maxwell's Equations
250(6)
6.2.3 Maxwell's Equations in Terms of Differential Forms
256(2)
6.3 Electromagnetism in Curved Continuum
258(13)
6.3.1 Maxwell's Equations and Constitutive Laws
259(3)
6.3.2 Variational Method and Covariant Maxwell's Equations
262(9)
6.4 Electromagnetism in Curved Continuum with Torsion
271(10)
6.4.1 Electromagnetic Strength (Faraday Tensor)
271(2)
6.4.2 Electromagnetism Interacting with Gravitation
273(8)
6.5 Einstein-Cartan Gravitation and Electromagnetism
281(16)
6.5.1 Reissner-Nordstrom Spacetime
281(4)
6.5.2 Schwarzschild Anti-de Sitter (A d S) Spacetimes
285(5)
6.5.3 Extension to Electromagnetism-Matter Interaction
290(4)
6.5.4 Geodesics in a Anti-de Sitter Spacetime
294(3)
6.6 Summary on Gravitation-Electromagnetism Interaction
297(4)
7 General Conclusion
301(2)
Appendix
303(14)
A.1 Lorentz Transformation
303(3)
A.2 Some Relations for the Connection
306(1)
A.3 Algebraic Relations for Bi-connection
307(3)
A.3.1 Identification of Coefficients
307(1)
A.3.2 Coefficients of Bi-connection
308(2)
A.4 Lie Derivative and Exterior Derivative on Manifold M
310(7)
A.4.1 Lie Derivative
310(2)
A.4.2 Practical Formula for Lie Derivative
312(1)
A.4.3 Exterior Derivative
312(2)
A.4.4 Stokes' Theorem
314(3)
References 317
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