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Part Ⅰ Tensors and Riemannian spaces 3

1 Preliminaries 3

1.1 Vectors in linear spaces 3

1.1.1 Three-dimensional vectors 3

1.1.2 General case 7

1.2 Index notation.Summation convention 9

Exercises 10

2 Conservation laws 11

2.1 Conservation laws in classical mechanics 11

2.1.1 Free fall of a body near the earth 11

2.1.2 Fall ofa body in a viscous fluid 13

2.1.3 Discussion of Kepler's laws 16

2.2 General discussion of conservation laws 20

2.2.1 Conservation laws for ODEs 20

2.2.2 Conservation laws for PDEs 21

2.3 Conserved vectors defined by symmetries 27

2.3.1 Infinitesimal symmetries of differential equations 27

2.3.2 Euler-Lagrange equations.Noether's theorem 28

2.3.3 Method of nonlinear self-adjointness 36

2.3.4 Short pulse equation 40

2.3.5 Linear equations 43

Exercises 43

3 Introduction of tensors and Riemannian spaces 45

3.1 Tensors 45

3.1.1 Motivation 45

3.1.2 Covariant and contravariant vectors 46

3.1.3 Tensor algebra 47

3.2 Riemannian spaces 49

3.2.1 Differential metric form 49

3.2.2 Geodesics.The Christoffel symbols 52

3.2.3 Covariant differentiation.The Riemann tensor 54

3.2.4 Flat spaces 55

3.3 Application to ODEs 56

Exercises 59

4 Motions in Riemannian spaces 61

4.1 Introduction 61

4.2 Isometric motions 62

4.2.1 Definition 62

4.2.2 Killing equations 62

4.2.3 Isometric motions on the plane 63

4.2.4 Maximal group of isometric motions 64

4.3 Conformal motions 65

4.3.1 Definition 65

4.3.2 Generalized Killing equations 65

4.3.3 Conformally flat spaces 66

4.4 Generalized motions 67

4.4.1 Generalized motions,their invariants and defect 68

4.4.2 Invariant family of spaces 70

Exercises 71

Part Ⅱ Riemannian spaces of second-order equations 75

5 Riemannian spaces associated with linear PDEs 75

5.1 Covariant form of second-order equations 75

5.2 Conformally invariant equations 78

Exercises 78

6 Geometry of linear hyperbolic equations 79

6.1 Generalities 79

6.1.1 Covariant form of determining equations 79

6.1.2 Equivalence transformations 80

6.1.3 Existence of conformally invariant equations 81

6.2 Spaces with nontrivial conformal group 83

6.2.1 Definition of nontrivial conformal group 83

6.2.2 Classification of four-dimensional spaces 83

6.2.3 Uniqueness theorem 86

6.2.4 On spaces with trivial conformal group 87

6.3 Standard form of second-order equations 88

6.3.1 Curved wave operator in V4 with nontrivial conformal group 88

6.3.2 Standard form of hyperbolic equations with nontrivial conformal group 90

Exercises 90

7 Solution of the initial value problem 93

7.1 The Cauchy problem 93

7.1.1 Reduction to a particular Cauchy problem 93

7.1.2 Fourier transform and solution of the particular Cauchy problem 94

7.1.3 Simplification of the solution 95

7.1.4 Verification of the solution 97

7.1.5 Comparison with Poisson's formula 99

7.1.6 Solution of the general Cauchy problem 100

7.2 Geodesics in spaces with nontrivial conformal group 100

7.2.1 Outline of the approach 101

7.2.2 Equations of geodesics in spaces with nontrivial conformal group 102

7.2.3 Solution of equations for geodesics 102

7.2.4 Computation of the geodesic distance 104

7.3 The Huygens principle 105

7.3.1 Huygens'principle for classical wave equation 106

7.3.2 Huygens'principle for the curved wave operator in V4 with nontrivial conformal group 107

7.3.3 On spaces with trivial conformal group 107

Exercises 108

Part Ⅲ Theory of relativity 111

8 Brief introduction to relativity 111

8.1 Special relativity 111

8.1.1 Space-time intervals 111

8.1.2 The Lorentz group 112

8.1.3 Relativistic principle of least action 113

8.1.4 Relativistic Lagrangian 114

8.1.5 Conservation laws in relativistic mechanics 115

8.2 The Maxwell equations 116

8.2.1 Introduction 116

8.2.2 Symmetries of Maxwell's equations 117

8.2.3 General discussion of conservation laws 119

8.2.4 Evolutionary part of Maxwell's equations 122

8.2.5 Conservation laws of Eqs.(8.2.1 )and(8.2.2 ) 129

8.3 The Dirac equation 132

8.3.1 Lagrangian obtained from the formal Lagrangian 133

8.3.2 Symmetries 134

8.3.3 Conservation laws 136

8.4 General relativity 137

8.4.1 The Einstein equations 137

8.4.2 The Schwarzschild space 138

8.4.3 Discussion of Mercury's parallax 138

8.4.4 Solutions based on generalized motions 139

Exercises 141

9 Relativity in de Sitter space 143

9.1 The de Sitter space 143

9.1.1 Introduction 143

9.1.2 Reminder of the notation 145

9.1.3 Spaces of constant Riemannian curvature 147

9.1.4 Killing vectors in spaces of constant curvature 148

9.1.5 Spaces with positive definite metric 149

9.1.6 Geometric realization of the de Sitter metric 152

9.2 The de Sitter group 153

9.2.1 Generators of the de Sitter group 153

9.2.2 Conformal transformations in R3 154

9.2.3 Inversion 156

9.2.4 Generalized translation in direction of x-axis 158

9.3 Approximate de Sitter group 158

9.3.1 Approximate groups 158

9.3.2 Simple method of solution of Killing's equations 161

9.3.3 Approximate representation of de Sitter group 163

9.4 Motion of a particle in de Sitter space 165

9.4.1 Introduction 165

9.4.2 Conservation laws in Minkowski space 166

9.4.3 Conservation laws in de Sitter space 168

9.4.4 Kepler's problem in de Sitter space 169

9.5 Curved wave operator 171

9.6 Neutrinos in de Sitter space 172

9.6.1 Two approximate representations of Dirac's equations in de Sitter space 173

9.6.2 Splitting of neutrinos by curvature 174

Exercises 175

Bibliography 177

Index 181