Tensor Calculus, Relativity, and Cosmology A First Course by M. Dalarsson Ericsson Research and Development Stockholm, Sweden and N. Dalarsson Royal Institute of Technology Stockholm, Sweden ELSEVIER ACADEMIC PRESS Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo
Contents 1 Introduction 1 Part I Tensor Algebra 3 2 Notation and Systems of Numbers 5 2.1 Introduction and Basic Concepts 5 2.2 Symmetric and Antisymmetric Systems 7 2.3 Operations with Systems 8 2.3.1 Addition and Subtraction of Systems 8 2.3.2 Direct Product of Systems 8 2.3.3 Contraction of Systems 9 2.3.4 Composition of Systems 9 2.4 Summation Convention 10 2.5 Unit Symmetric and Antisymmetric Systems 11 3 Vector Spaces 15 3.1 Introduction and Basic Concepts 15 3.2 Definition of a Vector Space 16 3.3 The Euclidean Metric Space 18 3.4 The Riemannian Spaces 18 4 Definitions of Tensors 23 4.1 Transformations of Variables 23 4.2 Contravariant Vectors 24 4.3 Covariant Vectors 24 4.4 Invariants (Scalars) 24 4.5 Contravariant Tensors 25 4.6 Covariant Tensors 26 vii
viii Contents 4.7 Mixed Tensors 26 4.8 Symmetry Properties of Tensors 27 4.9 Symmetric and Antisymmetric Parts of Tensors 28 4.10 Tensor Character of Systems 30 5 Relative Tensors 33 5.1 Introduction and Definitions 33 5.2 Unit Antisymmetric Tensors 34 5.3 Vector Product in Three Dimensions 36 5.4 Mixed Product in Three Dimensions 38 5.5 Orthogonal Coordinate Transformations 39 5.5.1 Rotations of Descartes Coordinates 39 5.5.2 Translations of Descartes Coordinates 41 5.5.3 Inversions of Descartes Coordinates 41 5.5.4 Axial Vectors and Pseudoscalars in Descartes Coordinates 42 6 The Metric Tensor 43 6.1 Introduction and Definitions 43 6.2 Associated Vectors and Tensors 46 6.3 Arc Length of Curves: Unit Vectors 48 6.4 Angles between Vectors 49 6.5 Schwarz Inequality 51 6.6 Orthogonal and Physical Vector Coordinates 52 7 Tensors as Linear Operators 55 Part II Tensor Analysis 59 8 Tensor Derivatives 61 8.1 Differentials of Tensors 61 8.1.1 Differentials of Contravariant Vectors 64 8.1.2 Differentials of Covariant Vectors 64 8.2 Covariant Derivatives 65 8.2.1 Covariant Derivatives of Vectors 65 8.2.2 Covariant Derivatives of Tensors 66 8.3 Properties of Covariant Derivatives 67 8.4 Absolute Derivatives of Tensors 69 9 Christoffel Symbols 71 9.1 Properties of Christoffel Symbols 71 9.2 Relation to the Metric Tensor 74
Contents ix 10 Differential Operators 79 10.1 The Hamiltonian V-Operator 79 10.2 Gradient of Scalars 79 10.3 Divergence of Vectors and Tensors 80 10.4 Curl of Vectors 82 10.5 Laplacian of Scalars and Tensors 83 10.6 Integral Theorems for Tensor Fields 85 10.6.1 Stokes Theorem 85 10.6.2 Gauss Theorem 86 11 Geodesic Lines 89 11.1 Lagrange Equations 89 11.2 Geodesic Equations 92 12 The Curvature Tensor 97 12.1 Definition of the Curvature Tensor 97 12.2 Properties of the Curvature Tensor 100 12.3 Commutator of Covariant Derivatives 103 12.4 Ricci Tensor and Scalar 104 12.5 Curvature Tensor Components 105 Part III Special Theory of Relativity 109 13 Relativistic Kinematics 111 13.1 The Principle of Relativity 111 13.2 Invariance of the Speed of Light 112 13.3 The Interval between Events 112 13.4 Lorentz Transformations 116 13.5 Velocity and Acceleration Vectors 119 14 Relativistic Dynamics 123 14.1 Lagrange Equations 123 14.2 Energy-Momentum Vector 125 14.2.1 Introduction and Definitions 125 14.2.2 Transformations of Energy-Momentum 128 14.2.3 Conservation of Energy-Momentum 130 14.3 Angular Momentum Tensor 131 15 Electromagnetic Fields 135 15.1 Electromagnetic Field Tensor 135 15.2 Gauge Invariance 140 15.3 Lorentz Transformations and Invariants 142
x Contents 16 Electromagnetic Field Equations 147 16.1 Electromagnetic Current Vector 147 16.2 MaxwelLEquations 149 16.3 Electromagnetic Potentials 154 16.4 Energy-Momentum Tensor 155 Part IV General Theory of Relativity 163 17 Gravitational Fields 165 17.1 Introduction 165 17.2 Time Intervals and Distances 167 17.3 Particle Dynamics 169 17.4 Electromagnetic Field Equations 173 18 Gravitational Field Equations 177 18.1 The Action Integral 177 18.2 Action for Matter Fields 182 18.3 Einstein Field Equations 188 19 Solutions of Field Equations 193 19.1 The Newton Law 193 19.2 The Schwarzschild Solution 195 20 Applications of the Schwarzschild Metric 207 20.1 The Perihelion Advance 207 20.2 Black Holes' 215 Part V Elements of Cosmology 223 21 The Robertson-Walker Metric 225 21.1 Introduction and Basic Observations 225 21.2 Metric Definition and Properties 227 21.3 The Hubble Law 234 21.4 The Cosmological Red Shifts 235 22 Cosmic Dynamics 239 22.1 The Einstein Tensor 239 22.2 The Friedmann Equations 250 23 Nonstatic Models of the Universe 253 23.1 Solutions of the Friedmann Equations 253 23.1.1 The Flat Model (k = 0) 255
Contents xi 23.1.2 The Closed Model (k = 1) 255 23.1.3 The Open Model (k = -1) 257 23.2 Closed or Open Universe 258 23.3 Newtonian Cosmology 260 24 Quantum Cosmology 265 24.1 Introduction 265 24.2 The Wheeler-DeWitt Equation 266 24.3 The Wave Function of the Universe 270 Bibliograhy 275 Index 277