Tensor Calculus, Relativity, and Cosmology

Transcription

1 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

2 Contents 1 Introduction 1 Part I Tensor Algebra 3 2 Notation and Systems of Numbers Introduction and Basic Concepts Symmetric and Antisymmetric Systems Operations with Systems Addition and Subtraction of Systems Direct Product of Systems Contraction of Systems Composition of Systems Summation Convention Unit Symmetric and Antisymmetric Systems 11 3 Vector Spaces Introduction and Basic Concepts Definition of a Vector Space The Euclidean Metric Space The Riemannian Spaces 18 4 Definitions of Tensors Transformations of Variables Contravariant Vectors Covariant Vectors Invariants (Scalars) Contravariant Tensors Covariant Tensors 26 vii

3 viii Contents 4.7 Mixed Tensors Symmetry Properties of Tensors Symmetric and Antisymmetric Parts of Tensors Tensor Character of Systems 30 5 Relative Tensors Introduction and Definitions Unit Antisymmetric Tensors Vector Product in Three Dimensions Mixed Product in Three Dimensions Orthogonal Coordinate Transformations Rotations of Descartes Coordinates Translations of Descartes Coordinates Inversions of Descartes Coordinates Axial Vectors and Pseudoscalars in Descartes Coordinates 42 6 The Metric Tensor Introduction and Definitions Associated Vectors and Tensors Arc Length of Curves: Unit Vectors Angles between Vectors Schwarz Inequality Orthogonal and Physical Vector Coordinates 52 7 Tensors as Linear Operators 55 Part II Tensor Analysis 59 8 Tensor Derivatives Differentials of Tensors Differentials of Contravariant Vectors Differentials of Covariant Vectors Covariant Derivatives Covariant Derivatives of Vectors Covariant Derivatives of Tensors Properties of Covariant Derivatives Absolute Derivatives of Tensors 69 9 Christoffel Symbols Properties of Christoffel Symbols Relation to the Metric Tensor 74

4 Contents ix 10 Differential Operators The Hamiltonian V-Operator Gradient of Scalars Divergence of Vectors and Tensors Curl of Vectors Laplacian of Scalars and Tensors Integral Theorems for Tensor Fields Stokes Theorem Gauss Theorem Geodesic Lines Lagrange Equations Geodesic Equations The Curvature Tensor Definition of the Curvature Tensor Properties of the Curvature Tensor Commutator of Covariant Derivatives Ricci Tensor and Scalar Curvature Tensor Components 105 Part III Special Theory of Relativity Relativistic Kinematics The Principle of Relativity Invariance of the Speed of Light The Interval between Events Lorentz Transformations Velocity and Acceleration Vectors Relativistic Dynamics Lagrange Equations Energy-Momentum Vector Introduction and Definitions Transformations of Energy-Momentum Conservation of Energy-Momentum Angular Momentum Tensor Electromagnetic Fields Electromagnetic Field Tensor Gauge Invariance Lorentz Transformations and Invariants 142

5 x Contents 16 Electromagnetic Field Equations Electromagnetic Current Vector MaxwelLEquations Electromagnetic Potentials Energy-Momentum Tensor 155 Part IV General Theory of Relativity Gravitational Fields Introduction Time Intervals and Distances Particle Dynamics Electromagnetic Field Equations Gravitational Field Equations The Action Integral Action for Matter Fields Einstein Field Equations Solutions of Field Equations The Newton Law The Schwarzschild Solution Applications of the Schwarzschild Metric The Perihelion Advance Black Holes' 215 Part V Elements of Cosmology The Robertson-Walker Metric Introduction and Basic Observations Metric Definition and Properties The Hubble Law The Cosmological Red Shifts Cosmic Dynamics The Einstein Tensor The Friedmann Equations Nonstatic Models of the Universe Solutions of the Friedmann Equations The Flat Model (k = 0) 255

6 Contents xi The Closed Model (k = 1) The Open Model (k = -1) Closed or Open Universe Newtonian Cosmology Quantum Cosmology Introduction The Wheeler-DeWitt Equation The Wave Function of the Universe 270 Bibliograhy 275 Index 277