Tensors, Relativity, and Cosmology

This book combines relativity, astrophysics, and cosmology in a single volume, providing an introduction to each subject that enables students to understand more detailed treatises as well as the current literature. The section on general relativity gives the case for a curved space-time, presents the mathematical background (tensor calculus, Riemannian geometry), discusses the Einstein equation and its solutions (including black holes, Penrose processes, and similar topics), and considers the energy-momentum tensor for various solutions. The next section on relativistic astrophysics discusses stellar contraction and collapse, neutron stars and their equations of state, black holes, and accretion onto collapsed objects. Lastly, the section on cosmology discusses various cosmological models, observational tests, and scenarios for the early universe.

Table of Contents:
1 Introduction Part I. TENSOR ALGEBRA 2 Notation and Systems of Numbers 2.1 Introduction and Basic Concepts 2.2 Symmetric and Antisymmetric Systems 2.3 Operations with Systems 2.3.1 Addition and Subtraction of Systems 2.3.2 Direct Product of Systems 2.3.3 Contraction of Systems 2.3.4 Composition of Systems 2.4 Summation Convention 2.5 Unit Symmetric and Antisymmetric Systems 3 Vector Spaces 3.1 Introduction and Basic Concepts 3.2 Defnition of a Vector Space 3.3 The Euclidean Metric Space 3.4 The Riemannian Spaces 4 Definitions of Tensors 4.1 Transformations of Variables 4.2 Contravariant Vectors 4.3 Covariant Vectors 4.4 Invariants (Scalars) 4.5 Contravariant Tensors 4.6 Covariant Tensors 4.7 Mixed Tensors 4.8 Symmetry Properties of Tensors 4.9 Symmetric and Antisymmetric Parts of Tensors 4.10 Tensor Character of Systems 5 Relative Tensors 5.1 Introduction and Definitions 5.2 Unit Antisymmetric Tensors 5.3 Vector Product in Three Dimensions 5.4 Mixed Product in Three Dimensions 5.5 Orthogonal Coordinate Transformations 5.5.1 Rotations of Descartes Coordinates 5.5.2 Translations of Descartes Coordinates 5.5.3 Inversions of Descartes Coordinates 5.5.4 Axial Vectors and Pseudoscalars in Descartes Coordinates 6 The Metric Tensor 6.1 Introduction and Definitions 6.2 Associated Vectors and Tensors 6.3 Arc Length of Curves. Unit Vectors 6.4 Angles between Vectors 6.5 Schwarz Inequality 6.6 Orthogonal and Physical Vector Coordinates 7 Tensors as Linear Operators Part II. TENSOR ANALYSIS 8 Tensor Derivatives 8.1 Differentials of Tensors 8.1.1 Differentials of Contravariant Vectors 8.1.2 Differentials of Covariant Vectors 8.2 Covariant Derivatives 8.2.1 Covariant Derivatives of Vectors 8.2.2 Covariant Derivatives of Tensors 8.3 Properties of Covariant Derivatives 8.4 Absolute Derivatives of Tensors 9 Christoffel Symbols 9.1 Properties of Christoff Symbols 9.2 Relation to the Metric Tensor 10 Differential Operators 10.1 The Hamiltonian r-Operator 10.2 Gradient of Scalars 10.3 Divergence of Vectors and Tensors 10.4 Curl of Vectors 10.5 Laplacian of Scalars and Tensors 10.6 Integral Theorems for Tensor Fields 10.6.1 Stokes Theorem 10.6.2 Gauss Theorem 11 Geodesic Lines 11.1 Lagrange Equations 11.2 Geodesic Equations 12 The Curvature Tensor 12.1 Definition of the Curvature Tensor 12.2 Properties of the Curvature Tensor 12.3 Commutator of Covariant Derivatives 12.4 Ricci Tensor and Scalar 12.5 Curvature Tensor Components Part III. SPECIAL THEORY OF RELATIVITY 13 Relativistic Kinematics 13.1 The Principle of Relativity 13.2 Invariance of the Speed of Light 13.3 The Interval between Events 13.4 Lorentz Transformations 13.5 Velocity and Acceleration Vectors 14 Relativistic Dynamics 14.1 Lagrange Equations 14.2 Energy-Momentum Vector 14.2.1 Introduction and Definitions 14.2.2 Transformations of Energy-Momentum 14.2.3 Conservation of Energy-Momentum 14.3 Angular Momentum Tensor 15 Electromagnetic Fields 15.1 Electromagnetic Field Tensor 15.2 Gauge Invariance 15.3 Lorentz Transformations and Invariants 16 Electromagnetic Field Equations 16.1 Electromagnetic Current Vector 16.2 Maxwell Equations 16.3 Electromagnetic Potentials 16.4 Energy-Momentum Tensor Part IV. GENERAL THEORY OF RELATIVITY 17 Gravitational Fields 17.1 Introduction 17.2 Time Intervals and Distances 17.3 Particle Dynamics 17.4 Electromagnetic Field Equations 18 Gravitational Field Equations 18.1 The Action Integral 18.2 Action for Matter Fields 18.3 Einstein Field Equations 19 Solutions of Field Equations 19.1 The Newton Law 19.2 The Schwarzschild Solution 20 Applications of Schwarzschild Metric 20.1 The Perihelion Advance 20.2 The Black Holes Part V ELEMENTS OF COSMOLOGY 21 The Robertson-Walker Metric 21.1 Introduction and Basic Observations 21.2 Metric Definition and Properties 21.3 The Hubble Law 21.4 The Cosmological Red Shifts 22 The Cosmic Dynamics 22.1 The Einstein Tensor 22.2 Friedmann Equations 23 Non-static Models of the Universe 23.1 Solutions of Friedmann Equations 23.1.1 The Flat Model (k = 0) 23.1.2 The Closed Model (k = 1) 23.1.3 The Open Model (k = -1) 23.2 Closed or Open Universe 23.3 Newtonian Cosmology 24 The Quantum Cosmology 24.1 Introduction 24.2 Wheeler-DeWitt Equation 24.3 The Wave Function of the Universe Bibliography Index