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.

* Clearly combines relativity, astrophysics, and cosmology in a single volume so students can understand more detailed treatises and current literature

* Extensive introductions to each section are followed by relevant examples and numerous exercises

* Provides an easy-to-understand approach to this advanced field of mathematics and modern physics by providing highly detailed derivations of all equations and results

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