# Tensors, Relativity, and Cosmology

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## Description

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.

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## Product details

- Paperback | 320 pages
- 154 x 228 x 22mm | 598.75g
- 29 Apr 2005
- Elsevier Science Publishing Co Inc
- Academic Press Inc
- San Diego, United States
- English
- New.
- 012200681X
- 9780122006814
- 1,415,745

## 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

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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

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## Review quote

"...it's absolutely perfect if you need help with the mathematical aspects of relativity...Many notions from tensor algebra and differential geometry are introduced and explained very clearly, and the main essential formulas are all presented in details."--BookInspections.com, May 27, 2013

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## About Nils Dalarsson

Nils Dalarsson has been with the Royal Institute of Technology, Department of Theoretical Physics in Stockholm, Sweden, since 1999. His research and teaching experience spans 32 years. Former academic and private sector affiliations include University of Virginia, Uppsala University, FSB Corporation, France Telecom Corporation, Ericsson Corporation, and ABB Corporation. He holds a PhD in Theoretical Physics, an MSc in Engineering Physics, and an MSc in Education. Mirjana Dalarsson is affiliated with the Research and Development program at Ericsson Corporation. She holds a Licentiate degree in Engineering Physics and has more than 25 years of research and teaching experience. Former affiliations in the academic and private sector include the Royal Institute of Technology, Belgrade University, Uppsala University, and ABB Corporation.

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