Tensors, Relativity, and Cosmology

by Nils Dalarsson, Mirjana Dalarsson MSc - Engineering Physics 1984 Licentiate - Engineering Physics 1989

Overview

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

Product Details

ISBN-13:	9780080575438
Publisher:	Elsevier Science & Technology Books
Publication date:	04/29/2005
Sold by:	Barnes & Noble
Format:	eBook
Pages:	320
File size:	5 MB

About the Author

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.

Table of Contents

1 IntroductionPart I. TENSOR ALGEBRA2 Notation and Systems of Numbers2.1 Introduction and Basic Concepts2.2 Symmetric and Antisymmetric Systems2.3 Operations with Systems2.3.1 Addition and Subtraction of Systems2.3.2 Direct Product of Systems2.3.3 Contraction of Systems2.3.4 Composition of Systems2.4 Summation Convention2.5 Unit Symmetric and Antisymmetric Systems3 Vector Spaces3.1 Introduction and Basic Concepts3.2 Defnition of a Vector Space3.3 The Euclidean Metric Space3.4 The Riemannian Spaces4 Definitions of Tensors4.1 Transformations of Variables4.2 Contravariant Vectors4.3 Covariant Vectors4.4 Invariants (Scalars)4.5 Contravariant Tensors4.6 Covariant Tensors4.7 Mixed Tensors4.8 Symmetry Properties of Tensors4.9 Symmetric and Antisymmetric Parts of Tensors4.10 Tensor Character of Systems5 Relative Tensors5.1 Introduction and Definitions5.2 Unit Antisymmetric Tensors5.3 Vector Product in Three Dimensions5.4 Mixed Product in Three Dimensions5.5 Orthogonal Coordinate Transformations5.5.1 Rotations of Descartes Coordinates5.5.2 Translations of Descartes Coordinates5.5.3 Inversions of Descartes Coordinates5.5.4 Axial Vectors and Pseudoscalars in DescartesCoordinates6 The Metric Tensor6.1 Introduction and Definitions6.2 Associated Vectors and Tensors6.3 Arc Length of Curves. Unit Vectors6.4 Angles between Vectors6.5 Schwarz Inequality6.6 Orthogonal and Physical Vector Coordinates7 Tensors as Linear OperatorsPart II. TENSOR ANALYSIS8 Tensor Derivatives8.1 Differentials of Tensors8.1.1 Differentials of Contravariant Vectors8.1.2 Differentials of Covariant Vectors8.2 Covariant Derivatives8.2.1 Covariant Derivatives of Vectors8.2.2 Covariant Derivatives of Tensors8.3 Properties of Covariant Derivatives8.4 Absolute Derivatives of Tensors9 Christoffel Symbols9.1 Properties of Christoff Symbols9.2 Relation to the Metric Tensor10 Differential Operators10.1 The Hamiltonian r-Operator10.2 Gradient of Scalars10.3 Divergence of Vectors and Tensors10.4 Curl of Vectors10.5 Laplacian of Scalars and Tensors10.6 Integral Theorems for Tensor Fields10.6.1 Stokes Theorem10.6.2 Gauss Theorem11 Geodesic Lines11.1 Lagrange Equations11.2 Geodesic Equations12 The Curvature Tensor12.1 Definition of the Curvature Tensor12.2 Properties of the Curvature Tensor12.3 Commutator of Covariant Derivatives12.4 Ricci Tensor and Scalar12.5 Curvature Tensor ComponentsPart III. SPECIAL THEORY OF RELATIVITY13 Relativistic Kinematics13.1 The Principle of Relativity13.2 Invariance of the Speed of Light13.3 The Interval between Events13.4 Lorentz Transformations13.5 Velocity and Acceleration Vectors14 Relativistic Dynamics14.1 Lagrange Equations14.2 Energy-Momentum Vector14.2.1 Introduction and Definitions14.2.2 Transformations of Energy-Momentum14.2.3 Conservation of Energy-Momentum14.3 Angular Momentum Tensor15 Electromagnetic Fields15.1 Electromagnetic Field Tensor15.2 Gauge Invariance15.3 Lorentz Transformations and Invariants16 Electromagnetic Field Equations16.1 Electromagnetic Current Vector16.2 Maxwell Equations16.3 Electromagnetic Potentials16.4 Energy-Momentum TensorPart IV. GENERAL THEORY OF RELATIVITY17 Gravitational Fields17.1 Introduction17.2 Time Intervals and Distances17.3 Particle Dynamics17.4 Electromagnetic Field Equations18 Gravitational Field Equations18.1 The Action Integral18.2 Action for Matter Fields18.3 Einstein Field Equations19 Solutions of Field Equations19.1 The Newton Law19.2 The Schwarzschild Solution20 Applications of Schwarzschild Metric20.1 The Perihelion Advance20.2 The Black HolesPart V ELEMENTS OF COSMOLOGY21 The Robertson-Walker Metric21.1 Introduction and Basic Observations21.2 Metric Definition and Properties21.3 The Hubble Law21.4 The Cosmological Red Shifts22 The Cosmic Dynamics22.1 The Einstein Tensor22.2 Friedmann Equations23 Non-static Models of the Universe23.1 Solutions of Friedmann Equations23.1.1 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 Universe23.

What People are Saying About This

From the Publisher

Easy-to-understand text combines relativity, astrophysics, and cosmology in a single volume, providing a simplified introduction to each subject followed by detailed mathematical derivations

Editorial Reviews

"…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

From the Publisher