Mathematics ofDifferential Geometry

and Relativity

TENSORS

Zafar Ahsan

ok UV Title only(TENSORS)

TENSORS Mathematics of Differential Geometry

and Relativity

TENSORSMathematics of Differential Geometry

and Relativity

ZAFAR AHSANDepartment of MathematicsAligarh Muslim University

Aligarh

Delhi-1100922015

TENSORS: Mathematics of Differential Geometry and RelativityZafar Ahsan

© 2015 by PHI Learning Private Limited, Delhi. All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher.

ISBN-978-81-203-5088-5

The export rights of this book are vested solely with the publisher.

Published by Asoke K. Ghosh, PHI Learning Private Limited, Rimjhim House, 111, Patparganj Industrial Estate, Delhi-110092 and Printed by Mudrak, 30-A, Patparganj, Delhi-110091.

Contents

Preface............................................................................................... vii

1. TensorsandTheirAlgebra....................................................................1

1.1 Introduction 1 1.2 TransformationofCoordinates 3 1.3 SummationConvention 5 1.4 KroneckerDelta 5 1.5 Scalar,ContravariantandCovariantVectors 7 1.6 TensorsofHigherRank 11 1.7 SymmetryofTensors 14 1.8 AlgebraofTensors 21 1.9 IrreducibleTensor 27 Exercises 28

2. RiemannianSpaceandMetricTensor.................................................31

2.1 Introduction 31 2.2 TheMetricTensor 33 2.3 RaisingandLoweringofIndices—AssociatedTensor 40 2.4 VectorMagnitude 41 2.5 RelativeandAbsoluteTensors 45 2.6 Levi-CivitaTensor 46 Exercises 52

3. ChristoffelSymbolsandCovariantDifferentiation.............................54 3.1 Introduction 54 3.2 ChristoffelSymbols 54 3.3 TransformationLawsforChristoffelSymbols 57 3.4 EquationofaGeodesic 58 3.5 AffineParameter 61 3.6 GeodesicCoordinateSystem 62 3.7 CovariantDifferentiation 72 3.8 RulesforCovariantDifferentiation 76 3.9 SomeUsefulFormulas 79 3.10 IntrinsicDerivative:ParallelTransport 87

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vi Contents

3.11 NullGeodesics 89 3.12 AlternativeDerivationofEquationofGeodesic 93 Exercises 95

4. TheRiemannCurvatureTensor..........................................................97 4.1 Introduction 97 4.2 TheRiemannCurvatureTensor 97 4.3 CommutationofCovariantDerivative:AnotherWayof DefiningtheRiemannCurvatureTensor 99 4.4 CovariantformoftheRiemannCurvatureTensor 101 4.5 PropertiesoftheRiemannCurvatureTensor 102 4.6 UniquenessofRiemannCurvatureTensor 107 4.7 NumberofAlgebraicallyIndependentComponentsof theRiemannCurvatureTensor 109 4.8 TheRicciTensorandtheScalarCurvature 111 4.9 TheEinsteinTensor 116 4.10 TheIntegrabilityofRiemannTensorand theFlatnessoftheSpace 123 4.11 EinsteinSpace 130 4.12 CurvatureofaRiemannianSpace 132 4.13 SpacesofConstantCurvature 134 Exercises 136

5. SomeAdvancedTopics...................................................................... 138 5.1 Introduction 138 5.2 GeodesicDeviation 138 5.3 DecompositionofRiemannCurvatureTensor 142 5.4 ElectricandMagneticPartsoftheRiemannand WeylTensors 152 5.5 ClassificationofGravitationalFields 154 5.6 InvariantsoftheRiemannTensor 156 5.7 LieDerivative 159 5.8 TheKillingEquation 167 5.9 TheCurvatureTensorandKillingVectors 171 5.10 CurvatureTensorsInvolvingRiemannTensor 175 Exercises 182

6. Applications....................................................................................... 183 6.1 Introduction 183 6.2 Maxwell’sEquations 183 6.3 SpecialCoordinateSystem 190 6.4 Energy-momentumTensors 192 6.5 KinematicalQuantities—Raychaudhuri’sEquation 196 6.6 SolutionsofEinsteinFieldEquations 202 6.7 SomeImportantTensors 214

Bibliography....................................................................... 219

AnswersandHintstoExercises........................................... 221

Index................................................................................. 229

Preface

Theprincipalaimoftensoranalysisistoinvestigatetherelationswhichremainvalidwhenwechangefromonecoordinatesystemtoanyother.Thelawsofphysicscannotdependontheframeofreferencewhichthephysicistchoosesforthedescriptionofsuchlaws.Accordinglyitisaestheticallydesirableandsometimeconvenient toutilise tensorsas themathematicalbackground inwhich these lawscanbe formulated.AlbertEinstein (1879–1955) found itanexcellenttoolforthepresentationofhisgeneraltheoryofrelativityandasaresult,tensorscameintogreatprominance.Nowithasapplicationsinmostbranchesoftheoreticalphysicsandengineering,suchasmechanics,fluidmechanics,elasticity,plasticityandelectromagnetism,etc.

The present book is intended to serve as a text for the postgraduatestudents of mathematics, physics and engineering. It is ideally suited forthestudentsandteacherswhoareengagedinresearchingeneraltheoryofrelativityanddifferentialgeometry.Thebookisselfcontainedandrequiresonlyaknowledgeofelementarycalculus,differentialequationsandclassicalmechanics as pre-requisites. It comprises six chapters, and each chaptercontainsa largenumberofsolvedexamples.Eachchapterendsupwithacarefullyselectedsetofunsolvedproblems,andtheanswersandhintsforthesolutionoftheseproblemsaregivenattheendofthebook.

Chapter 1 deals with an introduction of tensors and their algebra.The symmetryproperties of the tensorshave alsobeendiscussedhere. InChapter2,thenotionoftheRiemannianspaceisdefinedwhichleadtotheconceptof fundamental tensors.Relative,absoluteandLevi-Civitatensorsare also defined and discussed here along with related results. Christoffelsymbols, covariant and intrinsic differentiations and related results formthecontentsofChapter3alongwiththeequationofageodesicandaffineparameter.InChapter4,adetailedaccountoftheRiemanncurvaturetensoranditspropertiesisdiscussed.TheintegrabilityconditionanduniquenessoftheRiemanntensoralongwiththeRicciandEinsteintensors,theflatnessofthespace,theEinsteinspacesandthespacesofconstantcurvaturehavealsobeendiscussedinthischapter.Chapter5dealswithsomeadvancedtopicslike equation of geodesic deviation, the decomposition ofRiemann tensor,itsinvariantsandtheclassificationofgravitationalfields.ThischapteralsocoversadetaileddiscussionofLiederivativeandrelatedresults.

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viii Preface

The final chapter deals with the applications of tensors to relativitytheoryanddifferentialgeometry.Thebookconcludeswithalistofreferences.Mathematicalequationsappearinginthebookhavebeennumberedseriallyineachchapter.If,forexample,Eq.(12)ofChapter1isusedinanysubsequentchapter,itwillrepresentedbyEq.(1.12).

While preparing the manuscript of this book, I have consulted manystandardworks.Iamindebtedtotheauthorsoftheseworks.Iamthankfultomypublisher,PHILearning,inparticular,theeditorialandproductionteamsfortheirniceeffortsinbringingoutthebook.Finally,Iwishtothankmyfamilymembersfortheencouragementandpatiencetheyhaveshownduringthepreparationofthemanuscript.

Anysuggestionsorcommentsforimprovingthecontentswillbewarmlyappreciated.

ZafarAhsan

Chapter 1

Tensors and Their Algebra

1.1 Introduction

In many areas of mathematical, physical and engineering sciences, it is oftennecessary to consider two types of quantities. First, those which havemagnitude only. Such quantities are known as scalars. Mass, length,volume, density, work, electric charge, time, temperature, etc. are theexamples of scalars. The second type are those which have both magnitudeand direction. These are known as vectors. Some of the examples of vectorsare velocity, acceleration, force, momentum, etc.

Quite often the notion of vector is not sufficient to represent a physicalquantity. What happens when we need to keep the track of two (or more)pieces of information for a given physical quantity? For such situation, weneed a tensor. A tensor contains the information about the directions andthe magnitudes in those directions. Thus, for example, the stress at a pointdepends upon two directions; one normal to the surface and the other thatrepresents the force creating stress and thus stress cannot be described by avector quantity. As another example, the measurement of charge density willdepend upon the four velocity of the observer and thus can be represented bya vector, while the measurement of electric field strength in some directionwill not only depend upon this direction but also on the four velocity of theobserver and thus such measurement cannot be described by a vector quantityalone. These and similar other examples led to the generalization of a vectorquantity to a quantity known as tensor.

Life would have been miserable without tensors as we cannot walk across aroom without using a tensor (the pressure tensor), it is impossible to align thewheels on our car without using a tensor (the inertia tensor) and definitelyone cannot understand Einstein’s theory of gravity without using tensors.The word “tensor” itself was introduced in 1846 by William Rowan Hamiltonto describe something different from what is now meant by a tensor. Thecontemporary usage was brought in by Woldemar Voigt in 1898. The wordtensor is derived from the Latin word tensus meaning stress or tension. Inanatomy the word tensor means a muscle that stretches or tightens some partof the body.

The concept of tensors has its origin in the development of differentialgeometry by Gauss, Riemann and Christoffel. The tensor calculus (also

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Tensors: Mathematics Of DifferentialGeometry And Relativity

Publisher : PHI Learning ISBN : 9788120350885 Author : AHSAN, ZAFAR

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