Tensor Calculus, Relativity and Cosmology

Tensor Calculus, Relativity, and Cosmology A First Course by M. Dalarsson Ericsson Research and Development Stockholm, Sweden and

N. Dalarsson

Royal Institute ofTechrwlogy Stockholm, Sweden

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Preface The subjects of relativity, gravitation, and cosmology are frequently discussed areas of modem physics among the broad audience. Numerous popular books on these subjects are available for the general reader with no background in physics. At the same time a number of textbook~ and monographs for advanced graduate students and young researchers in the area is available as well. However, intermediate books, allowing senior undergraduate students or junior PhD students to enter this exciting areaofphysics in a smooth and pedagogical way, are quite rare. It is generally assumed that new concepts of both physics and mathematics involved in the subject are too complex to allow a simple and pedagogical introduction. The only way to master the subject seems to be a hard and timeconsuming effort to put together the intelligible pieces of advanced textbocks into an understandable set of notes, as experienced by onrselves and many of our colleagues as well. This approach has the advantage of making young students with the capability and perseverance to go through such a learning process very well trained for future research tasks. However, atthe same time, it severely limits thenumberofpeople whoeverreallymasterthe subject. This book is an attempt to bridge the gap between a regular university curriculum, consisting typically of conrses in calculus and general physics, and the more advanced books in tensor cakulus, relativity, and cosmology. The book has evolved from a set of lecture notes originally compiled by one of the authors, M. Dalarsson, but has been improved and completed with a few exciting new t<>pics over the past IO years. It is the intention of the book to give to the readers a high level of detail in derivations of all equations and results. The more lengthy and tedious algebraic manipulations are in general outlined in such detail that they can

vi

Preface

be followed by an interested senior undergraduate student or a junior PhD stu&nt with very little or no risk of ever getting lost. It is our experience that a common showstopper for a young university student trying to master a subject are phrases in the literature claiming that something can be derived from something else by some "straightforward although somewhat tedious algebra." If a student cannot readily reproduce such a "straightforward" algebra, which most often is the case. the usual reaction under the time pressure Of the studies is to accept the claim as

a fact. And from that point, throughout the rest of the course, the deeper undcrstanding of the subject is lost. There are a number of advanced books on relativity, gravitation, and cosmology, and we have benefited from some of those as well as from unpublished notes produced by someofourdistinguishedrolleagues. Some of these sources are listed in the hibliography at the end of the book, as well as a few other books that may be recommended as suitable further reading. However, in an introductory book such as this one, it is pussible neither to mclude an extensive list of all original references and major textbooks and monographs on the subject nor to mention all the people who have contributed to our understanding of this exciting subject. We hope that our readers will find that we have, at least partly, fulfilled the objective of bridging the gap between the regular university cuniculum and the more advanced literature on this exciting subject, and that they will enjoy reading this book as much as we did writing it. M. Dalarsson and N. Dalarsson

Stocklwlm, Sweden. Jawwry 2003

Contents Introduction

Part I Tensor Algebra 2

3

Notation and Systems of Numbers 2.1

Inrroduction and Basic Concepts

2.2 2.3

Symmetric and Antisymmetric Systems Operations wilh Systems 2.3_1 Addition and Subtraction of Sysrems 2.3.2

2.4 2.5

Direct Product of Systems

2.33 Contraction of Systems 2.3.4 Composition of Systems Summation Convention Unit Symmetric and Antlsymmetnc Systems

10 11

Vector Spaces

15

3.1

Introduction and Baste Concepts

32

Definitioo of a Vector Space

33 3.4

The Euclidean Metric Space The Riemannian Spaces

15 16 18 18

Definitions of Tensors 42

4.1

Transformations ofVariables Contravanant Vector.;:

4-3 4.4 4.5 4.6

Covanant Vectors Invariants (Scalars) Contravariant Tensors Covar1am Tensors

23 23

24 24 24 25

26

vii

Contents

viii

4.7 4.8 4.9 4.10

26 27 28 30

Relative Tensors

33

51

33 34 36 38 39 39 41 41

52 5.3 5.4 5.5

6

Mixed Tensors Symmetry Properties of Tensors Symmetric and Antisymmetric Parts ofTensors TensorCharacterofSystems

Introduction and Definitions Unit Antisymmetric Tensors Vector Product in Three Dimensions Mixed Productm Three Dimensions Orthogonal Coordinate Traru.-fonnations 5.5.1 Rotations of Descartes Coordinates 55.2 Translations of Descartes Coordinates 55.3 Inversions of Descartes Coordinates 5.5 4 Axial U!ctors and Pseudoscalars m Descartes Coordinates

42

The Metric Tensor

43

6.1 6.2 6.3 6.4 6.5

Angles between Vecton. Schwarz Inequality

43 46 48 49

6.6

Orthogonal and Physical Vector Coordinates

lntroduction and Definit10ns Associated Vectors and Tensors Arc Len,glh of Curves: Unit Vectors

Thnsors as Linear Operators Part II

Tensor Analysis

51 52

SS

59

Tensor Derivatives

61

8.1

Differentials ofTeitst'.n 8.1.1 Di.fferentialsofContravariantVectors 8.12 DifferentialsofCovariantVectors 8.2 Covariant Derivatwes 8.2.1 Covariant Derivatives ofVectors 8.2.2 Covariant Derivatives of Tensors 83 Properties of Covariant Derivatives 8.4 Absolute Derivatives of Tensors

61

Chrisroffel Symbols

71

9.1 92

71 74

Properties of Chri!>toffe1 Symbols Relation to lhe Metric Tensor

64 64 65 65 66 67

69

Contents

Ix

10 Differential Operators 10.1 10.2 10.3 10.4 10.5 10.6

The Hamiltonian V-Operator Gradient of Scalars Diver;gence of Vectors and Tensors Cnr1of\.ectorsLap1acian of Scalars and Tensors Integra1 TheorcrIB for Tensor Fields 10.6.1 Stokes Theorem 10.62 Gauss Theorem

11 Geodesic Lines 11.1 11.2

Lagrange Equations Geodesic Equatiori.s

12 The Curvature Tensor 12.l 12.2 12.3 12.4 12.5

Definition oflhe Curvature Tensor Properties oflhe Curvature Tensor Commutator of Covariant Deiivatives Ricci Tensor and Scalar Curvature Tensor Components

79 19 19 80 82 83 85 85 86 89 89 92

97 97 100 103 ]()4

105

Part Ill Special Theory of Relativity

109

13 Relativistic Kinematics

111

13.1 13.2 133 13.4 13.5

The Principle ofRelativity Invariance of lhe Speed of Light The Interval between Events Lorentz Transformations Velocity and Acceleration Vectors

14 Relativistic Dynamics 14.1 142

14.3

Lagrange Equations Energy-Momentum Vector 142.l Introduction and Definitions 14.2.2 Transformations of Energy-Momentum 14.2.3 Conservation of Energy-Momentum Angular Momentum Tensor

15 Electromagnetic Fields 15.1 152 153

Electromagnetic Field Tensor Gauge Invariance Lorentz Traru.-fonnations and Invariants

ll1 112 112 116 119

123 123 125 125 128 130 131

135 135 140 142

Contents

16 Eledromagnetic Field Equations 16_1 16.2 16.3 16.4

Electromagnetic Current Veaor Maxwell Equations Electromagnetic Potenti;tls Energy-Momentum Tensor

147 147 149 154 155

Part IV General Theory of Relativity

163

17 Gravitational Fields

165

17.1 17.2 17.3 17.4

Introduction Time Intervals and Distances Particle Dynamics Electromagnetic Field Equations

18 Gravitational Field Equations 18.l 18.2 18.3

The Action Integral Action for Matter Fields Einstein Field Equat10ns

19 Solutions of Field Equations

165 167 169 173

177 177 182 188

193

The Newton Law The Scbwan.sclrild Solution

193 195

20 Applications of the Schwarzsddld Metric

207

19.1 19.2

20.1 202

The Penhebon Advance Black Holes

WI 215

Eiaments of Cosmology

223

21 The Robertson-Walker Metric

225

Partv

2LI 2L2 213 21.4

Introduction and Basic Observations Mernc Defini.tion and Properties The Hubble Law The Cosmological Red Shifts

22 Cosmic Dynamics 22.1 22.2

The Einstein Tensor The Friedmann Equations

23 Nonstatic Models of the Universe 23.1

Solul:ions oflhe Friedmann Equations 23.LI The Flat Model (k = 0)

225 227 234 235

239 239 250

253 253 255

xi

Contents

23.2 23.3

23.L2 The Closed Model (k = 1) 23.L3 The Open Model (k = -I) Closed or Open Universe Newtonian Cosmology

24 Quantum Cosmology 24. I 24.2 24.3

lnlroduction The Wheeler-DeWitt Equation The Wave Function of the Universe

255 257 258

260 265 265

266 270

Bililiograhy

275

Index

277

... Chapter 1

Introduction

The tensor calculus is a mathematical discipline of relatively recent origin. It is fair to say that, with few exceptions, the tensor calculus was developed during the twentieth century. It is also an area of mathematics that was developed for an immediate practical use in the theory of relativity, with which it is strongly interrelated Later, however, the tensor calculus has

proven to be useful in other areas of physics and engineering such as classical mechanics of particles and continuous media, differential geometry, electrodynamics, quantum mechanics, solid-state physics, and quantum field theory. Recently, it has been used even in electric circuit theory and some other purely engineering disciplines. In the early twentieth century, at the same time when the tensor calculus was developed, a number of major breakthroughs in modem science were made. In 1905 the special theory of relativity was formulated, then in 1915 the general theory of relativity was developed, and in 1925 quantum mechanics took its present form. In the years to come quantum mechanics and the special theory of relativity were combined to develop the relativistic quantum field theory. which gives at least a partial explanation of the three fundamental fon:e.'> of nature (strong, electromagnetic, and weak). The remaining known fundamental force of nature, the force of gravity, is different from the other three fundamental force.'>. Although very weak on the small scale, gravity dominates the other three fon:e.'> over cosmic distances. This dominance, due to gravity being a long-range force that cannot be screened, makes it the only available foundation for any cosmology. The other three fundamental forces are explained through particle interactions

Chapter 1 lntroducllon

in the flat space-time Of special relativity. However, gravity does not allow for such an explanation. In order to explain gravity, Einstein had to connect it with the geometry of space-time and formulate a relativistic theory of gravitation. For a long time, general relativity was separate from the other parts of physics, partly because of the mathematical framework of the theory (tensor calculus), which was not extensively used in any other discipline during that time. The tensor calculus today is used in a number of other disciplines as well, and its extension to other areas of physics and engineering is a result of the simplification of the mathematical notation and in particular the possibility of natural extension of the equations to the relativistic case. Today. physics and a'\bOD.omy have joined forces to form the discipline called relativistic astrophysics. The major advances in cosmulogy, including the first attempts to formulare quantum cosmulogy. also increa'>e the importance of general relativity. Finally, a number of attempts have been made to unify gravity with the other three fundamental forces of nature, thus introducing the tensor calculus and Riemannian geometry to the new exciting areas of physics such as the theory of superstrings. In the first two parts of the book a pedagogical introduction to the tensor calculus is covered. Thereafter, an introduction to the special and general theories of relativity is presented. Finally an introduction to the modem theory of cosmology is discussed

Part I

Tensor Algebra

... Chapter 2

Notation and Systems of Numbers

~

Introduction and Basic Concepts

In order to get acquainted with the basic notation and concepts of the tensor calculus, it is convenient to use some well known concepts from linear algebra. The cullection of N elements of a column matrix is often denoted by subscripts as Xt.X2. •.• ,XN. Using a lower index i = I, 2, ... ,N, we can introduce the following short-hand notation: Xi

(i= 1,2.... ,N).

(2.1)

Sometimes. the same collection of N elements is denoted by corresponding superscripts asx 1 ,x2 , .• • ,x"". Using here an upper index i = 1,2, ... ,N, we can also introduce the following short-hand notation:

,! (i~ 1,2, ... ,N).

(2.2)

In general the choice of a lower or an uwer index to denote the collection of N elements of a column matrix is fully arbitrary. However, it will be shown later that in the tensor calculus lower and upper indices are used to denote mathematical objects of different natures. Both types of indices are therefore essentialforthedevelopmentof tensor calculus as a mathematical discipline. In the definition (2.2) it should be noted that i is an upper index and not a power of x. Whenever there is a risk of confusion of an upper index and a power, such as when we want to write a square of xi, we will

Chapter 2

Notation and Systems of Numbers

use parentheses as follows. x'-:J=(xi)2

(i=l,2, ... ,N).

(2.3)

A collection of numbers, defined by just one (upper or lower) index, will be called a first-order system or a simple system. The individual elements of such a system will be called the elements or coordinates of the system. The introduction of the lower and upper indices provides a device to highlight the different nature of different first-order systems with equal numbers of elements. Consider, for example, the following linear form: (2.4)

ax+by+c:z.

Introducing the labels llj can be written as

=

{ll,b,c} and xi= {x,y.z}, theexpession(2.4) 3

ll}X

1

+D2x2 +ll3X3 = LlliXi,

(2.5)

i=I

indicating the different nature of the two first-order systems. In order to emphasize the advantage of the proposed notation, let us consider a bilinear form created using two first-order systems xi and vi (i = 1.2,3). a1 1x 1y 1 + D12x1),2 + ll13x 1y3 +D31x3 y 1 + ll3z.x3y2

+ D21x2y 1 + a2zx2y2 + an.:x?-y3

+ ll3JX3y3 =

3

L

'

Llluxfyi

(2.6)

i=lj=l

Here we see that the short-hand notation on the right-hand side of Eq. (2.6) is quite compact. The system of parameters of the bilinear fonn (i.j= 1,2,3),

llij

(2.7)

is labeled by two lower indices. This system has nine elements and they can be repesented by the following 3 x 3 square matrix:

[

au

a12

ll21

a22

llJI

a32

a13] a23

(2.8)

.

a33

A system of quantities determined by two indices is called a second-order system Depending on whether the indices of a second-order system are upper or lower, there are three types of second-order systems: llif,

af,

llii

(i,j

= 1,2,

. . ,N).

(2.9)

Section 2 2 Synwneb1c and Antlsyrnrnetrlc Systems

A second-order system in N dimensions has N 2 elements. In a similar way we can define the third-order system.<;, which may be of one of four different types:

aijh

ajk•

af,

Jik

(i,j=l,2, ... ,N).

(2.IO)

The most general system of order K is denoted by (i1,i2, ... ,iK

Ui 1,ii•.. ,ix

= 1,2, ... ,N),

(2.11)

and, dependmg on the position of the indices, it may be of one of several different types. The Kth-order system in N dimensions has NK elements.

~

Symmetric and Antisymmetric Systems

Let us consider a second-order system in three dimensions Uij

(i,j

= 1,2,3).

(2.12)

The system (2.12) is called a symmetric system with respect to the two lower indices if the elements of the system satisfy the equality Uij =Uj1

= 1,2,3).

(i,j

(2.13)

Sirmlarly, the system (2.12) is called an antisymmetnc system with respect to the two lower indices if the elements of the system satisfy the equality aij

= -aii

(i,j

= 1,2,3)

(2.14)

The equality (2.14) indicates that an antisymmetric second-order system in three dimensions has only three independent components and that all the diagonal elements are equal to zero: UJJ

=0

(J

= 1,2,3).

(2.15)

Thus it is posstble to represent an antisymmetric second-order system in three dimensions by the following 3 x 3 matrix:

0 [-an

-a12

a12

0 -a23

a13] a23

.

(2.16)

0

In general, a system of an arbitrary ofder and type will be symmetric

with respect to two of its indices (both upper or both lower), if the corresponding elements remain unchanged upon interchange of these two indices. The system will be totally symmetric with respect to all upper (lower) indices, if an interchange of any two upper (lower) indices leaves the corresponding system elements unchanged. Elements of a totally

Chapter 2

Notation and Systems or Numbers

symmetric third-order system with all three lower indices satisfy the equality

•=%=•=·=~=~

ITTn

Analogously to the above, a system of an arl:iitrary order and type will be antisymmetric with respect to the two of the indices (both uwer or both lower), if the corresponding elements change signs upon interchange of these two indices. The system will be totally antisymmetric with respect to all upper (lower) indices, if an interchange of any two upper (lower) indices changes signs of the corresponding system elements. Elellrnts of a totally antisymmetric third-order system with all three lower indices satisfy the equality Uijk = -lljkj

= Ujki =

-Uj(k

= Ukij =

-Ulgi·

(2. J 8)

]) Operations with Systems Under certain conditions, it is possible to perform a number of algebraic operations with systems. The definition of these operations depends on the order and type of the systems. 2.3.1

Addition and Subtraction of Systems

The addition and subtraction of systems can be performed only with systems of the same order and same type. The addition (subtraction) of systems is performed in such a way that each element of one system is added (subtracted) to (from) the corresponding element of the other system (the one with the same indices in the same order). For example, the systems Afm and nfm can be added since they are of the same order and of the same type. The sum of these two systems is given by (2.19) and it is a system of the same order and type as the two onginal systems. Tiris definition is easily extended to addition and subtraction of an arbitrary number of systems. 2.3.2 Direct Product of Systems

A system obtained by multiplying each element of one system by each element of another system, regardless of their order and type, is called

Sectlon 2.3 Operations with Systems

a direct product or just a product of these two systems. Thus, for example, a product of two first-order systems ai and If is a second-order system

(2.20) For i,j = l, 2, 3, this operation can be written in the following matrix form:

(2.21)

In general, the set of upper (lower) indices of a system, created as a product of several other systems, is a collection of all upper (lower) indices of all of the constituent systems. For example, we have

(2.22) 2.3.3

Contraction of Systems

1bis operation is applicable to systems with at least one pair of indices of opposite type, i.e., at least one upper index and one lower index. The actual pair of indices of opposite type is then made equal to each other and a sum over that common index is performed. Thus, for example, by conrraction of a third-order system af, we obtain N

ai = LaJ = cil +aq +···+a~.

(2.23)

j=I

The contraction of a system of order k gives a system of order k - 2, which is easily seen from the example (2.23). The contraction of a mixed second-order system b} gives a zeroth-order system N

b= Lbf =bl +b~+· . +b~.

c2.24>

j=I

which is equal to the trace of the matnx [bj]. 2.3.4 Composition of Systems

The composition of systems is a complex operation consisting of a product of two systems and a contraction with respect to at least one of the indices of opposite type from each of the systems. The product of the two first-order systems cl'- and bi is a mixed second-order system d'-bj· By contraction

Chapter 2 Notation and Systems er Numbers

10

of this system. we obtain the composition of the systems the form

cl

and bi m

N

c=Lo!bj=a 1bi+a2bi+ ·-+d'bN

(225)

j=l

For N

= 3, the result (2.25) can be written in the following matrix form: (2.26)

~ Summation Convention At the very beginning of the development of the general theory ofrelativity, m order to simplify derivations of various results and expressions in the tensor cakulus, the summat:J.on convention over the repeated indices was introduced. According to this convention, when the same index m an expression appears twice, it is understood that the summation over that mdex is performed and no summation sign is needed. Thus, for example, we may write N

L

Oj:xi

= lljxi.

(2.Z/)

i=l N

N

LLajkxix.1' =ajkxix.1'.

(228)

i=lk=l N

N

LLvp:aik =D'j/,"aik_

(2.29)

j=lk=l

The repeated indices, over which a sununanon ts understood, are usually called dummy mdices When using the summation convention, the following rules should be kept in mind: 1. It is rnquired to know exactly which range of values aU indices can take H nothing else is specified, it is assumed that all indices in one expression or equat10n cover the same range of integers.

Section 2.5 Unit Symmetric and Antisymmetric Systems

11

2. When it is reqmred to represent any of the three diagonal elements a}, or a~, in order to avoid confusion with the summation convention, aM(M = 1,2,3)couldbeusedinsteadof0:::,(m = 1,2,3). In such a case the capital letters are only used for this purpose and are otherwise not used as mdices.

ai

3. In order to avoid confusion, whenever there are two or more pairs of dummy indices in the same expression, they shouW always be denoted by different letters and never by the same letters.

~

Unit Symmetric and Antisymmetric Systems

The unit synunetric system, called the Ci-symbol, is th~ symmetric second.order system, defined as follows: Ci!

= { 1,

1

0,

for 1 = J _ fori-::j:.j

(2.30)

Thus the Ci-symbol lij for (i,j = 1,2,3) is a system of nine elements such thatthe diagonal elements, with indices i and.j eqnal to each other, are equal to unity while the off-diagonal elements are equal to zero The matnx of this system is the following·

OJ [8j] ~ [0I 0 I 0 . 0

0

(231)

I

The Ci-symbol1soften called the substltut:Ion operator, smce by composition with the Ci-symbol 1t 1s possible to change the index label of any system. For example, it is possible to write

lijai=li}a1+li}a2+li]m=a1 .

(232)

The validity of the result (2.32) is obvious from the defimtion of the Ci-symbol (2.30). Since liM = 1 for M = 1,2,3, the trace of the Ci-symbol in three dimensions is given by (2.33) The use of the Ci-symbol may be illustrated by the following example. If a system of three independent coordmates is given by x 1 = {x,y,z}, these

Chapter 2 Notation and Systems of Numbers

12

coordmates, by definition. satisfy

ax' a;J = /'· o.

i)

fori = fori #j ·

(2.34)

Tue equation (2.35) can be simplified using the Ci-symbol as follows: (2.35) Another important system 1s the unit ant:lSymmetnc system. that is, the totally antisymmetric thtrd--order system m three dimensions, called the e-system. ltis denoted by tf!K or euk and is defined for (i,j,k = l,2,3) m the following way:

+1, ifijk is an even permutation of 123 etjk=

{

-1, ~fijki~anodd~errrnta~on~fl23 0, if1=1.1=k,J=korr=1=k

(2.36)

Let us now write down all 3! = 6 permutations of 123 in order: (123) (132) (312)

(321) (231) (213)

zeroth permutation first permutatJ.on second perrrntation third permutation fourthpermutation fifth permutation

(2.37)

The three cyclic pennutations (123), (312), and (231) are even perrnutabons alld the corresponding e-symbol elements are given by el23 = e312

= e231 =I.

(2.38)

Sinularly, for the odd pennutations the e-symbol elements are given by el32

= e321 =

e2l3 =

-l,

(2.39)

and all otrer elements of the e-symbol are equal to zero. Titis indicates thar thee-symbol has only six nonzero elements out of a total of 27 elements. Since thee-system in three dimensions is a third-order system, it is not possible to represent it by a two-dimensional matrix, but a three-dimensional ;;cheme is required. However, for better visibility a following schematic

Section 2.5 Unit Symmetric and Antisynwneblc Systems

13

representation is also possible 21

22

23

31

32

33

0 0 I 0 -1 0 0001·00· -1

0

0

0

l_ 0

;;_j

(2.40)

The most general Nth-order e-symbol is defined as follows:

£i'1i2-.iN

11, -1.

=

{

0,

an even permutation of 12 .. . N 1Nanoddpermutationof12 ... N.

i1i2 .. . iN

i]i2

.••

(2.41)

any parr of indices equal to each other

Using the e-symbols it 1s possible to write down the expression for the determinant of a matrix [ajJ, for (i,j = I, 2, 3), as follows:

a= det [ aj] = rnald;.a~.

(2.42)

In order to verify that the express10n (2.42) is tndeed equttl to the detenni-

::f=~=~~: [ajJ. we use the exP'e8sion for the determinant of a 3 x 3

a= det

[:f :~ j,] al

a2

(2.43)

~

a= a} (aia~ - aia1) - a1(af~ - ajaj) + aHafai - aja~) = a}a~a~ -a}aiai + a1afa1 - a~a~aj +a~~ai - a~aia~

(2.44)

From the definition of thee-symbol (2.36) we see that the result (2.44) can be written in the form

(245) which is identical to (2.42). The determinant of a matrix 1,2, . . ,N), has the form

[ajJ,

in a general case when (i,j = (246)

The definition (2.46) clearly mdicates the advantage of the system notation, since the expression for the detenninantof anN xN matrix in the expanded form, even for relatively small values of N, is quite complex.

~

Chapter 3

Vector Spaces

[[TI

Introduction and Basic Concepts

In general, a mathematical space ts a set of mathematical objects with an associated structure. 'flus structure can be specified by a number of operations on the Objects of the set. These operations must satisfy certain general rules, called the axioms of the mathematical space. In order to specify the operations and axioms used to define a vector space, it is first important to introduce some general concepts. The set of values (a1 ,a2 , .•. ,ct') of some N variables (x 1,x2 , .•. ,xN) is called a pomt. A set of all such pomts for all possible real values of the given N variables is called a real N-dimeusional space. A vector in such a space IS an ordered pair of points, such that it is specified which point is the first point (the origin of the vector) and which point is the last point (end of the vector). For example, if the point P(a 1, a 2 , .•. , ct') is the origin of the vector and the point Q(b 1,b1 , .. ,bN) is the end of the vector, then the vector pQ is the vector of displacement from the point P to the point Q. The system of values

or ma more compact notation c'=b1 -ai

(1=1,2,.

,N),

is called the system of coordinates of the vector

(3.2)

PQ

m the

N-dimeusionalspace

15

Chapter 3

16

Vector Spaces

The vector pQ m the N-dimensional space is thus fully determined whw we know all of the N coordinates of the point of origin P. i.e.. ai

(i=l.2.. ,N),

(3.3)

and all of the coordinates of the vector PQ. given by (3.2). However, m many cases only the coordinates (3 2) are used to spocify a vector, and it 1s ~med that they can be measured from an arbitrary point in the N-dimensional space as an origin. Such a vector is called a free vector. An arbitrary vector PQ = C will therefore usually be identified with the system of coordinates

c'

(i= 1.2•... ,N).

(3.4)

If we adopt a common point of origin for all vectors, defined in the N -dimensional space, then the position of every point in that space is determined by its position vector with respect to the adopted common pomt of origin. The point with coordinates (0, 0, .. , 0) is called the origin of coordinates. It is customary to adoptitasthecommon pointoforiginforall vectors in an N-dimensional space, although il 1s not mandatory The equality of two vectors 0 and brequires the equality of all of their components ai =b;

(i= 1.2.. .. N).

(3.5)

~ Definition of a Vector Space A general N-dimensional space in which the operation of addition (subtraction) is defined by the equation

c'=ai±b' (i=l,2., . . ,N),

(3.6)

and the operabon of scalar multiplication is defined by the equat:J.on

a' =Ab'

(i= 1,2, .. ,N),

(3.7)

for an arbitrary scalar A, is called the vector space in N dimensions The two vector operations satisfy the following axioms·

1. The commutativity of addition·

d+b'~h'+d

(i~I,2,.

,N)

(38)

Sectlon 3.2 Definition of a Vector Space

17

2. The associativity of addition:

(cl+ b') + c' = ci + (b 1 + c')

(i

= 1, 2, ... ,N)

(3.9)

3. The existence of a null-vector ri such that

cl +01 =01 +ll'

(i

= 1,2, . . N)

(3.10)

4. The distribution laws for scalar multiplication:

(A+v)ll'=Alli+vd

(i= 1,2, .. ,N)

(3.12)

5. The associativity of scalar multiplication: {>-v)d ~ ).{va')

{i ~ 1,2, .. ,N)

(3.13)

6. The existence of a unit-scalar I which sahsfies llli =ll1 (i= 1,2, ... ,N).

(3.14)

The foregoing definition of the N-dimeusional vector space implies cert.am properties of objects in it. The colinear or parallel vectors in the N-dimensional vector space are the vectors li and b which are linearly dependent, 1.e , such that

lli =J..b1 (i= 1,2, ... ,N).

(3.15)

A straight line in the vector space is defined by the equation

x 1 = lli +J..b1 (1=1,2, .. . ,N),

(3.16)

where A is the v
where A and v are variable parameters. The vector 0 detennmes one point on the plane, while the vectors band Care the direction vectors of the plane.

Chapter 3 Vector Spaces

18

~ The Euclidean Metric Space H the distance between two arbitrary points P(ll 1,a2, .. ,ct') and Q(bl, b2,. . , hN) of a vector space is defined by the equation

s=PiJ=

[

(b1-a1)2+(if-a2)2+.

]

+(hN-d')1

~

,

(3.18)

such a vector space is called the Euclidean metnc space. In the case of two infinitesimally close pomts P(y 1,y1, . . ,f") and Ql'-'1 +dy 1, j+dy2, .. . , ~ +d}N), thedistanceds is given by

a,2 ~ca/>' +(dy'J2+

. +{dy'')2,

(319)

The expression (3.20)is called thesquareofthelineelementds or the metric of the Euclidean metric space. The Euclidean metric space is a special case of the general vector space Thus the Operations of addition (subtraction) and scalar multiplicat:J.on, satisfymg the axioms (3 8)-(3.14), are defined in the Euclidean metnc space However, in a metric space there is another operation with vectors called the scalar product. defined as a composition ofvectorsai and bi, i.e.,

Ille Euclidean iretric space in N dimensions is usually denoted by EN

~ The Riemannian Spaces In the previous section, we have concluded that in a Euclidean metric space Erv. the metric is defined by m 2 ~ 81<jd/dy'

{k,j ~ 1,2, .. ,N),

(3.22)

where I are Descanes recr.angula.. coordinates. If, instead of the Descanes coordinates we introduce N arbitrary generalized coordinates x1'-, by means of the equations

I.

l~l<x 1 ,x'.

.. ,x")

{k~I,2. . . ,N),

(323)

Section 3 4 The Rtemannlan Spaces

19

then. due to the relations d/

~ a/ dx"' ax"'

(k,m

~ 1,2, .

.,N),

(324)

the metric can be written as follows: ds 1 =

f,l<j

a_! dx"'!!t_dxn =

axm

axn

f,l<j

a_!

ayj dX"dxn.

axm axn

(3.25)

If we introduce a notation

a/ ay;

(3.26)

8mn = fikj axm 8x". the metric can be written in the fonn ds1 =gmndx""dxn

(m,n=l,2, .. ,N).

(3.27)

From (3.26) 1t is clear that gmn, m general, is not equal to the Ci-symbol and that (3.27) cannot be reduced to the sum of squares of differentims of N coordinates Thus (3.27) is a general homogeneous quadratic fonn. Let us now, as an example, consider the usual three-climensional Descartes system of coordinates _I = {x, y, z} and the system of sphencal coordinates x"- = {r,O,
-y2 =x 1 sinx2sinx3

(3.28)

y3 = x 1 cosx2. The components of the system 8mn m sphencal coordinates are obtained using the definition (3.26) as follows:

811=(~)2 +(~Y +(~Y = (sinx2cosx3) 2 + (sinx2sinx3) 2 + (cosx2) 2 =I

(~) + (:~~y + (:~:) 2

822 =

2

= (x 1 cosx2cosx3)2 + (x1 cosx2smx3) 2 + (-x 1 sinx2) 2 = (x 1) 2

833 =(~)2+(:~:)2 r(~)2 = (-x1 smx2sinx3)2 + (x1 smx2cosx3)2 =

(x 1 sinx2) 2

(3 29)

Chapter 3

20

Vector Spaces

Similar considerations show that 8mn

=0

form "F- n.

(3.30)

Thus the metric (3 Zl) becomes ds2

= (dxl)2 + (x1)2(dx2)2 +(xi sinx2)2(dx3)2,

or using the usual notation for the sphencal coordinates ds 2 = dr2 + r 2de 2 + r 2 sin 2 0dqi1,

(3.31)

(3.32)

whtchis the well-known expression for the metric in spherical coordinates. The expression (3.31) or (3.32) is a homogeneous quadratic fonn and it is not a sum of squares of differentials Of the three coordinates. In general a space Of N dimensions, in which a metric is defined with respect to the Descartes system of rectangular coorilinates (3.22), does not cease to be a Euclidean metric space when the metric is expressed with respect to some other genenilized system of coordinates (3.27) and is no longer a sum of squares of differentials of N coordinates. The space remains the same Euclidean metric spaceandonlythesystem Ofcoordinates is changed. Let us now consider a case when the Descartes coordinates y'<-(k 1,2, ... ,N) can beex-riessecJ-in terms of a number M (M
=

y' = v'(x1,x', ... ,x") (k = 1.2... ,N).

(3.33)

In such a way we define an M-climensional subspace, denoted by [W, embedded in the original N-dimensional Euclidean metric space EN. The metric of the Euclide31l metric space has the form

J,2

= S'id/dyl

(k,j

= 1,2,

. .,N),

(3.34)

and because of the relation

a-'= ay' dx" Y

arr

'

(3.35)

the metric of the sqbspace RM can be written in the form

a,2 =

gofidx"dx'

(a,/J = 1,2,. .,M),

(3.36)

where

(3.37) In (3.37) and below the Greek indices run from 1 to M while the Latin indices run from 1 toN.

Section 3.4 The Rlemannlan Spaces

21

Tiws the square ofthe distance between two infinitesimally close points in the subspace RM IS defined by the homogeneous quadratic differential form (3.36). Let us now consider the M-dimensional subspace RM independently of the original N-dimensional Euclidean metric space EN, m which rt is embedded. Now we would like to know whether it is po8Slble to find M independent variables and use them as coordinates, such that the metric form (3 36) can be written as the sum of squares of differentials of these coordmates. If tlus is possible, the M-dimensional space fW is in its own right a Euclidean metric space. If this is not possible the M-dimensional space RM is called the Riemannian space. Let us, for example, consider the points on a sphere of a unit radms in the three-dimensional Euclirlean metric space £3, defined by the Descartes rectangular coordinates y; = {x,y, z}. The metric has the usual form

Let us now define a two-dimensional subspace R2 of the Euclidean metric space 6. in which the position of the points on the unit sphere is specified by the polar angle
(3.39)

The components of the system 8af3 are obtained using the definition (3.37) as follows:

811

(a/)' + (a>')' ax + (a>')' fuT

= fuT

1

= (cosx 1 cosx2) 2 + (cosx 1 sm.x2) 2 + (-sinx 1) 2 = 1 g22=

(~)2 +(~)2 +(~)2

= (-sinx 1 sinx2) 2 + (smx 1 cosx2) 2 = (sinx1) 2 g12=g21 =0.

(3.40)

Thus we obtain the metric of the subspace R1 m the form (3.41)

Chapter 3 Vector Spaces

22

ds1

=

d0 2

+ sin2 edq?

(3.42)

It turns out that it is not possible to find two real variables {u, v} such that the metric (3.42) can be written in the form ds2 = du1 +dv2. (3.43) Thus the surface of the unit sphere, as a two-dimensional subspace R1 of the original Euclidean metric space E3, does not have the internal Euclidean metric. In other words, there are no Euclidean coordinates on the surface of the umt sphere. It can, therefore, be concluded that in a Euclidean metric space there are some subspaces with anon-Euclidean metric. SuchsubSpaces are called the Riemannian spaces. In principle, the metric geometry can be genemlized by definmg the metric of a space in advance by choosing the functions 8mn in an arbitrary way, with only requirements that the system 8mn be symmetric and doubly cliffen:ntiable The metnc does not even have to be positively defined. Thus a space, with a metric wluch ts notposillvelydefined but m wluch the components Of the system 8mn are constants, is scmetimes called a pseudo-Euclidean space. Analogously, a space which is not positively defined and in which the components of the system 8mn are arbitrary functions of coordinates is sometimes called a pseudo-Riemonnitm sptice.

~

Chapter 4

Definitions of Tensors

@]]

Transformations of Variabies

Tensors. as mathematical objects, were originally introduced for an immediate practical use in the theory of relativity. The main subject Of the theory of relativity is the behavior of physical quantities and the laws of nature with respect to the transformations from one sys1em of coordinates to another. It was therefore important to introduce a new class of mathematiclll objects that are defined by their transformation laws with respect to the transformations from one system of coordinates to another. Such mat:hemal:ical objects are called tensors The systems that we call tensors have linear and homogeneous transformation laws with respect to the transformations of coordinates In order to define the main types of tensors, let us consider an arbitrary transfonnation of variables x1< into some new variables z!', defined by

z"~z"(x 1 ,?-,

. /

1 )

(k~ 1,2, .. ,N)

(4.1)

From (4.1) we may write J<

az"

a.._ =axmdx"' (k,m=l,2, . . ,N).

(4.2)

The differmtials of the coordmates d;:!- are linear and homogeneous functions of the differentials Of the old coordinates dx"'. The differentials of coordinates are by definition treated as the components of a special type of tensors, called contravarillnt vectors.

23

Chapter 4 Definitions of Tensors

24

Let G be a zeroth-order system. The system of N quantities iJG/iJil is lransformed acro£cling to the transformation law

ac ax"' ac al = az" axm

(k.m

=

I. 2, .. ,N).

(4.3)

The components aG /iJzk are linear and homogeneous functions of the components iJG/iJx111 • The components of the system iJG/iJx111 are by definition treated as the components Of a special type of tensors, called covariant vectors.

@]]

Contravariant Vectors

Generally speaking, any system of quantrues, defined with respect to the systems of coordinates {xk} and {z"} by N quantit1esAk ancL4k respectively, which is transformed acconling to the transformation law (k,m

~

1,2, . ,N)

(4.4)

is called a contravariant vector. The contravariant vectors are always denoted by one upper index.

@]]

Covariant Vectors

On the other hand, any system of quantities, defined with respect to the systemsofcoorrlinates {.xk} and {z':} by N quantities Bk and ilk. respectively, which 1s transformed according to the transformation law

.B,-

ax"'

~ az'

B,,,

(k,m ~ 1,2, . ,N)

(4.5)

is called a covariant vector. The covariant vectors are always denoted by one lower index.

@]]

Invariants (Scalars)

Let us nowformazeroth--ordersystem by composihon of one contravariant and one covariant vector (4.6)

with respect to a system of coordinates {xk}. Then, with respect to a new system of coordinates {z':} this system has the value

P~;>,kiJ ~ az' max" k

_ az' .iJ::'.'. m

8x"'A {lzkB,, -

axm (lzkA B,,.

(47)

Section 4.5 Contravariant Tensors

25

where the upper bar over the quantities F. A_k, and ih denotes that they are defined with respect to the new coordinates {z':}. On the other hand, by definition Of the ~-symbol, we have

az'ilX'~s" axm az" m

(k.m,n

= 1,2, .

.• N),

(4.8)

and the result (4 7) becomes (4.9)

(4.10) The result (4 IO) shows thatthe quantity F has the same valuem all systems Qfcoordmates. Such a quantity is, therefore, called an invarumt or a scalar A composition of one contravariant and one covariant vector is therefore called the scalar product since it behaves as a scalar with respect to an arbitrary transfonnation of coordinates

@]]

Contravariant Tensors

Letus consider asystemofN 2 products of components oftwocontravariant vectocs Bm and vn. denoted by (4.11) with respect to a system of coordinates {.xk}. In some other system of coordmates {f} this system will have values m accordance with (4.4), i.e., (4.12) In analogy with (4.12), any second-order system. defined with respect to the systems of coordinates {.xk} and {z':} by N 2 quantities Amn and A_ik, respectively, that is transformed accordmg to the transformation law (4.13) called a second-order comravariant iensor. The second-order contravariant tensors are always denoted by two upper indices.

IS

26

Chapter 4 DefinltionsorTensors

A third-ordercontraw1riont tensor 1s a system of N 3 quant1lles, denoted by Amnp, which is transformed according to the transfonnation law (4.14) Analogously with the definitions (4.13) and (4.14), tt IS possible to define contravariant tensors of arbitrary order. By convention, any contravariant tensor is denoted by a number of upper indices only.

@:§]

Covariant Tensors

A system defined with respect to the systems Of coordmates {xk} and {z':} by N 2 quantities Amn and Ajk, respectively, that is transformed according to the transformation law (4.15) 1s called a second-order covariant tensor. Second-order covdliant tensors are always denoted by two lower indices. A third-ordercot'llriLmt tensor is a system of N 3 quantities, denoted by Amnp, which 1s transformed according to t:re transformation law (4.16) Analogoosly with the definitions (4.15) and (4.16), it is possible to define covarianttensors Ofan arbitrary order. By convention. any covariant tensor is denoted by a number of lower indices only.

@1l

Mixed Tensors

Let us consider a system, defined with respect to the systems of coordinates {xk} and {z':} by N 4 quantities A;15n and 1• respectively, which is transformed acconling to the transformation law

Af

(4.17) the systemA;1sn is Ql]Jed afourth-order mixed tensor, i.e., a second-order contrav.niant and second-order covariant tensor The indices m, n are contrav.niant indices and indices p. s are covariant indices. In analogy with (4.17) it is possible to define mixed tensors with arbitrary numbers

Section 4.8 Symmetry Properties of Tensors

27

of contravariant and covariant indices. There are two more types of fourth-order mixed tensors

(4.18) Using the transfOrrnation law (4 .17) it is easy to construct the transformation laws fOI" arbitrary mixed tensors. The contravariant and covariant tensors can, of course. be treated as special cases of mixed tensors.

@]]

Symmetry Properties of Tensors

A second-order covariant tensor A1k is called a symmetric tensor if its components satisfy the equality

Aik =Aki

(j,k= 1,2, ... ,N),

(4.19)

and the tensor A]k is called an anti.symmetric tensor if its components satisfy the equality A,,~

-A;, (1.k ~ 1,2, .. ,N).

(4.20)

Analogously, a second-order contravariant tensor AJk is called asymmemc tensor if its components satisfy the equality Ajk

= Akj

(j,k

= I, 2,

.. ,N),

(4.21)

and the tensorAjk is called an llntisymmetric tensor if its components satisfy the equality tJk ~ -A'i

(1.k ~ 1,2, ... ,N).

(4.22)

It is 1mponam to note that the symmetry properties Of tensors are independent of the coordinate system in which the tensor components are defined. In other words, a tensor that is symmetric (antisymmetric) with respect to one coordinate system remainS symmetric (antisymmetric) with respect to any other coordinate system. In orckr to show that it is the case, let us assume Ihat an arbitrary second-order contravariant tensor, defined with respect to the coordinate system {xk} by N 2 componentsAmn, satisfies the equality Amn

= A"m

(m,n

= 1, 2,

.. ,N),

(4.23)

which means that it is a symmetric tensor in the coordinate system {.xk}. In some other coordinate system {z':}, this tensor is given by the

Chapter 4 DefinitlonsorTensors

28

components (4.24) However, by means of an interchange Of the dummy mdices m ++ n, which is just a change of notation that does not affect the result of summation, and using (4.23) we obtain

Nk=~a! AtlJll= a! ~Amn=A1g, axn axm

axm axn

(4.25)

which shows that the tensor A_;k in the new coordinate system {z':} is indeed symmetric as welL ht general, if a tensor of an arbitrary order and type is symmetric (or antisymmetric) upon interchange of one pair of its indices (both lower or both upper indices) in one system of coordinates {xk}, it remains symmetric (or antisymmetric) upon mterchange of the correSponding pair of indices in any other system Of coordinates {z':} In other words, the symmetry properties of tensors are independent Of the coordinate system. in which the tensor components are defined.

@]]

Symmetric and Antisymmetric Parts of Tensors

Let us consider a second-order covariant tensor Amn This tensor can always be written as a sum of one symmetric and one antisymmetric tensor as follows:

(4.27)

Amn =A(mn) +Armn]-

The symmetric tensor defined by the expression

(4.28) is clllled the symmetric part Of the tensor Amn, while the antisymmetric tensor defined by the expression A[mn]

= ~(Amn -

A,,m)

is called the tmtisymmetric part Of the tensor Anm-

(4.29)

Section 4.9 SymmetricandAntiaymmetrlcPartsolTensors

29

Analogously to (4.27), it is possible to use an arbitrary non-symmetric third-order covanant tensor Amnp to create a totally symmetric and totally antisymmetric part as follows: A(mnp)

= ~(Amnp +Ampn +Apmn +Apnm+Anpm +Anmp)

AlmnpJ

= J!
I

(430)

(431)

From (430) and (431) it ts easily seen tha1 any mrerd:tange ofindices m, n, and p leaves Atmnp) unchanged, while it reverses the sign of A[mnpJ- The expressions (430) and (431) can be rewritten in a more compact form by introducing a special label for the permutations of the indices m, n, and p, asfollQws: nj(m.n.p)

(j

= O. l.2.3.4.5).

(432)

The components of the system (432) are the 3! permutations of the three indJces m, n, and p, which can be listed as follows: no(m,n,p)

= mnp

n 1 (m,n,p)

= mpn

n2(m,n,p) = pmn

7r3(m,n,p)

= pnm

7r4(m.n.p)

= npm

(4.33)

n5(m,n,p) = nmp.

Using (432) and (433), the expressions (430) and (431) can berewntten in the followmg more compact form: l

A(mnp)

3!-1

= J! L

An1 (mnp)

(4.34)

1=0

(4.35) valid for the tlurd-order covariant tensor AmnpThe expressions (4.34) and (435) are easily generalized to the case of the Nth-order nonsymmetric tensor, where the totally symmetric and

Chapter 4

Definitions of Tensors

An,(•1i1-iN)

(4.36)

30

totally antisymmetric parts are given by

l A(iii2- iN) =

N!-1

L

Ni

"j=O

1 Ar£i•2--•NJ=M

N!-t

L<-O'Air1<•1i2.iNl·

(4.37)

p--U

(4Tij]

Tensor Character of Systems

Sometimes we do not readily know the transformation laws for all systems encountered in different expressions. In order to be able to determine whether a given system is a tensor or not, we need some criteria for determination of the tensor character of systems. These criteria can be based on the following statement: If an expression AkBk is invariant with respect to the coordinate transfonnations and we know that Ak transforms as a contravariant vector (or that Bk transforms as a covariant vector), then we know that the system Bk is a covariant vector (or that the system Ak is a contravariant vector). In order to prove the foregoing statement, let us begin with the assumption that the expression AkBk is an irw
(438) If we then know that N is a contrav-dllimt vector, we know that it transforms as

Ai=~Ak.

(439)

ax'

Substituting (4.39) into (4.38) we obtain

Bzl

k-

k

CJxkA B1 -A Bk=A

k(az'-

)

CJxkBj-Bk

=0.

(4.40)

As the equality (4.40) is valid for an arbitrary contravariant vector Ak, we have (4.41)

Section 4.10 TensorCharaclerolSystems

31

(4.42) The expression (4.42fis the transfonnationlawfor a covariant vector Bk. and it shows that Bk is indeed a covariant vector. In the same way it can be shown that, if the expression AmnB11'IY' is invariant with respect to the coordinate transformations and we know that 1111' and IY' are two different contravariant vectors, then Amn is the second-order covariant tensor. This is valid even if, as a special case, the expression AmnB11'B" is an invariant for an arbitrary contravariant vector Bk, pm.ided that it is known that Amn also satisfies the condition Of symmetry Amn =Anm

(m,n= 1,2.... ,N).

(4.43)

These conditions can be generalized to tensors of an arbitrary order and used as the criteria for det:ennination of the tensor character of systems. As an example of these rules, let us consider a mixed second-order system ~1- Let us hike the numbers ~1 as the coordinates of a second-order mixed tensor with respect to an arbitrary coordinate system {xk}, which is always possible. The question is whether they will keep their values, i.e.• v.ttether they will remain the coordinates Of a ~-symbol, after the transformation to some new coordinate system{~}. Thus we may write (4.44) or, since {~} is a system of mutually independent coordinates, we have

~!1 = ~ B~ = ~ =~~-1 Bx"' Bzl

BV

(4.45)

The result (4.45) shows that the ~-symbol is indeed a second-order mixed tensor, which has the same coordinates in all coordinate systems.

"" Chapter 5

Relative Tensors

ffi]]

Introduction and Definitions

The tensors defined m the previous chapter are sometimes also called the absOiute tensors. since there are systems of quantities which, upon the transformations of coordinates, transform accCTding to the similar but somewhat more general laws. Such systems are called relative tensors or pseudotensors. Thus a fifth-uder system, three times contravariant and twice covariant.

A;~~i3

(r1,fi,r3,it,J2=1,2, ... ,N)

(5_1)

is defined as a relatil:e tl'llSor orpseudotensorOfweightM, if it transforms according to the transfonnation law -i1i2i3 -

Alth

-

Iax IM Clz-'

Clz

m1m2m3

An1n2

11

Clz'

2

3

Cli Cl.xrli Ctr'2

(lx'1'1 c1-rm2 (l_xfflJ (lzi1 (lzJ2 -

<5 2)

In (5.2) we may introduce a notation (5.3)

for the Jacobian of transfonnat1on of the original coordinates {xk} imo the new coordinates {~}-In analogy with the definition (5.2), it is possible to define the relative tensors Of an arbitrary weight, order, and type. The concept Of relative tensors includes the absolute tensors, defined in the

33

Chapter 5 Relative Tensors

34

previous chapter, as a special case. The absolute tensors can be treated as relative tensors of weight zero. In particular, the relative tensors of weight M = -t-1 are called the tensor densities, while the relative tensors of weight M -1 are called the tensor capacities.

=

[g]

Unit Antisymmetric Tensors

As an important example of relative tensors, we consider the e-symbol in three dimensions with three upper indices., Le.,

e'Jk

(i,j,k

= 1,2,3).

(5.4)

If we assume that eiik is a relative tensor of an unknown weight M, then it transforms according to the transformation law

(5.5)

On the other hand, by definition Of the Jacobian Of the transfonnation 6., we have

(5.6)

The Jacobian Of the inverse transformation is given by

1:::..-1

=

lat"I =eiJk~

Bzzaz3_ Bx' ClxJ Bxk

Bx"

(5.7)

Let us now, for the moment, consider the following system: Arst

= eiikaiaji{

(5.8)

for an arbitrary mixed system a;:'. In the expanded form this system looks like Arst =tfit4a~ -a~aj~ +aJaja~

-aJ'4tfi +a2ajtfi

-ala{~.

(5.9)

From (5.9), we see that the system Arsr is a fully antisymmetric system with respect to its three indices, in the space of three dimensions. However, an arbitrary third-order antisynuneiric system in three dimensions bas only one independent component, Le., A123, v.ttereas the other five

Section 5.2 Unit Antisynmetric Tensors

35

nonzero components aredeterminedby the conditions offull antisymmetry. Thus we may write Arst =Al23erst.

(5.10)

In other words, any folly antisymmetric third--order system in three dimensions is proportional to thecorresponcling e-symbol Using (5.8) and (5.1 OJ we obtain

(5.lll Using(5.1 l)we may write

el]k~~Bz' =ersreijk~~Bz3 Bx' Bxl Bxk

Bx' Bxl Bxk'

(5.12)

or using (5.7)

eijk~~~ =ersr6.-t_ Bx' Bx.I Bxk

(5.13)

Substituting (5.13) mto (5.5) we obtam (5.14) From (5.14) we see that if we choose M = +1. the components of the system erst remain unchanged upon the coordinate transfonnations_ Thus the system eijk is a thtrd-orderrelative contravariant tensor with the weight M +1 (tensordensity). Thetransfonnationlawoftheunitantisymmetric system ei;k is therefore given by

=

erst=

IBz"I ax" Ieijk~~~ = Lleijk~~ Bz'. Bx' Cl.;V Bxk Bx' Bxl Bxk

(5.15)

In a similar way, we can consider thee-symbol with three lower indices in three dimensions eijk

(i,j, k = 1, 2, 3).

(5.16)

Again, if we assume that eijk is a relative tensor of an unknown weight M, then it transforms according to the transfonnation law irst

ax" IM

= IBz"I

ax' axi ax' eiJKBzr'BZS&'

(5.17)

Using an analog Of Equation (5.11) in the fonn (5.18)

Chapter 5 Relattve Tensors

36 we obtain (!xi iJxi (lxk

ey1cazrazs Clz' =

Bx' iJxl Bxk

e..-sreqk&T BZ2

(lz3'

(5_19)

(5.20) Substituting (5.20) into (5.17) we obtain (5.21)

From (5.21) we see that if we choose M=-1. the components of the system e..-st remain unchanged upon the coordinate transfonnations. Thus the system ei;k is a tlurd-order relative cornravarianr tensor with the weight M -1 (tensor capacity). The transformation law of the unit antisymmetric system ei;k is therefore given by

=

(5.22)

In gencraJ, the

~fonnation law Of an arbitrary tensor ts defined by

1. Order-the number Of indices 2. Type-the position and order Of indices 3. Weight--the exponent of the Jacobian of transformation.

[[ill

Vector Product in Three Dimensions

Let us consider two vectors Of the same type. e.g., two conrravariantvectors denoted by Ai and B;, in a three-dimensional space (i = 1, 2, 3). Using the relative tensor eyk. we can define a first-order system

which is a first-order covariant relative vector with the weight M = -1, sinceitisdefinedas a composition of one relative tensor ofweightM = -1 with two absolute vectors. The expression (5.23) can be expanded into

Section 5.3 Vector Product In Three Dimensions

three equations: C1 =A2B3-A3B2 C2=A3Bt-AIB3

(5.24)

C3=A1B2-A2B1 The transfonnation law for the vector C, is given by

tj= t:,._-•cr Cl~. Bzl

(5.25)

v.ttere the Jacobian Of the transfonnation is given by

"~1:~1

(5.26)

In a similar way we may define a relative contravariant vector C; of the weight M = + 1, starting with 1wo absolute covanant vectors A1 and Bj as well as the relative tensor d1k, i.e., Ci= e'ikAjBk

(i,j, k

= 1, 2, 3).

(5.27)

The expression (5.27) can be expanded into three equations: C 1 =A2B3 - A3B2 2 C =A3B1-A1B3 3 C =A1B2-A2B1-

(5.28)

The transformation law for the vector c; is given by (5.29) The product (5.23) or(5.27) is called the vector product of two contravariant or two covariant vectors in three dimensions, respectively. This definition includes the usual definition of the vector product of two vectors (assumed to be the position vectors) given by their Descartes rectangular coordinates:

2 AJ. 31 C=AxB= A11 A2 1Bi Bz 83 i.

(5.30)

In the definition (5.30), i, and 3are the unit vectors of the three mutually orthogonal ax:es of the Descartes coordinate system. It should be noted that, in the Descartes coordinate system, the contravariant and covariant

Chapter 5 Relative Tensors

36

coordinates of the vector product have the same numerical values, Le., (5 31)

[~

Mixed Product in Three Dimensions

The mixed product of three contravariant va,-torsN, Bj, and Cj is formed

by a composition of the vector product Dj =ejkmBkcm

U.k,m=

1,2,3)

(5.32)

with the vector N. It has the form

or, in the Descartes coordinate system, (5.34)

From the definition (5.33) it is evident that the zeroth-order system V is a relative invariant of the weight M = -1, or the scalar capacity, since it is composed of three absolute contravariant vectors and the tensor capacity e11un. The transfonnation law for this system is (5.35) In a similar way, the mixed product Of three covariant vectors Aj, Bj, and G is fonned by composition of the vector product

with the vector Aj, and it has the form

or, in the Descartes orthogonal coordinates,

c~1~:C1 ~~C2 ~~1C3

(5.38)

From the defim.t1on (5.37) it ts evident that the zeroth--0rder system G is a relative invariant Of the weight M +I, or the scalar density, since it is comprn>ed of three absolute covariant vectors and the tensor density efrm_

=

Section 5.5 Orthogonal Coordinate Translonnations

39

The transfonnation law for this system is (5.39)

G=!!i.G.

Unlike absolute scalars, the systems V and G, as relative scalars, in general change with respect to the transformations of coordinates. The behavior of these systems with respect to a special class of the orthogonal transfonnations of Descartes coordinates will be discussed in the next section.

ffi]]

Orthogonal Coordinate Transformations

In order to highlight some important properties of the vector products and mixed products in three dimensions, we will consider the orthogonal transformations of Descartes coordinates rotation, translation, and inversion. 5.5.1

Rotations of Descartes Coordinates

Let us observe two Descartes coordinate systems Kand K' with a common z-axis, denoted by 3 = 3', perpendicular to the plane of the paper. The system K' is obtained as a result of a rmarion in the positive sense of the system K about a common 3-axis for some angle (), as shown in Figure 1. The relation between the coordinates of a position vector of a given fixed point P, with respect to the coordinate systems K! and K, is given by

[z'~:] [-s~ne cose

=

sin() cose 0

/ ~AJxi

(j,k ~ 1.2,3).

(5.40)

(5.41)

The transformation (5.41) is a linear transfonnation with the Jacobian (5.42)

Thus the Euclidean metric, given by the analog of Equation (3.18) in three dimensions, is invariant with respect to the rotations of the Descartes coordinates. This can easily be shown by using the distance between the pointP(x 1,x2,x3 ) from the origin 0(0,0,0), which in the coordinate

Chapter 5

40

Figure 1.

Relative Tensors

Rotations of Descartes coordinate systems

system K is given by (5.43)

In the coordinate system K the distance between these points is given by

s' = [
(5.44)

Subsntuting (5.40) into (5.44) we obtain

[

S' = (x 1 cos() +x2 sin8) 2 + (-x 1 sin() +x2 cos()) 2 + (x3) 2

]

~

(5.45)

(5.46)

S'=S.

Thus the distance between the pomts m the Descartes coordinate system is invariant with respect to the rotation of coordinates about the 3-axis. The metric form of the space is also invariant with respect to the rotation of coordinates about the 3-axis, i.e., we have

di2 =ds2 .

(5.47)

It is easily shown that the foregoing results are valid for arbitrary rotations of coordinates in the Descartes coordinate systems.

Section 5.5 Orthogonal Coordinate Transformations

41

5.5.2 Translations of Descartes Coordinates

Let us consu1er two Descartes coordinate systems K and K' in three dimensions, where the coordinates in the system K 1 are denoted by {.fl and the coordinates in the system Kare denoted by {x"}. The translation of the coordinates is given by the equation

-! =,!< +,/

(k = 1,2,3),

(5.48)

where ak is some constant translation vector. The Jacobtan !::,,, of this transformation is given by (5.49)

Thus the metric fonn of the three.-Oimens1onal space m Descartes coordinates is invariant with respect to the translation of the coordinate systems, since we have

u!- = dx'

(k

= 1,2, 3).

(5.50)

From (5.50) we have

di2 =~jkdz 1 d!=~jkdx 1 dxk=ds2

(j,k=l,2,3),

(5.51)

which proves the invariance of the metric form with respect to the coordinate translations. 5.5.3 Inversions of Descartes Coordinates

Let us again consider two Descartes coordinate systems Kand K' in three dimensions, where the coordinates in the system K' are denoted by {/} and the coordinates in the system Kare denoted by {x"}. The inversion of the coordinates is given by the equation

i=-,!<=-SJx'

(k=l,2,3).

(5.52)

It can be shown that the invers10n of Descartes coordinates cannot be achieved by means of any rotation of the original coordinate system K. The metric form is invariant with respect to the inversmn. since we have ds' 2 = ~jkdzid/ =~jkd(-xi)d(-x") = ~jkdxldxk = ds 2. (5.53)

The Jacobian of this transformation is equal to

(5.54)

Chapter 5 Relative Tensors

42

5.5A Axial Vectors and Pseudoscalars in Descartes Coordinates

Thus, as aconclusmn of tlus section, we note that rotat10n, translation, and inversion constitute a group of orthogonal transformations which leave the metric form invariant. Furthermore, the transformation laws of relative vectors and scalars (e.g., vector products and mixed products in three dimensions), with respect to rotations and translations of coordinates, are the same as those for the absolute vectors and scalars. However, relative vectors and scalars (e.g., vector products and mixed products in three dimensions) do not transform like absolute VCL"tors and scalars, with respect to the inversion of coordinates. As we have concluded before, the transformation law of an absolute contravariant vector in three dimensions is given by

(j,k

= 1,2,3),

(5.55)

with respect to the coordinate transformations from {x"} to {/}. An example of this type of vector is the polar vector of the position of a certain point in a three-dimensional space. However, the vector product of two abso(ute covariant vectors Aj and Bj is a relative contravariant vector Cj, which transforms according to the transformation law

az! k c-1__"&kc

.

_

(J,k- 1.2.3).

(5.56)

Substituting (5.42) or (5.49), into (5.56), we see that the vector product ts transfonned in the same way as the absolute vectors (5.55), with re>pecl to rotations and translations. On the other hand, the vector product reverses sign with respect to the inversion of Coordinates and does not transform as the polar vectors. This difference between the vector product, as the relative vector, and the polar vectors, being the absolute vectors., was noted in the three-dimensional vector algebra before the developmem of the tensor calculus. In the three-dimensional vector algebra, the vector product is called the axial vector, as opposed to the position vector which is called the polar vector. From the tensor point of view this difference is easily understood, since it relates to the definition of the position vector as an absolute vector and the vector product as a relative vector. The mixed product in three dimensions, as a relative scalar, is invariant with respect to rotation and translation but it reverses sign with respect to inversion. In the three-dimensional vector algebra these scalars are usually called pseudoscalars.

"' Chapter 6

The Metric Tensor

@]]

Introduction and Definitions

As we have seen in the previous chapters, in the Euclidean metric space it is possible to reduce the metric fonn to a sum of squares of the differentials of the coordinates, i.e., we may write

di~E;,dy'd/

(j,k~l.2,

,N)

(6.1)

If, instead of the Descartes orthogonal coordinates {).k}, we introduce the arbitrary generalized coordinates {x"} by means of the equations

/ ~/(x'.x'. ... ,xN)

(k ~ 1,2, .. ,N),

(6.2)

-

(6.3)

we may write

;._a/

ff) - axmdx"'

(k,m - 1,2, ... ,N),

and the metric form can be written as follows:

ds2=8mndx"'dx 11

(m.n=l,2... ,N).

(6.4)

where (6.5)

is a symmetric second-order system. The square of the infinitesimal line element ds is by definition an invariant in all coordinate systems and dxm is an absolute vector. Thus, using the criteria for the tensor character of

43

44

Chapter 6 The Metric Tensor

systems and the fact that 8mn is a symmetric system, we conclude that 8mn is an absolute second-order covariant tensor. This tensor is called the metric tensor. The determinant of the matrix a<>sociated with the metric rensor is give'!. by (6.6)

where the multiplication rule for determinants has been used. Thus the determinant of the metric tensor is equal to the square of the Jacobian of the transfonnation from the given Descartes coordinates to the arbitrary generalized coordinates. Assuming that the Jacobian of the transformation from the given Descartes coordinates to the arbitrary generalized coordinates is a nonzero real number, the detenninant of the metric tensor is always a positive quantity, ie., we have g > O. As an example, using the results (3.29), the matrix form of the metric tensor in the system of spherical coordinates x1' = {r, (), q:i} is given by

(6.7)

and the determinant of the metnc tellSOI" is given by

(6.8)

As a system, the determinant g is a relative scalar invariant of the weight M = 2, which is easily shown as follows:

which proves that the determinant g is mdeed a relative scalar invariant of the weightM = 2. From (6.10) we see that ,Jg is a relative scalar invariant oftheweightM= t.i.e., (6.11)

Secllon 6.1

lnb'Oductlon and Ceflnitfons

45

Let us use GM to denote the cofactor of the element 8mn in the determinant 18mnl· Then according to the determinant calculation rules. we have (6.12)

g=8mnGM.

The adjunct matrix to the matrix [gmn], which is denoted by adj [gmn], is by definition the transposed matrix of the cofactors, i.e.• [GM]1' = [G11 m]. The matrix, inverse to the matrix fgm 11J, is therefore given by

inv[g~J ~adj [gm,]~ rcm'f g

(6.13)

g

By definition, a product of a matrix with its own inverse is equal to a unit matrix. Thus we may write (6.14) or, using the system notation. (Gkn)T

8m,-8

Gnk

=gmng =~.

(6.15)

Introducing the notation (6.16) the expression (6.15) becomes (6.17)

From (6.17), we see that the system g"k is a second-order contravariant tensor, which is usually called the contrai:ariant metric tensor. Analogously to the covariant metric tensor, the contravariant metric tensor is also a symmetric tensor. i.e.• we have (6.18) Using (6.17) and the nrultiplication rules for the determinants we find (6.19)

(6-20)

Chapter fl The Metric Tensor

46

The '"determinant of the contravariant metric tensor is a relative scalar invariant of the weight M = -2. which is easily shown using (6.10), (6.21)

(6.22)

Since the antisymmetric unit system e'1i 2 -··it1 is a relative contra.variant tensoroftheweightM = + J, it is possible to fonn anabsolutecontravariant antisymmetric tensor ei1i2 ••• it1, using ,Jg. as follows: (6.23) On the other hand, since the antisymmetric unit system ei1 i 2 ••. it1 is a relative covariant tensor of the weight M = -1, it is possible to form an absolute unit covariant antisymmetric tensor Eiiii---it1• using ,Jg. as follows: (6.24)

The absolute tensors defined by (6.23) and (6.24) are called the Ricci antisymmetric tensors in the space of N dimensions.

~

Associated Vectors and Tensors

In a metric space, the contravariant and covariant tensors can be transformed to each other using the metric tensors 8mn and In general the upper indices can be lowered and the lower indices can be made to be upper indices, using the metric tensors For example. a covariant vector

r.

Am =grrmA",

(6.25)

derived from a contravariantvector A" using the metric tensor g,,,,,, is called the associated vector to the contravariant vector A"_ In the same way, a contravariant vector (6.26) derived from a given covariant vector A,, using the metric tensor gm", is called the associated vector to the vector A,,.

Section 6.2 AssociatedVectorsandTensors

47

For the associated vectors. the following rules are valid: 1. The association relation of two vectors is reciprocal. If a vector Am = gmnAn is associated to the vector A", then the associated vector to the vector Am is the vector A". This can be shown as follows: gmnAn

= gmngmkAk = ~~Ak =An.

(6.27)

2. The absolute square of a contravariant vector Am or a covariant vector A,.,, is the scalar
= gnmgmkAk = ~ZAk =A".

(6.28)

3. From the preceding rule it is evident that the absolute squares of the associated vectors are equal to each otheL 4. The scalar product of the vectors Ak and Bk is equal to the scalar product of the vectors Am and Bm, and it is invariant with respect to an arbitrary coordinatetransfonnation. This can be shown as follows: gnmAm

= gmngmkAk = ~rAk =A".

(6.29)

These rules allow us to consider the vectors Am and Am as the covariant and comravariant coordinates of the same vector, which we may denote by A. From (6.29) we see that, in the metric space, it is possible to define the scalar product of two vectors and regardless of their type, i.e.,

A

B

A-'iJ=AkBk=AmBm=gmnAmBn.

(6.30)

The analogous rules apply to the tensors of an arbitrary order and type. By composition with the metric tensors 8mn and gmn, the upper indices are iowered and the lower indices are turned to the upper indices, respectively. All tensors, created from each other by composition with one or more metric tensors, are called associated tensors. Thus, for an arbitrary second-order covariant tensor llmn we can create three aSSO(':iated second-order tensors amn, a;:', and~. as follows: amn

= tf""g"jakj

a;:'=tf""aw

(6.31)

a;,= g"kamk. It should be noted that the tensors a;:' and a;:., associated to the tensor am"' as defined in (6.31), are in general not equal to each other. They are equal to each other only when the covariant tensor llmn is symmetric, i.e.,

48

Chapter 6 The Metric Tensor

when amn = anm. If we apply (6.3 i) to the metric tensor itself, we first note that the definition of the associated contravariant metric tensor turns to identity:

Secondly, since the metric tensor is a symmetric tensor, there is a unique mixed metric tensor equal to the corresponding i5-symbol, (6.33)

in agreement with the Equation (6.17).

~

Arc Length of Curves: Unit Vectors

Let us consider a Riemannian metric space inN dimensions, de&..Tibed by a system ofN generalized coordinates {xi}. If these coordinates are functions of an arbitrary parameter t, then a curve in this Riemannian space may be specified by N parameter equations ,!< ~,!<(t)

(k ~

1.2•... ,N).

(6.34)

The square of an infinitesimal arc length element of the curve between the points ii' and ii' + tb! is given by dSJ

= 8mndxmdxn

= 1,2, ... ,N).

(m,n

(6.35)

The infinitesimal arc length element itself is given by

ds

= Jgm

11

dxmdx 11

(m,n

= l,2,

.. ,N),

(6.36)

and its derivative with respect to the parameter is given by

ds

di=

J

dr"dx" 8mndtdt

(m,n=l,2, ... ,N).

(0.37)

The arc length of a curve from some reference point, with a parameter value of to, to some arbitrary point, with a parameter value oft, is then equal to s(t)

=

l

<~mdx" 8mn-d -dt.

to

t

dt

(6.38)

Section 6.4 Angles between Vectors

49

From (6.35) we see that we may write 8mn ~Sm~= f,mnAmJ..." = AmAm =

i · J:. =I,

(6.39)

where Am is a vector defined by the expression (6.40)

From (6.39) we see that va,"l:or Am has the absolute square equal to unity. Thus the absolute value of the vector Am, denoted by IAI is also equal to unity: (6.41)

A vector with the absolute value equal to unity is called a 1mit vector. The vector Am is a unitcontravariant vector. As xm are the coordinates ofa point on a given curve ands is the length of the arc of that cmve.. the vector Am is the tangent un.it vector to this curve.

~ Angles between Vectors Let us now consider two polar unit YeL"l:ors Am and µm with a common origin and with the end points A and B, as shown in Figure 2. From Figure 2 we see that Em=µm-Am

(m=I,2, ... ,N).

Flgure2. The angle e between l.lnit vectors A"' and u"'

(6.42)

Chapter 6 The Metric Tensor

50

Using the cosine theorem. the absolute square of the vector written in the fonn 2

lt::l =1µ1

2

t::m

can be

+ p,.i2-21µ11Alcose = 2-Zcose = 2(1- coseJ. (6.43)

On the other hand, by definition, the absolute square of the vector t::m has the form

Since A andµ are unit vectors, we have

(6.46) and the result (6.45) becomes llE 12

= 2(1 -

gmnA mµn).

(6.47)

Comparing the results (6.43) and (6.47) we find that the cosine ofcm nngle lx·tween two unit vectors is given by the expression (6.48)

If we have two arbitrary contravariant vectors Am and Bm, then they define two directions with unit vectors

(6.49) The angle between these two directions is, according to (6.48), given by (6.50) From (6.50) we also see that the orthogonality condition for two contra~ variant vectors Am and Bn has the form 8mnAmB"=O

(m,n=l,Z, ... ,N).

(651)

The angle formed by two curves at their in~on is an angle between their tangent unit vectors at the intersection point. For two curves given by the parameter equations x"' = lfrm(t) and X' = q::in(t), the angle formed by

Section 6.5 Schwarz Inequality

51

these curves at their intersection point is given by

where gmn, dlf/'1 /dt, and dq;m jdt are all calculated at the inlersection point Of the two curves. From (651) we see that the orthogonality condition for

the two cun•es at their intersection point is given by

[§]]

Schwarz Inequality

Let us consider two arbitrary contravariant vecto£s, denoted by Am and Bm, and let us form a linear combination of these two vectors as follows: (6.54) where a is an arbitrary real absolute scalar parameter. The linear combination cm itself is also a contravariant vector The absolute square of the vector cm is given by the positive definite form

ICl 2 = gmncmcn

ICl 2 ~ g=(Am + aB"')(A" + aB") ICl 2 = 8mnAmAn + 2agmnAmB" + a 2gmnBmB" 2

ICl = IA1

2

-L-

2agmnAmB"

2

+.a IBl

2

(6.55)

•

For the quadratic form (6.55) to be nonnegative for all values of the parameter a, the following inequality must be satisfied: (6.56) Using the notation A

= IAI and B = IBI. the inequality (6.56) gives (6.57)

The inequality (6.57) is known as the Schwarz inequality and it is of importance in a number ofbranches of mathematics and physics.

Chapter 6 The Metrlc Tensor

52

[§&]

Orthogonal and Physlcal Vector Coordinates

In this section we consider an important special case when the three-dimensional Euclidean metric space is defined by some set of orthogonal curvilinear coordinates {:x!'} (k = 1, 2, 3). For such a system of

curvilinear coordinates. in every point of space the following conditions are satisfied: 812

= 823 = 831=0.

(6.581

Thus, we may write 8mn = htf>mn

(m.n.M

= I. 2. 3).

(6.59)

where hM = hM(:x!') are some functions of coorrlinates {xk"}. In this case the metric of the space can be written in the form ds2

= (h1d.x1)2 + (h.i.dx2)2 + (h3fil3)2,

(6.60)

and the matrix of the metric tensor is a diagonal 3 x 3 matrix [g,,,,,]

=[

(h,) 0 0

2

0

(h,) 2 0

OJ 0 .

(6.61)

(h3J'

Let us now, as an example, consider the vector of generalized velocities in this coordinate system. The contravariantcoonlinates of the velocity vector are given by dxm

V"=-.

(0.62)

dt where dxm are the coordinates of a contravanant polar vector in any system of coordinates and dt is an absolute scalar parameter. Using (6.61) we can calculate the covariant coordinates Of the velocity vector as follows: (6.63)

As the gener:alized coordinates do not neccessarily have the dimension of length (e.g., they can be the angular coordinates), the func:ions ~ = ~(.yk") are notneccessarily dimensionless (e.g., they may have the dimension oflength). From (6.63) it follows that the dimensions of the contravariant and covariant coordinates of the velocity vector are not the same, and they are not the same as the expected dimension of the velocity vector (i.e., length/time). The coordinates of the velocity vector that do have the dimension of velocity (i.e., length/time) are called the physiml

53

Section 6.6 Orthogonal and Physical Vector Coordinates

coordino.tes of the velocity vector. The physical coordinates can, in the special case under consideration, be obtained from the contravariant or covariant coordinates using the fonnulae

(6.64) As an illustration, let us consider the velocity vector in the spherical coottlinales, where

x1

= r,

:XJ- =

e,

x3

= 'f',

(6.65)

The contravariant coordinates of the velocity vector in the spherical coordinate system are given by (6.67) The associated covariant coominates of the velocity vector in the spherical coordinate system are given by

v1=~, v2=r2~,

v3=r2sin

2

ej,-.

(6.68)

From the results (6.67) and (6.68) we see that neither all the contravariant nor the covariant coordinates of the velocity vector have the dimension of velocity (i.e., length/time). The physical coordinates of the velocity vector in the sphencal coonli~ nate system, i.e., the prOJeCtions of the velocity vector to the directions of the curvilinear axes in a given space point, according to the definition (6.64) have the fonn

vcn

dr

de

dq;

=di' va) = rdt, vrn = rsmedi.

(6.69)

In a similar way, we can consrruct the physical coordinates of an arbitrary

vector with respect to an arbitrary orthogonal system of coordinates in three dimensions. This approach can also be extended to an arbitrary N-dimensional space, but it is most commonly used in three or sometimes four dimensions.

~

Chapter 7

Tensors as Linear Operators

Second-order tensors can be described as linearoperatCJ£S acting on vectors in metric spaces. An operator in the N-dimensional metric space is defined by the way it acts on different vectors in the space. For example. in the expression

Bm

= O'/:An

(m,n= 1,2, .. . ,N),

(7.1)

the tensor a,;' represents a linear operator, which in a unique way relates a vector Bm to the original vector Am. This operator is called a linear operator since it satisfies the conditions of linearity and homogeneity, i.e.,

O;;'(A" +B") ~ O;;'A"+O;;'B" O;;'(~A") ~ ~O;;'A",

(7.2)

where An and B" are arbitrary contravariant vectors and f3 is an arbitrary absolute scalar. Eigenvectors of an operator a,;' are defined as those vectors sn that retain the direction and only change the absolute value as a result of the action of the operator o;:. In other words, for the eigenvectors S", we have O;;'S"~>S"'

(m,n~l,2,.

(0;;'->S;;')S" ~o

. .,N),

(m,n~ 1,2,. . .,N).

(7.3)

(7.4)

55

Chapter 7 Tensors as Linear Operators

56

Tue system of homogeneous linear Equations

(7.4) has a non-trivial solution for S" only if its determinant is equal to zero. i.e., (7.5)

The N solutions for the parameter A of the algebraic Equation (7.5) are called the eigenvalues of the operator o;:. The Equation (7.5)is smietimes called the seculn.r equation. As an example of the concepts we have just defined, let us consider a tensor o;:, defined in a three-dimensional metric space by the matrix

[O;;'l=

I 0 [5

5]0 .

0 -2 0

(7.6)

I

Substituting (7.6) into (7.5) we obtain the equation for the parameter A in thefonn

1-1.

0 1 5

0 -2-1. 0

5

0 1=0.

(7.7)

L-A

By expanding 1he determinant in (l.7) we obtain (I -A)[(-2-A)(l -A)]+ 5(-5(-2-A)] =O-~~+~~-m+~~+~

~Q

= (A+2)[25-(l.-1)2] = 0. The solutions of the Equation (7.8) for the parameter A are the following:

J...+2=0

~

A=-2

25-(A- 1)2 =0 ~ A=l±5.

(7.9)

Thus the three eigenvalues of the operator o;: are equal to

The corresponding eigenvectors S". pn. and Qn are obtained from the matrix equations

[I 0 5] [S'] = [s'] o -2 o s o 1

s2 sJ

-2 s2 ,

s3

(7.11)

Chapter 7 Tensors as Linear Operators

57

[~ ~2 ~][~~] = [~l

(7.12)

-4

[~ ~2

fl rnn

=

6

+

mil

(7.13)

By solving the matrix Equations (7 .l l}-(7.13). we obtam the norrnahzed (unit) eigenvectors of the operator o;: as follows:

(7.14\

(7.15)

(7.16) It is easy to show by direct substitution that the eigenvectors (7 .14}-(7.16)

satisfy the above matrix Equations (7.11)-(7.13), respectively. It should also be noted that the eigenvectors sn, pn and Qn, which correspond to different eigenvalues, are orthogonal to each other. In other words, it is easy to show by direct calculation that these eigenvectors satisfy the orthogonality conditions

[S"J'[P"] = [S"J'[Q"l = [P"J'[Q"l =

o.

(7.17)

where the notation MT is used for a transposed matrix of an arbitrary matnx M. Furthermore, the three eigenvecto£s are normalized so that their absolute values are equal to unity. In other words, they are unit vectors of the three independent directions. Thus, it can be shown by direct calculation that the eigenvectors satisfy the nonnalization conditions

[S"J'[S"]

= [P"J'[P"] = [Q"J'[Q"] =I.

(7.18)

The set of normalized eigenvectors orthogonal to each other is called the set of orthononnal eigenvectors. In general, the eigenvectors of an operator a,;', corresponding to tht": different eigenvalues, are nrutually orthogonal ifthe tensor = tf'1'o;:

omn

Chapter 7 Tensors as Linear Operators

58

is symmetric. This can be shown by using the definitions O';S"~AsS"'

O'/:Pn=ApPm

O';:S"Pm O':PnSm

= gnk0"1"S"Pm = 0"1"SkPm = AsS"'Pm = gnk0"1kpnsm = 0"1"PkSm = ApPmSm.

(7.19)

(7.20)

Using the symmetry of the tensor om1', i.e., the equality om1' = cfan, we can rnterchange the dummy indices m+*k to see that the left-hand sides of both of the Equations (7 20) are equal to each other. Furthermore, we note the equality S"'Pm = pmsm. With these properties, we can subtract the second equation from the first equation in (7.20) to obtain

0 =(As - Ap)S"'Pm.

(7.21)

From the result (7.21) we see that whenever the two eigenvalues are not equal to each other, i.e., whenever As ¥- Ap, the two corresponding eigenvectors are indeed orthogonal to each other: (7.22) The e1genvect01S of an operator are in general determined up to an arbitrary multiplication constant. However, in most cases it is convenient to have normalized eigenvectors of an operator, i.e., to use unit vectors as eigenvectors. Thus we require, as a convention, thatthe eigenvectors should satisfy the normalization conditions (7.23) Tius implies that the eigenvectors of an operator o;: in the threedimensional metric space (e.g., sm, pm, and Qm above) define three muwally orthogonal direction'i, provided that the corresponding eigenvalues are not equal to each other, i.e., As #=- Ap #=- Ao. These three directions are sometimes called the mnin directions of a second-order

tensoro;:. Iftherootsofthe secular Equation (7.5)arenotdistinct, wehaveadegeneYUC).•. In the three-dimensional case, iftworootsare equal to each other, the

second-order tensor has only one main direction and a plane perpendicular to it, where all directions are main directions of the tensor. If all three eigenvalues of a second-order tensor are equal to each other, such a tensor does not distinguish any particular directions in the three-dimensional metric space. In other words, all directions in the three-dimensional metnc space are the main directions of such a tensor.

Part

n

Tensor Analysis

._ Chapter 8

Tensor Derivatives

~

Differentials of Tensors

In the curvilinear coonlirud:es the differential of a covariant vecror Am, denoted by d.Am, is not a vector. It is easily shown using the transformation law of covariant vectors (4.5). If a covariant vector is defined with respect to the systems Of coordinates {xk} and {zk} by N coonlinatesA,,, and Am. respectively. then it transforms according to the transformation law

-

ax"

(m,n

Am= Bz"'A,,

= 1,2, ... ,N).

(8.1)

(Bx") Bz"' A,,,

(8.2)

Using (8.1) we obtam

-

Bx"

dAm= azmd.An--t-d

(8.3) The result (8.3) clearly shows that d.An does not transform as a vector. except in a special case when

azx" azmazk

= 0•

(8.4)

i.e., :in the case of a linear orthogonal transfonnation of Descartes COCJ£dinates into some new Descartes coonlinates. As the differential Of

61

Chapter 8 Tensor Derivellves

62

a covariant vector

dAm =

~~ dxn

(m, n

= 1, 2, ... , N)

(8.5)

is a system that ts not a covari<mt vector, while the systemd.xn is a covariant vector, the second-order system BAm/B.x'I is not a tens01: The observation that the differential d.Am is not a vector is related to the definition of the differential, i.e.,

dAm ~ A,,,(x' + dx'J-A,,,(x')

(k,m ~ 1,2, ... ,N).

(8.6)

By definition (8.6), dA,,, is adifterence between two covariant vectors with the origins in different, infinitesimally close points of the metric space. On the other hand, the multiplier in the tr.ms formation law (8.1) is of the form (8.7) and it is in general a function of coordinates. The multiplier (8. 7) is different in different points of the metric space, except in the special case of linear transformations. It is therefore evident that vectors, in general, transform differently in different points of the metric space, and consequently the differential (8.6) cannot transform as a covariant vector. In order to construct a differential of a covariant vector in curvilinear coordinates, which is a covariant vector itself, it is required that both vectors on the right-liand side of (8.6) be situated in the same point of the metric space. This can be achieved by moving one of the infinitesimally close vectors entering the right-hand side of the Equation (8.6) to the same point where the other vector is situated. The operation of moving one Of the infinitesimally close vectors to the point where the other vector is situated must be rerformedin such a way that in the Descartes coordinates the resulting difference is reduced to the ordinary differential lfAm. Since dA,,, is a difference of the components of the two infinitesimally dose vectors, the comp0nents of a vector to be moved to the point where the other vector is situated must remain unchanged This can only be achieved by a parallel displacement of the vector between the two infinitesimally close points, using the Descartes coordinates. Let us therefore consider an arbitrary contravariant vector with the coordinates Am at a point Of the metric space with coordinates x"1 The coordinates of this contravariant vector at the point of the metric space with coordinates x"1 +rum are denoted by Am + d.Am. An infinitesimal translation of the vector Am to the pointofthemetric space with coordinates x"1 + rum generates a translated vector denoted by Am at the point Of the metric space with coordinates x"1 + dx"' Thus the differential Of the

Section 8.1

Dlffen!ntlalscfTensors

63

contravariant vector Am can be written in the form

dAm ~Am(x'+dx')-Am(x'J d.Am =Am(xk+tb:k)

A_m(_x1< +tb:k)

+Am(x'+dx')-Am(x') ·; b d.Am

f.> (8.8)

= DAm(_x1< + tb:k) +Mm.

In Equation (8.8) we have mtroduced a differentral DAm between the two vectorsAm(.xk +tb:k) andA.m(xk +tb:k) situated atthe same point.xk +m:k of the metric space, which itself behaves as a cootravariant vector with respect to the coordinate transformations, as follows:

Furthermore, in Equation (8.8)we have introduced an increment Mm, due to the parnllel translation of the vector Am to the point of the metric space with coordinates x"1 + dx"', as follows: (8.10) As Mm is a difference between the translated vector Jim at the point.xk +m:k and the nontranslated vector Am at the infinitesimally close point xk, the system l'JAm is not a vector. The difference l'JAm vanishes in the Descartes coordinates and DAm reduces to d.Am, as required. In order to calculate the increment l'JAm, we note that it is a function Of the coordinates of the contravariant tensor Am themselves. This functional dependence must be linear, since a sum of two vectors must transform according to the same transfunnatJon law as each of the vectors. Furthermore, it alsolias to be alinearfunctmn of the coordinate differentials. Thus, we may write (8.11)

r::;,

where is a system of functions of coordinates, which are usually called the Christoffel symbols of the second kind. Composition with the metric tensor gives the Christoffel S)Wlbols ofthe first kind, as fbllows· (8.12)

r::;,

The system is dependent on the coordinate system and in Descartes coordinates all of its components vanish, i.e., = 0. It is therefoce clear that is not a tensor, since a tensor that is equal to zero in one coordinate system must remain equal to zero in any other coordinate system. In a Riemanruan space it is not possible to find a coonlinate system to satisfy

r::;,

r::;,

••

Chapter 8 Tensor DerivaUves

the condition r~ ~ 0

(k,n,p ~ 1, 2, ... ,N)

(8.13)

in the entire metric space. 8.1.1

Differentials of Contravariant Vectors

Substituting the result (8.11) into the definition (8.8), we obtain the result for the differential of a contravanant vector Am, as follows· (8.14)

(8.15)

8.1.2

Differentials of Covariant Vectors

In order to derive an expression analogous to the result (8.15) for covariant vectors, let us coosider an absolute covariant vector Am and an absolute contravariant vector Bm. The ccmposition of these two vectors gives an absolute scalarAmBm. As thescalarsareinvariantw1threspecttotheparnllel translation, we can write

BmliAm

= -AmfJBm = .+-Anr~Bmdxp,

(8.17)

where we have made an interchange Of the dummy indices m ++ n on the right-hand side of Equation (8.17), such that it can be rewritten as (8.18) As the equality (8.18) is valid for an aibitrary cootravariant vector Bm, we have

<8.19) On the other hand, analogously to the case Of the contravariant vectors, we may write

Section 8.2 Covariant Derivatives

65

)dx" ~ DxP

DA,,,~ (aA,,, -r"mp An BxP

DAm
(8.21)

~ Covariant Derivatives 8.2.1

Covariant Derivatives of Vectors

From Equation (8.15), we conclude that in the tensor analysis the differential Of a contravariant vector d.Am, which does not have a tensor character, is replaced by the tensor differential DAm. Comparing the results (8.5) and (8.15), we see that, following the same approach, we need to replace the partial derivative of a contravariant vector with respect to a coordinate, which does not have a tensor character either, by the covarillTlt deril•atil>e ofa controvariant vector with respect to a coordinate, as follows:

aAm BxP

~

DAm DxP'

(8.22)

wliere the covariant derivative of a contravariant vector is defined by (8.15). as follows: (8.23)

:d:ee~~~=s;::~~;~Xat~e~·

the covariant derivative

From theresult(8.21), weseethatthecovariantderivative ofa covariant vector is defined by the expression

DA,,,~ BA,,, -r" A. DxP

(lxP

mp

n

(8.24)

The results (8.23) and (8.24) show that the covariant differentiation of both contravariant and covariant vectors gives the corresponding second-order tensors. In general, by covariant differentiation we preserve the tensor character of an aibitrary tensor, but we increase its order by one. It should be noted that for both contravariant and covariant vectors the order is increased by one additional covnrillllt index. Because this operation always increases the number of covariant indices of an arbitrary tensor by one, the corresponding derivative lias been named the covarinnt derivative.

Chapter B Tensor Derlvallves

65

8.2.2 Covariant Derivatives of Tensors

Let us consider a special case of a second--order contravariant tensor, obtained as a product of two contravariant vectors Am and Bm, denoted by = AmBn. By par.tllel translation. we obtain

cmn

fJ(AmBn) =Aml'JBn +BnMm.

(8.25)

Using the definition <8.11). this expression becomes

S(AmB') ~ -Amr";,Jfdx.P - B'r",;;,A'dx" S(AmB') ~ -(rz,AmJJ + r:';,A'B') dx1'

(8.26J

= -(r~Cmk--t- r;:;,ckn)ruP.

fJC"'n

Because of the linear character of the operation of par.tlle1 translation, the result (8.26) is also valid for an arbitrary contravariant tensor As the differential of a contravariant tensor cmn is by definition given by

cmn.

DCmn

=

dC"m - l'JC"'n'

(8.27)

substituting frqm (8.26), we obtain

DC"m=

) acm' ( ~+r~cmk+r;;;,ckn dx.P.

(8.28)

From (8.28), the covariant derivative of a second-order cootravariant tensor cmn is defined as

~n

=

a~ + r~cmk + r;:;,ckn

(8.29)

The same approach can be used for a second-order covariant tensor Cmn A.nBn. where we may write

l)(AmBn)

= BnMm +A,nl)Bn = +Bnr~kdx1' + A,,,r~pBkdx1'

l)(AmBn)

= ( r~kBn + r~AmBk) dx1'

l)(AmBn)

l)Cmn

=

(8.30)

= (r!.i,c1:n + r~cmk)M.

Again, because Of the linear character of the operation of parallel translation, the result (8.30) is also valid for an arbitrary covariant tensor Cmn· As the differential of a covariant tensor Cmn is by definition given by (8.31)

Section 8 3 Properties cf Covariant Derivatives

67

substituting from (8.30), we obtain (8.32) From (8.32), the covariant derivative of a second-order covariant tensor Cmnis

~; = a~;

-

r!pckn -

r~cmk.

(8.33)

By simple consideration of Equations (8.29) and (8.33), it is easy to define the covariant derivative of a mixed second--order tensor C;:' as

~; =

aa; + r;:;,c! - r~c;;.

(8.34>

Analogously to the definitions (8.29), (8.33), and (8.34), it 1s possible to define covariant derivatives Of tensors of an arbitrary order and type. As an example, let ns write down the covariant derivative Of a mixed third--order tensorC~:

(8.35) The example (8.35) dearly represents the general method for construction of the covariant derivatives of arbitrary tensors.

[§

Properties of Covariant Derivatives

There are a number of important rules and special cases that apply to covariant differentiation. These rules and special cases are frequently used in tensor calculations, and the most important ones are listed next. I. The covariant derivation is a linear operation and the following linearity condition is fulfilled:

D DxP(aAm

DAm

DBm

+ fJBm) =a DxP + f3 DxP.

(8.36)

2. The covariant denvative of a product of two rensors is defined m the same way as the usual derivative of a product of two functions. As an example, for the product AmCmn. we may write

~p(AmCmn) = ~~: Cmn +Am~;n.

(8.37)

Chapter 8 Tensor Derivatives

68

3. The covariant derivative of the metric tensor is equal to zero. In order to prove this proposition, let us observe that for a covariant vector DAm, in the same way as for any other vector_ we have DAm = 8mnDAn.

On the otherhand,usingA,,,

(8.38)

= gmnAn, we may write

As An is an arbitrary vector, by comparison of Equations (8.38) and (8.39) we immediately conclude that

In other words. with respect to the operation of covariant differentiation, the metric tensor can be treated as a constant, regardless of the coordinate dependence of its components.

4. The covariant derivative of the li-symbol is equal to zero. In order to prove this proposition, we use the earlier results

Dg., ~o,

Dg"* ~ 0.

(8.41)

Using (6.17) and (8.41), we may write Dll;:

= D(gmkgnk) = gnkDg,,k + 8nkDgnk = O.

(8.42)

Using (8.34). it is also possible to prove this proposition by direct calculation:

~~; = :~ -r r::;,/j! - r!/;;:' = r::;, - r::;, = o.

(8.43)

5. The covanant denvative of an mvariant scalar funcllon 1s equal IO its ordinary partial derivative, and we may write D¢

_ilj_

DxP

BxP

(8.44)

The parual denvative Of a scalar function 1s therefore transformed as a regular covariant vector. In order to prove this proposition, let us observe a scalar¢ = AmBm. where Am is some absohtte contravariant vector and Bm is some absolute covariant vector. The covariant derivative of the scalar fllllction ¢ is given by

~! = ~(AmBm) = ~~:Bm+Am~8;.

(8.45)

Section 8.4 Absolute Derivatives ol Tensors

69

Z! = ~~:Bm+r~kBm+Am~~;-r~mB,..

(8.46)

By interchanging the dummy indices k ~ m, we see that the second term on the right-hand side of Equation (8.46) is cancelled by the fourth teilll, and we obtain

Z!= ~~:Bm+Am~~;

=

a:P(AmBm)=~.

(8.47)

which proves the proposition (8.44-).

~

Absolute Derivatives of Tensors

Let us assume that a curve C in the N--dimensional metric space is given by means of N parameter equations x"'=x"'(s)

(m=l,2, ... ,N),

(8.48)

where s is some scalar parameter. Let us now consider a covariant vector Am defined along the curve Casa function of the parameters. The absolute derivative of the vector Am with respect to the parameter sis defined using (8.21) as follows:

DAm = (aAm _ rn n) ~. ds

fJxP

ds

DxP ds

~

ds

(8.49)

(8.50)

In the same way we define the absolute derivative of a contravariant vector Am, as follows: (8.51) The definitions of the absolute derivatives (8.50) and (8.51), for covariant and contravariant vectors, respectively, are easily generalized to the case of an arbitrary tensor. For example, the absolute derivative of a tensor C:Jc 1sgiven by (8.52)

70

Chapter 8

Tensor Derivatives

where the covariant derivative of the tensorC;l is given by Equation (8.35),

i.e., (8.53) The properties of the absolute derivative are the same as the properties of the covariant derivative and will not be repeated in this section.

.,. Chapter 9

Christoffel Symbols

[[I]

Properties of Christoffel Symbols

In order to define various tensor derivatives in the previous chapter, we have introduced a system r~. called the Christoffel symbol of the second kind. The objective of this chapter is to specify this system and ontline its most important properties Let us first prove that the Christoffel symbols are symmetric with respect to their lower indices. In order to prove this statement we note that if A,,, is an arbitrary covariant vector, then the quantity (9.1) is a second-order covariant tensor. Let us now assume that the vecto£ A,,, can be given by the expression D¢ Am= Dx"'

=

8 8.x"''

(9.2)

where¢ is some arbitrary scalar. Then the expression (9.1) can be written in the form

71

Chapter 9

72

DA,,, - DAp __!21_ DxP

Dxm (lx:P(lxm

- __!21_ + (r' fJx"'fJxP

pm

Christoffel Symbols

- r'mp) ~ ()xi<

(9.4)

The first and second term in Equation {9.4) cancel each other and we obtain

~~;-~:~=(r;m-r~)~-

(9.5)

As the left-hand side of Equation (9.5) is a second-order covariant tensor, we conclude that the right-hand side of Equation (9.S), i.e., (9.6)

is also a second-order covariant tensor. However, we J...-now by defimtion of the Christoffel symbols lhat Ibey are identically equal to zero in all points of the Euclidean metric space described by the Descartes coordinates. Thus the second-order covariant tensor (9.6) is identically equal to zero in the

Desautes coordinates. But a tensor that is identically equal to zero in one coordinate system must be equal to zero in any other coordinate system. Thus we conclude that

~-:;- ~:~ = (r;m- r~) ~ =0.

{9.7)

As the vector Am. gtven by (9.8)

is an arbitrary covariant vector and ¢ is an arbitrary scalar function, we conclude from Equation (9.7) that

r;m = r!.i,

(k,m,p

= 1.2..

.N),

(9.9)

and the Christoffel symbols of the second kind are indeed symmetric with respect to their lower indices. From (9.9)it isevidentlhat, for the Christoffel symbols of the first kind, we have

rn.rm

= rn,mp

(m,n,p

=

1,2, ... ,N).

(9.10)

Let us now derive the transformation law of Christoffel symbols with respect to the transformations of coordinates. If a contravariant vector is defined with respect to the systems of coordinates {.J!} and {t'} by N coordinates A"' and Am, respectively, then it transforms according to

Section 9.1

Properties ol Chistorfel Symbols

73

the transformation law (9.11)

Usmg (9.11) we may wnte (9.12) or, after calculating the derivative in the parentheses,

aA.m OzP

Ox" 0 2 zm

ax" O:.:"' OAj

.

= azP axi ~ + 8zP ax'
<9-13 )

The transformation law (9.13) is just a direct confirmation of the fact that the partial derivative of a contravariant vector is not a tensor, as we have shown indirectly m the previous chapter. On the other hand, the co van ant derivative of the contravariant vector is a mixed second-order tensor and it transforms according to the transformation lav.

DA.m

ax'< Ozm DAj

D;:f' =

""iizP Oxj Dxk·

(9.14)

Here we recall the definitions of covanant derivatives of contravariant vectors (8.23), in the coordinate system {t'}, (9.15)

and in the coordinate system {x"}, i.e.,

DN Dx'<

OAj

j

= ihf + rlkA.1

(9.16)

Substituting (9.15) and (9.16) into(9.14), we obtain

aA.m OzP

+ tm A,n nr·

Ox" Ozm OAj +Ox'< Ozm rj Al 8zP Oxi lk ·

= 8ifi Oxi aJ"

(9.17)

Substituting (9.13) into (9.17) we obtain

Ox'< Ozm ON Bxi ~

""iizP

ax" 02zm

.

_ _

+ ""iizP OxkOxiA1 + r;~n

= ~azm~+~Ot'" r1 A' OzP Oxj Ox'<

OzP Oxi lk

.

(9.18)

Chapter 9

74

Ctwistdlef Symbols

The first term on the left-hand sule cancels the first term on the right-hand side in Equation (9. I 8), and by using (9.19)

we obtain from (9.18)

l a.xi i0. (l'Ji~~+tm~)A'=~azmr1 l (JzP Oxk(lxi

np

(JzP axJ

(9.20)

As Equation (9.20) is valid for an arbitrary contravariam vector A 1, Ibis tensor can be omitted. After omitting A1 and multiplying lhe remaining equation by (9.21) we finally obtain (9.22) Equation (9.22) 1s the reqmred trans.formation law for the Chnstoffel symbols, and it shows that the system rfk behaves as a tensor only with~ to the linear transformations of Descartes coordinate systems, when we have (9.23)

In general. as we have coocluded before, the Christoffel symbol is not a tensor.

~ Relation to the Metric Tensor For practical calculation of the Christoffel symbol we need to find its relation to the metric tensor. In order to find this relation, we start wilh the :identity ?.~ ·;

rt

" D!~n = a:;n - r!u,gkn - r~gmk =VO,

(9.24)

a:; = r

(9.25)

n,mp

+ r m,np·

Section 9.2 RelatlontotheMetricTensor

75

The permutations of indices in Equation (9.25) give the equation

a:;:

= r m.pn + rp.mn = 0,

(9.26)

and, with reversed signs, the equation

- ~~';;: = -rp,nm -

rn,pm

=0

(9.27)

By adding together Eqmtions (9.25), (9.26), and (9.27) and using the symmetry property (9.10), we obtain

r

-

_!_

m,np - 2

(a8mn (lxP

+

agpm - Clgnp) (l_xt! (lxm .

(9.28)

The result (9 .28) is a definition of the Christoffel symbol of the first kind as a function of the metric tensor. The Christoffel symbol of the second kind is then given by

rm l1P

= 8....m1..r

k,np

= _!_ 8 rnk (88kn-+- Clgpk 2

axP

axn

_ Clgnp) ax'<

(9.29)

The result (9.29) is a definition of the Christoffel symbol of the second kind as a function of the metric tensor. Using the definition (9.29), we can calculate the contracted Christoffel symbol of lhe second kind r:;im, as follows: (9.30)

Because of the symmetry of the metric tensor and the symmetry of the Christoffel symbol with respect to the interchange of its lower indices, the first term in lhe parentheses of Equation (9.30) cancels lhe lhird term, and we obtain (9.31)

The determinant of the metnc tensor is given by the expression (6.12) as follows: (9.32) where G"'k is a cofactor of the determinant /Emn I that corresponds to the element 8mn· The differential of the determinant g is, in view of Equation (9.32), equal to \9.33)

76

Chapter 9 Christoffel Symbols

Using now the definition of lhe contravariant metric tensor (6.16). i.e., (9.34)

mEquation (9.33), we obtain

dg = 8g""ag"" =>

fJ!.; = 8g""

a;;.

(9.35)

Substituting {9.35) into the result (9.31), we obtain

rm _ _!_~ - (lln.J8 mn- 2gaxn axn .

(9.36)

If we now recall the result (6.17) in the form (9.37)

gmkgm=li::1, we may contract it to obtain gmkgmk=li:::=N

(k,m=I,2, .. ,N).

{9.38)

Thus by differentiation with respect to the coordinate xn. we have (9.39)

(9-4-0) Thus there is an alternative expression for the contracted Christoffel symbol of the second kind obtained by substituting {9.40) into (9.31 ): (9.41)

gknr;:;,,

since it frequently appears in various calculations in the tensor analysis. This quantity is, by definition, given by

It is also of mterest to calculate lhe quantity

(9.42)

Using the symmetry of the metric tensor and mterchangmg the dummy indices k ~ n, we see that lhe first two terms in the parentheses of Equation (9.42) are equal to each other Thus we may rewrite {9.42)

n

Section 9.2 Relatlon to the Metric Tensor

as follows: (9.43)

The first term on the right-hand side of Equation (9.43) can be rewritten as follows: kn ,aEkl - kn a ( I ) kn (Jg"'I 8 8"' axn -8 ~ 8"'Ek1 -g Ek1--a;n

~t'"_a__(~m)-~"ag"'' =-agmn_ axn

k

l axn

axn

(9.44)

Combming the results (9.31) and (9.36). we may write I J.nfJEkn _

I fJg _ fJln.JE

2" a,;i- 2ga:;J - ---;g-·

(9.45)

and the second term on the right-hand side of Equation (9.43) becomes

_!g"'' 2

8

,,,ag,,, - -g"''aln.Jji _ - 8 mn_l_ 8.fii ax'-

ax'-

.Jiiax"·

(9.46)

Substituting (9.44) and (9.46) into (9.43) we obtain g

"'rmkn-- (agmn ..mn_I_ ".fii) axn +l> .JE axn _ I ( agmn mna.fii) - --:.;g ha;;;-+ g ax" .

and using the rule for the derivative of the product the expression for the quantity gknr;::, as follows:

(9.47)

.JEgmn, we finally obtain (9.48)

~

Chapter 10

Differential Operators

I10.1 I

The Hamiitonian V-Operator

In the ordinary three-dimensional Euclidean metric space the Hamiltonian V-operator is defined by lhe expression Vm

a = Bm =Ox"'

(m

= 1,2,3).

{IO.I)

However. by acting with this operator on an arbitrary vector or tensor we obtain a system that does not have the tensor character, as shown by, e.g., Equation {9.13). In an arbitrary N-dimensional generalized curvilinearcoordinaJ:e system, insteadofVm. we therefore use the covariant Hamiltonian operator Dm, defined by Dm

=

D Dxm

(m

= 1,2,

.. ,N)_

(10.2)

Using the operator Dm. we define the operators called gradient, divergence, curl, and Laplacian in the following four sections.

I10.2 I

Gradient of Scaiars

The gradient of a scalar function¢ is a covariant vector with its covariant coordinates defined as follows: (10.3)

Chapter 10 Dlfferentlal Operators

80

The contravariant coordinates of the gradient of a scalar¢ are obtained as follows: (10.4) In the orthogonal curvilinear coordinates, with the metric form given by (6.60), i.e.,

the physical coordinates of the gradient of a scalar¢ are given by I

D(m)ef>

I

8¢

= h,;Dmef> = h,; axm

{m,M

= 1,2,3).

{I0.6)

The definition {I 0.6) gives the correct physical coordinates of the gradient in the we11-known cases ofcylindrical and spherical coordinate systems. As an illustration, let us consider the spherical coordinates where, using {6.65) and {6.66). we may write =r,

xl=fl,

x3=
h 1 =1,

h1=r,

h3 =rsine.

x1

(10.7)

The gradient of a scalar function¢ is then. according to {10.6). given by 1

grad
where

I10.3 I

roe

rsmBBtp"'

cuun

er, ee, and e
The divergence of an arbitrary contravariant vector Am is a scalar obtained by the composition of the covariant Hamiltonian operator Dm with that contravariant vector Am. Le., (10.9) Using here the result {9.36), we may write DmAm

=~~=+a~!An.

(IO.IO)

Section 10.3 Divergence ol Vectors and Tensors

81

Since n is a dummy index in the second term on the right-hand side of Equation {IO.I 0), it can be changed tom. Tlrus we may write DmAm

= 8Am + -2._ 8,J8Am axm

DmAm

I

,J8 axm

(

aAm

="""""Ji ./if axm +

8,fii )

axmAm .

{I0.11)

Finally, the expression for the divergence of the contravariant vector Am is given by (10.12) If a vector is defined by its covariant coordinates A,,., then the divergence of such a vector is written in the form

{I0.13) where we have used the property that the metric tensor behaves as a scalar with respect to covariant differentiation. In such a case, using {I0.12), we obtain

DmAm ~ 7.a!m(,fiig~An)-

(10.14)

Analogously to the case of a contravariant vector {I0.12), tt tS possible to define the divergence of an arbitrary tensor with respect to one of its upper indices. Tlrus, the divergence of the tensor T;r' is defined by

{I0.15) In the orthogonal curvilinear coordinates. with the metric form given by {10.5), we have ,Jg = h1h2hJ, and the physical coordinates of the divergence of a contravariant vector Am are given by (10.16) where the physical coordinates of the contravanant vectOI" Am are given by A(m)=hMAm

{m,M=l,2,3).

(10.17)

The definition (10.16) gives the correct physical expressions for the divergence in the well-known cases of cylindrical and spherical coordinate systems. As an illustration, let us again consider the spherical coordinates

Chapter 10 Dlfferential Operators

82

defined by {I 0.7)_ The physical expression for the divergence of the vector Ain the spherical coordinates is given by

divA=~-J;(r2A.)+ rsilnea°e-(sm6'Ae)+ rs:ne a~-

I10.4 I

{10.18)

Curl of Vectors

In an arbitrary N-dimensmnal menic space the curl of a vecrorfunctionA,,. is a second-order covariant tensor Fmn. defined by the expression Fmn=DmAn-DnAm

(m,n= 1,2, ..• N).

(10.19)

In the special case of a three-dimensional metnc space. it is possible to construct a contravariant vector ck related to lhe tensor {10.19) using the three-dintenSional totally antisymmetric Ricci tensor E:kmn. defined acrording to the general definition {6.23) as follows: E:kmn

= ~J:mn

(k,m,n

= 1,2,3).

{10.20)

Thus the curl in the three-dimensional metric space IS defined by

In the orthogonal curvilinear coordmates, with the metric furm given by {10.5), we have ,Jg= h1h2hJ, and the physicaj. coordinates of the curl of a covariant vector An are given by

where the physical coordinates of the covariant vector An are given by A(n)

=~

(n.N

=

1,2,3).

(10.23)

As the expression {1022) is somewhat more complex and generally not readily understood, we expand it for the three components of the curl in

Section 10.5 Laplacian ol Scalars and Tensors

83

three dimensions: corl(l)A

= _!___ (8(hJAc3» _ h2hJ

8(h2Ac2)))

ax1

ax3

cm'-,, A=_!___ (8(h1Acn) _ 8(hJA(3»)

~

ax3

h1h,

ax 1

A _ _!___ (a(t12Ac2» _

curl (J)

h1 h1

-



B(h1Acn>) 8x2 .

8x1

The expressions (I 0.24) are usually structured into a determinant defined as fbllows:

curtA=h1 ~2h3 1t

h1Acn

t

h2Ac2)

ti·

(1025)

hJAo>

The definition (I 0.25) gives the correct physical expressions fbr lhe components of lhe curl of a vector function in the well-known cases of cylindrical and spherical coordinate systems_ As an illustration, we can write down this detenninant for the spherical coordinates defined by (10.7). Tirus we may write

e,

reo

curl A= - 2- .- iii r sme A,

' iiii

~

rAe

rsine~

-

a

1

I

curl

rsinee
a

,

(1026)

A=-.'-[_'.'_ (smeA,o)- BAa]e, rsme ae 8tp aA, a(r~ >]~eo-1-;'[aa;-ae aA,]_efl'. +-; sine iltp -a; '[ 1

{I0.27)

[10.5 [ Lapiacian of Scaiars and Tensors TheLaplacian of a scalar function in the three-dimensional vector analysis is defined by the expressiOli fl¢= div{grad¢).

(10.28)

84

Chapter 10 Dltferential Operators

In the tensor analysis this expression becomes

From {I0.29) we see lhat the Laplacian of the scalar function ¢ is a divergence of a vector given by its covariant coordinates (10.30) Using here the definition (10.14) of the diveigence of a vector defined by its covariant coordinates, we may write

" ¢- _I__a_ ( ~'!±_) ,fij8x"' ,jig 8x' ·

(10.31)

In the orthogonal curvilinear coordinates, with the metric furm given by{I0.5), we have ,Ji= h1hi_h3 and the Laplacian of the scalar function¢ becomes (10.32)

where

(10.33)

Substituting (I 0.33) into ( 10.32) we obtain

[_8_ (h2h31!1_) hi ax 1

1'¢ _ _ I_ - h1h2h3 8x 1

+~(h~:3~)+a~3 (h~2~)]-

00.34)

The definition {10.34) gives the correct physical expressions fur the Laplacian in the well-known cases of cylindrical and spherical coordinate systems. As an illustration, in the spherical coordinates defined by (I 0.7). the expression for the Laplacian becomes

~¢= .!__~ (r2 ~) r28r

Br

+r2sineae_il_ (sine~)+-fl2¢_ ae eaqll _ l_

1 r2sin2

(10.35)

Section 10.6 Integral Theorems for Tensor Fields

J

85

10.6 I integrai Theorems for Tensor Fieids In the three-dimensional vector analysis there are two important integral theorems, called the Stokes theorem and the Gauss lheorem.. These theorems remain valid in the tensor analysis and their formulation is generalized in such a way that they can be applied to integrals in arbitrary N-dimensional metric spaces_ These theorems in the tensor notation will be discussed in the rest of this section. 10.6.1

Stokes Theorem

In the usual three-dimensional vector notation the Stokes theorem is defined as

(10.36) Tix! integral on the right-hand side of Equation ( 10.36) is a surface integral over a surface S bound by a closed contour C. Tix! integral on the left-hand sideofEquation(l 036) is alineintegralroundthe dosedcontourCrunning along the boundaries of the surface S. In the tensor notation the furmulation of the Stokes theorem has lhe fotlowing furm: (10.37)

where Fmn 1s the curl tensor of the vector Am. defined by Equation (l 0.19), i.e.,

(1038)

On lhe other hand, (10.39) is the contravariant tensor of an infinitesimal element of the swface S. From lhe definition (I 0.37) we see lhat lhe Stokes theorem in tensor notation is valid for arbitrary generalized metric spaces. Let us now prove that Equation (10.37) is equivalent to Equation (1036) in the three-dimensional metric space. We first note that in the three-dimensional space we have (10.4-0)

From (10.40) we see that the integrals on lhe left-hand sides of Equations (10.36) and (10.37) are indeed equivalent to each olher. In order to prove that the right-hand sides of Equations (10.36) and (1037) are also

Chapter 10 Differential Operators

85

equivalent to each olher, it is corwenient to introduce a three-dimensional covariant suiface vector as follows:

dSk

= 4skmndxmdxn = ~..Jie1unndxmdxn.

{10.41)

In the Descartes coordinates, where ,J8 = I. the components of this vector are given by dS1

=dx2 dx3 ,

dS2

=d.x1dx3 ,

dS3

=dx1dx2 •

(10.42)

Furthermore, analogously to (10.21), it is convenient to introduce lhe contravariant curl vector, denoted by Ck and related to the tensor F mn by the expression Ck=

~ek11Fjt =eki1DjAJ =

-Jge'41DjA1 (curlA.r. =

{10.43)

Using {10.43) we may write

curl.A -dS = (curtA)kdSk = ;;:jlehnnD1A1dxmdxn_

{10.44)

On the other hand, the e-symbols satisfy the identity

(I0.45) Substituting {10.45) into {I0.44) we obtain

curtA-dS=~(llkll~

~,i;!i)DjA1dxmdxn

curl.A ·dS = ~ (D,,.An - DnAm)dxmdxn

(10.46)

curlA.-dS=!Fmntf.Smn. From {10.46) we see that the integrals on the right-hand sides of Equations {I 036) and {1037) are also equivalent to each olher in the three-dimensional metric space 10.6.2 Gauss Theorem

In the usual three-d1mens10nal vector notation the Gauss theorem is defined by means of the equation

fsA-dS=

L

divlid!J.

(10.47)

Tre integral on the nght-hand side ofEquat10n (10.47) is a volume integral over a volume !J bound by a closed surface S. The integral on the left-hand

87

Section 10.6 Integral Theorems for Tensor Fields

side of Equation {I0.47) is a surface integral over the closed surface S enclosing the volume Q_ The volume element d!J is a relative scalar with the weightM=-1. If the volume element is defined with respect to the systems of coordinates {x"} and {z~'} by d!J and dfJ.m. respectively, then according to the Jacobi theorem it transforms as fbllows:

dii = l!~ld!J =

l!;.-

d!J.

1

(10.48)

Therefore. in the tensor fbrmulation of the Gauss theorem. we use the itwariant volume element ,Jgd!J to obtain

fsAmdSm=

l

DmAm,Jgd!J.

(10.49)

The tensor fbrmulation of the Gauss theorem {10.48) is valid for arbitrary generalized metric spaces. The equivalence of Equations {10.47) and {10.48) in the Descartes coordinaJes, \.Vhere ,J8 1 and Dm = Vm. is evident. The Gauss lheorem can be extended to the case of an arbitrary tensor with at least one uwerindex. As an example, fora mixed lhird-ordertensor Tkmn. lhe Gauss lheorem is given by

=

t

T;:"'dSm

~

l

DmT;:"' ,/ii d
(!0.50)

.,. Chapter l1

Geodesic Lines

In Euclidean metric spaces the shortest distance between two given points is a straight line. However, even the simplest example of a two-dimensional Riemannian space on the swface of a unit sphere shows clearly that in the Riemannian spaces there are in general no straight lines. It is therefore of interest to find the curves of minimum (or at least stationary) arc length connecting the two given points in a Riemannian space. In Riemannian space~ we in general do not impose the requirement to find the lines of minimum arc length, and we replace it by the requirement to find lines with a stationary arc length between the two given points. We can illustrate this on the simple example of the two-dimensional Riemannian space on the surface of a unit sphere. For both arcs on the same circle, connecting the two given points on the unit sphere, we can only say that their lenglh is of a stationary character. but we cannot say that they are of mIBimmu length. The curves with stationary arc length between two given pcints A and B are called the geodesic lines, and they are determined by the solutions of the corresponding geodesic differential equations. In order to construct these differential equations, we will use the variational calcnlns and Uigrange eq11ations. which will be derived in the next section.

[[iJJ

Lagrange Equations

Let ns observe two fixed points A arrl B in an arbitrary metnc space. Their coordinaJes are given by the parameter equations (11.1)

89

90

Chapter 11

C"

A(

c

' Figure3.

Geodesic Lines

)B

C' Curves~thel1xedpointsA

and B

where sis the arc-length parameieI: Berween the fixed points A and B we can draw a family of curves of various arc lengths, as shown in Figure 3. Within the family of curves, which connect the two fixed points A and B as shown in Figure 3, lhere is a single curve C with a stationary arc length. which we call lhe geodesic line. All other curves between the two fixed points A and B, e.g., C' and C", do not have a stationary arc length and they deviate from lhe geodesic lines by some variations llx"'(s) for all values of the parameters. It should be noted lhat points A and B are fixed and common for all the curves connecting lhese two points. Therefore, the variations lix111 at these two points are equal to zero:

(11.2) Let us now define a function,

(IL3) along each of the lines connecting the points A and B. This function may depend on the coordinates x"' and their first derivatives with respect to the parameter s, as well as on the parameter s itself, as indicated in the definition (I 1.3). This function is usually called the Lagrange function or Lagrangian. The integral of this f\Inction between the points A and B is given by (11.4) and it is usually called the action integral. According to the Hamiltonian variational principle, the integral (11.4) has a stationary value along the geodesic line C. Using the variational principle, we can derive the differential equation satisfied by the Lagrangian function (11.3). The variation of the action integral{] J.4) along the geodesic line C i.s equal to zero, and wemaywrlte

U=

1s'(dr") SA

lSL x"',-,s ds=O. ds

(11.5)

Section 11.1

Lagrange Equations

91

By definition of the variation we have 111.6) Since the variations of the coordinates and their derivatives with respect to the parameter are small, we can expand the first term on the right-hand side of Equation (I l .6) into a Taylor series and keep only the zeroth- and first-order tenns. Thus we obtain

Since the differentiation with respect to the parameter is independent from the variation, the order of these two operations can be interchanged. Thus we may write

aL aL d ( lSL=-i5x"'-I - ( ) - lSx axm () ~ ds

m) ,

(11.8)

Subsntuting (1 I .9) into (I I.5) we obtain

(II.IO)

The second term of Equation (I I.IO), which is calculated at lhe boundary points Of the geodesic lines A and B, vanishes due to the condition (11.2). Thus we obtain

92

Chapter 11

Geodesic Unes

Since the variations lSxm are arbitrary and in general different from zero, the result (I LI I) requires that the following equations be satisfied:

a~;,-~[at~)]~o (m~1.2

•...• NJ

(11.12)

The system of N Equations (11.12) is a system of differential equations that must be satisfied by the Lagrangian function (11-3) along lhe geodesic line C. These equations are usually called the Lagrange equations.

I11.2 I Geodesic Equations In order to construct the geodesic equations that define lhe curve with a stationary arc length, we may choose the arc length itself as the action integral with zero variation. Using the expression (6.38) we write

01.13} From(l l.13) we see that the Lagrangian function is given by the expression

(11.14) and it is equal to unity along the geodesic line C, where

ds2 =g1md:ltb!'

(k,n= 1,2, ... ,N).

(II.15)

Along the geodesic line, where the metric condition (1 I.15) is fulfilled, we have

and

Section 11.2 Geodesic Equations

93

By interchanging the dummy indices kB- l. we may rewrite the second term on the right-hand side Of the result (l I.17) as follows:

1 rJgm1dx d:J!<

axk

1 (Ogm/

dsds = 2

rJxk

;Jgmk) dx

1

dx'

+ ~ dsds·

(11.18)

Substituting (I LI 8) imo (l I .17) we obtain

d [

aL

d; a(~)

]

2 d x' =8mlds2

I (a8ml

Ogmk) dx

1

dx'

+1 a;F+a;r dsd;"

(ll.19)

On the other hand we have

Using the metric condition (l I.15}, we obtain

rJL

!rJg1k~~

rJx"'

2rJx"' ds ds

(11.21)

Substituting (l I .19) and (11.21) into the Lagrange Equations (l 1.12), we obtain d

2

x'-

~ (;Jgml ;Jgmk - rJg1k) ~~ - 0 ax"+ rJxf rJx"' ds ds - .

8mld.s2 +2

(11.22)

Now, usmg the definition of the Christoffel symbols of the first kind (9.28), i.e.,

r

_ m,lk-

_!_ (rJgm1

2

;J_xk +

rJgkm _ Oglk) :Jxf

rJx"'

,

(11.23)

we obtain from (l I .22) (11.24) The composition of Equation (I I.24) with the contravariant metric tensor If"" gives the most commonly used formulation of the geodesic equations, as follows: (1125)

Chapter 11

94

Geodesic Lines

If a set of parameter equations of some curve C in a generalized N-dimensional metric space is given by x"'~x"'(s)

(m~l,2, ...

,N),

(11.26)

then the tangem va"tcI to this curve is defined by

u"'

~ ".;;'

(m

~ 1,2,. . .,N).

(11.27)

Using (11.27), Equation (I L25) can be written in lhe form

(11.28) Using Equation (l I .22) we can also write

Interchanging the dummy indices l ++kin the second term on the left-hand side of Equation (I I.29), we obtain

On the other hand, using the definition Um

_!!._(

dum ds - ds 8m/U

= gm1u1, we may write

')--8ml{fS du dgmlJ~ ;Jgm/ l k + ds -8ml ds + rJ.0 UU 1

(11.31)

Substituting (I 1.3 I) into (I I .30), we obtain (11.32)

(11.33)

Section 112 Geodesic Equations

95

The equation (I 1.33) is an alternative fonn of the geodesic equations, which does not explicitly involve the Christoffel symbols. Using the geodesic Equations (1125), we may also write du" ax' ~+rfrJ-= ds ds

(au" ) dx' ~+r:J:J ax-

-d =O.

s

(IL34)

or, using the definition of the covariant derivative of a contravariant vector (823), (11.35)

we find that the absolure derivative of the tangent vector (I L27) with respect to the par.uneter s is equal to zero: (11.36) This means that if we translate the vector ii" along the geodesic line from the pointx"' to lhe point x"' + dX", it will coincide with the tangent vector ii"+ dum at the point x"' +dX". This is a specific pruperty Of the tangent vector um along the geodesic line. In the Descartes coordinates. where = 0, the geodesic differential equation becomes

r;k

d2yn ds2 =0.

(11.37)

The solution of this equation is a set of linear equations (11.38) which represent the parameter equations of a straight line, which confinns our earlier statement that the geodesic lines in the Euclidean metric spaces are straight line~.

~

Chapter 12

The Curvature Tensor

I12.1 I

Definition of the Curvature Tensor

In lhe Riemannian space the parallel translation of a vector between two given points is a palh-t.lependent operation. i.e.. it gives different results along different palhs. In particular, if a vector is parallelly translated along a closed path back to the same point of origin, it will not coincide with the original vector. Let us lherefore derive a general formula for the change of a vector after a parallel translation along some infinitesimally small closed contour C. This change, denoted by Mm. is obtained using the result (8.I 9), in the following manner:

Using the Stokes theorem, this line integral over the closed contour C can be transfonned into a surface integral over a surface S bound by the closed contour C i.e., ·

where we use the covariant operator

,, ;:, >' (12.3)

97

Chapter 12 The Curvature Tensor

98

Let us also, for the sake Of simplicity, introduce the non-tensor operator (12.4)

Using the definition of the covariant derivative of a secood--0rder covariant tensor (8.33), we may then write

D,.Cmp DµCmk

= rJkC.np - r!m:C1p = apcmk - r~c1k -

r!kCm1

r~Cmi·

(12.5)

Subtracting the two Equations (12.5) from each other we obtain

D,.Cmp-DµCmk

= akcmp - r~C1p - r!kCm1 - apcmk + r~c1k + r~Cm1 = akcmp - OpC,nk + r~c1k - r!m:c1p + rfq,Cm1 -r!kCm1.

(12.6)

Using here the symmetry of the Christoffel symbols with respect to their lower indices, we see that the last two terms in Equation (12.6) cancel each other. Thus, using lhesymmerric surface rensor dSkp = dsPk, we may write ( DkCmp - DµCmk) dSkp

= (akcmp-apcmk +r~c1k- r~c1p)ds1:;p.

02.7)

By interchanging the indices k -++pin one oflhe last two terms in parenlheses and using lhe symmetry of the surface tensor dSkp, we see that these two terms cancel each other. The result (12.7) therefore becomes (DkC.np - DpCm;JdSkp = (()kCmp - ()pCmk)dSkp

Applying (12.8) with Cmp =

r~,,

Mm~ HJa.(r;;,_A.)-a,(r;:,,A.)]ds"', ~=

i£(akr~,, -

(12.8)

in Equation (12.2). we obtain

aPr:;,kA"

(12.9)

+ r~akAn - r~apAn) dSkp. (12.IOl

On the other hand, the change of the vector A,, along the contour C is due to the parallel translation (8.19) and is given by c5A,,

= r~~1dx1'

=::}

apAn

= r~~l-

(12.l l)

Section 12.1

DefJnitionoftheCurvatureTensor

99

Substituting (12.11) into (12.10), we obtain Mm=

~

fc {akr~n

-

aPr~n + r~r!,01 - r:;,kr~1 ) dS1cp (12.12)

As lhe labels of the dummy indices are irrelevant, we can interchange the indices n ++ l in lhe last two terms ofthe integral (12.12). Thus we obtain

Introducing here the notation R1:,,kp

= a,r~ - apr~ + r~rfk - r~r~.

c12.14)

the result (12.13) becomes (12.15)

Since lhe closed contour C is infinitesimally small, it is possible to replace lhe integrand of the integral (12.15) by its value at some point enclosed by the contour C, and to bring it outside lhe integral Thus, we finally obtain a general formula for lhe change of a vector after a parallel translation along some infinitesimally small closed contour C in lhe fOrm (12.16)

The mixed fOurth-order tensor defined by the expression (12.14) is called the curvature tensor of a given metric space. The tensor chamcter of the curvature tensor is evident from (12.16), since An is a cordriant vector, 6.Skp is a contravariant surface tensor, and Mm is a difference of two covariant vectors at lhe same point on the contour C. The cunrature tensor plays lhe key role in lhe lheory of gravitational field and is very important in tensor analysis in general. The formula, analogous to (12.16), for a contravariant vector Am can be obtained using the fact that a scalar AmBm is invariant under parallel translations. Thus we may write 6.(AmBm)

= MmBm +AmfiBm = MnBn +Am6.Bm = 0.

(12.17)

Chapter 12 The Curvature Tensor

100

Using here the result (12.16) we obtain "A"Brl +Am_!Rnmkp BfiSkp=(fiA•+!R". A.mt:,,skp)Bn =0. n nuqr·

LV1.

2

2

(12.18) As the covariant vector B,, is arbitrary, we obtain the formula (12.19)

In a given Euclidean space lhe curvature tensor is identically equal to zero, since it is possible to choose lhe cocnlinates where r~ =::::= 0 and therefore ~P 0, in the entire metric space. Because of its tensor character, the curvature tensor is then equal to zero in any olher coordinate system defined in the Euclidean metric space. It is related to the observation lhat, in the Euclidean metric space, lhe parallel translation is not a path-dependent operation and lhe parallel translation along a closed curve C does not change lhe translated vector. The reverse argument is valid as well If lhe curvature tensor is equal to zero in lhe entire metric space, i.e., R:kp = 0, then such a space is Euclidean.. Indeed, in any metric space, it is possible to construct a local Descartes sysrem in an infinitesimally small ponion of thar space. On lhe other hand, if R~ = 0 in the entire space, then lhe parallel translation is a unique path-independent operation by which this infinitesimally small portion of a given space can be tnmslated to any other portion of that space. Thus we may construct Descartes coordinates in the entire space and lhe given space is Euclidean. As a summary, the criterion for determining the character of a spare is that a metric space described by a metric tensor 8mn is Euclidean if and onlyifR:!,1;p =0.

=

I12.2 I

Properties of the Curvature Tensor

From lhe definition of the curvature tensor

R~

= a,.r~ - apr:;,k

t- r~ra

- r~kr;;,.

02.20)

it is easily seen lhat it is antisymmetric with respect to the interchange of lhe last two lower indices k -++ p, such that we have

R~=-R~k-

(12.21)

Section 12.2 Properties ol the Curvature Tensor

101

It is also possible to show that the curvature tensor satisfies lhe following identity: R~1R;,rn,.+R1/q,,,,=O.

(12.22)

In order to prove lhecyclic identity (12.22)., let us rewrite (12.20)asfollows:

~1:p = akr~ - apr::..i. + r~rt1- r~r~ R;mk = amr;k- akr;m + R1/q,m

r!krtni- r!mrik

(12.23)

= aPrkm - amr~ + rLnr;;, - r!i,rbn.

By inspection of Equations (1223), using lhe symmetry of the Chnstoffel symbols with respect to lhe interchange of the two lower indices, we see that for each tenn in these three equations there is a countertenn with the owosite sign. Thus, by adding together lhe three equations (12.23) we obtain lhe cyclic identity (12.22). In addition to the mixed curvature tensor (12.20), it is sometimes useful to define the covariant curvature tensor by

Rmnkp

= 8m1R~1:p·

(12.24)

or, using the definition (12.20),

Rmnkp = 8mlakr!v, - 8m10pr~ -+- r!v,rmJk - r~rm,lp·

(12.25)

In order to obtain a more symmetric fonn of the covariant l-'UI'Vature tensor (12.24), we use Equation (9.25) in the form

Opgm1 ak8ml

= n.mp + r m.lp = r,,mk + r ....lk

(12.26)

and lhe definition of the Christoffel symbol Of the first kind (9.28) in lhe fonn I

r m.np = 2(apgmn + angpm - Omgnp)·

(12.27)

Using (1226) and (12.27) we can calculate

gm/akr!v,

= 8mlilk (g1srs.np) = (gm1&:-g1s) r ...np +ll~akrs.np = -(g'"'ak8m1)rs.np+akrm.np = -r!v,akgm1 +&:-rm.np = -r!v, (r1.mk + r mJk) + ~ (~Opgmn + l:l-Bngpm -

l:l-Omgnp) (1228)

Chapter 12 The Curvature Tensor

102

By interchanging the indices k ++pin (12.28) we also obtain

8mlapr!,;. = -r!i. (r1,mp + r m,lp)

+ ~ (apakKmn-+- rJpBngkm -

Opamgnk).

(1229)

Substituting (12.28) and (12.29) into (12.25) we obtain

Rmn1p

=

-r!v, (r,.mk + r m,1k) + r~ (r,,mp + r m,!p) + ~ (&cap8mn + i:lkBnKpm - akamgnp) - ~ (apak8mn + llpfl,,gkm - apamgnk)

+ r!i.vrmjk - r!.x,rm,lp.

Rmnkp

1

= 2(0,.Cl,,gpm + apa,,.gnk -

+ r!.i.r1.mp -

(12.30)

akamgnp - apBnKkm)

r~rt.mk-

(12.31)

Fmally we obtam the alternative expression for lhe covanant curvature tensor in the fonn

Rmnkp

= ~ ( d,.:B,,gpm + ilpl:lmgnk +K1,;

OkiJ,,,gnp - rJpBnKkm)

(r~r!.v.- r!v,rJm1;)-

(12.32)

From lhe expression (12.32) we see that the covariant curvature tensor IS antisyrrmetric with respect to the interchange of both the first two indices (m ++ n)and lhe last two indices (k++p). Thus we have

From the expression (12.32) we also see lhat lhe covariant curvature tensor is symmetric with respect to lhe interchange of lhe first two mdices and the last two indices (mn-++ kp), i.e., (12.34) Furthronore we note that the cyclic property (12.22) also remains valid after the contraction wilh lhe covariant metric tensor, such lhat we have Rmnkp

+ Rmpnk + Rmkpn = 0.

(12.35)

Section 12.3 Commutator o1 Covariant Dartvattvas

103

Using (12.33) and (12.34), it can also be shown that lhe cyclic properties analogous to (12.35) are valid not only for the last three. but also for any three indices oflhe covariant curvature tensor.

J

12.3 I Commutator of Covariant Derivatives In the Descartes coordinates, with Dk = Ot. it is allowed to change lheorder of the covariant differentiation with respect to the two different coordinates. In other words, we may write

where AP is an arbitrary contravariant vector. Since AP is an arbitrary contravariant vector, it can be omitted and we can write lhe corresponding operatCI" expression as follows:

The quantity [Dm, Dk], introduced in (12.38). is called the commutator of lhe two operatms Dm and Dk. This commutator is equal to zero in the Desc.artes coordinates. Let us now calculate the commutator [Dm,Dkl in lhe arl:i.trary l-'"llTVilinear coordinates. In order to calculate this commutator, we first calculate

DkDmAP

= Dk(OmA.P + r~A")

DkDmAP = Ok(OmAP + r!:mA") + rf1(0mA 1-+- r~") + rLnca,AP + rfnA") DkD,,,AP = &,OmAP + r!;,,,&,An +Anakr!;,,, + rLamA1+ rLr!..nA" - ri,,,a1AP - ri,,,rfnA"

(12.39)

Analogously, we may write

DmDkA!'

= OmilkAP + r:',,amA" +A"Omr:',, + r~ 1 &,A1

+ r~ 1 rbiA" -

r~<JtAP - r~rfnA".

(12.40)

Ifwe now form lhe difference between the expressions (12.39) and (12.40). we see that several terms cancel each other, i.e.• the first term cancels the first term, lhe second term cancels lhe fourth term. lhe fourth term caocels

Chapter 12 The Curvature Tensor

ll4

the second term, the sixth term cancels the sixth term, and lhe sevenlh tenn cancels the sevenlh term. Thus we obtain
= (akr!;,,, - Omri,, +r~r!.n- r~,r~)An. (12.41)

Now, substituting the definition of lhe curvature tensor (12.14), i.e.,

Jfnkm

= ai:r!;,,, - amri:, + r~,,rf1 -

r~r~ 1 •

(12.42)

into Equation (12.41), we obtain (DkDm - DmDk)AP = ~~"·

(12.43)

n line with the criterion fur determining lhe character ofa space, discussed earlier, we conclude from (12.43) that the changeoflhe order oflhe covariant differentiation is allowed only in lhe Euclidean metric spaces with the zero curvature tensor, i.e., with Jfnkm = 0.

I12.4 I

Ricci Tensor and Scalar

from the curvature tensor (12.14), by contraction of the single contravariant index wilh the second covariant index, it is possible to construct a covariant second-
Rmn = Bnr~m

-

Bpr::,,, + r~r~1 - r!nr:P-

(12.45)

It is easily verified lhat the Ricci tensor can only be defined as in (12.44).

Let us for example consider lhe alternative contracted tensor can be written as follows:

iekmn = ~fRi..n = gPkgp1Ri..n = gPkRpkmn = 0.

R!mn, which (12.46)

Rk..n

from lhe definition (12.46) we see that is a scalar product of lhe symmetric metric tensor gPk and lhe antisymmetric covariant curvature tensor Rp1cmn. which is by definition equal to zero. Using the symmetry properties (12.33) and (12.34) in the definition of lhe Ricci tensor (12.47)

Section 12.5 Curvature Tensor Components

105

and interchanging the dummy indices k ++ p, we may write Rmn

= rfkRnpkm = /f'Rnkpm = rfkR1mmp = Rmn.

(12.48)

Thus the Ricci tensor is symmetric with respect to its two indices, i.e., R,,,,,~R,,,,,

(m.n~l,2,

.. ,N).

(12.49)

Using the Ricci tensor (12.44), we can define lhe Ricci scalar as follows:

(12.50) The Ricci tensor and Ricci scalar are extensively used in the general lheory of relativity and in cosmology.

I12.5 I

Curvature Tensor Components

The components oflhe curvature tensor are related to each other by means of the symmetry relations (12.33Hl2.35). Thus lhe components of lhe curvature tensor are not all independent, and lhenumberof lhe independent components of the cl.IT'larure tensor in lhe N-dimensional space is equal to

112.51) As this general rerult for the N-dimensional metric space will not be extensively used and its derivation involves long and complex combinatorial manipulations, it will not be considered in detail here. We will only consider lhe two simplest special cases, a two-dimensional metric space (N = 2) and a three-dimensional metric space (N = 3). In a two-dimensional metric space, lhe indices of the cov.uiant curvature tensor Rkmnp• i.e., (k, m, n,p) can assume the values 1 and 2. Thus lhe components of lhe cov-.uiant curvature tensor can 5e presented by lhe following rectangular scheme: km=ll km-12 km=2l km=22

np= ll

"f'=l2

0 0 0 0

0

0

R1212 -R1212

-R1212

0

0

np=2l

R12q_

np=22 I

0 0

I I

(12.52)

--%-j

From the scheme (12.52) we see lhat out of possible 2 4 = 16 compooents, lhe 12 components with k = m or n = p or both are equal to zero because of lhe antisymmetry Of lhe covariant curvature tensm: The remaining four

Chapter 12 The Curvature Tensor

106

nonzero components with k "# m and n "# p are related by antisymmetry relations as well Thus, ac.cording to the expession (12.51), we see that in a two-dimensional metric space lhere is just one independent component of the oovariant curvature tensor. i.e.•

n= 4(4-1) 12

=

L

(12.53)

The Ricci scalar is lhen defined by R R

= g"'1'rfkR1annp = 2 g 11 g22R1212 11

= 2(g g2

2

-

12 g g21)R1212

2 g

12 21

g

R1212

= 2 lg"'1'j R1212 = ~R1212g

(12.54)

The quantity R/2 is equal to lhe Gauss curvature of a surface, or lhe inverse of lhe product of the main radii of lhe curvature. Let us now consider a special case of the surface of the unit sphere, wilh the metric of the furm

ds2 =d(P+sin2 0dq}

(12.55)

Thus, lhe covariant metric tensor can be written in the matrix form [gmnJ =

[~ (sin~1)2]•

(12.57)

and lhe contravariant metric tensor can be written in the matnx form

[I

[g= )= 0

0]

(sinx 1)-2 ·

(12.58)

In this case, the quantity R/2 should be equal to unity. In order to show lhat, we need to calculate the only independent component of the curvature tensor, i.e.,

R1212 = -~a1a1g22 + gucr:Zrh - r:.rl2) ·+ g22cri2ri2 - rf.riz).

(12.59)

On lhe other hand, the Christoffel symbols of the first kind are defined by

I

r m,np = 2(Op8mn + On8pm -

Omgnp)-

(12.60)

107

Section 12.5 Curvature Tensor Components

Substituting (12.56) into (12.60), we obtain r1.11 =O r1,12 = r1.21 = o r1.22 =

-~C:l1g22 =

1 -sinx1 cosx (12.61)

r1.11 =O r1,12 = r121 =

+~C:l1g22 =

1 +sinx cosx1

r2.22=o The Christoffel symbols of the second kind are defined as (12.62) and in our example, lhey are given by 12 r: 1 = g 11 r1.11 + g r2.11 = o 12 rb = rJ1 =gt1r1.21+ 8 r 221 = o 1 rJ 2 = g 11 r1.22 ;- g 12 r2.22 = -sinx1 cosx rf1 = 8 21 r1.11 +i2r2.11 =O 1 1 1 rf2 = ri1 = g 21 r1.12 + g22r2,12 = +(sinx )- cosx 21 ri2 = 8 r1.22 + Cr222 = o.

(12.63)

Substituting lhe results (12.63) into (12.59), we obtain R1212 =

-~C:l1C:l1g22 + g22(1''fz) 2

R1212 = (sinx 1) 2 -(cosx1) 2 + (sinx 1) 2(sinx1r 2(cosx 1) 2 (12.64) Finally the Gauss curvature of lhe unit sphere is given by

which proves that lhe Gauss curvature of lhe unit sphere is indeed equal to unity.

Chapter 12 The Curvature Tensor

108

In lhe case of three-dimensional space (N = 3), in which lhe curvature tensor has a total of 3 3 = 27 components, according to (12.51) lhere are six independent components of the covariant curvature tensor:

n= 9(9-1) =6. 12

(12.66)

There are also six independent components of the Ricci tensor Rm11- Using the suitable Descartes coordinates in a given point, it is always po&<>ible to make three of these six components equal to zero. In particular, it is possible to diagonalize the Ricci tensor and define the curvature in any given point of'fl three-dimensional space by three independent quantities. In four-dimensional space there are 20 independent components of the covariant curvature tensor:

n

16(16-1) 12

= - --

= 20.

(12.67)

The case of a four-dimensional space is particularly important in the general theory Of relativity and in cosmology.

Part III

Special Theory of Relativity

... Chapter 13

Relativistic Kinematics

I13.1 I

The Principie of Reiativity

For the description of the motion of particles, it is necessary to have a system of reference. By a system of reference we mean a coordinate system to which we attach a clock. The coordinate system is used to determine the positions of particles in space and the clock is used to mesure the times at which the positions of particles in space are measured. There are systems of reference where the free motion of a particle, i.e., the motion of a particle on which there is no action of any external forces, is such that the particle has a constant velocity. SuchsystemsofreferencearecalledinertiaI systems. If two systems of reference are translated with a constant velocity with respect to each other and one of them is an inertial system, then the other system of reference is also an inertial system. Any free motion in that .system of reference will also be along a straight line with constant velocity. Using these definitions, we can write down the statement of the special principle of relatfrity as follows: All laws of nature are equal in all inertial systems of reference. In other words, the equations that express the laws of nature are invariant with respect to the transformations of spatial coordinates and time from one inertial system of reference to another.

111

Chapter 13 Relativistic Klnematic6

112

I13.2 I

Invariance of the Speed of Light

Let us consider two bodies interacting with each other, and let us assume that on one of the bodies some event has occurred (explosion, distance increase, or some other change). Then this event will be noticed at the position of the other body after the lapse of some time. If the distance between these two bodies is divided by this time interval, we obtain the maxinuon .<;peed of interaction. In nature. the motion of bodies with a speed higher than the maximum speed of interaction is not allowed, since by means of such a body it would

be possible to achieve the intei.L"tion with a speed higher than the maximum speed of interaction. Based on the special principle of relativity, we conclude that the maximum speed of interaction is invariant with respect to the transformations of spatial coordinates and time from one inertial system of reference to another. Thus, it is a universal constant of nature. It is shown that this universal constant is equal to the speed of light in the vacuum, and its numerical value is given by

c = 2.99776 x 108m/s.

(13.1)

The principle of invariance of the speed of light is therefore stated as one of the basic principles of relativistic mechanics. According to this (1'inciple the speedoflight is independent of the motion of the light source, i.e., of the choice of the system of reference in which the motion of light is described. In the next section, the mathematical formulation of this principle is pesented.

113.31

The intervai between Events

An event in nature is determined by four coordinates. These include the three space coordinates of the position where the event has taken place and the time coon:linate when the event has taken place. Each event fa represented by a point, called the world point of the event, with the coon:linates

X"= {x0 ,x1 ,x2,x3 }

(m=O,I,2,3).

(13.2)

The first coordinate is a lime coordinate defined by ;fl = ct, and it has the dimension of length. The other three coordinates are the spatial Descartes coordinates

X' ~ {x,y,z)

(a~

1,2,3).

(13.3)

Section 13.3 The Interval between Eventa

113

We will from now on use Latin indices for the ooon:linates of the four-dimensioual space-time (13.2), and Greek indices for the usual three-dimensional spatial coordinates (I 3.3). The set of all world points constitutes a four-dimensional manifold called the world. To each particle in such a world, there corresponds a line called the world line. The world line of a particle at rest is a line parallel to the time axis. In order to express the principle of invariance of the speed of light mathematically, let us now consider two systems of reference denoted by K and K', which move with respect to each other along the common x-axis with some constant velocity V. These coordinate systems are shown in Figure 4. The times in coordinate systems K and K' are denoted by t and t 1 , respec1ively. Let one event consist of a signal sent at time t1, with a constant velocity equal to the speed oflight, from the point with coordinates [x1 ,Yt. z1} in the co
Because of the principle of invariance of the speed of light, in the coordinate system K' the coordinates of these two events are related by the

y

j

J__, I Flgure4.

The coornmiate systems Karid K'

x=x'

Chapter 13 Relatlvistlc Klnemalics

114

analogous equation

If the coordinates (t1,Xt.Y1.z1) and (t2,x2.n.z2) are the coordmates of any two events, then the quantity

is called the interval between these two events. Analogously to (13.6}, the squareoftheinterval between twoinfinitesimallycloseeventsisdetennined by the metric

=

where (m, n 0, I, 2, 3), and the contravariant space-time coordinates in this four-dimensional metric space are given by

X" = [x0 ,x 1,x2,x3 } = [ct,x,y,z}.

(13.8)

The components of the covariant metric tensor 8mn are given by the following matrix:

[g~J~

[

I

0

0

-I

0

0

0

0

0 0] 0

0

-I 0

0 -I

(13.9)

-

From (13.9) we conclude that the world is a four-dimensional psendoEuclidean metric space. The determinant of the metric tensor 8mn is g = -L Thematrixofcofoctor.;ofthematnx (13.9) ts

[Gm']-

-I 0 0 OJ [ 0 0 0

I 0 0 0 I 0 0 0 I

(13.IO)

Using (6.13), the inverse matrix to the matrix [gmn} is given by

[g""']

~ inv [g~J ~ adj [g~J ~ [G""']T _ g

g

(13-11)

Section 13.3 The Interval between Events

115

We conclude that the components of the contravariant metric tensor gnn are the same as the components of the covariant metric tensor (13.9), i.e., [g""']~

[

I 0 0 0

0 -I 0 0

0 0 -I 0

(13.12)

From (13.9) and (13.12) we see that

!f""&>n~s;:

(k,m,n~O,l,2,3).

(13.13)

The covariant space-timecoordinatesm this four-dimensional metnc space are given by

Xm =gmnx"

= [xo,xi.x2.x3} = [ct,-x.-y,-z}.

(13.14)

Using (13.8) and (13.14) we see that

x,,,x"'=gmnx"1x"=c2t2-x2 -);L-z2.

(13.15)

Using expressions (13.4) and (13.5) and the principle of invariance of the speed of light, we conclude that an interval that is equal to zero in one inertial system of reference is also equal to zero in any other inertial system of reference. On the other hand, the quantities ds and ds' are two infinitesimally small quantities of the same order. Tlrus we may write

ds=tlds',

(13.16)

wherethecoefficienttlmaydependonlyontheabsolutevalueoftherelative velocity V of the two inertial systems of reference K and K 1 • It may not depend on the spatial coordinates or time because of the assumption of the homogeneity of space and time in the inertial systems of reference. It may not depend on the direction of the relative velocity V of the two inertial systems of reference K and K 1 either, because of the assumption of the isotropic nature of space Hlld time in the inertial .systems of reference. Thus, as we may write (13.16), we may also write

ds'=cul.s.

(13.17)

From the equations (13.16) and (13.17) we obtain tl2 = 1 orll = ±1. On the other hand, from the special case of the identity transformation with ds = ds' we conclude that ll = +I, so that we always have

ds'=ds

(13.18)

Thus we obtain the mathematical formulation of the principle ofinvariance of the speed of light (13.18), which implies the invariance of the interval

Chapter 13 Relatlvt&ac Kinematics

116

with respect to the transformations from one inertial system of reference to another.

j 13.4 j

Lorentz Transformations

The objective of the present section is to derive the formulae for transformations of coordinates from one inertial system of reference to another. In other words, if we know the coordinates of a certain event X" = [ct'.x',y',;:'} in some inertial system of reference K', we need the expressions for the coordinates of that event = [ct,x,y,z} in some other inertial system of reference K As X" is a contravariant vector, it is transformed according to the transformation law

z:n

(13.19)

The transformation (I 3.19) is a linear transformation and the coefficients A':( areindependent of coordinates_ The system A'; is a mixed second-order tensor in the pseudo-Euclidean metric space defined by the metric (I 3.7), since it is defined with respect to the linear transformations (13.19). In order to calculate the components of the tensor A:;', we now introduce an imaginary time coordinttte T

= ict

(i

= -J=i),

(13.20)

and an imaginary metric

dn =ids

(i

= .J=i).

(13.21)

Thus the metric of the space becomes

dn 2 =dr2 +d:x2+dy2 +tfl-.

(13.22)

and we thereby define the four-dimensional Descartes coordinates. The linear transformation (13.19) must keep the metric ds or dn invariant. As we have ru:gued before, the only transformations of Descartes coordinates that keep the metric form invariant are the transformations of translation, rotation, and inversion. Parallel translation of four-dimensional Descartes coordinates is not a suitable candidate for the transformation (13.I 9), since it merely changes the origin of the spatialcoordinates and the origin for measurement of time. Similarly. the inversion of the four-dimensional Descartes coordinates is not suitable either, since it merely changes the sign of the spatial coordinates and time.

Section 13.4 Lorentznansformation&

r' \

117

'l

Figure 5.

Rotations in the rx plane

Thus we conclude that the only suit.able candidate for the transformation (13.19) is a rotation of four-dimensional Descartes coordinates We are looking for the fornrulae for transformation from the coordinate system K' to the coordinate system K as shown in Figure 4_ In that case we have y = y' andz = t and we are only interested for the rotation in the rx plane, as shown in Figure 5. FromFigure5. we immediately obtain the remaining two transformation fornrulae: x=x'cosl/f-r'sinl/f r =x'sinl/f+r'cosl/f.

(13.23)

According to Figure 4, the coordinate system.K' is moving with respa.'1: to the coordinate system K with a constant velocity V_ If we then consider the motion of the origin of the coordinate system K' we then have x' = 0. Then from (13.23) we obtain tanl/I

=-~ =i~ '

(13.24)

c

Now. using the trigonometric fornrulae I

cosl/I =

J1 +tan2 l/f'

(13.25)

Chapter 13 Ralalivtsllc Kinematics

118

we obtain

I

costfr=---,

J•-Y/r

. smt/f

=

JiV/c

v2.

(13.26)

·-~

Substituting (13.26) into (13.23), we obtain

r

=

r' + i(V /c)x'

J1-r; x= x'-i(V/c)r'

(13.27)

J1-'{: y~y'

y~y'

z=z

z=<..'

The results (13.27) are the well-known fornrulae for transformations of coordinates from one inertial system ofreferenceto another. They are called the Lorentz transfonnations. In the special case when V « c, the Lorentz transformations are reduced into the so called Gallilei transformations of nonrelativistic mechanics t=t'

x=x'+Vt

(13.28)

y~y'

z~t.

Although the Gallilei transformations are closer to our everyday experience than the Lorentz transformations, they are not in accordance with the principle of relativity and they do not leave the metric form of the four-dimensional space-time invariant. Using (13.27), it is easy to construct the tensor of Lorentz transformations A'; in the following matrix form:

[ .:] ..'!f£_

(A;]~

Pi M ' Pi Pi 0 0

..'!f£_

0

I

0

I 0 0 I

(13.29)

Section 13.5 Yelocity and Acceleralion Vectors

119

The inverse of the tensor A';. denoted by (A';)- 1, is given in the matrix form

g oo]

_ ____!:'.K_

_,_

g

0

0

0

0

0

I

.

(13.30)

and it is obtained from the tensor (13.29) by reversing the sign of the relative velocity V, i.e., by putting V-+ -V. Thus we may write

ax"'= CAm)- 1(Vl = Am(-\1)_ Clz'I n n

(13.31)

It 1s easily shown, by du'ect multiphcatlon of the matnces (13.29) and (13.30), that

(13.32)

I13.5 j

Velocity and Acceleration Vectors

In the special theory of relativity, the time differential dt 1s not a scalar invariant, and the usual definition of the three-dimensional velocity dx" iP =(it

{a=

1,2,3),

(13.33)

is less useful since it does not behave as a vector with respect to the transfonnations from one inertial system of reference to another_ Therefore we introduce a/our-velocity, as a contravariant vector ,/" =

Dx"'

dxm

--;b = d;

(m = 0.1,2.3).

(13.34)

On the other hand, by definitton, we have 2 ds2 8mndT"'dxn = c d? + 8afJd.t?dxfi

=

(13.35)

and the square of the intensity of the three-dimensional velocity is given by

120

Chapter 13 Relativistic Kinematics

From (13.35) with (13.36). we obtain

v')

8aflv"v')

ds2 =c?dt2 ( I-ii

=c2dt2 ( 1+-c2- .

(13.37)

Substituting (13.37) into (13.34), we OOtain

.£" =

dX"

Cm ~ 0. I. 2. 3).

,,--;I

cdtvl-~

(13.38)

The temporal zeroth component of the four-velocity is then given by

u=n· o

I

(13.39)

and the three spatial components of the four-velocity are given by

if~-""--.

cR

(13.40)

From(l3.39)and(l3.40)weconcludethatthecomponentsoffour-velocity are not independent of each other, but satisfy the equality

since we have (13.42)

Thefour-ncceleration of a particle is defined as a contravariant vector

w"'

=

D:: = d;~

(m

= 0, I, 2, 3).

(13.43)

Here, using (13.38), we may write I

cR

dum

W"=----

(13.44)

dt'

W" ~ _ _l_ _ ~_l_dxm

c2Ratnat·

(13.45)

Section 13.5 Veloclty and Acceleratlon Vector&

121

The temporal zeroth component of the four-acceleration is then given by

w"- __l_'!.__I_

cpd1g·

(13.46)

and the three spatial components of the four-acceleration are given by

W'=--·--'!._~

c2RdtR

(13.47)

From (13.41), we conclude that the vector of four-acceleration is always orthogonal to the vector of four-velocity. This can be shown by differentiating (13.41) with respect to the metric, i.e.,

~ (gmnu"'u") = 8mn d~m u" + 8mnu"' ':n = 0, dum dun dum 8~-;i;-u"+g~-;i;u"'=2g=-;i;-u"=0, 8mnWmu"

= W,,u" = 0.

Thus the vectors W' and u" are indeed orthogonal to each other.

(13.48)

~

Chapter 14

Relativistic Dynamics

I14.1 I

Lagrange Equations

Letusconsiderafreeparticle,i.e.,aparticlethatisnotundertheinfluenceOf anyfurces. Theequationsdescribingthemotion of this particle are obtained using the variational principle. The action integral is defined by (11.4), ie., I=

Ls, ( dx") ls' SA

L x",- ds= ds

L(xn.u'1)ds.

(14.1)

SA

where u'1 is the tour-velocity of the particle. The equations of motion of the particle are obtained using the variational principle, i.e., the condition that the variation of the action integral is equal to zero, l)I = 0. The Lagrange equations of motion of the particle are given by (11.12) in the form aL ~

d

(aL)

d; iJu" =0 (n=O,l,2,3).

(14.2)

Let us now define the form ofthe Lagrangian function for a free particle. The Equation (14.2), derived using the variational principle applied to the action integral (14.1), are equal in all inertial systems of reference and the action integral is a scalar invariant with respect to the Lorentz transformations (13.29). Tirus, the Lagrange function Litse1f is also a scalar invariant. Because of the homogeneity Of the four-dimensional space-time in the inertial frames of reference, the Lagrange function cannot be a function of space-time coordinates x"_ Thus we may write L=L(u")

(n

= 0, 1,2,3).

(14.3)

123

Chapter 14 Relalivislic Dynamics

124

The only invariant that can be created using the tour-velocity vector u'1 is given by

Un.I'= g1m.f.!' =I

(k,n = 0, 1,2,3).

(14.4)

Thus the action integral of a free particle is simply proportional to the arc length in the four-dimensional space-time manifold, i.e.•

l =-me

s8

1

ds =-me

1J

dx' dx" g1:n--ds. dsds

s,

SA

I'!.

(14.5)

where mis a mass parameter of the free particle. From (14.5) we see that the Lagrangian function is given by the expression

L

= -mcJgknu''·u"

(14.6)

Usmg the Lagrangian ( 14.6) we may write

~ = ~ [-mc(gl<J,/ui)l/2] iJu"

iJu"

= -~mc(gjjifui)-ll2 (gknif + 8njUi)_ Here, using glrjiful weobt.ain

(14.7)

= I and the S}'IIIllletry of the metric tensor 8nj =

:~ = -~mc2gknif = -mcg1:nif = -mcun-

8jn.

(14.8)

Thus we may write

_!!_~=mcdun ds Clu"

(14.9)

ds

Using the Lagrangian (14.6) we may also write

aL

(14.10)

-=0.

ax'

Substituting (14.9) and (14.10) intotheLagrangeequation(14.2), we obtain

me~=!!:_ (mcu,.) = ds

ds

0.

(14.11)

The equations of motwn of a free particle are then given in the following form:

du' w"=d>=O

(n=0,1,2,3).

(14.12)

125

Section 14.2 EneJQV-MomentumVector

From Equation (14.12) we see that a free particle moves along a straight line with constant velocity. Let us now investigate the nonrelativistic limit Of the action integral (14.5) when the particles are moving with low velocities (v«c). If we substitute (13.37) into (14.5) we obtain

I= -mc

2

[•ag2 ['' 1- -;idt =

,

,,

LT~. iP)dt

(14.13)

where LT is a Lagrangian defined with respect to the nonscalar time variable t, given by 2

Lr(~. iP) =-me

VU 1-

8aftV"vP Zi =-me2J1-t- -c2-.

(14.14)

In thenonrelativisttc limit of particles moving with low velocities (v we may use the approximation

U v v•-zz"'l-222.

« c),

2

(14.15)

Substituting (14.15) into (14.13) we obtam

I~

f

(-=2+~mv2)dt.

(14.16)

Thus the nonrelativistic approximation of the Lagrangian Lr can be written as follows: (14.17) As the nonrelativistic Lagrangian is defined up to an arbitrary additive constant, the second term in (14.17) does not contribute to the nonrelativistic equations of motion and can be dropped in the nonrelativistic applications. Its physical significance in the special theory of relativity will be discussed later in this chaJ:(er. Thus the action integral (14.5) has the proper nonrelativistic limit

I14.2 I 14.2.1

Energy-Momentum Vector Introduction and Definitions

In nonrelativ1stic mechanics there are a number of constants of motIOn. These include the energy and the momentum of the particle. The objective of the present section is to define the energy and the momentum of a particle

Chapter 14 Relatlvlsllc Dynamics

126

in the framework of the special theory of relativity. Let us start with the action integral (14.13). in the form

,,

I=

[,

LT(x",v")dt.

(14.18)

The Lagrange equations of motion of a particle, analogous to Equation (1L12), defined in terms of the Lagrangian Lr with the nonscalar time variable t as the parameter, are given by

-fJLr - -d (aL - 7) = 0 ax'1' dt ava

(a= 1 2 3).

•·

(14.19)

Let us now calculate the total time derivative of the Lagrangian Lr as ful1ows:

(14.20) Using here the equations of motion (14.19) and the definition of the threedimensional velocity (13.33), we obtain

dL,dt

= ~ (aLT) v" + fJLT dva = dt

ava

ChP dt

!!:____

dt

(fJLr ChP

v").

0 4 .Zl)

The result (14_21) can be rewritten in the fonn


(14.22)

Thus the quantity

£ =

!;:

v" - Lr

= Constant

(1423)

is a constant of motion. The quantity (14.23) can be recognized from nonrelativistic mechanics as the total energy of the particle. Let us also define the nwmentum ff', conjugate to the coordinate~. as follows:

where

[p")

= {Px.Py,p,),

{Pa)= {-p,,-p,,-p,).

(1425)

Substituting (14.24) into (14.23) we obtain E

= -pav°'- -Lr= -gaf!PavfJ -

Lr=

p ·V- LT.

(14.26)

Section 14.2 Enetgy--Momantu Vector

127

Using the definition (14.14) of the Lagrangian Lr, we can calculate Pa= - fJLr

ava

= mc2~

(1 + 8ap~vfJ)l/2.

av"'

e,-

(14.27)

·

2 1( &/lv"vfJ)-1/ I a p =mc2- I + - -&;:fJ-(v"vfl)_ a 2 c2 c1 av°' Using the symmetry of the metric tensor &/J

(14.28)

= 8pa. we obtain (14.29)

Using Va = 8afiVfJ, we finally obtain

mva

a

pa=--~p

nnP

=---

(14.30)

RR

Substituting (14.30) and (14.14) into (14.26), we obtain the total energy of the particle as follows:

R2

2R'

mv.v" mv' E=---+mc2 l--=-+mc

M mC'

c2

G

R

G

= R-mc2y•-ii-+mc2v•-Zi=

1--

c'

n· mc2

(14.31)

Tlrus the total energy of the particle is given by (14.32)

The parttcle at rest with v

= 0 has the so-called rest energy Eo given by Eo=mc2 .

(14.33)

Chapter 14 Reiativislic Dynamics

126

The kinetic energy of the particle is obtained if the rest energy (14.33) is subtracted from the total energy (1432). Thus we obtain

EK=E-Eo=m?\g-1). For the particles moving with low velocities (v

(14.34)

« c), we may Wtite

I v2 --R>l+zc2·

R

(14.35)

Substituting ( 1435) into (14.34) we obtain £K =

In the same approximation (v the particle are given by

~mv2 •

(14.36)

« c), the ccmponents of the momentwn of (14.37)

The approximate results (14.36) and (14.37) are the well-known nonrelativistic expressions for the kfiletic energy and the momentum of a moving particle, respectively. 14.2.2 Transformations of Energy-Momentum

The results for the momentum of the free particle (14.30) and the energy Of the free particle (1432) were derived from the Lagrangian Lr, defined with respect to the nonscalar time variable t, in a noncovariant way. Now we want to define the energy and momentum as the constant-. of motion using the covariant Lagrangian L given by (14.6). fu analogy with the definitions (14.24), we now define the energy-momentum fuur-vector Pn with the following covariant and contravariant components: Pn = _ __!!_£ iJu"

= mClln ==} p" =

_

__!!_£ = ma/1 iJlln

(14.38)

From Equation (14.ll) we see that the four components of the energy-momentum of a free particle satisfy the equations

d~"

=0

(n=0,1,2,3).

(14.39)

Thus forafreeparticlethe fourcomponems of the energy-momentum four va'tol" are conserved. Using (13.39) here, we obtain the temporal zeroth

Section 14.2 EneJgV-Momentum Vector

129

component of the energy-momentum vector, which is proportional to the energy £of the particle, in the form

o o me £ p =mcu =--~c·

R

(14.40)

where tlr energy of the particle is given by (14.32). Furthermore, using (13.40), we obtain the three spatial components of the energy-momenrum four-vector, which are equal to the components of the three-dimensional momentum of the particle ffX given by (14.30). i.e.• (14.41)

From the results (14.40) and (14.41 ), we see that the energy and momentum ofaparticlearenottwo independent quantities as innonrelativisticmechan ics. lnrelativistic mechanics they are components of the same four-vector. It should be noted that the constant of motion analogous to the nonrelativistic result (14.23), is identically equal to zero:

:~u"-L=-pnu"-L=:O.

(14.42)

The energy-momentum vector as a four-vector transforms with respect to the transformations from one inertial system of reference to another. according to the transformation law(14.43)

where pk are the components of the energy-momentum tensor in the inertial system of reference K, while pk are the components of the energy-momentum tensor in the inertial .sys1em of reference K' moving along the common x-axis with a velocity V With respect to K. Here, using the explicit form of the tensor A~ given by (13.29), we obtain ' ') £=Ji I \fl (£+Vpx --;;r

Px~ g~:+~E') py=py pz=p~.

(14.44)

Chapter 14 RdatMsUc Dynamics

130

Using (14.38) with (13.41) we obtain

P'Pn

= m 2c 2 U'un = nf-c2 •

(14.45)

From (14.45) we may write

£

2

2 2

71 +f1Xpo:=mc.

(14-46)

From the results (14.25) for the components of the three-dimensional momentum vector, we may write ffXpa=8af3ffXp 13 = -P-P=-p2. (14.47) Substituting U4.47J into (14-46), we obtain (14.48) The relation between the energy and the three-dimensional momentum of a particle then becomes

£ = cJp2 +mzcz.

(14.49)

The function (14.49) is usually called the Homiltonian of the particle and is denoted by H. Thus we may write

H=cJp2+m2c2=Jp2c2+m2c4.

{14.50)

being the most usual definition of the Hamiltonian of a particle.

14.2.3 Conservation of Energy-Momentum The conservation law of the momentum of a particle is a consequence of the homogeneity of space and the conserv-dtion law of the energy of a particle is a consequence of the homogeneity of time. Because of the homogeneity of space-time, the mechanical properties of a free particle remain unchanged after the translation of a particle from a point with coordinates x" to a point with coordinates x" + An. where An is an infinitesimally small constant four-vector. Thus the Lagrange function must be invariant with respect to this translation and its variation must be zero. Therefore we may write

ljL=

:~l)xk= :~Ak=O.

ll4.51}

Here, using the equations of motion (14.2), we obtain

/jL=

_!!___

[.!1!:,]Ak =0-

ds au

(14-52)

Sectron 14.3 Angular Momantum Tansor

131

Since J... n is a nonzero infinitesimal.four-vector, the expression (14.52) gives

-fs[:~] =0.

(14.53)

Usmg {14.38) we now obtain

_!!._~=mc'!!!!;_=~=O. dsaun

ds

ds

(14.54)

The expression (J 4.54) then gives

'!!!:_ = __l_'!!!:_ =O°"p"=ConstanL ds

(14.55)

cHdt

Thus we obtain the three-dimensional momentum conservation law if.=O=>fi=Constant,

(14.56)

and the energy conservation law dE

dt = 0 => £

I14.3 I

= Constant.

(14.57)

Angular Momentum Tensor

In relativistic mechanics the angular momentum tensor is defined by the expression (k,n

Mnk =XnPk-Xkf'n

= 0, 1,2,3).

(14.58)

Only the spatial components of the angular mome mum tensor with (k.n = l, 2, 3) have a physical meaning and coincide with the usual definition of the angular momentum in noorelativistic mechanics. In nonrelativistic mechanics it is customary to form an axial three-dimensional angular momentum vector Ml!= ~el!a,B.Map

= emf3XaP,B

(a_f3

= 1.2.3),

04.59)

or, in the classical vector notation,

M=l!1 ! !l=rxp. Pt

P2

(14.60)

P3

The conservation law of the angular momentum tensor (14.58) is a consequence of the isotropic nature of the four-dimensional space-time_

132

Chapter 14 Relatlvistic Dynamics

The three-dimensional angular momentum conservation law is a consequence of the isotropic nature of the three-dimensional space. Because of the isotrDpic nature of the four-dimensiona] space-time, the mechanical properties of a free particle remain unchanged after rotations in the fourdimensional space-time. Let us consider the special case of a rotation for some angle (} about the 3-axis in three-dimensional space. Then the relation between the coordinates zm in the rotated inertial system of reference K' and the coordinates :x!' in the original inertial system of reference K is given by

t' ~ O.jxi (j,n ~ 0, 1,2,3).

(14.61)

Using (5.40) we may rewrite (14.61) in the matrix form

[

x 0 0 OJ00 ['"] x2 -

0t cose sine 0 -sine case 0 0 0

1

If we now consider the rotatton for some mfimtesimal angle {j() we have cos/jlJ ~

1,

sinc58

~

(14.62)

x3

1

c5fJ.

~

0, then

(14.63)

Substituting ll4.63) into l]4..62J we obtain

0 I _;ie

~~ g] [~] 1 0

0

0 I

x2 x3

'

(14.64)

wherec5!2j is a mixed tensor defined by the following antisymmetric matrix:

[80."]-[g0 j

-

0

0 0 -88

0

00 oOJ0 -

~e

0

0

(14.66)

Section 14.3 Angular Momentum Tensor

133

Here, using (13.9) and (13.12), we obtain the covariant coordinates Xk of the four-dimensional space-time in matrix form:

· [xk]= (gjk][xl]

=

[I 0 0 OJ[~ = [""J 0 0 0

-1 0 0

0 -1 0

0 0 -1

x1 x2

-x1 -x2 ·

x3

-x3

(14.67)

We can also calculate the COlll(lOPents of the antisymmetric contravariant tensor c5Qnl: in matrix form as follows: (14.68)

0 -I

0

0

-I

0

0

JJ [~

OJ

88 0 --1!8

0

0 0 .

0

0

0

(14.69)

Thus we finally obtain the matrix form of the antisymmetric contravariant tensor MJ.nk as follows:

00 0 0 [8Q"'] ~ [ 088 0

0

00] 0

-88

o o· 0

(14.70)

0

For an arbitrary infinitesimal rotation, the contrdvariant tensor MJ.nk has a more complex form compared to the simple matrix (14.70). but it is always an antisymmetric tensor. Thus we always have

MJ.nk=-MJ.kn

(k,n=0,1,2,3),

fl4.71)

and the most general infinitesimal rotation of the coordinates is given by the trnnsforrnation relations

(14.72) The most general infinitesimal rotation of four-velocity vector is given by the following transformation relations:

um=u"+c5un,

c5u"=MJ.nku1:.

(14.73)

The Lagrange function of a particleL()?!, un) nrust be invariant with respect to this infinitesimaJ rotation, i.e., we must have c5L = 0. Let us now calculate the variation of the Lagrangian L with respect to the infinitesimal

134

Chapter 14 Relatlvlstic Dynamics

rotations (14.72) and (14.73). i.e.• /JL

= :~/jxn +:~/Ju"= 0.

(14.74)

/JL

= ~/jgnkXk + ~MJ.nkUk = 0.

(14.75)

Substituting the equations of motion (14.2) into (14.75) we obtain IJL

aL)

= {jQnk ['1ds_ ( (Ju"

Xk

+ ~uk] = 0. aun

(14.76)

Using the definition of the energy-momentum four-vector 04.38J and the definition of the four-velocity (13-34), we obtaill

IJL=

-lJQ"k[~xk+Pn~] =-!!_ (MJ.""p.,xk) =0. ds ds ds

(14.77)

Using here the antisymmetry of the tensor lJO.""· the Equation {]4.77) becomes IJL=

-~ljQ""fs(p.,xk-PkXn) = ~ljgnk~;k = 0.

(14.78)

From (14. 78). as a direct consequence of the isotropic nature ofspace-time, we obtain the conservation law of the angular momentum tensor. i.e.•

d~;k = 0::::}

m;nk

= 0::::} Mnk = Con.
The spatial part of this conservation law for (n, k

=

(14.79)

1. 2. 3), i.e., (14.80)

is the usual conservation law of the three-dimensional angular momentum vector.

~

Chapter 15

Electromagnetic Fields

I1s.1 I

Electromagnetic Field Tensor

The electromagnetic field in the four-dimensional space-time is described by a four-vector A"~A"V).

(15.1)

which is called the potential of the electromagnetic field. The temporal compooent of this four-vector is defined by (15.2)

where 'l/f(x'<) is the elecrric scalar porenrial. The three spatial components of this four-vector, A" ~A"(x'),

(15.3)

constitute the so-called magnetic vector potential. Thus we may write An= {

~ifr,Ax.Ay.Az}

An =gnkAk

= {~'lfr,--Ax.-Ay,-Az}.

(15A)

135

136

Chapter 15 Elactrmnagnetlc Fl&lcls

The action for a particle with mass m and charge q moving in the electromagnetic field defined by An(,!") is then given by (15.5)

where ls is the action for a free particle given by ls

= -me

(SB

ls,

J

g1mukunds,

(15.6)

and IQ is the action describing the interaction of the charged pru:Ucle with the electromagnetic field defined by the four-vector An(_xk). The simplest invariant action that can be constructed using four-vectorsAn(,!") and U', describing the electromagnetic field and the motion of the particle, respectively, is given by 513 IQ= -q {

lsA

gknAkU'ds = -q

r

13

lsA

Anu'1ds.

(15.7)

Substituting (15.6) and (15.7) into (15.5) we obtain

I~ -

t:

(mcJg.,,u'u"+qA,,.£')ds.

(15.8)

The Lagrangian of the charged particle moving m the electromagnetic field defined by An(,!") is then given by

L

= -mcJg1mukun -

qAnU'.

(15.9)

Using (15.9) we obtain

tJL

ax'

BAn

~ -q ax'"'·

(15.10)

and

d ( -aL) =--(mcuk+qAk) d

-

ds

auk

ds

(15.11)

Subst:itutmg (15.IO) and (15.11) into the Lagrange Equations 04.2) gives

~ (mcuk +qAk) =q ~~U',

(15.12)

q (8An me ~= ds ax'

05 · 13>

_8Ak) ·". ilx" u

Sectron 15.1

137

Electromagnetic Field Tensor

In (15.13) we define the covariant elechomagnetic field tensor F1cn as follows: F1cn

8An

8Ak axn =~An - 811Ak.

= axk -

(15.14)

By defirution (15.14). the covariant electromagnetic field tensor F1m is an antisymmetric tensor and it satisfies F1m = -Fnk

(k,n = 0, 1,2.3).

(15.15)

Substituting (15.14) into (15.BJ we obtain the equatJ.ons of motion of a charged particle in the electromagnetic field

me duk ds

=

qF1mU'.

Using the definitJ.on of four-momentum Pk write

(15.16)

= mcuk in (15.16). we may (15.17)

Let us now demonstrate that the three spatial Equations (15.17) in the three-dimensional vector notation are equal to the well-known expression for the Lorentz force. Equation (15.17) for the three spatial components (a= 1. 2. 3) has the form

'!:°' =qFo:nU'=qFaou +qFa{Juf3_ 0

(15.18)

Using here (13.37). (13.39), and (13.40), we obtain (15.19) We then define the three-dimensional electric field vector denoted by Ea and the three-dimensional magnetic induction vector denoted by Bv as follows:

Fao=~.

F°'o=!!:._

c

where

(E"J = (E,.Ey,E,),

(Eal= (-E,, -Ey, -E,)

{Ir}= {Bx.By,B.;}.

{Ba}= {-Bx, -By. -Bz}

(15.21)

136

Chapter 15 Electromagnetlc: Flelds

Substituting (15.20) into (15.19) we obtain

~;

(15.22)

= qEa - qeafJ\JV(JBv_

Using here Pa={-Px·--PY·-pz}, iP=(v.,,vy.vlJ and the expressions (15.21) for the components of the vectors i: and B, Equation (15.22) gives

- ~ = -qEx -

-if- = -

~=

q(vyBz - VzBy) (15.23)

-qEy - q (vi.Bx - VxB.i)

-qE,, - q(vxBy -vyB_,).

Thus we obtain the familiar expression for the Lorentz force in the threedimensional vector notation, as follows:

§_=qi+qVxB.

(15.24)

dt

Using the defirutions (15.20), the covariant electromagnetic field tensor

F1m can be written in the following matrix form:

0 E_,/c E,/c [FAnl = -E,:fc 0 -Bi. -Ey/c Bz 0 [ -&/c By Bx The mixed eJectromagnetic field tensor

F!=i'JFjn

E,fc] By -B_"(

.

(15.25)

0

F! is given by

(j,k,n=0,1,2.3).

(15.26)

and can be written Jn the following matrix form:

l

0 E,/c Ey/c E,/c] [F~ ~ '-'i][F ] ~ E,/c 0 B, -By n u;. Jn Ey/c -Bz 0 Bx Ede Bv -Bx 0

(15.27)

The contravanant electromagnetic field tensor Fkn is given by

Flcn=Ff1Fjlg1n=Ffg"' (j,k,l,n=0,1,2.3),

(15.28)

Sec:tion 15.1

Eloolromagnolic Field Tensor

139

and it can be written in the following matrix fonn:

-E,/c -Ey/c

0 [Flcn]

= [Ff][gln]

=

Ex~c

0 B., -By

Ey/C [ Ez.!c

-Bz 0 Bx

-E,jc] By -Bx

-

(15.29)

0

As the next step, let us now calculate the electric field vector Ein terms of the potentials of the electromagnetic field (15.l). By definition (15.14) of the tensor F fen, we have

Ea Using now 1/F

=

=CFao=c(:~ - ~~ )-

(15.30)

cAn and xD = ct we obtain

Ea~:!01/f £"' =gafJ axfJ

0:;.

aAa

(15.31)

OAa

-ai ={gradifr)a -Tt-

{15.32)

The contravariant components of the four-dimensional gradient of a scalar function ifr are given by

05.33) Using {15.33) for (a = 1. 2, 3), we obtain the relation between the electric field vector E and the potentials Of the electromagnetic field (15.1). in the t:hree-rlimensional vector notation

-

al\

E=-ar-gradifr.

(15.34)

Let us now calculate the magnetic induction vector Bin terms of the potentials of th~ electromagnetic field {15.1)_ By definition {15.14) of the tensor F1m. we have

Fae~ ilaAe - a,Aa ~ (s~s- -si;s~) a.A,

115.35)

where we have used the Identity {10.45) in the form (15.36)

140

Chapter 15

Elecirornagneti Flelds

From (15.35) we obtain

(15.37) Using {15.37) and the definition of the curl of the vector A, we may write

B.x

= -[0(-Az) _ ily

i:J(-Ay)]

oz

=

0A4 ily

_

= -[o(-A,J _ o(-A,)] = OA, _

B

Oz

Y

B~ •

=

Ox

Oz

-[0(-Ay) _ 0(-Ax)] Ox oy

=

0Ay

= curl.xA

OA,

= cwclyA

oz

Ox

0Ay _ OAx =curl ox ily "

A.

0 5 .3S)

Equations (15.38). written in the three-rlimensional vector notation give

'B=cur1A.

(15.39)

The result (15-39) is the familiar relation between the magnetic induction vector B and the magnetic vector potential Of the electromagnetic field(l5.3). The temporal zeroth component of Equation {15.16) gives

me?~: =qcFoarf'.

~ c~~) = *' (;;-mr) =qcFfu~ From (15.20)we see thatcF0atP

=

(15.40)

(15.4!)

E· V. Thus we finallyobt.ain

d!K =qE-ii.

(15.42)

Equation (15 .42) is the statement that the change of the kinetic energy EK of a charged particle in the electromagnetic field is equal to the work done by treelectric field£. The work done by the magnetic field is identically equal to zero, since the force of the magnetic fieldqV x Bis always perpendicular to the direction of the velocity v_

[gJ

Gauge Invariance

From the result {15.24) we see that, by measurements of the forces acting on a chaiged particle in the electromagnetic field, we can measure the

Section 15.2 Gauge Invariance

141

components of the three-dimensional electric field vector E and the threedimensional magnetic induction vector B. Thus the components of the electromagnetic field tensor are observable physical quantities that are uniquely defined. On the other hand, the components of the potential of the electromagnetic field (15.1) are not uniquely defined. From the definition of the electromagnetic field tensor {15.14) we see that its components are invariant with respect to the gauge tnmefonnntions of the potential of the electromagnetic field:

(15.43) where is an arbitrary scalar function. Substituting (15.43) into {15.14). we obtain

(15.44) Tirus we need to impose an additional condition on the components Of the potential of the electromagnetic field in order to make it more precisely defined. Such a condition is usually called the gauge condition or simply the gauge of the theory. In relativistic electrodynamics the most commonly used gauge is the Lorentz gauge, which requires that the components of the potential of the electromagnetic field satisfy the equation OAn OAn Oxn = Oxn

=

OA 0 OA°' .. 1 01/f O.x<J +Ox"'= div A+ -;Ji.iii =0.

{15.45)

When the Lorentz gauge (15.45) is adopted, then using {15.43) we obtain OAn= OAn+~. Oxn Oxn OxnOxn

(15.46)

As both potentials A(.:0) and A(xk) satisfy the Lorentz condition (15.45), the arbitrary function (.xk") must satisfy the wave equation of the form

(15.47) Thus (xk) is no longer an arbitrary scalar funruon but a solubon to the wave Equation {15.47). It should be noted here that, even when we impose the Lorentz condition (15.45), the components of the potential of the electromagnetic field are still not uniquely defined and only the class of the allowed gauge transformations (15.43) is significantly reduced.

142

I15.3 I

Chapter 15 Electromagne1ic Fields

Lorentz Transformations and Invariants

The potential Of the electromagnetic field Ak, as a four-vector. transforms with respect to the transformations from one inertial system of reference to another according to the transformation law Ak

= A!A'n

(k,n

= 0, 1,2, 3),

(15.48)

where Ak are the components of the energy-momentum tensor in the inertial system of reference K, while A'n are the components of the energy-momentum tensor in the inertial system of reference K' moving along the common x-axis with a velocity V with respect to K. Using here the explicit form of the tensor A~ given by {13.29), we obtain

o/f

= - 1-(w' +VA;) Ji-;]

A,=

~(A;+~w')

yl-;fr

(15.49)

c

The transverse components of the vector Aremain unchanged, and only the longitudinal component is affected by the transformation from the inertial system of reference K to the inertial system of reference K', moving along the common x-axis with a velocity V with respect to K. Theelectromagneticfield tensor Fkn transforms fromoneinertialsystem of reference to another, according to the law

F1 1=AJAjFfn

{j,k,l,n=0,1,2,3),

{15.50)

where F11 are the components of the energy-momentum tensor in the inertial system of reference K while Ffn are the components of the energy-momentum tensor in the inertial system of reference K' moving along the common x-axis with a velocity V with respect to K. Using here the explicit form of the tensor A~ given by (13.29), we obtain

Fm= A~A~Ffn = A8A:F~. + AbA?Fio Foo

= ~AiF~ =

F03

= A~A;Ffn = AgA~F~3 + AfiA~Fi3

F13

= A~AjFk,,

= ~A~F~ 3

= A~A~Fk,, = F12 = A~AiFk,, = F31

A8A~Ffu + AlA~F;2

A~A:FJ 1 + A~A?F; 0 A:AiFi2 + A?AiF~.

(15.51)

Section 15 3

143

LorentzTIW1Sfonnalfonsand Invariants

Introducing here the notation I y= ~~y

2 (

,;•--;r

2

v ) 1-2

=1.

(15.52)

wemaywrit.e

Ag= A}= y,

A~= A~= ~y, A~= A~= 1.

(15.53)

Substituting (15.53) mto (15.51), we obtam

Fm=r 2 (1-;;.)F~1=Ffi1 Fo2=yF~+fyFli Fm= yFfi3 + fyFi3

(15.54)

F13=F;_, FJ1 = yFJ1

F12

+ fyFJo

= yF; 2 + -¥rFfii.

Now, using the explicit form of the tensor F1m given by (15.25), we have E,~E;

Ey=Y(E;-vs~)

Ez=r(E;+vs~)

(15.55)

Bx=B~

By= y (B~ + ~E;) B.:=r(B~-5E;). Thus, we finally obtain the tnmsfurmation laws for the three-dimensional electric field vector E and the three-rlimens1onal magnetic induction vector Bin the fonn E;-vs~ Ez= E;+vs; Ex=E;, Ey=g·

,-

Bx=B~.

By=

B~+ ~E;

g · 1-;o

g

B' - VE' z-g·

B-~

E

B

(!5.56)

(15.57)

The longitudinal components Of the vectors and remain unchanged, and only the transverse components are affected by the transformation from

144

Chapter 15 Electromagnellc Flellls

the inertial system of reference K to the inertial system of reference K', moving along the common x-axis with a velocity V with respect to K. In order to find the invariants of the electromagnetic field, following the discussion leading to Equation {7.5). we write the secular equation for the electromagnetic field tensor in the form det (F! - AO!) = O.

(15.58)

where A is by definition a scalar invanam. Using here {1527) and introducing tor simplicity a vector Ef c, we obtain

e=

-A

I

ex ey ez

I

e,

e,

-A

B, -J...

e, -By =O. Bx

-Bx

-A

-Bz By

(15.59)

Expanding the expression (15.59) and adding together the terms of the same order in the scalar parameter A, we obtain -A[-A(J.. 2 + B'1:)- H,p.. B,,_ - BxBy) -B.,,(Bi.Bx +A.By)]

- eAex(A2 +B'1:) - Bi.(-Aey - Bxe:i) - By(-eyBx +Ae,:)]

+ ey[ex(ABz -

BxBy) --t- A(-Aey - Bxei.) - By(eyBy + ei;Bi;)]

- ei;[ex(BzB., +Ally)+ J...(-eyBx

+ J...e~) + Bi(eyBy + ezBz)] = 0 (15.60)

4

2

2

J.. +J..2 B'; +>.. B;- ABxByBz +J.. B; + J..B"ByBz

- J..2 e';-e';B;- AexeyBz - exBxezBz - exBxeyBy + J..e_'
J..

+ J..2 (B; + B; + B; - i ; - e; - e~)-

(i;B'; + e;B;

+ e;.B'; + 2exBxeyBy + 2exBxe-,,B-,, + 2eyByezBi) = 0.

{15.62)

Sac:tion 15.S Lorentznanstormadonsand Invariants

145

In the three-dimensional vector notation. (15.62) becomes A4 + A2 (B2

Using here

-

e2)

-

(e -·iJ)2 = U.

(15.63)

e= Ejc, we obtain (15.64)

Equation (15.64) can also be written m terms of the electromagnetic field tensor F1cn as follows·

A4 +A2 (~F1mFkn)-(~eJkinFjkF1n)

2

=0.

(15.65)

Since A is an invariant absolute scalar, the quanbty !F1mFkn = B 2 - !!'!2_ = Invanant (15.66) 2 c is an absolute mvariant of the electromagnetic field. The quantity !_eJ"klnFjkFtn

8

= E B =Relative invanant c

(15.67)

is a relative invariant. since B = curlA. is a relative axial vector. However, from (15.65) we see that the square of the quantity (15.67) is also an absolute invariant of the electromagnetic field. These are the only two scalar invariants that can be c011Structed using the electromagnetic field tensorF1m.

~

Chapter 16

Electromagnetic Field Equations

I16.1 I

Eiectromagnetic Current Vector

Let us consider asystemconsistmgofanumberofduuged particles moving in an electromagnetic field specified by the electromagnetic potential fourvector An=An(x")

(k,n=0,1,2,3).

(16.I)

As the action of the system is an additive quantity, the total action of the interaction between the charged particles and the electromagnetic field is a sum of terms {15.7),given by (162)

where M is just a label for the Mth particle and not a tensor index. An explicit summatmn sign is therefore required in Equation (16.2). In electromagnetic field theory it is usually assumed that there is a continuous distnbution of charges in three-dimensional space. with the charge density defined by dq

a=dv.

(163)

147

145

Chapter 16 Eleclromegnetic Field Equallons

where the differential dV is an mrinitesimal volume element of the threedimensionalspace. In such a case Equation (16.2)can be written as follows:

s,

IQ=-

1 L v

adV

SA

dX'dxo An--dt c

(16.4)

Let us now introduce the four-dimensional volume element. defined by the expression dfl.=dx0 dV=dx0 dx 1 dx2 dx3. (16.5) Since four-dimensional space in the special theory of relativity is a pseudoEuclidean space. the four-dimensional volume element {16.5) is a scalar invariant. Substituting {16.5) into (16-4), we obtain

.

dX'

c Jn

dt

I IQ= - -

J a-AndQ_

(16.6)

Since the action integra] (16.6) and the four-dimensional volume element (16.5) are both scalar invariants andAn is a covariant four-vector, the system defined by (16.7)

is a contrdvariant four-vector called the electromagnetic currn1t vector. The temporal component of the four-vect:o£ (16.7) is proportional to the charge density a and is given by J 0 =ca.

(16.8)

The spatial components of the four-vector {16.7) constitute a threedimensional current density vector. It is the flux of the charge q through the element of the surface n that surrounds the volume domain V of the three-dimensional space in which the charges are distributed.

Jo:=alP=.!!!.!i_ (16.9) dCTdt The components of the electromagnetic current vector can therefore be structured in the form J" ~

{w,J,,J,.J,j = {ccr,-Jx,-Iy, -Ii:}

Jn= 8nlclk

(16.10)

Substituting (16.7) into {16.6) we obtain (16.11)

Section 16.2 Maxwell Equations

149

where CQ is the interaction Lagrangian densi"fy. Because of the electric-charge conservation law, the charge density a and the current density} satisfy the contilruity equation. According to the electric-charge conservation law, the negative increment of the electric charge q within a three-dimensional volume V is equal to the total flux of electric charges through the boundary surface n of the volume V in the unit of time. Thus in the three~dimensional vector notation, we may write

di

1,·

~ - - adV= J-dn. dt v n

(16.12)

Using the three-dimensional Gauss theorem and regrouping, we obtain (16.13) From (16.14), we obtain the differential continuity equation in the form

oa ot

•

-+divJ=O.

(16.14)

In tensor notation, the result (16.14) becomes ill"

ox"

~o

(n~O,l,2,3).

(16.15)

The continuity equation (16.15) is therefore related to the conservation law of the total charge in the entire system, defined by

q

~1 adV ~ v

!1

c v

f'dV,

(16.16)

where Vis the entire available volume Of the system Since there is no flux of electric charges through the boundary surface n of the volume V of the entire system, the surface integral in (16.12) vanishes and we see that the time derivative of the quantity (16.16) is equal to zero.

I16.2 I

Maxwell Equations

The objective of this section is to derive the differential equations satisfied by the electromagnetic field tensor and its components. From the definition of the electromagnetic field tensor (15.14), i.e., OA,, OAk Fkn = Oxk - 0.X" = OkAn ~ OnAk.

(16.17)

150

Chapter 16 Electromagnetic Field Equations

we see that it satisfies the cyclic equation a}Fkn + O,,F)k + i:JkF11j

+ 011B1Ak -

= ajakA11 -

aja,,Ak

a,,~AJ 4- DkOnAj - akajA,, =

(16.18)

0

Thmi the first differential equation satisfied by the elecuomagnetic field tensor is (16.19) When all three indicesj, k, and n are equal to each other (j = k = n). Equation (16.19) is a trivial identity since Fkn = O fork= n. When two of the indicesj, k, andn are equal toeachother(j = k orj =nor k = n), Equation (16.19) is also a trivial identity due to the antisymmetry Of the electromagnetic field tensor F1m = -F,,k. The only equations of interest are the four equations obtained for j of- k of- n. Thus we may write OFm

OF20

OF12

OFoi

£tF30

OF13

OF02

OF30

OF23

OF12

OF31

OF23

a;y+~+ axO =O

a;y+a;m+

axO =O (16.20)

~+a;y+ axO =O ~+ ax1 +a;m=O. Using here the components of the covariant electromagnetic field tensor (15.25), we obtain

OE.x _ OEy Oy ax

+ OBz

OE"'_ OEz Oz O:x

+ OBy

=O

Ot

=O

Ot

(16.21)

OEy - OEZ +OB;,; =0 Oz Oy Ot OBr. Oz

+ OBy + OBx Oy

=O.

O:x

In the tlrree-, i.e.,

-

oB

curlE+at=O,

divB=o.

(16.22)

Section 16.2 Maxwell Equations

151

This Equation (16.19) is the four-dimensional form Of the first pair of Maxwell equations. lnordertoderive the secondpairofMaxwellequationswc need to define the action of the electromagnetic field. The total action for a system consisting Of a continous distribution of charged particles in the electromagnetic field is given by (16.23) wh:re ls is the action for the free particle distribution that does not include the electromagnetic fields or potentials. Its explicit form is therefore not needed in the pref.Cot section The term lQ is the action describing the interaction of the charge distribution with the electromagnetic field and it is given by (16.Il). The tcrm[p is the action of the free electromagnetic field that is an integral of an invariant scalar function called the Lagrangian density £p over the entire three-dimensional volume Vin the time interval [tA, tBJ. Tlrus the action of the electromagnetic field is of the form

The invariant Lagrangian density function £p cannot depend on the potential:, A 11 , as they are not uniquely defined. It may only depend on the space-time derivatives of the potentials ~An. or in other words on the electromagnetic field tensor FIm- Thus we may write

Furthermore, ~eofthelinearity oftheelectromagnetic field equations, the invariant Lagrangian density function £p rrmst be at most a quadratic function of the electromagnetic field tcnwr. The only field iINariant that satisfies this condition is given by (15.66). The invariant Lagrangian density £p is therefore given by (16.26) The invariant (15.66) can be multiplied by an arbitrary constam, and we have chosen this constantto be equal to (-2µ 0)- 1 inordertosecurc the cor(µoc 2 1 rect physical dimensions. In (16.26) the quantities µ 0 and t<:o are the vacuum magnetic permeability and the vacuum electric pennitivity,

=

r

152

Chapter 16 Electromagnetlc Field Equations

respectively. Substituting (16.26) into (16.25) we obtain

fr=-~ { FknF1mdfi = 'T{;µo Jn

1i's (" v

2 0 E -

t.A

2

82

2µo

) dV dt.

(16.27)

Snbstitnting (16.11) and (16.27) into (16.23) we obtain the total action Of the system in the form

l=ls-_2_ { (µof 11 An+!_FknF1m)dfi=!_ { £dfi. cµo Jn 4 c Jn

(16.28)

During the derivation of the equations of motion of a charged particle in the electromagnetic field (15.16) in the previous chapter, we assumed that the electromagnetic field is defined by four given functions fonning the fourvector A,, = A,,(xk)_ We have therefore only varied the Lagrangian (15.9) with respect to the quantities describing the motion of the partide, i.e., the space-time coordinates xk and coordinates of the four-velocity ,/. On the other hand, in the pref.ent derivation of the electromagnetic field equations, we assume that the motion of the charged particles or a continuous distribution of charges in space is defined by given space-time coordinates xf< and coordinates of the four-velocity ,/ _We will therefore calculate the variations of the action (16.28) with respect to the quantities describing the electromagnetic field, ie., the electromagnetic potential four-vector An and the electromagnetic field tensor F1m. Thus the variation of the first term in (16.28) is zero and this tcnn does not contribute to the derivation of the cla-"'tromagnetic field equations. It can therefore be omitted from the action of the system. Furthermore, the electromagnetic current four-vector as a function of a given charge distribution and its motion in space, is also not varied. The variation of the action (16.28) is then given by

.r,

(16.29) with

From Equation (16.29) we may write

153

Section 16.2 Manvell Equations

{I

-a.[~]J~A.dfl a (Cl.i.An) +~c Jri{ ak[~SA 11 ]dfi=O. O(OkAn)

M=! c

Jri

iJ£

aAn

06.32)

Applying the Gauss theorem to the second integral on the right-hand side of Equation (16.32). it is reduced to the integral over ire hypersurface that encloses the given domain fi of ire four-dimensional space-time. On the other hand, ire variation Of the electromagnetic field variables is assumed to be zero on the boundary of ire domain fi, and this integral vanishes. Equation (16.32) iren becomes

Jri {~-a,[~]JM.dfl=O. OA,, O(BkAn)

M=! { c

(1633)

Since ire vanabonSAn is arbttraiy, from (16.33) we obtam ire electromagnetic field equations in the fOllowing fonn:

(1634) Now, using the expression (16.30), we obtain iJ£ n iJA,, =-J'

(1635)

and

~=!!:._ OFjl 0 (OkAn)

=

OFj1 0 (OkAn)

=

O£_il_(81A1-0JAjl OFjl 0 (OkAn)

:~1 (sJs;-t/ltJ) =a~ - a~k = 2 a~

= 2~ (-__!_pilpil) OF1m

4µo

=

-~~ (FjlFjl) 2µo OF1m

= _ _!__pil(sksn -!ls1:) = -~(pkn _pnk) = _ _!_pkn 11 11 2µo

2µo

µo

(16.36) Substitutmg (16.35) and (16.36) into the field Equations (16.34), we obtain -Jn -8k ( -µoF 1 "") =0,

(1637)

154

Chapter 16 Elecbomognetlc Fleld Equations

or, after regrouping.

aF"'

Oxk =µof!

(16.38)

(k,n=O,I.2,3).

The temporal component Of Equation (1638) gives

aF = ~~- = ~div E = µofJ = µ 0 ca

a.xa

c

a.xa

c

ruvE =

µo
·

= ~­
(16.39)

(16.40)

The three spatial Equations (16.38) give

aF"•

aF'"

~+fhT=µol'X,

-~ O!" _t13uw:~;

=

-~ O!" +t?f3w:~; =µor.

(16.41)

(16.42)

In the three-dimensional vector notation Equation (16.42) is given by 1 aE curlB- 21-fii =µof.

(16.43)

Th: Equations (16.40) and (16.43) are the f.eeond pair of Maxwell equations. Thus the complete system of electromagnetic field equations in the four-dimensional notation is given by

a;;+:::+ °:J

=0,

(16.44)

-~o

(16.45)

with the contmuity equation

a1" ar<

The four-dimensional formulation Of Maxwell Equations (16.44) with (16.45) is quite compact and it nicely emphasizes the relativistic nature of these equations.

I16.3 I

Electromagnetic Potentials

The objective of the present section is to fonnulate the differential equations for the electromagnetic potentials An = A,,(.xk") and to outline

155

Section 16.4 Energy-Momentum Tensor

their solutions. In order to derive the differential equations for the electromagnetic potentials, we substimte the definition of the electromagnetic field tensor (15.14) into Equation (16.38). Thus we obtain ~((fAn - anAk)

= akatAn -

an~Ak = uor.

06.46)

Using the Lorentz gauge (15.45), the second tenn on the left-hand side vanishes, and we obtain the differential equation for the pct:entiab of the electromagnetic field in the fonn

ii'A" axkaxk =µor.

(16.47)

By expanding Equation {16.47) we obtain

~ a2An

- V2An =µa.In

(- a12

V2An - ~ a2An =-µa.In. c1 at2 The solution of this equation has the fonn

A"(x')

~ '::'! { 4ir

}y

J"(x" - R,f) dV,

(16.48)

(16.49)

(16.50)

R

whcrethemtegral isoverthedomam V where thesourcesoftheclectromagnetic field are distributed. The quantity R in (16.50) is the three-dimensional distance between the position of the sources and the position where the pctential is calculated, i.e., (16.51) Thus. for a given electrcmagnetic current tour-vector Jn(xk), we can calculate the electromagnetic potential four-vector A"tx1'°) using the result (16.50). The electromagnetic field tensor i~ then obtained using the definition (15.14).

I16.4 I

Energy-Momentum Tensor

We have shown earlier that the momenrum co~tion law i~ a consequence of the homogeneity of space and the energy conservation law is a consequence of the homogeneity of time. In the four-vector language, due to the homogeneity of space-time, the ph).'l)ical properties of a free field

156

Chapter 16 Electn:lmagnsllc Fleld Equations

remain unchanged with respect to the space-time translations. Tlrus, the energy-momentum tensor of the electromagnetic field is a field invarianr defined with respect to the ~pace-time translations. The Lagrangian Of the free electromagnetic field is given by (16.26), Le.,

£ (F1m)

~ £F(F1m) ~ _ _!_F""F1m. 4µo

(16.52)

Furthermore, it is assumed that there are no sources of the electromagnetic fieW, i.e., that the electromagnetic current four-vector is equal to zero, = 0. Thus the interaction Lagrangian CQ, defined by (16.11 ), vanishes. The second pair of Maxwell Equations (16.38) then gives

.r

(16.53)

In order to define the energy-momentum tensor of the electromagnetic field, let u~ first calcnlate the ~pace-time derivatives of the Lagrang:Ian density £ as follows: iJ£

iJ£

a,,£= aA}OnAj + O(Cl.i.Aj) a,,(DkAj).

(16.54)

Using the electromagnetic field Equations (l 6.34l. we obtain

On£

~ 0. [ 0 (iJ£ ) OnA1]Cl.i.Aj

(J 6.56)

On the other hand, by definition we may write

On£= h!ak£.

(16.57)

From (16.56) and (16.57) we obtain

1

1 -h!£] ~ iJ,T: ~o.

0. [o(~ )iJ.A

(16.58)

T!

where is the mixed energy-momentum tensor of the electromagnetic field, defined by (16.59)

Section 16.4 Energv-Momenlls Tensor

157

Substituting (16.36) and (16.52) into (16.59) we obtain k I k. T,, = -~F ~OnAi

+ 4 µI

k

0

·1

h,,P Fit-

(16.60)

The result (16.60) can be put into a more convenient form by writing pkjOnAj =Flg(OnAj-O;An)+FlefOjAn,

(16.61)

The third term on the nght-hand side of Equation (16.62) vanishes because of the result (16.53). The f.eCOnd term on the right-hand side of Equation (16.61) makes no contribution to the conservation law (16.58). since we have (16.63)

~~~:~::!~UJ!!~:::::;~:i~~:e~~rt=~~:~:~~~t Thus the second term does not contribute to the field invariants obtained from the energy-momentum tensor (16.60). Therefore we may replace pki OnAj by plefFn1 in (16.60) to obtain the final result for the energy-momentum tensor of the electromagnetic field in the form

T! = _ _}__pkip.,i + _!_h!F11F_J1· µo 4µo

(16.64)

The contravariant energy-momentum tensor i-. then given by

and the covariant energy-momentum tensor is given by Tnk

= _ _!_FiFnj + _2_g,,kF11Fj1. µo

4µo

(16.66)

Using the differential torm ot the energy-momentum conservatton law (16.58), we may write down the integral form as follows:

1 v

°"TnkdV =

11ilT"' +1

-

-dV

cvOt

v

a,,T'30'-dV = O.

(16.67)

158

Chapter 16 Electromagnetic Field Equat1cms

Applying the three-
f ~Tnkdv=!~[f T"0dv]+i. yfJa.dnu=O lv cdt Jv fa

(16.68)

where TI is Ihe closed surface surrounding the volume V. If we let the field domain V grow to infinity (V--+ oo), tre surface integral in (16.67) vanishes and we have

{ O,T""dV='!_[~ { T"'dV]=d/f'_=O. & ch &

h

(16.69)

Thus we obtain the definition of the conserved energy-momentum vector of the electromagnetic field:

JI'=_!_

f

cJv

T"0 dV = ConstanL

(16.70)

The energy of the electromagnetic field is then

E=cp0 = [TXldV,

(16.71)

and the components of the three-dnncnsional momcntnm of the electromagnetic field are given by

JI'

= !_ { yao dV =Constant. cJv

(16.72)

In order to calculate the energy (16.71) we first use {16.65) with {16.27) to calculate

.!!._).

TJ0 = _ _l_f"'"F<J -(
22µo

µoa

(16.73)

Using the definitions of the mixed and contravariant electromagnetic field tensors, (15-27) and (15.29), respectively, we obtain

TJO = _ _I_ (-~) 1 c

µ0

= lOoE1 -

yW

(
2

2

(£oE2 - 2µo 2

_

B2 ) ,

=

£0£2

2

+ !£__ 2µo

B

)

2µo (16.74)

(16.75)

Section 16.4

~Tensor

159

Substituting (16.75) into (16.71) we obtain the expression for the total energy of the electromagnetic field,

E = 1 (t0oE v

2

2

+!f___)dv=1 EdV, 2µo v

(16.76)

where E is the elec.,nomagnetic energy density given by E

= yflO = E"oE2 2

+ !£___

(16.77)

2µo

Inordertocalculatethemomentum(l6.72)wefirstuse(16.65)with(16.27) to calculate 2

ro = _ _!_p01F'!1 _ 8 °0 (t0oE µo

_

!!:___) .

(16.78)

_!_~pv_

(16.79)

2µo

2

Using g 00 = 0, we obtain T..,o

= _ _!__FJPFj; = µo

_

_!__p0!38pvF".., =

_

µo

µo

~=-~Ev,

F".., = -eva."'B,,,,

Using (15.20) in the form (16.80)

we obtain

Try_Q

= _

_2_evr.>wEvBw = +-2...e"vwEvBw. (16.81) cµo cµo D=E"oE and (cµor 1=ao. the result (16.81)

Here, using the notation becomes

T..,o = c(D x foa..

(16.82)

Substituting (16.82) into (16.72) we obtain the expression for the total momentum of the electromagnetic field·

1'=1(DxB)dV=1PdV. ,. v

(16.83)

where P is the electromagnetic momentum density gtven by (16.84) In order to calculate the spatial components of the electromagnetic energy-momentum tensor in terms of the three-dimensional electric field

160

Chapter 16

EledrornaSJ1etk: Flel_d Equations

vector Eand magnetic induction vector B. we use Fkj to rewrite the definition 06.64) as follows: T!

= __!___ [FnjF1k - ~li!(e2 2

l'O

=

= -Fik and (16.26)

B 2 )]

~ [ e!- ~li!(e2 - B 2 ) J,

(16.85)

where we introduced the vecmr e=E/c and the tensor e!=Fnjp1k to simplify the calculations. The contravariant electromagnetic energymomentum tensor is then given by ynk

= g"JTk1 =

__!___

µo

[enk _~2 8nk(e2 _ B2)]

(16.86)

We can now use the matrix forms of the covariant and contravariant electromagnetic field tensors, (15.25) and (15.29). respectively, to calculate the components of the tensor E>!:

The spatial pan of the tensor

[BP]=

2 e; + B; - B eyex+ByBx [ ezex + BzBx

E>! is then given by the following mauix: exey +BxBy exez + BxBz ] e:+B;-B2 {ez~ByBz e;.ey -t- BzBy ez + Bz - B

2

.

(lb.88)

The contravariant components of the three-dimensional system 06.88) are obtai~dfrom

(16.89)

161

Section 16A Energy-Momentum Tensor

or, in the matrix form,

[SUP] =

-e; - B~ + B2 -exey - BxBy - ByBx -e; - ~ + B [-eyex -ezex - BzBx -ezey - BzBy

2

From the matrix (16.90) we obtain

Ef'P =-t?ef3-JPBf3 -gaf3B2 E"'EfJ =-~-JPBf3-gr43B2 .

(16.91)

Substituting the result (16.91) into the spatial pan of Equation (16.86), we obtain yafJ =

_!__ [-t?ef3 -IPB/3 - gr:if3IJ2 - !gaP(e 2 - B2 )] l'O

= _

2

_!_ [t?ef3

+IPB13 + _!.RafJ(e2 + B2 )]

""

(16.92)

2

Thus we finally obtain the spatial spatial components of the electromagnetic energy-momentum tensor in terms of the three-dimensional electric field vector Eand magnetic induction vector B, as follows: (16.93) where we define the Maxwell stress tensor 7"P by the equation (16.94) Usingtheresuhs (16.77) with (16.84)and (16.93) we may write the explicit matrix form of the electromagnetic energy-momentum tensor as follows:

[T"'] -- [;'p2 !';n -T2• -;., _'J12 p3

_731

_732

!';"]

_'J13 ·

(16.95)

_733

From the definition (16.94) we see that the three-dimensional Maxwell stress tensor is a symmetric tensor. By examination Of(16.95) we conclude that the complete four-dimensional electromagnetic energy-momentum tensor is also a symmetric tcn50r, i.e., we have ynk

= Tfm,

Tnk

= Tk.n·

(16.96)

162

Chapter 1£1

~FleldEquatlons

If the electromagnetic field is defined in a domain V with the boundary surface n. the expression

F" ~

i aT"• f, --dV = v axfl

a

T"fJ dilp

(16.97)

defines the force due to the electromagnetic field pressure on the boundary surfacen.

Part IV

General Theory of Relativity

... Chapter 17

Gravitational Fields

117.1 I Introduction The special theory of relativity discussed in the last four chapters is based on thcconceptofinertialframcs of reference. 'An inertial frame of reference is defined as a system of reference where a free particle, i.e., a particle on which there is no action of any external forces, moves along a str.tight line with constant velocity_ On the other hand, the gravitational interaction, as one of the fundamental interactions in nature, is a long-range interaction that cannot be screened. The concept of inertial frames of reference is, therefore, not compatible with gravitational phenomena. The only way to define an approximately inertial frame Of reference is to visualize it as being far away from any matter_ Given the influence of the force of gravity on the observations of various physical phenomena, both on Earth and in the universe as a whole, the concept of inertial frames of reference as a foundation for the fonnulation of the laws of nature is clearly not sufficient Since the force of gravity is an unscreened long-tange force, it can be considered as an intrinsic property of space-time and related to space-time geometry. In the absence of matter we can define the inertial frames of reference where the geometry of space-time is pseudo-Euclidean. The space-time metric is then given by

165

166

Chapter 17 Gravllallonal Fields

such that the components of the metric tensor 8mn are constant and given by the following matrix:

(gm"]~

[

0I 0

0 -I 0

00 -I

0

0

0

0 0 ] 0 .

(17.2)

-1

In the presence of matter it is not possible to find the four space-time coordinates xi such that the metric of the space-time manifold is reduced to the pseudo-Euclidean expression (17.1). Tlrus the geometry Of space-time in the presence of matter must be a psendo-Riemannian geometry, where the components Of the metric tensor 8kn are functions of the coordinates xi. i.e., we have 8kn=8kn(Xj)

(j,k,n=O,I,2,3).

(17.3)

The components of the metric tensor Of the given space-time manifold (17.3) can therefore be considered as the gravitation field potential.'>; and the gravitational effects are described by the metric itself. From (17.2) we note that the detenninant g=-1 Of the pscudoEuclidean merric tensor Of special theory of relativity is a negative constam number. In the general theory of relativity g g(xl) < 0 is a negative definite function of space-time coordinates. In the ordinary positive-definite Riemannian metric spaces, we have frequently used the function ,,J8 to define the absolute tensors and invariants. In the pseudo-Euclidean and pseudo-Riernannian :.pace-time:. of the theory of relativity with g < 0, this function is not a real function and it is generally replaced by ,J=g, which is a real function. In the pseudo-Euclidean special theory of relativity we have Fg = I, so the relative tensocs and invariants were transfonned as the absolute scalars and invariants. and the explicit appearance of the factoc Fg was not necessary. In the presence of gravity it is not possible to reduce the metric Of space-time to the pseudo-Euclidean expression (17 .I) in the entire space. It is important to note that the gravitational effects are understood as a dcviationOfthespace-timemetricfrom the pseudo-Euclidean metric (17.1). The metric (I 7.3) is therefore a function of the local distribution of matter as a source of the gravitational field and cannot be fixed arbitrarily in the entire space. The metric tenwr, as the gravitation field potential, is in this context a solution of the gravitational field equations, which will be derived in the next chapter. Although we cannot transform away the gravity in the entire space, we can select the local frames of refereoce falling freely in the gravitational field where the gravitational effects are locally tranMormed away. In a

=

Section 17.2 TimalntervalsandDistances

167

small laboratory falling freely in a gravitational field, the laws of nature are therefore the same as those observed in an inertial frame of reference in the absence of gravity. The laboratory has to be small ~ince the gravitational fields are functions of coordinates and vary in magnitude and direction in different points in space_ The force of gravity can therefore be considered as uniform and transfonned away by a suitable cfloice of local frame Of reference only in a sufficiently small space domain. Thus we are in most cases able tocoverthe space-time with a patchwork of local inertial frames of reference. The concept Of local inertial frames Of reference is very useful in the general theory of relativity.

I17.2 I

Time Intervals and Distances

In the general theory of relativity, the choice of the generalized coordinates

used to describe the four-dimelll)ional space-time manifold is not restricted in any way. Thus the coordinates xl are in general not equal to the distances and time intervals bctWeen events in the same way as in the special theory Of relativity_ Therefore, given a set Of generalized coordinates xi, we need to relate these coordinates to the actual distances and time intervals between the observed events. In the general theory of relativity we denote the proper time by r. Let us now consider two infinitesimally separated events that take place at exactly the same point in space. Then we have dx 1 =dx2 =dx3 =0, and we may write

Tlrus we obtain the relahon between the element of the proper tune dr and the coordinate differential dx0 as follows_ (17.5)

or for the proper time between any two events occuring at the same point in space, T

=

'! ~

.JiOQdx0 .

(17.6)

In the special theory of relativity the element di of spatial distance is

defined as the distance between two infinitesimally separated events taking place at the same time. In the general theory of relativity we cannot use

166

Chapter 17 Gravltatlonal Fields

this definition, since the proper time is a different function of coordinates at different points in space. In order to find the element dl of the spatial distance, we consider two infinitesimally close points A and B with coordinates y} and xi+ dxi, respectively. Let us suppose that a light signal is directed from point A to point B and then back from point B to point A along the same path. The time required by the light signal to travel from point A to point B and back, observed from one point in space and multiplied by c, is double the distance between the two points. For the two events representing the departure of the light signal at one point and the arrival at the other point, we know from (13.4) that the square of the interval ds2 is equal to zero. Thus we may write ~ ds2 = g1mdxkdxn = goo(dx°) 2 + 2g0adxndxo + 8afJdx"dxfJ = 0. (17.7)

The equation (17.7) is a quadratic equation with respect to dx0 and its solutions are given by

0 (dx°)± = - 80adx"dx ± _!_J(g0o:8op - 8oo8utJ)dx"dxP. 800

(17.8)

800

The coordin~te time required by the light signal to travel from point A to point B and back is then given by (dx0)+ -(dx0)_

~ 2)(g0agop -gooi1ap)dx"dxP.

(17.9)

800

Using (17.5) we obtain the proper time required by the light signal to travel from point A to point B and back, multiplied by c. as follows:

cdT

=

.JiciO[(filO)+ - (dxO)_]

~_'!:__J(80a8op-goog.,,)dx"dxP. .JiOO

(17.10)

Thus the spatial clement di between the two pomts A andB is a half of the proper time interval (17.10) multiplied by c, i.e.•

I df=-cdT=

( -8r:tJJ+-g0o;go.13) dx"tb;I-'. "

2

(17.11)

800

Tlrus the square of the space metric d/2 can be written in the form

dz2

=

Yu.13tb:" tb: 13

Yr:tJJ

=

-8uJJ +

g~!op.

(17.12)

It should be noted that the metric tensor 8kn in general depends on the time coordinate. The space metric dt2 is therefore also time dependent, and in

Section 17.3 Particle Dynamics

169

general it does not make sense to integrate the element of the spatial distance di, as such an integral would depend on the world line chof.en between the two end points. Only when the metric tensor g1m is not time dependent and the distance can be defined over a finite portion of space does the integral of the element of the spatial distance dl along a space curve have a definite meaning.

117.31

Particle Dynamics

In the special theory of relanvity Ihe motion of a free particle of mass mis defined by the action integral (14.5). The action integral of a free particle is simply proportional to the arc length in the four-dimensional space-time manifold, i.e.,

I= -me

Ls,

ds =-me

SA

Ls" J SA

dxk,ix" g1m--ds, ds ds

(17.13)

where mis amass parameter of the free particle. From (17.B)we~ee that the Lagrangian function is given by the ex~ion (17.14)

The equations of motion of a particle in the gravitational field are obtained by variation of the same action integral (17.13). since the introduction of the field of gravity is nothing but the change of the space-time metric to a pseudo-Riemannian metric with a coordinate-dependent metric tensor (17.3). On the oilier hand, except for the overall COlll)tant multiplier -me, the Lagrarigian(l 7.14) is the same as the Lagrangian (11.14) used to derive the geodesic equations (11.25). Thus we conclude that a free particle in the gravitational field moves along the geodesic lines and that the equations of motion are the geodesic equations (11.25), i.e.,

d1X1 ds2

axt axk

+ radsds =

0

¥s-+r;1u 1,J=o

(k.l,n = 0,1,2,3)

(k.l.n=O,I,2,3).

(17.15)

(17.16)

In order to find the nonrelatiVll)tiC limit of bquatJons (17 .16). we first note thatthenonrelativistic limit (v « c) of the components of the four-velocity

170

Chapter 17 Gravllattonalfields

(13-39) and (13.40) is given by

o I 1 u= ~l l.:og:;'

y1-;;r

if'=-"'--~".'._ r:--7 cv•-~

(17.17)

c

Thus the leading term on the right-hand side of the expression (17 .16) is the one with I = k = 0, such that we may write

du"

ds

-r&

=

Cn =

For a static gravitational field with (\)goo in (17.18) are the spatial equations

o. I.2,3).

(17.18}

= 0, the only nontrivial equations

du"

ds = -r& ca= 1.2.3}.

(17.19)

Using (17.17) and the zeroth-order nonrelativistic approximation for the differential of the parameter ds ~ cdt, we further obtain

1 dv"

iidt = -r& ca= I,2,3}

(17.20)

Let us now recall the definition of the Christoffel symbols of the first kind (9.28),i.e.,

r __ k,Jp -

! (ag1g agpk _ ag1p) 2

axr + ax1

For a static gravitational field with 8o8Jp

r

axk

•

(17.21)

= 0, we obtain

- ~ (ag1;0 8gok - 8goo) - _!_ 8goo k.00-2 axO + axO axk 2axk·

(17.22)

Now using lhe definition of the Christoffel symbols of the second kind (9.29),i.e.•

r~

-

1P - 8

!

nkr . nk (aglj kJp - 2 8 axr

+ 8gpk a.v

- agjp) axk •

(1723)

we obtain

r&i = gnkrk.00 = -~gnk ~~~.

(17.24)

The spatial components of Equation (17.24) give

(17.25)

Section 17.3 Particle Dynamics

171

For a static gravitational field with iJugoo = -0 we then obtain

r~ = -~ 8afJ~~~-

(17-26)

Substituting (17-26) intu (17-20), we obtain

~ ~t" = ig«P~!~

(a,p

= 1,2,3).

(1727)

In order to find the nonrelativistic approximation for the component goo of the metric tensor, letus consider the nonrelativistic Lagrangian for a particle moving in the gravitational field defined by the gravitational potential ¢, given by

We may always add a constant term to the nonrelativistic Lagrangian without affecting the dynamics. Thus, following the result (14.17) we add a constant -mc2 to the Lagrangian (17.28) and obtain

L

= -me?-+ imv2 -

mt/J

=

-mC1 - ~mgapv"vP -

mt/J,

(17.29)

where gap are the spatial components of the pseudo-Euclidean metric tensor. In the Descartes coordinates they are given by EaJJ = -f.ap. Using (1729) we obtain the nonrelativistic approximation to the action integral (17.13) in the form I~-mc

£'• ( !A

'"dx')

¢ I c+-+-gap-dt, c 2 c dt

(17.30)

or, after some regroupmg,

I= -me

l

s,

s,

ds (17.31)

From Equation (17.31) we obtain the nomelativistic approximation for the line element ds as follows. ds~

) 1 ,!" ( 1+ c2 tbP+-gap-dxfJ. 2 c

(17.32)

Thenonrelativ1stic approximation to the metric of the space-time manifold is then obtained by squaring the expression (17.32) and dropping the terms

172

Chapter 17 Gravitational Fields

of the order v2/c2 and higher. Thus we obtain ds

2

~ (1 + c~)

2+ (1 + ~) Ea/J ~dxfJ'!!_

2

(tbP)

c

dt

c

,!" )' +4I ( Ea/J-;;-dxfJ

(1733)

Here we may drop the last term as thetennofhigherorderin v2 jc2 to obtain

Assuming that the gravitational field is relatively weak, we may use the approximations 2

(

I

L_) ~1+~ c2 c2

(17.35)

ruid

(1+*)8apdx'-'dxf3

~ 8afJdx'-'dxf3.

(17.36)

Substituting (1735) and (17.36) into (17.34), we finally obtain

ds2

~ (1+¥i)(lhQ} 2 +8apdxadxfJ.

(1737)

From the result (1737) we obtain the nonrelativistic approximation to the quantity goo in the form goo~

2¢

I +(!i.

(1738)

Substituting (17.38) into (17 .27). we obtain

d;; EfP ~ =

(a,{3 = 1,2,3).

(17.39)

Using here the pseudo-Euclidean spatial metric g«fJ, consistent with the result (17.37), we obtain in the three-dimensional vector form

~=

-grad
q,

=

¢(i).

(17AO)

173

Section 17.4 Electromagnetlc Field Equations

If we now recall the nonreiativistic result for the gravitational potential¢. wemaywrite q,(i) = -

G:-1 ==> gradq, =

~~ T,

(17-41)

where Mis the mass of the source of the gravitational field, r is the threedimensional radial coordinate given by

r = Jx2 +J.2 +z2•

(17.42)

and G is the gravitational constant given by (17.43) Substitm:ing (17-41) into (17.40) and multiplying by the mass of the test particle m, we obtain the result for the nonrelativistic gravitational force between two bodies with masses Mand m as follows: -

dV

GMm ..

Fe= mdi = --;.r-r-

(17.44)

Tll.is in thenonrelativisticlimitthe equations of motion of a particle (17.I 6) are reduced to the corresponding Newtonian nonrelativistic equations of motion.

I17.4 I

Electromagnetlc Fleld Equations

The objectlve of the present section is to generalize the elecnornagnetic field equations, derived in the previous chapter, to the pseudo-Riemannian metric space in the presence of gravity. In the special theory of relativity the electromagnetic field tensor is defined by Equation (15.14), i.e.,

F1m

=

i3kAn - i3nAk

(k, n

= 0, I, 2, 3)_

(17.45)

AI> the partial derivatives do not transform as vectors in lhe Ricmannian metric spaces, the expression (17-45) has to be modified to the covariant expression

Using the definition of the covariant derivatives (8.24). i.e.,

DkAn

=

i3kAn - r!.rcA-j.

DnAk = i3nAk - f'1'knAj,

(17.47)

174

Chapter 17 Gravlta1klnlll Fields

we obtain Fkn =<\:An -

P,,0; -

Cl,,Ak + riknAj-

(17.48}

Usingthe~ymmetryoftheChristoffelsymbolsofthesecondkind(9.9),Le.,

r'nk ~ r'1m

(k,m,p ~ 0, 1.2,3).

(17.49)

the second and fourth terms on the right-hand side of Equation (17-47) cancel each other and we reproduce the pseudo-Euclidean result (17.45). Even in the presence of gravity the definition of the electromagnetic field tensor(l7-45)remains valid. Using thedefinitionofthecovariantderivative of a second-order covariant tensor (8.33), we have DjFkn = 81Fkn- rLjF1n- r~Fk/

D,.Fik = 8nFjk - r)nFtk - rL,f"it

(17.50)

DkFnj = <1kFnj - r~kFli - rjkFn1Now, using the antisymmetry of the electromagnetic field tensor F1n = -F,.1,

Fk1

= -F1k.

F1r = -Ftj

(17.51)

and the symmetry of the Christoffel symbols of the second kind

r~

=

rJk• r!1 = rJn• ri,, = r!.t.

(17.52)

we obtain from (17.SO) the cyclic expression DjFkn

+ D,,Fik + DkFnj = BjF1m + lJ,,Fjk + akFnj =

Q_

(17.53)

Thus we conclude that the first pair of MaxweJl equations remains unchanged in the presence of gravity and has the form

a:;+~;: +a;~ = o

(j,k,n = 0.1.2,3).

(17-54)

In order to derive the second pairofMaxwell equations, let us consider the interaction Lagrangian (16.6) in the form

IQ=-~ c

{

Jf'/.

a~A,,dQ. dt

(17.55)

The four-dimensional volume element, defined in the pseudo-Euclidean space-time of special theory of relativity by the equation df2=dxOdV=dxOdxldx2tf2, (17.56) is no longer an absolute scalar. Instead of (17.56) we define the irwariant volume element as FE dQ_ Thus we rewrite the interaction Lagrangian

Section 17.4 Electromagnetic Field EquationS

175

(17.55) as follows:

IQ~_!

{

_""__~A• .j=gdQ ~ _!

cfnFEtb!l

{ l'A• .j=gdQ.

cfn

(17.57)

Since the action integral (17.57) and the four-dimensional volume element ,,/=gdQ are both absolute irwariants and Rn is a covariant four-vector, the system defined by

(17.58) is by definition the contravariant electromagnetic current four-vector. The temporal component of the four-vector (17.58} is proportional to the charge density a and is given by

(17.59) and the three-dimensional current density vector is

(17.60) Given the definition (17.58) we can generalize the second pair ofMaxwell equations (16.38) by replacing the partial derivative ak by the covariant derivative Dk. The second pair of Maxwell equations then becomes DkFm = µoJ"

(k,n = 0, 1,2.3).

(17.61)

Using the definition of divergence of an arbitrary tensor with respect to one of its contravariantindices (10.15), we obtain

In the same way we generalize the equation of continmty (16.45) to obcain

1 a (y~g/ r-o ') Dnl, =FE~ =0.

(17.63)

Thus the complete system of electromagnetic field equations in the pseudoRiemannian space-time manifold, which includes the gravitational effects, is given by 8F1m

8Fjk

8Fnj

axi + axn + axk

= 0,

I

A

a(

"') = µo/ ,

fuk ,,/=gF

(17.64)

175

Chapter 17 Gravlta11on111Fields

with the continuity equation

(17.65) The psendo-Riemannian four-dimensional formulation ofMaxwell equations in the presence of gravity given by (17.64) with (17.65) is quite compact, and it nicely illustrates the general prescription for the introduction of gravity in the laws of physics. In general, the laws of physics must be valid in all systems of reference and expressed as covariant tensor equations. The field equations of physics, which irwolve the derivatives of the field variables, must therefore be rewritten such that the ordinary derivatives are replaced by the covariant derivatives. Even if we are working in a fiat space-time described by some curvilinear coordinates, we need to use the covariant derivatives if we want the field equations to be valid in all coordinate systems. If we assume that a physical object described by a set of pseudo Euclidean equations does not represent an appreciable source of the gravitational field, then it does not significantly influence the components of the metric tensor Ekn(xl) as the potentials of the gravitational field. In such a case the geometry of the four-dimensional space-time manifold is rigidly detennined by some massive external sources of the gravitational field. Then the effect of the physical object under consideration on the geometrical structure may be neglected. Under these circwm.1ances the set of pseudo-Euclidean equations describing the physical object can readily be generalized to incorporate the effects of gravity by making the substitution

d-+ D,

& -+Dk.

drl.-+ ,J=i;d!:l.

(17.66}

Thus the prescriptions (17.66) can be used to generalize any pseudoEuclidean equation, used to describe a physical object that is small in comparison to the sources of the gravitational field, to incorporate the gravitational effects and to be valid in all systems of reference.

~

Chapter 18

Gravitational Field Equations

I1B.11

The Action Integral

In the previous chapter we concluded lhat the companents of the metric tensor of a given space-time manifold are the potentials of the field of gravity, and the gravitational effects are described by the metric itself_ The metric tensor is therefore a system of functions of the local distribution of matter as a source of the gravitational field and cannot be fixed arbitrarily in the entire space. The metric tensor as the field potential is therefore a solution of the gravitational field equations. The objective of the present chapter is to derive the gravitational field equations from the suitable action integral of the gravitational field The total action integral for a system consisting of a continuous distribution of matter as the source of the gravitational field and the gravitational field itself is given by l=lo+IM.

(18.1)

where lo ts the actmn of the gravMtional field m empty space, where there are no field sources, and IM is the action describing the interaction of the matter distribution with the gravitational field. The action of the gravitational field in empty space can be written in the form lo

~~

{ £c(g.,,, a}g.,,).J=ii dQ, cf~

(18.2)

where £c is the mvariant Lagrangian density of the gravitational field that is integrated over the invariant four-dimensional domain Q. The action describing the interaction of the matter distnDution with the gravitational

178

Chapter 16 Gnnritallor1lll Reid Equations

field can be written as follows: IM=

~ f

c Jn LM (g1m. a;81m) Fgdn,

(18.3)

where CM is the invariant Lagrangian density of the source fields. For example, if the SOUICe of the gravitational field is the electrcmagnetic field. then CM is the Lagrangian density of the electromagnetic field (16.52), i.e., LM(gkn) =

-~g1;;g.,F"'FP. 4µo

(18.4)

It should be noted that the electromagnetic field F kn in bquation (18.4) is taken as the given SOUICe of the gravitational field, and the Lagrangian £M is varied with respect to the gravitational field potentials 8knIn order to construct the invariant Lagrangian density of the gravitational field Cc, we note that it is a function of the metric tensor g1m and its first derivatives 8jg1m only. On the other hand, by using (9.25) in the form 8j8kn

= r n,kj + rk,nJ = Enlr~ + Eklr~i•

(18.5)

we see that the first denvahves of the metric tensor ai8kn can always be expressed in terms of the suitable metric tensors and Christoffel symbols of the second kind_ Thus the invariant Lagrangian density of the gravitational field£ccan be seen as afunctionofthemetric tensor g1m and the Christoffel symbols of the second kind

rl,. Thus we may write (18.6)

The only nontrivial tensors that can be created from the metric tensor Ekn and the Christoffel symbols are the curvature tensor R/1m. defined by (12.14) as

rl,

Rl,1 = a,,rf1 - a,rfn + rf1rj,, - rr,,r;1;

(18.7)

the tensors obtained by contractions of the curvature tensor, e.g., the Ricci tensor R1m. defined by (12.44) as

Rm

=

RL,j = a,,rk -

a.;rL, + r'rJn - rr,,r;1 ;

and the Ricci scalar R, defined by (12.50} as R = i'7'RknThus, a suitable candidate for the invariant Lagrangian density proportional to the Ricci scalar R. In such a case we may choose

(18.8)

£o is (18.9)

where G is the gravitational constant given by (17 .43). The factor of proportionality in Equation (18.9) is added to ensure the correct dimensions

Section 16.1 TheAcilonlntegral

179

of the physical quantities. The action integral (18.2) may then be written in the form

c' [

lo=-~-

16:rrG n

RFgdQ_

(18.10)

However, from the preceding definitions we note that the Ricci scalar R depends not only on the metric tensor g1m and its first derivatives ~i81m· but also on the first-order derivatives of the Christoffel .i.-ymbols a1rfn and thereby also on the second-order derivatives of the metric tensor. On the other hand, the gravitational field equations must contain up to the secondorder derivatives of the metric tensoc as the gravitational field potential. AJ; these equations are obtained by the variations of the action (18.2). the Lagrangian density £v must not contain the derivatives of the metric tensor of the higher order than the first. In order to resolve this difficulty, we need to analyze the structure of the Ricci scalar R. If it can be shown that the terms containing the second order derivatives of the metric tensor do not contribute to the variations of the action integral (18.10), then we could proceed using the action integral (18.10) in the derivation of the gravitational field equations. Since the Ricci scalar is the only available nontrivial scalar rnvarrant creai:ed_ using metric tensor 8kn and the Christoffel symbols of the second kind rt,,, it is important to show that the preceding assumption is indeed valid. Let us therefore calculate the quantity ,J=gR as foDows:

FgR = Fgg'"R.,, = Fgg"'a,,rh

-,J=gg'
=&(F.t"'rl)-a;(F..'"rt,,) a,,(F.F/"')rl +ai(,J=ig1m)rfu.-,J=ig1m(r:,,r~ - r~rjn)· (18.11) Interchanging the dummy indices n ~ j in the first term on the right-hand side of Equation (18.11), we obtain

FgR =a, (F..''rb.- Fgg"'rt,,)

+ ,,,r=grf,, )=.a, (F.t"') - F.rl

)=.a,, (F.l'") (18.12)

180

Chapter 16 GravltatlonM Reid Equations

Here we note the result (9-48) in the form (18.13)

and we calculate

)=ga1 (RI")= l"a1 (ln.j::g) + a;l"-

(18.14)

Using the result (9.36) in lhe form

rJ;, = aj(lnFi),

(18.15)

)=gaj (Ht"')= lnrJ;, + aig1m_

(18.16)

we obtain

Using further the result (8.29) applied to lhe metric tensor glm and the property that the covariant derivative of the metric tensor is equal to zero, we have

aj,r = -r;ilP- r;1g"P_

(18.18)

Substituting (18.18) into (18.16) we obtain

)=.ai {Ht"')= 8knr};,- r;ilP -

r;ig"P.

(18.19)

Substituting (18.13) and (18.19) into(18.12) we obtain

FiiR = a1 ( Fiii''r'i,, - Fiii'"rl) + ,J=grl, (t"'rZ, - r;ilP - r!it'P) + .J=Krt,g1nrjn ,J=gglm (rk:,r~j- rtrJn)

(18.20)

or, after regrouping,

,,,!=gt<= •1 (Fiii'1r1,,-

Fiii'Tl)

+ F8 { rL,lnrft, + r~ 1nrjn) - ...t-irl, {r;ilP + r;ig'P) (18.21)

Section 18.1

The Action lnteQral

futerchanging the dummy indices k hand side of Equation (18.21) gives

181 +:>-

j in lhe second term on the right-

~=a; (F
Htm (rlrJ;, + rJ;,rl)-Fs(rfnr;1/'P+ rfnr!jt'P) (18.22)

Now, interchanging all dummy indices in the third term on the right-hand side of Equation (18.22), we obtain

~=a, (F./'ir;,,-F.i/"'rf,,) +2-Fi.'"rf,,r'j, -FB(rtrki"'+ Iftr;!,,1")-F~

(rr,r:i -rtrtz) (18.23)

Using the symmetry of the Christoffel symbols of the second kind with respect to its two lower indices (9.9), the result (18.23) becomes

,J=iR = ai {F81'1 r~ - F~rt,,) + 2,J=gg1"'rl,rJ;, -

2-FBtmr~rJ,,, -Hi"' { rr,,r;i - r~rt,,)

FgR =a, (Fgg'irk,, +2,J=i,r1

(18.24)

F.F!"rt,,)

(rlrJ;,-r~rfn)-FKi"'{rr,,r;i -rtrJ,,,) (18.25)

Nowwenotethatthesecondtermontheright-handsideofEquation(l8.24) is twice the third term with a positive sign. Thus we obtain

FgR =a, ( Fgg'irk,, -

F..'"rt,,)

+Hi"'{rr,,r;1 -

r~,,rJn)-

(18.26)

The final result for the quantity ,J=gR can therefore be written in the form (18.27)

182

Chapter 16 Gravnatlonal Fleld Equations

where we define

(18.28) and

wi = ,J=g {s'<jrk,,- 8

1mrl).

(18.29)

Here we note that the quantity r depends only on the metric tensor 8kn and its first derivatives C1j8kn· while the terms containing the second-order derivatives of lhe metric tensor in lhe Ricci scalar R are collected into an expression that has the form of the divergence of a vector, i.e., iJ,;wi. Substituting (18.29) into (18.10), we obtain 3

l o = -c- - [

3

16:rrG n

c [ r.J=gdO--16:rrG

i;;i

a-widQ. 1

(18.30)

According to the Gauss lheorem, lhe second integral on lhe right-hand side of Equation (18.30) can be transfbrrred into an integral over ahypersurface surrounding the domain Q over which lhe integration is carried out in the first integral. Thus the variation of the second integral vanishes because of the variational principle, which requires that lhe variations of the fields at the limits of the domain Q are equal to zero. Consequently, we have 3 [ c Mc=---~

16:rrG

n

3 [ c RFgdD.=---~

16:rrG

n

rFgdD..

(18.31)

It should be noted that lhe variation of qie action (18.3 l) is a scalar invariant, although the integral

c' [ rFgdo.

---

16:rrG n

(18.32)

and the quantity r defined by (18.28) are not scalar invariants. Thus we have shown that the terms in the Ricci scalar R containing the second-order derivatives of the metric tensor do not contribute to the variations of the action integral (18.10) and that we may use lhe action integral (18.10) in the derivation of lhe gravitational field equations.

I1a.2 I

Action for Matter Fields

The variation of the total action (18.l) of the gravitational field in the Jl£esence of matter is given h}I

8J =8Jc+8JM =0.

(18.33)

Section 18.2 Action for Matter Fields

183

Thus we need to calculate the variations 810 and 8/M. In this section we consider the variation of the action integral of matter fields IM. i.e.•

8/M = !

c Jn{ 8 (../=ii£M)dQ,

81M = !

f Jf'/.

c

(18.34)

[a (../=iit:M) st"+ a(../=iit:M) s ( 8"')] Jn. ;;gkn a (aigkn) &.

(18.35)

'J

Integrating the second term in lhe integral (18.35) by parts and dropping lhe integral over the hyper.rurface boundary of the domain of integration n, following the same steps as in the case of the electromagnetic field, we obtain

Let us now define lhe energy-momentum tensor Tkn of lhe matter fields as follows:

a(../=ii£M) v-.,.,, -__a, [a (../=ii£M)Ja (a;•"') ag"' .

! =~ 2

(IR.37)

Substituting (1837) into (1836) we obtain

8/M

= __!_ I 2c

Usingheretheresult8 (gkngJn)

Tkn8,r1=8k/TljgnJ8,r1 =

-8lTfj8gnj

Jn

Tkn8gkn ..J=idO.

(18.38)

= 88; = 0, we may write = -gk1Tljgkn88nj

= -Tnj88nj =

-Tkn88kn·

(18.39)

Substituting (1839) into (18.38) we obtain

8/M

= _!_ I 2c

Jn

T""8gkn..J=id0.

(18.40)

Let us now calculate the energy-momentum tensor for the electromagnetic field with CM given by Equation (18.4). In the case of the electromagnetic field the Lagrangian £Mis a function of the metric tensor 8kn only and not of its derivatives ;;j8kn· Thus using (1837) we obtain lhe contravariant energy-momentum tensor in the form

T"'

= __2__ a(../=ii£M) = -2a£M _ _2__ a,f=g LAf. ,,;=g a8 .,, a8 .,, ...r=g a8 .,,

(18.41)

184

Chapter 16 Gravltatlonal Reid Equations

Using the result

2 a,,;=g 1 a8 d" I" ,J=gag;;=gag1m =g= '

(18.42)

weobtainfrom(18.41) (18.43) Using (18.4), we may write fJCM = _ __!_Fpqpil~(g ·g i) ag1m 4µo ag1m Pl q

= +__!__Fpqpli 2ok~ng 1 1 4µo

= +__!_pkqplng t = +_!__FkqP.'. 2µo q 2µo q

P

q

(18.44)

Substituting (18.4) and (18.44) into (18.43), we obtain

Tm= _.!._pk1p!I + _!__,rpjlp.11_ 1 µo

(lE.45)

4µo

Theresult(l8.45) is identical to the result (16.65) obtained by other means in lhe special theory of relativity with pseudo-Euclidean geometry_ As the next example, let us now consider a continuous distribution of noninteracting particles with a total mass m and a rest energy £o in a threedimensional domain of volume V. If we denote the mass density of the matter distribution by p, we may write

The action integral of the given matter distribution can be written in the form

IK = -me fsB ds

lsA

lK =

--C

=

-c

1v 1:JJ ~

f p dV fB J8j1dxidx1 lv JA

(18.47)

8jl~~.,;=gtIV dt.

(18.48)

Introducing here a system q-in. (18.49)

Section 16.2 Action for Matter Fields

185

the action intEgral (18A8) can be written in the form IK =

~c}g I £MKF8dO.,

£MK= -cJg;1¢i¢l_

(18.50)

Since £MK is the invariant Lagrangian density, we see from (18.50) that the system (J;Jn is a comravariant four-vector. The temporal component of the vector cl>n is proportional to the mass density p and is given by (18.51) The spatial components of the vector n are proport10na1 to the flux of mass through a unit of the boundary surface dll of the volume V in the

unit of time, i.e., (18.52) The definitions (18.51) and (18.52) are fully analogous to their respective

electromagnetic counterparts (17.59) and (17.60). In analogy with the electromagnetic result (17.63), the mass conservation law gives the following continuity Equation:

Dn"~ y-gaxn ~~(-Fi")~o.

(18.53)

In the three-dimensional vector notation the continuity Equation (18.53) has the form

%'i -r div (pV) = 0.

(18.54)

Integrating over the volume of the three-dimens10nal domain V and using the Gauss theorem, we obtain the intergal form of the mass conservation law:

_!!._ I dt

lv

pdV =

I pV-dfI_ Jn

(18.55)

Acconling to the mass conservation law (18.55), the negative increment of mass m within a three-dimensional volume Vis equal to the total mass flux through the bmmdary surface I1 of the volume Vin the unit of time. Using Equation (18.48), we may write F8£MK

=-pc~;;;;;;= -pc~.

(18.56)

186

Chapter 16 Gravttatk>nal Field Equations

Using (18-41) we obtain lhe energy-momentom tensor of the given mass distribution as follows: (18.57) In the pseudo-Euclidean case we note lhat the component correct result for the energy density,

pc'

Too=--.

R

T>0

gives the

(18.58)

and the components Tfu give the correct result for the momentum density, (18.59) For a weak gravitational field at low velocities, the following Umits can be used: 800< = 8a0 --+ 0,

8a/J --+ -f,aJJ,

(18.60)

F8 --+ -.fiOO.

(18.61)

and ds--+ -.fiOOc dt,

Substituting (18.61) into (18.57), we obtain (18.62) The energy-momentum tensor (18.62) can be considered as the kinetic

energy-momentom tensor, which does not include the contribution from the internal energy of the given maner distribution. In order to include the contribution from the internal energy of the matter distribution, we may use lheresultforthedifferential of the internal energy dU = -pdV, where p is the iressure within the matter distribution. Thus we obtain the action integral corresponding to the internal energy in the form lu=-

[ " Udt=-I[ CMU,J=idO., tA

C

(18.63)

Q

where the Lagrangian density CMu is given by

CMu=-~=p.

(18.64)

Section 18.2 Action for Matter Fields

187

As the Lagrangian density £Mu does not include any dependence on the metric tensor g1a,. we use the corurt:raint (18.65)

l -;;;;:;;; = 0 to create a Lagrangian function/

=

f

(uk ,A) wilh the constraint

/(if.A)=p~+J..(1-~)·

(18.66)

Using the methodofLagrangemultipliers, theequationsfortheparameter A and the variables if are given by

*=l-~=0=>~=1

(18.67)

and

~=

i

(p - A) ;;;;;;;2g1gu1

= 0 =>A =p.

(18.68)

Using (18.68) we obtain the suitable Lagrangian density forthe calculation of lhe energy-momentum tensor £MU= p

+p

(1

-J

811uiu1)

= p.

(18.69)

The contribution to the energy-momentum tensor from the Lagrangian density (18.69) is obtained using (18A3) as follows:

T<'iJ) ~ -2a£M -g'"£M ~p(,/,f' -F!"),

ag.,,

(18.70)

where we used (18.69) to calculate (J£M = _!p r;;;;;;;;uk,f' = _!_pif,f'. (Jgkn 2 'tjK1lUJW, 2

(18.71)

Putting together the results (18.62) and (18.70). we obtain the total energy-momentum tensO£ of a matter distribution in the form Tkn = T(~)

+ rml =

The result (18.72) -is

(p

+ pc2}lfU' -

grmp.

(18.72)

lhe most commonly used form of lhe energy-momentum tensor of a matter distribution in the general theory of relativity. It should be noted lhat the quantity T>0 is always positive, and in most practical calculations the contribution from lhe pressure terms can be neglected.

185

Chapter 18 GnMtatlonal Field Equations

Irn.3 I

Einstein Fieid Equations

In this section we consider the variation of the action integral Io in (18.33). which describes the gravitational field itself, and derive the gravitational field equations. Using the result (18.31) we have

Mc= _ _!!_8 f RFgdO._ 16.TrG Jn

(18.73)

In order to calculate lhe variation (18.73) we need to calculate

8 (,J=8R) ~ 8 (-F8.'"Rkn) ~8(,./=i)R+,J=88g"'R.,,+,J=8g'"8R.,,.

(18.74)

Using the results (9.35) and (9.40), we may write

(18.75) and we have

1

8(,.f=i)~ -1 -s 8 ~ - -(-88.,,sg"'). 2,Fg 2,Fg 8 (,.f=i) ~ - ..;;g.,,8g"'.

(18.76)

(18.77)

Subsritming 08.77) into (18.74), we obtain

8 (,J=8R)

~ Fg(Rkn - ~g.,,R) 8g"'+,J=8g"'8R;,,.

08.78)

Nex1 we calculare the expression glm8Rim as follows:

g1m8Rkn

= g1m8 (a,,rl +ajrl,+ rirJn - rf,.,rjj) =,r18 ( a,,rk- ajrfn) -g1msrr,,rjj -tmrfnt.rji + g1mf.r~rj,,, + ,r1rf}rtn + .{"r~}rt, - l,.r~_jjrt, (18.79)

where we have added the term

l,.r~}r~ - lT~srt, =

o

(18.80)

189

Section 18.3 Einstein Fleld Equations

at the end of Equation (18.78). Regrouping the terms, Equation (18.79) becomes

/«'t.Rim = /«'t. (anrk- a1 rl,) +ta' (r~t.rfp- r:irhz) + /«'rtt.rtn + /«'t.r~rjn -/«'rh,t.r:1 - g'«'r~}rfp (18.81) Rerunning the dummy indices, we obtain

g'«'t.R1m= /«'t. (anrfp-aprfn) +gkn {r!1t.rfp- r~ 1 8rhz) + g'«'r~8r~1 + g'«'8r~r~, - g'«'rf,,t.r!1 - /«'r!18rfp

(18.82)

/«'t.R1m=Ff«'t. (anrfp-aPrh,) +ta' (r!1t.rfp- r!1t.rfn) + 8 jnrft,t.r~k + gkjr;;,8rf,, - 81nr}nt.rfp - gkjr;nt.rfp. (18.83) Using the symmetry of the Christoffel symbols with respect to the lower two indices, we may rewrite (18.83) as

/«'t.R1m =Ff«'t. (anrfp-Bprfn) +ta' {r!18rfp-r!1t.rfn) -(-gjnr!,-gkjr;;,)t.rf,, + (- 8 1nr;-1<1 r;n)8r~. (18.84) From the results Dp/«'

= 0 and Dn/«' = 0, we have Bp/«' = -r_!,8 1n - r;;,g:1 antr' = -rj,.gftz - r;s'<j.

(18.85)

Substituting (18.85) into (18.84) we obtain

/"8R,,, ~ /"Bn (sr:;,)- /"ap(srf,,) -t.rf,,ap/" + t.rfpangM +ta' (r!18rfp- r~1 t.rfn) (18.86)

190

Chapter 18 Gravttatlonel Fleld Equations

/"8Rm ~a. (/"sr:;,)- ap (/"sr:;,) + r~, (/"~r~)- r!1 {/"~rfn)-

(18.87)

Interchanging rhe dummy indices k ++ p in the first and third term on the right-hand side of Equation (18.87), we obtain

/"8Rm ~ aP (/'sr;:,,-/"sr:;,) + r!, {g1'P ~r~ - Ff«' ~rr,,) .

(18.88)

Introducing here a four-vector wP,

wP=/'Pl!rZ.,-/«'lirf,.,. and u"ing

r!1

=

(18.89)

ap In A. the result (18.88) becomes

/"SR,,,~ -'-ap (.j=gwP).

A

(18.91)

Substituting (18.91) imo the result (18.78) we obtain

8(FgR) =A(R,,,-~g,,,R)sr+ap(NwP).

(18.92)

Substituting further (18.92) into (18.73) we obtain

Uc~ ---"'{ (Rm - ~g,,,R) 8/" .j=gdfl. I6rrG}n 2 - -c'- [ a (.j=gwP)dfl.. I6nG n

'P

(18.93)

The second integral in (18.93) can, by means of the Gauss theorem, be transformed into an integral of wP over the hypersurface surrounding the entire four-dimensional domain Q. When we vary the action Io, the varj ation of the second integral vanishes because of the variational principle, which requires that the variations of the fields at the limits of the domain Q be equal to zero_ Thus the second integral in (18.93) can be dropped and we obtain the final result for the variation of the action integral for the

191

Section 18.3 Einstein Fleld Equadons

gravitational field in the form 8/c

~ - l~G

k(R.,, - 4g.,,R) sg"' Adfl..

(18.94)

Using now the result (18.33), i.e., -Uc =UM with (18.38) and (18.94). we obtain

_c3__ f 16rrG]n

(Rm- 2~•.,,R)srAdn

~ _ _I_ f T.,,sg"'Adfl.. 2c Jn

(18.95)

Since the variations of the metric tensor are arbitrary, t:re result (18.95) gives Einstein field equations for the gravitational fields in the form 1

8rrG

Rkn-28knR= -7Tkn. The field Equations (18.96) with mixed tensors

(18.96)

J?! and T,,k are given by

8

R~ - ~O!R = - ~GT!.

(18.97)

Contracting Equation (18.97). we obtain

R~ - ~8~R=R-2R= -R= - 8 ~GT.

(18.98)

Substituting (18.98) into (18.96). we obtain an alternative form for the gravitational field equations: 8

Rkn=-

~G(Tkn-~gknT).

(18.99)

In empty space we have Tkn = 0. and Equations (18.99) give Rm = 0. The result Rkn = 0, however, does not mean that space-time i.s flat (ie., pseudo-Euclidean). The con4i_tion for space-time to be pseudo-E~lidean is that the curvature tensor Rf1,, is identically equal to zero, i.e., Rf1,, 0.

=

.,.. Chapter 19

Solutions of Field Equations

In the previous chapter we derived the gravitational field equations. The gravitational field equations are nonlinear equations, and they include the self~interaction, i.e., the interaction of the gravitational field with itself. The principle of superposition, which is valid for electromagnetic fields in the special theory of relativity, is therefore not valid for gravitational fields. However, in most practical cases we work with weak gravitational fields for which the field equations can be linearized and the principle of superposition ~ approximately valid. In thh chapter we discuss the nonrelativistic limit of gravitational field equations and the simplest solu~ tion of the complete gravitational field equations for the static spherically symmetric f:teld produced by a spherical ma~ M at rest, known as the Schwarzschild solution. The Schwd!ZM':hild solution has played a major role in the early development of the general theory of relativity and is still regarded as a solution of fundamental importance.

Irn.1 I

The Newton Law

In order to find the nomelativistic limit of the gravitational field Equations (18.96) or (18.99), we start with the definition of the Ricci

tensor (18.8), i.e.,

Rm= Bnrfj -

airl, + r~r;n -

r~r:j-

(19.1)

193

194

Chapter 19 SoluUons of Fleld Equatians

In the nomelativistic limit in the static gravitational field, with the approximate metric given by (17.37), the only nontrivial Equation (19.1) is the one with k = n = 0. Thus we calculate

For the static gravitational field with 8og1m = 0 we may use the approximation (17.26) with (17.38), such that we have

roo = --iti"apgoo =-~a"¢.

(19.3)

Substituting (19.3) into (19.2), we obtain (19.4)

Fr001 the result for the energy-momentum tenser of a matter distribution (18.72). in the static nonrelativistic case under consideration. we obtain

Timsweobtain Too -

~gooT = ~pc2 .

(19.6)

Substituting the results (19.4) and (19.6) into the gravitational field Equation (18.99) with k = n = 0, we obtain 1

Roo=~aaa

a

8rrG ( I ) 4rrG
(19.7) (19.8)

Thus we obtam the nomelativistic gravitational field equation for the gravitational potential

The soh1tion of Equation (19.9) is given by .p(i)=-G

pdV

1

-, v R

(19.10)

Section 19.2 TheSchwarzschildSokJllon

195

where R is the three-dimeru.ional distance between the position of the sourc~ and the position where the potential is calculated, i.e., R=

J1xa - X'Xl

2.

(19.11)

For a unifonn mass distribution over the volume V, we obtain


v

(19.12)

R

The result{l9.12) agrees with the result (17.41), which leads to the Newton law Of gravity (17.44). Thus the gravitational field Equations (18.96) or (18.99) have the correct nonrelativistic limit ( 19.9). which leads to the

Newton law of gravity.

I19.2

f

The Schwarzschild Solution

The Schwarzschild solution is a solution of the complete gravitational field equations for a static spherically symmetric field produced by a spherical mass Mat rest. The sratic condition requires that all the components of the metric tensor g1m be independent of x° or time t_ Furthermore, we must have gao = g0a = 0. The spherical symmetry su,ggests the choice of the spatial coordinates as the three-dimensional spherical coordinates (r, (}, q:i). The most general spherically symmetric metric satisfying these conditions can be wrillen in the form

di'-= £v{r)c2 dJ2 - e2). {r)dr 2 -

r 2 (de 2

+ sin2 Bd'l,2)

(19.13)

=

where v = v(r) and A A(r) are yet wispecified functions Of the radial coordinate r_ In the metric (19.13) we are using the exponential functions in order to secure the right signature of all team.. Thus the covariant metric tensor of a static and spherically symmetric gravitational field is given by 2

"00 --e 0

[gmn]=

[

OJ 0

e0 0

0

- r2

0

0

0

0

-r2 sin2 0

2 J..

(19.14)

-

The contravariant metric tensor is then given by

000 -r-2 sin- 2 e

l .

(19.15)

Chapter 19 SoluUons of Fleld EquaUons

196

The Chrjstoffel symbols of the first kind for the metric (19.13) can now be calculated using the definition (9.28), i.e.•

(19.16) The results for the Christoffel symbols of the 6rstkind for the metric ( 19.13) are summarized in the following list:

ro,oo = l
= ro,10 = !Ci3,-goo+ 8og10 -CJogo1J = v'(r)exp(2v) ro.zo = !caegoo + itoK:!.11 - CJog02) = o r o,o3 = ro,Jo = !
ro,01

ro,02

=

r1.oo = !CC\>810 + 8ogo1 - a.-goo) = -v'(r)exp(2v) r1.01 = r1.10 = !ca..-g10 + Bog11 - a.-gm) = o r1.02 = r1,20 = !
Section 19.2 The Schwerzschld Solutlon

rz.02 = r1,20

= 4
= r1.30 = !
197

r\lgzz

Bag02)

=O

r1,m

i1Qg32 - Bago3) = O

rz.11

Bagn)

r1,l3 = r1,31 = f(Bq£21

=0

BrK22 - Bag12)

= -r

+ Br832 - &lgn) = o

rz.:!!

= !WaK!! +Bag:n -

Bag22) = 0

rz,23

= r1,32 = !(Bqig!! +

Ba832 - Bag23)

ru3

= f(
r3.oo = fCBog30 + Bogm - Clq:igoo) r3,01 = r3.IO

= !CBr830 +

=o

lJog13 - Clq:igo1)

= !
=O

=0 =0

r3,02 = f\20

Clq:igo2)

r3,m =

Clq:ig03) = o

= fCCir83t + Br813 - Clq:ig11) = 0 = r3,21 = f
ru1

rJ,12

The Christoffel symbols of the t.eeond kind for the metric (19.13) can be calculated using the definition (9.29), ie.,

(19.18) The results for the Christoffel symbols of the second kind for the metric (19.13) are summarized in the following list:

r& = tf1ri.oo = g 00r o.oo = o r81 = rf0 =g0ir1.U1 = g 00 ro,m = v'(r) 00 0 r82 = r~0 = c iri!J2 = c ro.02 = o fi:3 = r~0 = g0iri,m = g00 ro,m = O

196

Chapter 19 SolutlonsotFieklEquallons

r~1 =

g

01

r1,11 = g 00ro.11

r~2

= r~1 = r~3 = r~ 1 =

=

o

g rj.12 = g 00 ro 12 = 01

g

01

ri,B

=

gOGro.13

o

=o

r~ = g01 r1 ;>;>, = g 00 r o.n = o = r~2 = g 0irj,2J = gu0ro,23 = O

rt

r~3

= g0ir1.33 = c00r 033 = o

rAo = g1irJ.oo = g 11 r1,oo

= v'(r)exp(2v -

2A)

rl1 = r:0 = g1irJ.01 = g11 r1.01 =0 rJ2 = rl0 =gUrJ.02 =g11 r1,02=0 rJ3 =rA0 =g•ir;.m =g 11 r1.m =0 r: 1 = g11ri.11 = g11 r 1,11 = A'(r) r:2=r~1 =g11 r1.12=g11 ru2=0

r/3 = rl 1 =g1ir;.n =g 11 r1.1J =0 rb = g11 r;.!2 =g11 r1,22 = -rexp(-U) rb = rlz = g11r;,23 = g11 r1,23 = o r13 =g11 r;.33 =g11 rt.33 = -rsin2 6exp(-2A) r~=0r;.oo =.i2r2.oo=0 r51 = ri0 = g 21 r1.01 = g22r2,01 = o r62 = ri0 = ,/ir,JJ2 = ?-r2.02 = o

r63 = rio =iir1.m =i2r2,m =0 rf1 =g21 r1 .11 =i2r2.11

=0

rfz = ri1 = g 21 r1.12 = rr2.12 = ~

rf3 = ri1= ifr;,B = ?rz.13 = o ri2 =

2

g ir;22

=Cr221 = o

rt = ri2 = g 2ir;,23 = ?rz23 = o ri3 =g2ir.1. 33 =i22 r2.33 = -sin19cose rbo =

3

g irJ.oo = g

33 r3,oo = o

199

Section 19.2 The Sc:hwarDdllld Solution

rfi1 = rio = g3ir1.01 = g33 r3,01 = o r&_ = rio =g31ri.02 =g33r1.1c =0 rij3 = rj0 =g3iriJJ3 =g33 r3.03 =0 ri1 =glir1.11 =g33 ru1 =0 r~ 2 = ri1 = g31rj.12 = g33rJ.12 =O rj3 = rj1 = g 3jrj,l3 = 8 33 r 3.n = ~ r~z = 8 31 r 12! = g3 3 r 32! = o r~ = rj2=g3iri23 =g33r3.23 =:~::=cote

3

rj3 = gll'ri.33 = g33 r3,33 = o.

(19.19)

From the results listed in (19.19) we can make the following conclusions:

rJo r~0] rio rj0 """"o rio rio rjo 0

[r;;ft] ~ [r$o] ~ [r/rio

[r~,

r~2

rr,] =O,

(r~]=[r~a]= r~1

r~2 r~3

r"

r~2 r~3

where (a, f)

(19.20)

(19.21)

= 1, 2, 3). Furthermore, we can calculate r~1

= r8o+ rJ1 +r6i + rfi3=O

r{i

= r~0 +r: 1 + ri2+ri3= v' +A'+~

r~1 = r~0 + r~ 1 + r~ + r~3 =cote

r~i = r~o+ r~1 + ri2 +rj3 =O. Using the results (19.22) we note that

(19.22)

200

Chapter 19 SolutionsofFieldEquadons

apr~j = 0

for

a =fa {J.

(19.24)

From the resuhs listed in (19.19) we also have

r:2 = rh =r~3 =0 => Brr!_s =O

for

r~z = ~· ri3 = ri3 = 0 => Cler!a = 0 ri2 = rj3 = ri3 = o =>


a~r!,s

O for

=

o

a -=f=.{J

for a-=/=for

a

-=f=.

fJ fJ

a =fa {J.

(19.25)

(1926)

The Ricci tensor for the metric (19.13) can be calculated using the definition (18.8). i.e., Rkn

= Bnrb- airfn + r;rtn -

rf:,,r~r

(19.27)

From the result (12.48) we see that the Ricci tensor is a symmetnc tensor Rm = Rnk and that it has only 10 independent components The formulae for the components of the Ricci tensor for the metric (19.13) are summarized in the following list:

Roo = aorl1 - airj0 + rgjr~ - rg 0 r~1 Rm =Rio= Brrlj- C1jrl1+ r~jr:1 - rg1r;j Rm = R20 = aerlj - a1rfn + rgjr~2 r~r;j R 03 = R 30 = a~rl; - a 1 rb 3 + rgir;3 - rg3 r;i Rn =Brrt-air{1+rfjrt1-rf1r;; R12 = R21 =Cler{; - a;r{2 + rfir;2 - rf2r:; R13 = R31 = a~r{j - ajr{3+ r~;r~3 - rf3rtj R12 = aar~j -
(19.28)

The results for all 10 individual components of the Ricci tensor for the metric (19.13) can be obtained from the list (19.28). However, the

Section 19.2 The Schwarzschild Solution

201

calculation ~s can be facilitated using some of the general results (19.20)-(19.26). Thus for a 1,2.3 we may use (19.27) to calculate

=

Rao= aor!1 -

air!0 + r!1 r~ - r:0r£r

(19.29)

a rl == 0. we obtain

Using the condition for the static metric 0

Rao =

-apr!0 - r20 rij -

r!0 r~1 + r!irk.

(19.30)

rtj = 0 from <19.20) and (19.22), respectively, the

Using r!0 = 0 and result (19.30) becomes

Rao= r!1 r~ = r~0 rg0 + r~rg0 + r!0 rp 0 + r£wr,B'0

(19.31)

Using rg 0 = 0, r!0 = 0. and r~p = 0 from (19.19), (19.20), and (19.21), respectively, we see that all fourtenns on the right-hand side of Equation (19.31) vani."h. Thus we obtain

Rao=R0a==O (a=l,2,3). Furthermore fora, fJ

(19.32)

= 1, 2, 3 and a f- {), we may use (19.27) to calculate

Rap = aprLi - a1r-!_e + r!1r£P - r~r~ . (19.33) From (19.24) and (19.26) we see that the first two terms on the right-hand side ofEquation (19.33) fora f- {)vanish. Using (19.22) we then obtain

1

Rap = r!1rtP - r~.sr~j- r!.sr{1 - r~.sr~j· r!_a = 0 for a 1- fJ from (19.21) and

Using r~.8 = 0 and respectively, we have

(19.34) (19.25),

(19.35)

Thus the second term on the nght-hand side of Equauon (19.35) is not equal to zero only for (a{J) = (12). Let us now calculate the first term on the right-hand side of Equation (19.35). as follows:

r:1 r~tJ = r:0r2P + r~wr:'ti· r!_,r~P = r~ 0 r2p + r:;0 r~JJ + r2wrO'p + r:;wr:w

(19.36)

(19.37)

From (19.19) we see that the only nonzero term of type r~ 0 or rgtJ is rg1 = v'(r). Thus for a f- fJ the first term on the right-hand side of

202

Chapter 19 Solutions of Fleld Equations

Equation (19.37) is zero. Furthennore, using (19.20) and (19.21). we see that the second and third term on the right-hand side of Equation (19.37) are also equal to zero. Thus we have

r~ir~tl = r~wr:;tJ = r!.,.rf,a + r~wr2'.s + r~wrJp =

r! 1 r:ti + r!2 riti + r!3 ri" + r~ 1 r~.s + r~ 2 r~.s + r~3 ri.s (19.38) + r~.r~.B + r~2 ri.a + r~ 3 r~ti·

Using the result 09.38) we can show by direct calculation that

r!jr~,, = 1J!1J~ri3 r~2 = f>!l>~

co;e.

(19.39)

Substituting (19.39l into (19.35) we finally obtain 11,,p~R,.=o

(19AO)

(af'fJ).

From the results (19.32) and (19.40) we conclude that all off-diagonal components of the Ricci tenwr are identically equal to zero. (19.41) Thus the only nontrivial components of the Ricci tem.or are the diagonal components. Using the static property of the metric Clorf,, :=: 0 and the results (19.20)-(19.26), they can be calculated as Roo

=-BrrJo - rJ0 r{i + 2rJ 0 r~0 = (-v" + v'A' -v'2 - ~v') exp(2v -2A)

Ru

=Brr{_,-arr: 1 -r: 1 r{i+(r~0 )2 +{r: 1)

2

+(ri2 }2 +(ri3 }2 =v"-v'A'+v 12 -~A' Cler£-arrb +2rhri2 + (ri3}2 -rhr{i = (1 + rv' - rA') exp(-2A) - 1 R33 = -arrh-aeri3 - r~ 3 rfi- ri3 r~_,+2r~ 3 ri3 + 2r~3 ri 3 = sin2 () [(1 +rv' - rA')exp(-2A)- I].

R:'.2

=

(19.42)

From the results (19.42) we see that R 33 = R 22 sin 2 e. As sin2 () is in general different from zero, substitution of these two components of the Ricci tensor into the vacuum gravitational field equations R1m = 0 gives the

Section 19.2 The Sd1warzschild Solution

203

same differential equation for v and A. Tirus the vacuum gravitational field equations R1m = 0 give only three independent equations for v and A. i.e.•

-v" + v'A' -v12 2

v" -v'A' +v' (1

+ rv' -

-

~v' =O ~A'= 0

rA')exp(-2A) =I.

(19.43)

Adding togetrer the first two equations in (19.43) gives

-~(v'+>')=O

=>

~(v+A)=O.

(19.44)

From (19.44)weseethatthequantity v+ A must be a constant, On the other hand, for truge values of radial coordinate r the space-time is approximately flat and both v and A tend to zero as r --J> oo. Thus the constant v +A must be equal to zero, ~d we have

v+J..=0 =>A= -v,

(19.45)

Substituting (19.45) into the third of Equations (19.43). we obtain

~ [rexp(2v)j =I.

(1+2rv')exp(2v) =

(19.46)

Integrating (19.46) we obtain

=r-

rexp(2v)

re,

(19.47)

where the integration coru.tant re is called the gravitational radius of the body, Tirus we obtaln

goo =exp(2v)

= 1-

~

gu = exp(2A) = (1-7)-I

(19.48)

The gravitational radius of the body is defined using the Newtonian

limit(l7.38) with(I9.12)as goo--+ 1+"!!!!. c1

= 1- 2GM_ c1r

(19.49)

Comparing Equation (19.49) with the first of Equations (19,48) we find re =

2GM

----;;'l'·

(19.50)

204

Chapter 19 SolutionsofFieldEquations

Thus the final result for the SchwarzM:hild spacHime metric is given by

ds

2

= (1- ~)c2 di1- (1- ~)--

1

dr

2

-Y1 (de 2 +sm 2 Bd~ 2 ). (19.51)

From the Schwarzschild solution (19.51) we conclude that the empty space outside a spherically symmetric distribution of matter can be described by a static metric. Furthermore, it should be noted that the solution (19.51) is also valid for moving masses as long as the motion Jttserves the required symmetry, e.g., a centrally symmetric pulsation. We note also that the metric (19.51) depends only on the total mass of the body that is the source of the gravitational field, just as in the case of the Newtonian theory. Ifwe put dt = 0 in the metric (19.51) we obtain the three--dimensional space with a line element:

As the Schwan~hild metric tensor gkn is not time dependent, the dis~ can be defined over a finite portion of space and the integral of the element of the spatial distance di along a space curve has a definite meaning. In other words it is ~ible to split tre space-time into the space and time with definite meaning. If we can tum the mass M of the source down to zero, the metric (19.51) reduces to the pseudo-Euclidean metric of the flat space-time in the spherical coordinates:

Turning the M on again, we introduce a distortion into both the fourdimensional space-time continuum and the three-dimensional space itself, and neither of them is flat any more. The level of distortion is proportional to the dimensionless quantity rc/r. In the flat space-time described by the metric (19.53) the radial coordinate r is the mea"ure of the radial distance from the origin of the coordinates. In the curved space-time it is no longer the cru>e and r is just a space coordinate that does not mea"ure the radial distance from the origin. From the lineelement(l9.52)weobtsin the square of the infinitesim31 radial distance dR for dB = drp = 0, as follows: (19.54)

205

Section 19.2 The Schwarza:hlc:I Solution

Thus we see that the actual radial distance increment dR is larger than the coordinate differential dr, i.e., dR > dr. The distance between two points r1 and r2 is then obtained as R11

=

i" (1- ~)-1!!

=

[Jr(r- re) +reln(.Jr+ .JT=TG)

"

dr

r

r:

> r2 - r1.

(19.55)

From the result (19.55) it can be shown that for small re/r the distance between the two points r 1 and r2 tends to r2 - r1. If we take the sphere in space with r =constant, then the three-dimensional space metric (19.52) becomes the metric of a two-dimensional sphere of radius r embedded in the Euclidean space, i.e., (19.56) The infinitesimal tangential distances are there fore the ~eas in Euclidean space: (19.57) Let us now consider the element of the proper time dr given by (17.5) as follows: dT

=

-/ifiidt=

vrrc 1 -

-;dt < dt.

(19.58)

The proper time between any two events occuring at the same point in space is then given by

[''Mc

(19.59) T= 1-~f
=

=

206

Chapter 19 SolutlonsofFleldEquattona

Nevertheless for a few bod~ in the universe that are actually smaller than their gravitational radius, some physically interesting things do happen at the boundary r = re. For example, matter and energy may fall into the region where r < re but neither matter nor energy (including the light signals) can escape from the region where r < re. Such a region is called a hlack hole and will be diM:u.SMrl in the next chapter.

~

Chapter 20

Applications of the Schwarzschild Metric

In the previous chapter we derived the static SchwaTZM:hild solution of the gravitational field equations for a static spherically symmetric field produced by a spherical mass M at rest. In this chapter we discuss hvo applications of the SchW3IZSChild solution to explain two physical phenomena that cannot be explained within the frameworlc of the classical Newtonian theory of gravitation.

/20.1

j

The Perihelion Advance

According to the Newtonian theory of gravitation. the orbit of a planet around the Sun is a closed ellipse with the Sun at one of the two foci. Thus the point, which is called the perihelion and where the planet is closest to the Sun, is fixed. However, experimental evidence shows that the perihelion of the planets is not fixed. but gradually rotates around the Sun. This rotation. although very slow, is cumulative and can be measured over a long period of time. In classical mechanics the Lagrangian of a planet in the central field of the Sun is given by

L=Et-mrJJ(r)=m1.! +GMm_ 2

(20.1)

r

207

208

Chapter 20 Appllcadons of the Schwarzschild Metric

The conse1ved energy of the planet is

E

=

BL t.f' - L

ava

= mv2 -

GMm

2

r

= Constant.

(20-2)

Fr(lll (20.1) we may write in the spherical coordinat~

L=m[~(r2 +r2 B 2 +r2 sin 2 B~2 )+

c;].

(20.3)

where the dots denote the time differentiation of coordinaies. Since the classical motion of the planet in the central field of the Sun is confined to the plane of the orbit, which we take to be the equatorial plane of the spherical coordinate~, we have 6 = n/2 and iJ = 0. From (20.3) we then obtain (20.4) The Lagrangian equation with respect to the angular variable rp is given by

'!_(~)=~ dt aq; arp·

(20.5)

Substituting (20.4) into (20.5) we obtain

~(nu2 ~) ~ 0, mr\O

= h =Constant

::=}

(20.6)

r 2 ri;

= ~·

where h is the conserved angular momentum of the planet. ing (20.7) into (20.2) in the polar coordinates, we obtain

E= '!2!_ [;2 + It - 2GM]. ,,,.-r2

r

(20.7) Substitut~

(20.8)

From (20.8) we may write ;2 _

-

"1!!_ _ _!j2_ + 2GM m

ni2r2

r

·

(20.9)

Dividing now ;2 from (20.9) by r 4 r;J from (20.7), we obtain

2_

('!'_)2

,Ad'P

=

~2 ('!:!__ - _±_ + 2GM) 2 2 h

m

m r

r'

(20.10)

Section 20.1

The Perihelion Advance

209

[£;(~)f =2:-(~)2 +2G~m2 (~)lntroducinghereanewvariable u

'1':)2 +,; = 2mE + 2GMm2 u ( dcp , h2 h2

2 '1':) = 2.mE h2 ( dcp

(20.11)

= l/r, we obtain (20.12)

2 2 4 2 M m -(GMm h4 h2

+G

_

u)' _

(20.13)

Introducing here the notation e2

2mE

G2 M 2 m4

pi=J?-r~h-,-.

p

GMm2

--,;2•

(20.14)

where

2Eli1-

h2

P= GMm2'

(20.15)

(~Y=~-(~-uf.

(20.16)

e=l+ G2_M2m3' we may rewrite (20.13) as follows:

The well-known solution of the differential Equation (20.16) is given by

u= l+ecoscp_ p

(20.17)

The validity of (20.17) is easily confirmed by direct substitution. Using (20.17) we can calculate (20.18) Substituting (20.18) into (20.16) and using cos2 cp + sin 2 cp = 1 we obtain the identity. Thus (20.17) is a correct solution of the differential Equation (20.16). Using here u = 1/r we finally obtain r ~ _ _P___

1 +ecoscp

(20.19)

210

Chapter 20 Applications of the Schwerzschlld Metric

This is the equation of an ellipse in polar coordinates. The length of the major axis of this ellipse. denoted by 2a. is given by

2a=r(~=0)+r(~=n)=

l:e

+ 1 ~e= 1 ~e2 .

(20.20)

Thus we may express the parameter p as

(20.21) Thus according to the Newtonian theory of gravitation the orbit of a planet around the Sun is indeed a closed ellipse with the Sun atone of the two foci, and the perihelion is fixed. Let us now consider the motion of the planet in the Schwarzschild space-time within the framewoik of the general theory Of relativity. The equations of motion are the geodesic Equations (11.25), ie.•

d2X' ds2

dxlw!

+radsds=O (k,l,n=0,1,2,3).

(20.22)

Using the nonzero Christoffel symbols of the second kind in the Schwarzschild metric (19.19), we obtain the four equations Of motion:

In Equations (20.23) the factor0f2 appears before each of the off-
Section 20.1 The Perihelion Advance

211

equations are reduced to:

Substituting the actual values for the Christoffel symbols of the second kind from (19.19) in the first and the third of Equations (20.24), we obtain

~-r2v'(r)~~=O

ds2 dsds d 2cp 2drd(f -+---~o, ds2 rds ds

(20.25)

exp(-2v)~ (exp(2v)~) ~0 ds ds _!_~ (r2':!!!..) =0. r 2 ds ds

(20.26)

Thus we obtain two constants of motion:

dt

exp2v~ r2

= (1-

'!!!.. =!!.__=Constant. ds

r>Jf

'G)dt I --; ds =~VI+;;;?.= Constant (20.27)

The fonn of the constants on the right-hand side of Equations (20.27) is chosen in such a way to secure the correct classical analogy. From Equations (20.27) we may write

(20.28)

212

Chapter 20 Applicatrons of the Schwarzschlld Metric

Now, using the result for the Schwav.schild metric (19.51) with()= rr/2 and() = 0, we have 1 ds =

(1 7)C1dr2 -(1- ~)

1

dr1 -T1d'P 2

(20.29)

Substituting (20.28) into (20.30) we obtain

2£) ( re)-' (l+mc2 re)-' (dr)' h - (1----,: ds - m2c2r2'

1= 1----;

2

re

1 - ---; = 1 +

2E

mC1 -

(20.31)

2 (dr)' ds - h ( re) m2c2r2

1 - ---; .

(20.32)

Using (20.27) we may calculate

'!!. _ 1 !!!._r2'!:!£_ - _!!_.!!..._ (') ds-f-'1.dcp ds- mcd

(20.33)

Substituting (20.33) imo (20.32) we obtain 2

2

h [ d (' )] m'c2 ~ ;

2

h

+ m2 c2 r2 ~

2E re me'+ 7

2

h rc

+ m2 c2'3-

(20.34)

Using here theresult(19.50)forrc andintroducinganewvanableu = 1/r, we obtain

(

du) dii

2

2mE

2GMm

2

3 +u2 =M+~u+rcu.

(20.36)

Section 20.1

ThePerihelionActvance

213

Using again the resuh (19.50) for re, we finally obtain

~) ( d'P

2

2

+u2 = 2m.E + 2GMm u+ 2GMu3. h2 h2 c2

(20.37 )

Comparing Equation (20.37) with the Newtonian Equation (20.12) we see that the only difference is an additional nonlinear term proportional to u3 that vanishes for c---+ oo. Th1s term can therefore be considered as a small perturbation of the Newtonian Equation (20.12) that iutroduces the generalrelativistic corrections to the Newtonian resuhs. Differentiating both sides of Equation (20.37) with respect to the angular variable 'f, we obtain

2~ (d2u d'P

d
+ u)

= 2!!!: (GMm2 + 'JGM u2). h2

d'f

c2

(20.38)

=

If we disregard the sol11tiondu/d'P 0, which represents the circular orbit with r = constant, we obtain the nonlinear differential equation for u in the fonn

d 2u

GMm2

3GM

2

dq}+u=~+~u

(2039)

The solution to the corresponding Newtonian linear equation, GMm 2

d 2 u<0J

diT +

--,;z-•

U(O) =

(20.40)

is given by Equation (20.17), i.e.,

1+ecosq:i_ p

u
(20.41)

Using the perturbation method, we may construct an approximate sol11tion of the nonlinear Equation (20.39) in the form u

= u
u<1),

u(l)

«

u<0 ).

(20.42)

Using (20.42) with (20.41) we may write d 2u d'f 2

d 2u
d 2u
d 2 u<1)

e

= d;;T +-;J;pl = d;;F- pcosq:i,

(20.43)

~cosq:i+~-

(20.44)

and u

=u<0>+ull)

= u(IJ

+

214

ApplcallonsoflheSchwarzschlldMetrlc

Chapter 20

Combining the results (20.43) and (20.44) and using the defimtion (20.14) we obtain

d 2u d2u
(20.45)

Subst1tutmg the resuh (20.45) into Equation (20.39) and puttmg u in the nonlinear term, we obtain the equation for the pertwbation the form

d2u~> + u(I) = 3G:1 (u(0))2. d
= u
u<0

in

(20.46)

c

Substituting the result (20.41) into (20.46) we obtain d 2 ul 1)

~+u

d 2 u
~

(1) _

+u

3GM

2

- c 2 p 2 (l+ecoscp),

en _

3GM

- c 2p 2

+

6GM 3GM 2 2 c 2p 2 e cos cp + c 2p 2 e cos 'I'·

(20.47)

(20.48)

Considering oJbits with small eccentricity e the term of the order e 2 can be neglected. Thecontributionfromtheconstanttermisnegligibleas well. The only term that produces an observable effect is the one proportional to cos cp with the contribution which increases contirruously after each revolution. Dropping the constant term and the tenn proportional to e 2, we obtain

dd
(20.49)

The solution of differential Equation (20.49) is given by

(20.50) which is easily shown by direct subsbtubon mto (20.49). Substitut ing (20.41) and (20.50) into (20.42), we obtain l+ecOS(/'

U = --p--

+

3GMe . c 2p p
(20.51)

Let us now introduce the increment 8.'P as follows:

3GM

6.'f

= 2i}'P =

3GM c2a(1 - e2) 'f,

(20.52)

215

Section 20.2 Black Holes

where we have used the resuh (20.21) for the parameter p. Thus we may rewrite (20.51) as follows: 1 +ecos(/' +e l:!..'fsinq:i p

(20.53)

Using the trigonometric formula cos('!'- l:!..'f) = cosq:icosl:!..q:i-rsin'Psinl:!..'f

(20.54)

and the approximations for the trigonometric functions of a small angle l:!..'f, given by cos l:!..'f ~ 1,

sin 6.q:i

~

l:!..'f,

(20.55)

we obtain COS'f-+- L\'fsinl:!..'f ~ cos(q:i- l:!..'f)-

(20.56)

Substituting (20.56) into (20.53), we finally obtain 1+ecos('P-8.'f) p

(20.57)

From Equation (20.57) we see that while a planet moves through an angleq.i, the perihelion advances by a fraction of the revolution angle equal to (20.58) For a complete revolution (IP= 2rr) the perihelion advances by an angle 6rrGM 6.q:i= c 2a.(l-e2 ).

(20.59)

For the planet Mercury the theoretical result for the relativistic perihelion shift is equal to l:!..'f = 43.03 seconds of arc per century, while the observed perihelion shift is equal to l:!..'f = 43.11 ± 0.45 seconds of arc per century. The perihelion shift of the planet Mercury could not be explained in the Newtonian theory of gravitation. This remarkable agreement of the theoretical result from the general theory of relativity with the observational dsta was the first major experimental confirmation of the theory.

120.21

Black Holes

In the previous chapter we mentioned the singularity of the Schwanschild metric (19.51) at the gravitational radius of the spherical body r = re and the possibility of the existence of bodies in the universe that are actually

216

Appllcationsolthe~Metrlc

Chapter 20

smaller than their gravitational radius ra. The region where r < ro around such bodies is called a bla(* hole. A number of physically interesting phenomena ocCUT at the boundary r = re. For example, we have argued that matter and ener,gy may fall into the region r < re of a black hole, but neither matter nor energy (including the light signals) can escape from that region. [n order to prove this assertion, let us consider a particle falling radially into a black hole with a radial velocity u 1 = dr /ds .As the particle is fulling radially, we have u2 = u3 = 0. The motion of the particle is described by geodesic Equations (11.25), ie., (20.60) Using u2 = u 3 = 0 and the results for the Christoffel symbols of the second kind (19.19), we may write the temporal equation of motion as follows

':!.!!!_ ds

=

du0

-r~u'if: = -2r~0u 1 u0 , dvdr

dv

o

ds = -2-;J;d;u0 = -2dsu ,

(20.61)

(20.62)

where the factor Of 2 appears because the term propot11onal to r~o = appear twice in the sum. Thus we obtain duo

ds

+2~u0 =exp(-2v)~(exp(2v)u0 )=0. ds

ds

rgl

(20.63)

Equation (20.63) can be integrated to give exp (2v)u0 = goou0 = K =Constant,

(20.64)

where K is the integration constant that is equal to the value of goo at the point where the particle starts to fall toward the black hole. Using the identity gknukil' = 1, we may also write

1 = gkn,}un = goo(u0 ) 2

+ 8ll(u1) 2 .

(20.65)

Multiplymg by goo and using (20.64)as well asgoog11 = -1, weobta.J.n

goo~ (goof'(.Pi'

+ goog11(u1i' ~ K2 -

(u1)2 =K2-goo = K2-1

+

7-

1 (u )',

(20.66)

(20.67)

Section 20.2 Ellacl
217

For the radially falling body we have u 1 < 0, and we may write

u1 =-(K2-

t-r7)I/2.

(20.68)

Using (20.64) and (20.68) we may calculate the ratio dt/dr as follows:

'!!_

dr

= !!!___1 = -~ (K2 -1 + ~)- 112 , cu

cgoo

~ = -~ (1

(20.69)

r

-7)-I (K2 -1+7)-1/2.

(20.70)

If we now assume that the particle falling radially into a black hole is close to the gravitational radius re, then we may write r = re+ E with E « re. Equation (20.70) then becomes

Using the approximation

( E)-I 1+re

E

~1--,

(20.72)

re

in Equation (20.71 ), we obtain

!"!!.. ~ dr

dt

Kre

[xi - ~]-112 ~ -~. re

CE

dr ~ -

(20.73)

CE

re c(r-re)°

(20.74)

Integrating (20.74) we obtain t ~-rein

(r- re)+ Constant.

(20.75)

From Equation(20.75) we note that as r--> re we have t---+ oo. Let us now consider an observer traveling with the particle. The proper time measured

218

Chapter 20 Appllcatlons of the Schwarzschlld Metric

by the observer in its own rest frame is given by (17.5), ie., I

= ;ds =

dr

,JgOOdt.

(20.76)

Thus we may write Ids 1 ;J;.dr = ~r,

(20.77)

= -~ (x2 -1 + ~)- ' .

(20.78)

dr =

1 ;ds =

1

dr

2

If we again assume that the observer traveling radially into a black hole is very close to the gravitational radius r0 , then we have r = rG + E with dr = dE. Equation (20.78) then becomes

de ( e dr=-- K 2- - -

c

+1:

rt;

de ( K 2 - e

dr~-c

)-l/2 .

)-l/2

re

(20.79)

(20.80)

Integrating from the starting point E to the point E = 0 where the obSttVeT reaches the gravitational radius re. we obtain r

~ _ _!_

1" (K2 - 2._)-1/2

c "

= 2rcK c

(i _~)'''I" 2 K rc

-

2 rcK [1 -J1 -

~

i=

-c

r

dE

re

r ~:G

E

"'

E

J

K 2rc'

« re.

(20.81)

(20.82)

From the result (20.82) we conclude that the observer reaches the point r re after the lapse of a finite proper time according to his own clock.

=

219

Section 20.2 Black Holes

Thus the singularity atr = rG is not areal unphysicalsmgularity, but merely a coordinate singularity that is a consequence of the choice of the coordi nate system. Let us now assume that, during the radial fall, the observer is sending light signals to a distant counterpart at precisely regular time intervals according to his own clock. Using the definition of the differential of the proper time (17.5), the cbfferential of the coordinate time is given by

dt

= _!!!_ = dr (1 - ~)-1/ , 2

,J8iii

(20.83)

r

and from the point of view of the distant receiver the light signals are red-shifted by a factor proportional to 120.84)

Thus, although the traveling observer passes the point r = rG after the lapse of a finite proper time, from the point of view of the distant receiver the time intervals between light signals become longer and longer as the traveling observer approaches the point r = re. The distant receiver never sees the traveling observer after his passing below the point r re, which is in line with the initial assertion that neither matter nor energy (including light signals) can escape from the region with r < re. The lack of communicahon with the outside world is a basic property of black holes. The boundary r =re is called an event honzon. Just as the curved Earth creates a horizon limiting the range of vision of an ocean navigator, the strongly curved geometry in the vicinity of a black hole creates an event horizon hiding tre space-tune of the interior of the region with r :::; re from the external observer. Nevertheless, the black holes do exert influence on the external observers by their gravitatJ.onal effects on external bodies arising from the Schwarzscluld metric for r > rG. Another interesting feature of the black hdes is that inside the regron r
=

= =

wl = c 2dr2= c1 (1 - ~)dt1 < o.

(20.85)

At this point we want to examine how bodies move inside a black hole. In particular it is ofinterest to find out if they are moving toward or away from the central mass. We first note that by virtue of the principle of invariance of the speed oflight. the world line elementds of a bght ray is always equal

220

Chapter 20 ApplcaliollB of the Schwa~hlld Metric

to zero. The equation ds = 0 defines the two light cones. As an tllustration consider a pseudo-Euclidean metnc described by Descartes coordmates, where dy = dz = 0. The light cones are then defined by (20.86)

XQ-XP =±c(tQ-tp).

(2087)

Equation (20.87) 1s the equation of the cross section of two cones with the y-plane or the z-plane. All world lines of a light ray in four-dimensional space-llme, passing through an arbitrary world point P, must lie on these two light cones where ds 0. All other permissible world lines of arbitrary bodies with d? > 0 must lie within the two light cones, since otherwise tre slope of the actual world line would be larger than c, indicating a body moving faster than the speed of light with d? < 0. All world lines within one of the two cones point into the future compared to the world point P, while all world lines in the other rwo cones point into the past compared to the worldpointP. As we have argued before, if we want toexplam how bodies are moving inside a black hole, the usual set of four coordinates (t, r, (),IP) is not adequate and we need to introduce a more suitable new set of coordinates. The simplest way to achieve this goal is to keep the three spatial coordinates and just introduce a new time coordinate w, defined by

=

Ir I

YG W=t+-ln --1

C

(20.88)

rG

From (20.88) we obtam

dt=dw-~(1-7)- dr.

1

(20.89)

The square of the onginal coonilnate time differenttal dt as a function of the new coordinate timed1fferent1aldw defined by Equation (20.89) is then given by

di1 =dw2 -

2 ::

(1-7)-I dwdr

'b (1--;:'G)-2dr.2 + c2r2

(20.90)

Section 202 Black Holes

221

On the other hand for the light cones of a radial motion, where we have d() = drp O. the resuh for the metric gives

=

ds

2

dt2

= (1 - ~) c2dt 2 ~ (1-7)-

(1-

~)-I tJr2 = 0,

(20.91)

2

=

d?.

(20.92)

Substituting (20.92) into (20.90), we can remove the dependency on and obtain a quadratic equation in dw/ dr as follows:

(a,,_.)' _ 2roer
dt2

<20·93l

The rwo solutions of the quadratic Equation (20.93) are given by

~=~(7±1)(1-~)- •

(20.94)

dw dr

(20.95)

1

The first of Equations (20.95) shows that outside the black hole for r > re, some world lines have a decreasing coordinate distance r with increasing coordinate time w, i.e., the light rays move toward the central mass. The second of Equations (20.95) for r > r 0 shows that some world lines have iocreasing r with increasing coordinate time w, i.e., the light rays move away from the central mass On the other hand, inside the black hole for r < re both Equations (20.95) show that all world Imes have decreasing coordinate distance r with mcreasing coordinate time w, i.e., the light rays always move toward the central mass. Based on the earlier conclusion that all other permissible world lines of arbitrary bodies with ds2 > 0 must lie within the two light cones, we see that all matter and energy (includmg tre light signals) within a black hole for r < re can only move toward the central ma'IS and can never escape from the region where r < re. The preceding discussion of the properties of black holes would be only academic unless there were reasons to believe that such objects exist in the universe. It is generully believed today that black holes do exist and that they are created by gravitational collapse in the final stage of the evolution of massive stars with a mass greater than I 0 times the solar mass. This type of black hole is expected to have a mass in the range of

222

Chapter 20

ApplicaUons of the Schwarzschlld Metric

two to three solar masses. Some stellar objects that may be candidates for

this type of black hole have already been studied. Super-massive black holes comprising thousan:ls, millions. or billions of solar masses may also exist. It has also been suggested that in the early stages of the creation of the universe, because of the high density of the hot matter, some small material

oi!iects could have been squeezed sufficiently to form the so-called mini black holes.

Part V

Elements of Cosmology

~

Chapter 21

The Robertson-Walker Metric

In the previous two chapters we have shown that the general theory of relativity provides the solutions for the space-time structure created by any given matter distribution. Thus, if we can specify the average disbibution of matter in the entire universe, the general theory of relativity provides the solution for the average space-time structure of the entire universe. The study of such a solution for the average space-time structure of the entire muverse 1s a pare of the subjec1 of cosmology.

I21.1 I

Introduction and Basic Observations

In the present chapter we study phenomena on a cosmological scale. For that purpose we need to develop a suitable model of the universe and to make suitable assumptions about the physical processes that are dominant on the cosmological scale. The two basic observations that allow us to address the large-scale structure of the universe are the expansion of the universe (the Hubble law) and the cosmic microwave background radiation. The first basic observation that allows us to address the large-scale structure of the universe is the discovery that the spectral lines of distant galaxies are shifted toward the red end of the spectrum. If we inteipret tlus red shift as a Dappier shift, this observation leads to the conclusion that all the distant galaxies are receding from us. Thus we conclude that the universe as a whole is expanding. This expansion implies a finite age of the Wliverse. In other words. by observing the sky we can only see stars

225

226

Chapter 21

The Robertson-Walker Metric

that are close enough for the radiation originating from them to reach us in such a finite time. It has been shown by Hubble that the velocity of the distant galaxies v increases linearly with their distance r. This proportionality is known as the Hubble law. Using simple Euclidean geometry, the Hubble law can be fonnulated as follows:

v=Hr (H=(55±7)~_2_), 'Mpc

(2Ll)

where the quantity His called the Hubble constant. The Hubble constant is a constant inasensethatitdoesnotdependon themagnitudeorthedirection of the vector r. However, it may depend on time, and the trumerical value given in (21.1) is its present-time value. According to the Hubble law the universe is in uniform expansion, which means that there are no privileged positions in the lllllverse and that an observer traveling with any galaxy sees the surrounding galaxies as receding from him. If we observe from Earth two different distant galaxies G 1 and (h_, then according to the Hubble law their respective velocities observed from our galaxy are given by (21.2)

The relative velocity of the galaxy G1 as observed from the galaxy (h_, denoted by V. is then obtained as follows:

V = V1 -V2 = H =HR,

c21.J)

where R = i 1 - 12 is the relative distance between the two galaxies Ci and (h_. Thus an observer traveling with galaxy Ch. sees galaxy Gi, and indeed any other galaxy, as receding from him. Although we have used nonrelativistic approximation in Euclidean geometry and ignored the time dependence of the Hubble constant, these general conclusions are nonetheless valid, i.e., the obsener traveling with each galaxy sees all the other distant galaxies as receding from him. Since the distant galaxies are receding from us with a velocity directly proportional to their distance, there must be a point at which each of them will approach thespeedoflightand therelativisticeffects will become dominant. The set of all those points is called the world horizon. The radius of the world horizon is approximately equal to 'WH = c/H = 2 x lOJU lightyears, if the present-day value of the Hubble constant is used. The galaxies that are located at or beyond the world horizon are invisible to us. The second basic observation that allows us to address the large-scale structure of the universe is the discovery of the extremely isotropic cosmic 1nicrowave background radiation. This radiation has the same intensity in all directions with a precision better than one part in a thousand. Regardless of the origin of the radiation, this observation is convincing evidence for the

Section 21.2 Metric Definition and Properties

227

assumption that any acceptable model of the universe has to bean isotropic model. Another interesting feature of the microwave background radiation is its relatively high energy density. It is the .component of diffuse radiation in the universe with by far the highest energy density. The universe today is matter dominated. but there are reasons to believe that the universe was radiation dominated in the early phases of its history and that the observed microwave backgroun:l radiation originates from these early phases of the history of the universe.

121.2 I Metric Definition and Properties The basic observations described in the previons section indicate that the large-scale structure of the universe is both homogeneous and isotropic. This assumption is often called the cosmological principle. We imagine that the matter in the universe is on the large-scale evenly distributed in the form of a cosmic fluid, with no shear-viscous, bulk-viscous, or heatconductive features. This is a good approximation of the actual univw;e as long as we take the large-scale point of view. Clearly, on a smaller scale, the matter in the universe is unevenly distributed and the universe is highly nonhomogeneous. We denote an observer at rest with respect to the cosmic fluid as a fimdamental observer. AE the universe expands, the cosmic fluid takes part in the expansion and the fundamental observer is co-moving with the fluid. Every co-moving observer in the cosmic fluid sees the same isotropic and homogeneous image of the univen>e. The objective of the present section is to define the space-time metric in the co-moving frame of reference. We cannot use the static Schwarzschild metric to describe the expanding universe, which is certainly nm static. We need a nonstatic, homogeneous, and isotropic metric to describe the entire universe. Let us take the proper time measured by the fundamental observer co-moving with the cosmic fluid as the time coordinate, which we here denote by w. In such a case the space-time metric has the following gener.tl form:

d
(21.4)

Applying the condition of isotropy and spherical symmerry to the metric (21.4), we obtain

d,<,2

= c?-t!v.J. -

D(r, w)m-2 - cE(r, w)dnl"'

- F(r, w)(d0 1 + sin2 Odqi).

(21.S)

228

Chapter 21

The Robart9on-Walke Metric

In Equation (21.5) it should be noted that we have chosen the dimensionless radial coordinate r = Rf L, where R is the usual radial coordinate with the dimension of length and L is a yet-unspecified reference length of the universe. ThefunctionsD(r, w) andF(r, w)thereforehavethedimensionof the square of length and the function E(r, w) has the dimension of length to secure the proper dimension of the metric (21.5). Let us now use the geodesic equations of a particle given by (11.32), in the form

~ = _!_ Og1~i/J'. ds

(2L6)

2 OxJ

For a particle at rest in the co-moving frame of reference we have

u0 =t,

rfX=O

(a=l.2.3).

Subsumting(21.7) into(21.6)andusinggoo

=

~ = ~(gjlli) = fs (8J1JUO) = ~dgjo =_!_ Og~ =O. ds dw

2

(21.7)

c 2 , we obtain

(JxJ

(21.8)

Thusweobta.J.n

dg;o = 0=} !!_E(r. w) dw dw

= 0.

(2L9)

From (21.9) we conclude that E = E(r) is not a function of time coordinate w, and the metric (21.5) becomes

d,<,2

=

c 1dW1- - D(r, w)dr2 - cE(r)drdw - F(r, w)(d0 2 + sin2 Odq}).

(21.10)

The tenn proportional todrdw can be eliminated by introducing a new time coordinate: t =

w-

If'

2c

E(r)dr=?dw

I = dt + 2cE(r)dr.

(21.11)

Using (21.11), we may write

+ cE(r)drdt + ~ [E(r)f dr1,

(21.12)

= -cE(r)drdt- ~ [E(r)fdr2.

(21.13)

c?dw1 = C1di1 and -cE(r)drdw

229

Section 21.2 Metric Definition and Properties

Substituting (21.12) and(21.13) into (21.10), we obtain ds1

= C1dt 1 -A(r,t)dr2 -B(r,1) (de 2 +sin2 0dqi2)

(21.14)

where 8ll =A{r,t) =Dfr,w(r,t)l+ g22

~ [E(r)]2

(21.15)

= B(r, 1) = F[ r, w(r, 1)1.

The functions A(r, t) and B(r, t) defined by (21.15) have the d1mens10n of the square of length to secure the proper dimension of the metric (21.14). In order to further specify the functions A(r, 1) and B(r, 1), we now apply the condition of homogeneity of space-time and consider the following coordinate transfonnation: ~=~+ca(r,8,qi)

(a=l,Z,3).

(21.16)

Using the imtial assumption that c 0 == 0 and Equation (21.16) we obtain Bt' BxfJ = sp

Bea

+ Bxfi'

Bz0 Bxfi

= 0.

{21.17)

The gener.il transfonnation law for the covariant metric tensor, defined with respect to the systems of coordinates [xk] and {f] by 8kn and gj/, respectively, is given by 8kn =

ad az1 _

8,J a;;n811·

(21.18)

Using Equations (21.17) and (21.18) we may write 8af3

~:~::.ii.a·

(21.19)

Substituting (21.17) into (21.19) we obtain 8aft =

(s! + :;:) (sp + :::) Cvu

(21.20)

Expanding the expression (21.20) and dropping the term quadratic in the derivatives of the small translational parameters ca. we obtain 8aft(x1°)

=Capt~)+ :;8vp(/) + :;;8va(z'').

(21.21)

The homogeneity of space-time now requires that the metric be mvariant with respect to the transformation (21.16), i.e., that we have

ii.pt!)~

g.p<.t')

(a,{!~ 1,2,3).

(21.22)

230

Chapter 21

The Robertson-Walker Metric

Substituting (21.22)into(21.21), we obtain

On the other hand, expandmg 8uJJ (.::") into the Taylor senes, we may write

Substituting (21.24) into (21.23) and dropping the tenns quadrabc m the small translational µuameters ca and their derivatives, we see that only the zeroth-order term of (21.24) contributes to the first-order Equation (21.23) Tirusweobtain

Using agam the homogeneity condition (21.22) and comparing Equations (21.26) and (21.24) with each other, we obtain

B::: Fma = f3 becomes

Ev+

:~8v,B(x") + ::;8va(>!) =

0.

(21.27)

= landusingthemetric(2Ll4)with(2I.I5), Equation(21.27) (21.28)

For a = 1 and f3 = 2 and using metric (21.14) with (21.15), Equation (21.27) becomes (21.29)

= = 2 and using the metric (21.14) with (21.15), Equation

Finally for a f3 (21.27) become.ct

1 +2~B=0. ~e Br ae

(21.30)

Section 21.2 Metric Definition e.nd Propertlet.

231

From the differential Equations (21.29) and (21.30) it is possible to eliminate e 2. We first rewrite (21.29) and (21.30) as

Be1 Br

Be 1 A Be B

From (21.31) we may write

B1e1 B1e• A - Brae= 00 2

B

2 2

2

B e B 1 2) 1 Brae= (1nB- f e

a,:z

ae' +a;a (1nB-112) ---a;·

(21.32)

Addition of the two Equations (21.32) gives

a2el ~+ ~ (1nB-1/2)e• BfP B Br1

+ ~ (inB-112) ~ = 0. Br

Br

(21.33)

Now, using Equations (21.28) and (21.30), we may also wnte

1 BA

B

2 Be 1

1BB

B

2Bt=: -;rae·

Aar=a;.OnA)=-;ra,.-

BBr =a; onB) =

1

(2L34l

As the small translation parameters t=:a are by definition independent of the time coordinate t, we conclude that Br(lnA) and Br(lnB) are not the functions of the time coordinate t, either. But the functions A(r,t) and B(r,t) are dependent on the time coordinate t. This is only possible if the functionsA(r,t) and B(r,t) are factcriz.ed as follows:

A(r,t)

2

= a(r)R (t),

B(r,t)

=

b(r)R 2(t),

(21.35)

where R(t) is some yet 1mspecified function of the time coordinate t that has the dimen.~ion of length. Titis function in some sense plays the role of the radius of the universe. The function R(t) should not be confu.'«!d with the Ricci scalar R used in the gravitational field equations, and whenever there is a risk ofconfusion the distinction between the two will be explicitly stated. On the other hand, we note that the dimensionless radial coordinate r is not uniquely defined. 'lbe metric (21.14) ts invariant with respect to the transformation from the radial coordinate r to some new radial coordinate p,

232

Chapter 21

The Robertson-Walker Metric

defined by r=

JP

(21.36)

F(p)dp-=*dr=F(p)dp,

where F(p) is some arbitrary function of the new radial coordinate p. Indeed, if we substitute (21.36) into the metric (21. I 4). we obtain ds1

= c1dt2 -A.(p,t)dp2 -iJ(p,t) (dtP + sin2 8d~,:i2)

(21.37)

where

[JP F(p)dp.t] [F(p)f 8ll =B(p,t) = B [JP F(p)dp,i]gn =A(p,t) =A

(2L38)

The metric (21.37) has the same form as the metric (2Ll4) and they are fully equivalent to each other. 'Ibus we are free to make an arbitrary choice of the form of one of the functions a(r) m b(r). Hwe set b(r) r 2, the metric (21.14) becomes

=

ds1

=

2 1 1 c 2dt - R (t)a(r)dr -Jil(t)r2

(de 2 + sin1 8d~l).

(2L39)

From the metric (21.39) we see that the surface area of a sphere with the relative radius r is equal to 4irr2R 2(t). H, at a conveniently chosen time instant t, we have R(t) = L, then the surface area of a sphere with the relative radius r = R/L is equal to 4nR2 and the angular coordinates() and 'P become the usual spherical angular coordinates. In order to specify the function a a(r), we use Equation (21.28) and the factorized form of the functionA(r,t) given by (21.35) to obtain

=

~~~=~~K4=~~'4

m~

'The solution of the differential Equation (21.40) can be written as (21.41) Substituting the factorized form of the functions A(r, t) and B(r, t), given by (21.35) with b(r) = r 2, into Equation (21.33), we obtain B2e1 !!_

Be2

r2

= !:__ (lnr)El + ~ (lnr) Be1' Br2

Br

Br

(21.42)

Section 21.2 Metric Definition and Properties

a2e

1

fJ()2

233

= ~a (!~ __!_e1) = ~~ (~). rfJr r2 afJr r

(21.43)

Substituting the solution (21.41) into (21.43) we further obtain

2 8(')

2

18 e r ;8ff1.=a112B;:

ral/2

=C

(21.44)

The expression on the right-hand side of Equation (21.44) is a function of the angular variables (} and

~=Ce.

(21.45)

ae2

H the translation (2L 16) is chosen to be along the polar axis (z-axis) in the corre.~nding Descartes coordinates, we have e(O,
,2

a(

a•128r

I

)

ral/2

=-L

(21.46)

The solution of the ilifferent1al Equation (21.46) is I

a(r)

= l-kr2·

(21.47)

It is easy to verify by direct calculation that (2L47) satisfies the differential Equation (21.46):

~a

2 1rv'l-kr Br

(J•-k'") ---

r

=-L

(21.48)

Substituting (2L47) into (21.39) we obtain the final result for the Robenson-Walker Metric in the form

dil = c 2dt1 - R 2 (t)

(___!!i!___ + r2de 2 -+- r 2 sin2 (}dq}) l -kr1

(21.49)

where R(t) is a dynamic function of the time coordinate t with the dimension of length, which will be calculated as a solution to the gravitational field equations, and k is the so-called curvature constant, which describes the geometry of the three-dimensional space at any particular time instant. The positive values of the curvature constant (k > 0) correspond to the

Chapter 21

234

The Robertson-Walker Metric

tbtee--
cdt=R(t)da.

da

1

=

dr 1 l -kr1 +r2d(J1+r1-sin2 ()dql

(21.50)

where da 1 is the metric of the three-dimensional space for a fixed value of the time coordinate! that is usually called the cosmic time. The cosmic time is a universal time equal for all observers at rest with respect to the local matter. Using Equation (21.50) we may write cl:i..t = R(t)l:i..a for finite proper distances. All mea.
~ The Hubble Law In the present section we discus.11 the Hubble law in the framework of the Robertson-Walker geometry. It has been concluded before that each galaxy has similar coordinates (r, (),
Section 21 4

The Cosmologlcal Red Shifts

235

'The integral (2L51) is elementary and gives

D,=

{

R(t)arcsinr R(t)' R(t) arcsinh r

(k

= 1)

(k=O) (k

(2L52)

= -1)

where it should be noted that r = 'R/L is the dimensionless radial coordinate. Thus the proper distance is proportmnal to the scaleradiusR(t), which changes with time. Keeping in mind that the radial coordinate r is a fixed co-moving coordinate, the proper velocity is obtained by differentiating the proper distance Dr with respect to the cosmic time t, i.e., (2L53) where the dots denote the time differentiatiorL Thus we obtain the Hubble law saying that at any given cosmic time t the speed of any distant galaxy relative to our own galaxy is proportional to its proper distance from our galaxy. The Hubble constant is given by H(t)=R(t)

R(t)'

(21.54)

and we see that the Hubble constant is indeed a function of the cosmic time as we have anticipated earlier in this chapter. The quantity Ho measured by the a
121.41

The Cosmological Red Shifts

In thepresentsectron wediscusscosmologicalredshifts as one of the means of measuring the proper distances to distant galaxies and test the validity of the Hubble law in the framework of the Robertson~Walker geometry. Let us consider a di
=

(21.55)

236

Chapter 21

The Robertscn-Walker Metric

Tirus we may write

dt R(t) =

I

dr

(2L56)

±; 1-krz·

Since the beam of light ts movmg toward us, the radial coordinate r decreases as the time coordinate t increases along the nnll geodesic, and it is awropriate to use a rmnus sign in Equation <21.56). Thus we may integrate Equation (21.56) to obtain 10

1

I

dt

t.J R(t) = ;

dr

f"d

Jo

I - kr2

(Zl.5?)

and to+.6to dt

1~+•~

R(t)

1 {rd

dr

Jo

I - fr2"

~;

2 ( 1.SB)

For all types of radiation received from distant galaxies, the time intervals 6.td and 810 are tiny fractions of a second, and over that tlme R(t) remains effectively constant. Subtracting Equatron (21.57) from Equation (21.58), we obtain

ru0

_

R(to)

6.td R(td)

= 0 ::::!>

6.to = R(to}_ 6.td R(td)

(Zl.S9 )

'lbe observed wavelength Ao and the emitted wavelength Ad are related to the time intervals 6.to and 6.td by the following definitions.

Ao= cl:i..to, Ad= cl:i..td-

(21.60)

Tims the red shift of the received light waves can be obtained in tenns of the function R(t) as follows: Ao-Ad

R(to)

Ad

R(td)

z~--=--L

(21.61)

In the expanding universe we have R(to) > R(td) and the red shift z is positive in agreement with the empirical observations. The red shift (21.61) is a consequence of the hght traveling in curved space-time and is not a result of the Doppler effect. Titis red shift is called the cosmological red shift. Most observed cosmological red shifts are rather small and Id is relatively dose to to. It is therefore possible to expand R(td) in a Taylor series around to as follows: R(td)

= R(to) +(Id- t0)R(to) + ~(Id -

2

to) R(to) + -

(21.62)

Section 21.4 Th&Co&mologicelRedShlfts

•

237

I

2

R(td) =R(to) [ I +Ho(Id-lo)R(to)- 2q0Ho(Id-lo)

2

+ ..

]

(21.63) where Ho is the present value of the Hubble constant given by 11,

R(to)

o= R(to)'

(21.64)

and qo is a dimensionless deceleration parameter given by qo=-

R(to)R(to) R2 (to) ·

(21.65)

'lbe deceleration parameter qo is pos.Ittve when R(t) is negative. i.e., when the expansion of the universe is slowing down. Substituting (21.63) into(21.61). we obtain

r

z ~ I+ Ho(td -

I 2 ']-' - I to)R(to) - 2,qoH 0 (td - to)

(21.66)

(21.67) Titls formula is sometimes very useful, but we must keep in mind that it is only valid for small cosmological red shifts where Id is relatively close to to. When we observe a galaxy with a red shiftz = 1, it means that the scale radius R(td) of the universe at the cosmic time Id when the radiation was emitted was one-half of the present scale radius R(to)- In other words, the size of the universe at the time Id was half of the present size of the universe. Unfortunately, we do not know the coSI111c time Id when the radiation was emitted. If we did, we could directly measure the function R(t). We therefore need some theory of the cosmic dynamics in order to determine the scale radius R(t). Such a cosmic dynanncs theory is the sul!ject of the next chapter.

~

Chapter 22

Cosmic Dynamics

The nonstatic models of the universe based on the Robertson-Walker metric (21.49) are described by their scale radius R(t) and the curvature constant k. The analysis in the previous chapter did not determine the scale radiu.11 R(t) as a function of the cosmic time t. Tlr knowledge of the function R(t) is essential for detenruning the rate of expan,.ion of the universe and other physical properties of the expanding universe. In order to find the solution for R(t), we need a theory of co..'mlic dynamics ba.'«!d on the gravitational field equations. We therefore combine the homogeneous isotropic Robertson-Walker metric with the gravitational field equations to obtain the dynamic equations satfafi.ed by the scale radiu.11 R(t). Trese equations are called the Friedmann equations.

I22.1 I

The Einstein Tensor

In the present section we use the Robertson-Walker metric (21.49) given by

(t)( 1-kr dr

ds 2 =c2dt2-R2

2

2

+r2 de 2 +r2 sin 2 ()dq}).

(221)

The covariant metric tensor for the Robertson-Walker metric is given by the following matrix:

(22.2)

239

240

Chapter 22 Cosmic Dynamics

The contravariant metric tensor for the Robertson-Walker metric is then given by

Using the matrix (22.2) we may write

goo= I. El2

811

= --R

2 2

r

,

=g33

Jil

1

-kr2'

= -R2 r 2 sin2 ().

(22.4)

The coorclinate differentials of the metric tensor components (22.4) can be calculated as

i\goo = o

(k

= O, 1,2,3), 2k:rR2

2Rk

= -1-kr2' llog11 = a"'g11 = O, Bog;u = -2R.k.r2 , aoc22 = a"'g22 = o,

8og11

ar8ll

= -(1-kr2)2'

8r8ll = -2R 2r,

Clog33 = -2Rkr2 sin2 (), 8r833 = -2R2rs'in 2 e, lJog33=-2R2r2 sinllcosll, a'Pg33=0.

(225)

In the results (22.5) and in the following calculations, we temporarily define by dots the derivatives Of the scale radius R(t) with respect to the temporal coordinate x° = ct rather than with respect to cosmic time t to simplify the calculations and to make the results compatible with the results obtained elsewhere in the literature using the units with c = I. Thus we have . R(t)

dR(t)

JdR(t)

= d}J = -;;--;Jt·

(22.6)

The Christoffel symbols of the first kind for the metric (22.1) can now be calculated using the definition (9.28), i.e.,

r- = ! (aEik + ag,,i - ag~). J.lm

2

CJx"

axk

(}xl

(22.7)

Section 22.1

241

The Einstein Tensor

Tlx!results for the Christoffel symbols of the firstkindforthe metric (22.I) are summarized in the following list: ro,oo = 4caogoo + iJogoo-iJogoo) = O I

rom = ro,10 = zC8rcoo + iJog10 - Cloco1) = o ro,02 = ro,20 =

I

2caogoo +

Cloczo - Cloco2) = O

ro,03 = ro,3o = 4cawcoo + iJog30 - aoxmJ

=o

RR

I

ro,11=zC8rg01+8rg10 - Clog11) =+I -kr2 ro,12 = ro,21=4caogo1+8rc20- Clog12) = O ro,13 = ro,31

=

4 (Clw801 + 8rR30 - iJog13) = 0

ro22 = 4caogo2 + llog20 ro,23 = ro,32

=

Cl0g:n) = +RRr

2

I

z
2 2 r o,33 = 4 (Clw803 + Clw83o - iJog33) = +RR r sin e I

r 1,00 = 2(iJog10 + iJog01 - 8r800) = 0 I r1.01 = ruo = zC8rc10 + Clog11 - Clrg01J = -

RR

1

-kr2

I

r1.oz = r1.20 = 2 = O I r1.11=zC8r811+8rg11 - 8rR11J

R 2kr

= - (l -kr2)2

I

r1,12 = r1.21=2
=

~@og12 +aoc21 -

Clr822)= +Iflr

242

Chapter 22 Cosmic Dynamics

I

r1.23

= r132 = 2(l13g12 + l19g31 -

rt.33

= ~(Clw813 +a'Pg31

- 8rg33)= +R2rsin 2 ()

rz,oo = 4caogzo + 8ogo2 rz.01

8r823) = 0

llogoo)

=O

= rz.10 = 4carg20 + Boc12 -

aoco1) = o

rz.02 = ruo = 4caog2o+Bog22-lloco2)= -RRr2

rz.ro = r2.30 =

4cawc20 + Bog32 -

llogm)

=o

I

rz,11=zC8rc21+8rc12 - llog11) = o r2,12

= rz.21 = ~(l10R21 + 8r822 -

rz.n = rz.31 = 4«:J"g21 t- 8r832 -

llog12) = -R2r llogn)

=o

rz.22 = 4caog12 + aog12 - aoc22> = o rz,23

= rz,32 = 4
r2,33

= ~(a.,,g23 + av>g32 -

r3.oo

= 2
r\01

= rJ,10 = 4car83o + Clog13 -

I

llog23)

= +R2r 2 sin() cos()

a'Pgoo)

=0 Clqigo1) = o

I

£'\02 = r3.20

=O

llog33)

= 2
Clwgo2)

=0

rJ,03 = ruo = 4
= 4car83t ;- 8r813 -

r3,12

= rJ.21 = zCllog31 + arg23 -

Clwc12) = o

r3.B

= r3,3t = 4
Clqign)

Clw811)

=O

I

= -R2r sin2 ()

Sectlon 22.1

The Einstein Tensor

rJ.22 = r3.23

~Wog32 + lJog23 -

243

a"'gll) = o

= r3,32 = ~(;J"'g32 + lJog33 -

r3.33 =

I

2ca"'833 + a"'g33 -

lJ"'g23) = -R2r 2 sin() cos()

a"'g33) =

o.

(22.8)

The Christoffel symbols of the second kind for the metric (22.1) can be calculated using the definition (9.29), i.e., (22.9)

The results for the Christoffel symbols of the second kind for the metric (22 I) are ~arized in the following list:

r&i = g01 rj,oo = c 00 ro,oo = O

r21 = r1o = gfU·ri!Jl = g°'Tom = o r82 = ~ = goirim = gooro,02 = o rg3 = ~0 = g0iri.03 = c 00 ro.o3 = o O' 00 RR •11 =c 9rj,11 =g ro,11 =+l-kr2

...,{)

rrz = r~I = g01 r 1,12 = g00 ro,12 = 0 rr3 = r~I = g0jrj,13 = c 00 ro,13 = 0 r~ = Et1rj,22 = g00 ro,22 = +Rkr2

2

r~3 = ~2 = c 0jr1.23 = c 00 ro.23 = o r~3 = gOir 33 = g00 ro.33 = +Rkr2 sin2 () 1

rlxi = g 1ir1 .oo = g 11 r1,oo = o - ljr1m-c . - II r1.01-+R - k r101- r110-c 1 11 rJ2 = r}o =g irj,02 =g r1m =O rJ3 = rl 0 =g 11 rj.m =g11 r1,m =0 1 11 r:1 = c irj,11=g r1.11=+ ~:r 1 11 r:Z=r~ 1 =c r 1.12=g11 ru2=0

244

Chapter 22 Cosmic Dynamics

r:3 = rl 1 =g•ir1,l3= 8 11 r1.B = o r~ 2 =gurj.22 =g11 r1.22 = -r(l -kr2) r~3

= rl2 = g•iri.23 = g 11 r1.23 = o

r~3 =

11

g rJ,33

= g 11 rt,33 = -r(1 -

21

kr2) sin2 £1

22

rlio = c r.1.oo = g r2.oo = o r51 = ri0 =c21 r;,01 =rr2.01 =O 2_2_2J __ n _k r 02 _ r 20 - 8 r_,.oz-rr2.02- +R

rl3 = rj0 = g2ir1m = g22r203 = o rf1 = g1ir1.11 = g22 r2.11 = o rf2

= rI1 = i11r;,12 = g22r2.12 = +~

rf3 = rj. = g2irj,13 = rrz.13 = 0 ri2 = g2ir1.ll = g22r2.n = o r~3 = rj2 = i11r1.23 = g2 2r2.2J = o rj3 =i'ir_,33 =g22r2.33 = -sinllcosll rlio = 1:31 ruJO = g 33 r3,oo = o rc)1 = ri0 =g31 r;,01 =g33 r3,01 =0 rJ2 = r1i = g3ir;m = g33r3,02 = o 3 3 r 03 =r10 =

R ;m=g33 rJ,m=+R

3Jr

8 ·

ri1 = g 31 r;,11 = g33rJ,11 = o ri2 = ri1 = g 3_,r;.12 = 8 33 r3,12 = o ri3 = rj. = g3ir;JJ = g33r3.B = +~ ri2 = g3ir;.22 = g33r3..n = o

r~3 = rj2 =c31 r_,.23 =g33 r3.23 =:::=cote rj3

= g 31 r;.33 = g33r3,33 = o

(22.10)

Section 22.1

The Einstein Tensor

245

From the results listed in (.22.10) we can make the following conclusions: (22.11) and

where (a,{3

= 1,2,3). Thus we further obtain

r2_s=-O, r.!_s.=O

foraf={3.

r~ =li!li~;. r!p =li!li~; +li~li~cot()

fora

t= f3

(22.13)

Using the results (22.12) and (22.13) we note that

air:S = o

fora

t= {3,

apr~ = o.

(22.14)

Furthennore, we can calculate the following sumsJ rooo+ro1+ • r202+ rJm=+3k roi= ii.

r{i = rro+rf1 +rf2 + r~J = 1 ~~,-2 + ~

r£ = r~ + rl1+ ri2+r~3 =cote ri1 =r~0 + r~ 1 +ri2 +rj3 =0.

{22.15)

Using the results (22.15) we note that

The Ricci tensor for the metric (22. l) can he calculated using the defirntion (18.8),i_e.,

From the result (12.48) we see that the Ricci tensor is a symmetric tensor = Rnk and that it has only l 0 independent components. The formulae for the components of the Ricci tensor for the metric (22.l) are SIUilllI3Iized

Rim

246

Chapter 22 Cosmic Dynamics

in the following list:

Roo =

aor&i - air/xi+ rg,r~ - r~r~i CJrrtj - 8jrJ1 + rgir~ 1 - rg 1 r~i

Rot= Rm=

Rm= Rm= aorti - air&2+ rg1 r;2 - rt;2 r;i Rm = R.30 = a"'r~i - a1 rj3 + rg1 r~3 - rg3 rJi

8rr{_,-a_,r{1 +rfir~ 1 - rf1 rJ1 = R21 = aor{i - air{2 + rfir~2 - rf2 r~1

Rn= R12 R.13

= R.31 = a,,,rfi - a1 r{3 + rfirl3 -

R22

=

R.23

= R.32 = awr4J - aJr~3 + r~Jr~3 - r~3rlj

R.33 =

rf3 rli

aor~i - air4 2 -1- r~ir~2 - r~2 r~i

a"'ri1 - a1 r~3 -r r~ir~3 -

r~3 r~1 .

(22.18)

The results for all I 0 individual components of the Ricci tensor for the metnc (22.1) can be obtained from this list (22.18). However, the calcu lation process can be factbtated using the general results (22.11 l--(2216). Thusfora,f3 = 1,2,3,anda -=fa {3, we may calculate

Rap= apr!i - air!JJ + r~_,r~P - r~f:lr~r

(22 I9J

From (22.14) and (22.16) we see that the first two tenns on the right-hand s1deofEquatton (22.19)fora -=fa f3 vanish and we may write Rafi=

Usmghere r2_s

r!ir~/J - r~8 rlj - r!8 r{; - r~8 r47

(22.20)

= 0 ancJT!_e = 0 fora f= f3 from(22.13). we obtain Rafi

= r:jr~/J - li!liE co:()

for a

t= /3-

(22.21)

Thus the second tenn on the right-hand side of Equation (22.21) ts not equal to zero only for (a{3) = (12). Let us now calculate the first term on the right-hand side of Equation (22.21), as follows: (22.22)

Section 22.1

The Einstein Tensor

247

r!jr~fJ = r2 0rgJJ + r;;0 r~P + r2wrOti + r;;wr;p.

C2223)

It is shown by direct calculation that the first three terms on the right-hand side Of Equation (2223) vanish. Thus we obtain

r:1 r~tJ = =

r;;wr;p = r!wrfp + r,;wrfp + r,;wrJ'p r! 1r:P + r!2 rfp + r! 3 r~P + r~ 1 r:le + r,;2 ri_s + r,;3 rts (22.24) -1- r; 1 r~P + r~ 2 rip -1- r!3 rjtJ

Using the result (22.24) we can show by direct calculation that

r:ir~/J = 8!8~r~3 rj2 = 8!8Ec:e.

(22.25)

Substituting (22.25) into (22 21), we finally obtain (22.26) Furthermore fora= 1.2,3, we may also calculate

Rao= Bor~i -

Bor20 - apr!0 + r!jr~ -

r:Or~1 .

(22.27)

Using here the results (22.11), (22.14), and (22.16), we see that the first three tenns on the right-hand sideOfEquation(22.27) vanish. We then have

Rao= r:Or~ + r!/Jr:O- r20 rli -

r!0 r;r

(22.28)

Using (22.11) further, the first and third term.~ on the nght-hand side Of Equation (22.28) vanish. Thus we obtain

Rao= r2.erfio + r~/Jr~0 - r!0 r~ 0 - r~rp".

(22.29)

Using once again the results {22.11). the first and third t:errru> on the righthand side of Equat:ton (22.29) vafJlsh. We may then write

Rao = r~Pr~0 -

r!0 r;;".

(2230)

Using the result (2224) we can show by direct calculation rhar R.u~o

(a~

1,2,3).

(22.31)

From the results (22.26) and (22.31) we conclude that all off-diagonal components of the Ricci tensor are identically equal to zero· Rkn=Rllk==O

(k#n).

(22.32)

248

Chapter 22 Cosmic Dynamics

Thus the only nontrivial components of the R1cc1 tensor are the diagonal components. Using the results (22.IIJ-{22.16), they can be calculated as

Roo = =

Bor~; + q;ir~

3Bo (~) + (rJ.)2 + (r62}2 + (rJ3)2 = 3~

Rn= arr{; - Bor~1

-

arr: 1 + ri;r~ 1 - r~1 r~;- r: 1r{;

=ar(. ~~r2 + ~)-ao(i ~r2)-ar( 1 ~:r2) k2r2

2R2

2

3R_2

+ l-kr2 + (l-kr2)2 +71-1-kr2 fr

(

-1-krZ R12

(Ji

fr 2) Rz Y/2+1.k) 1-krz+-; =-I-kr2 R+~

= aor~J - aor~2 - arr.h + ri;r~2 - r~2 r~; - r~ 2 r{; = lJo (cotll) - Bo (Rkr2 )-ar [-r (1 -kr2 )] +2R2 r2 - 2(1 -kr2 ) 22

=-Rr

+ cot2 () -

3k. 2r 2 ;-2- kr2

2

(.RR+~ 2k +2k)

R33 = -Bor~3 - 8rr~ 3 - llorj3 + r~;r;3 - r~3rl; - r~3r{;

- rj3ri; =-Bo (Rkr2 sin2e)- 8r [-r(l -8a (-sin() cosll)

+ 2R.2 r 2 sm.l() -

l

2 2 kr ) sin ()

2(1 - kr2) sin

2

()

- 2cos2 () - 3R2r 2 sin2 () + (2- kr2) sin2 () +cos 2 () (22.33)

Multiplying the covariant components of the Ricci tensor (22.33) by the corresponding components of the contravariant metric tensor given by (22.3), we obtain the components of the mixed Ricci tensor as

Section 22.1

The Elnslein Tensor

249

follows:

The RicCI scalar Rj is then given by

(22.35) Let us now fonn a mixed tensor G~, entering the gravitational field Equations (18.97) as

G~ = R~ - ~iS!Rj = - S~G T!

(22.36)

The ten.wr G~ is usually called the Emstern tensor. Substituting the results (22.34) and (22.35) into the definition (22 36), we obtain the components of the Einstem tensor for the Robertson-Walker metric as follows:

(22.37) At this stage it ts appropnate to reverse the tempomry convention (22.6) and to return to the usual definition of dots as the derivatives of the scale radius R(t) with respect to the cosmic time t, i.e., R(t)

~ d~~t).

(2238)

250

Chapter 22 Cosmic Dynamics

This change of convention introduces an additional factor of 1/c before each derivative defined by the dots. Thus lhe results (22.37) become

(22.39)

Equations (22.39) are the final results for the components of the Einstein tensor in lhe Robertson-Walker metric.

I22.2 I

The Friedmann Equations

In order to construct the gravitational field Equations (22.36), we now need the energy-momentum tensor of the cosmic matter. The assumption is that the matter in lhe universe is, on lhe large scale, evenly distributed in the form of a cosmic fluid with no shear-viscous, bulk-viscous, or heatconductive fe3tures. Thus we may use the mixed energy-momentum tensor of an ideal fluid, which is in the covariant form given by (18.72). We may lhenwrite

T!

(22.40)

where p is the pressure and p is the matter (or rest energy) density of lhe cosmic fluid. In the co-moving frame the cosmic fluid is at rest and we have

u0 =I,

if=O (a=l,2,3).

Using the results (22.41) and observmg thar p

T8 =

pc2,

Tf

=

«

(22.41)

pc2 , we obtain

T'f :::: Tj :::: -p.

(22.42)

Thns the energy-momentum tensor of lhe cosmic fluid in the co-moving frame can be structured in the following matrix:

0 0] 0

-p 0

0

0 . -p

(22.43)

251

Section 22.2 The Friedmann Equations

Substituting lhe results (2239) and (22.42) into (22.36), we obtain the gravitational field equations in the form

i.?.2+kC2

8rrG

(22.44)

~::::Jp

R R2 +kc2

RnG

2R+~=-7P

(22.45)

Equations (22.44) and (2245) are called the Friedwwnn equations, and lheir solution describes cosmic dynamics. Combining Equations (2244) and (22.45), we may write

R

8rrG

8rrG

2R+3P=-7P-

(22.46)

Furthermore, from Equation (22.44) we have R2+kc2

= 8~G pR2.

(2247)

Differentiating Equation (22.47) with respect to the cosmic time t, we obtain

2kR = ~G pR2 + S~G p2RR,

R

2R.

~

8:nG R

]P"Ji +

8rrG

3 2p.

(2248)

c22.49J

Substituting Equatlon (22.49) into (22.46), we obtain

8~G

pi+ 8~G3p = - 8~G 3~

p~+3(p-r~)

=0

(22.50)

(22.51)

Observmg that the universe as a system of galaxies in lhe smooth fluid approximation behaves as an incoherent dust, we can again use the approximation p « pCZ to obtain (22.52)

252

Chapter 22 Cosmic Dynamics

Imegrating t22.52) we obtam

pR 3

=

Pori6 = Constant,

(22.53)

where Po and Ro are the matter density and scale radiua at lhe present epoch (t = to). From the result (22.53) we see that the matter density varies in time asR- 3 and lhat the quantity of matt er in a co-moving volume element is constant during the expansion of the universe. This conclusion is relevant to the present matter-dominated epoch in lhe history of lhe universe. In the early phases of its history when the universe was radiation dominated. this conclusion was not valid because at that stage the assumption p « pc2 was not valid either. In the radiation-dominated universe, Equation (22.51) becomes

- (P+2p)kR=O. p+3

(22.54)

and the pressure component cannot be neglected. Using (22.53) and neglecting the ixessure, we obtain the Friedmann equations for the II131ter-dominated epoch of the universe as follows:

R R2 +kc2

2R+~=o

k.2 +kc2

8nGPo~

----w:- = -3-Rl

(22.55) t22.56)

The solutions of the Friedmann equations for the matter-dominated epoch of the universe given by (2255) and (22.56) will be the subject of the next chapter.

"" Chapter 23

Nonstatic Models of the Universe

In the previous chapter we derived the cosmic dynamic equations satisfied by lhe scale radius R(t), known as the Friedmann equations. As we concluded earlier, the nonstatic models of the universe based on the Robertson-Walker metric (21.49) are described by their scale radius R(t) and the curvature constant k. The oljective of the present chapter is there~ fore to study the appropriate solutions for the scale radius R(t) and the appropriate values of the curvature constant k and theirimplications for the future evolution and other physical properties of the expanding universe.

J23.1 J Solutions of the Friedmann Equations In order to solve the Friedmann equations (22.55) and (2256), we first introduce some useful quantities. Let us recall that the Hubble constant (21.54) is given by R(t)

H(t)~ R(tf

(23.1)

Substituting (23. I) into (22.56) we obtain kc2 87rGPoRb 2 H +}?2=~3-RJ·

(23.2)

253

254

Chapter 23 Nonstatlc Models of the Universe

In the pre;ent epoch (t = to) with R =Ro and H = Ho, Equation (23.2) becomes kc2 8rrGPo 2 (23.3) Ho+~=-3-

k 8rrGPo HJ ~ =~--;z

8:nG =

JcZ (Po-Pc),

(23.4)

where Pc is the critical (or closure) matter density of lhe universe. defined by

Pc= 3Hl_

(23.5)

8,,-G

Wilh the range of the estimated values of the Hubble constant Ho in the present epoch, the numerical value of the critical density is given by

Pc= 2

x

I0-26n2 ~

(0.5 S

11

:5: I).

(23.6)

Let us also recall the definition of the deceleration parameter q(t) defined according to Equation (21.65) as follows: q(t) = -

R(t)R(t) i?Z(t) =

I

ii

-JiiR·

(23.7)

From the Fnedmann equattons (22.55) and (22.56) we may write

R

4:n:GPo~

R=--3-w·

(23.8)

Substituting (23.8) into (23.7) we obtain q(t)

4:n:GPoR/i

= ----yj2[i3.

(23.9)

The present-epoch value of the deceleration parameter (23.9) is given by qo

=

4nGPo Po 3HJ = 2pc.

(23.IO)

From (23.10) we see that the present-epoch value of the deceleration parameter can be expressed in terms of the present and critical matter densities of lhe universe.

Section 23.1

23.1.1

Solutions of the Friedmann Equations

255

TheFlatModel(k...-0)

For lhe flat model with k = 0, Equation (23.4) gives Po :::: Pc and from Equation (23.10) we find qo :::: 1/2. Using Po :::: Pc and (23.5) we may write

H6=

8rr~Po­

(23.11)

Friedmann Equation (22.56) then becomes .2 8nGPo~ H6~ R=~=~·

(23.12)

i.?2 =~.A:::: Hoi!/ 2.

(23.13)

R

Equation (23.13) can be rewritten as follows:

R=~=AJr 1 f2=> f-JRdR=Afdt;-C.

{23.14)

Performing lhe elementary integration in (23.14) we obtain

2

JR 312 =At+ C

(3A)

=> R(t) = T

2 3 / 2 3

11 ,

{23.15)

where we used the initial condition R(O) = 0 to set the integration constant C in (23.15) equal to zero From Equation (23.15) we may also write

2

t = JAR3/2

2 (R) Ro

312

= 3Ho

(23.16)

The i:resent-epochEquation(23.16)fort =to has the form

2 3Hu

to=-.

(23.17)

Thesolution(23.15)forthescaleradiusR(t)withthechoiceofthecurvature constant k = 0. is. in the literature. sometimes called the Einstein-de Sitter Solution. 23.1.2

The Closed Model (k

= 1)

For the closed model with k = I, Equation (23.4) gives Po >- Pc. and from Equation (23.10) we find qo > 1/2. The Friedmann Equation (22.56)

255

Chapter 23 Nonstatlc Models of the Universe

then becomes

i?.2 -;;I+ 1 =

8rrGPoR3 B Jc2R Cl= R•

(23.18)

where we mrroduce a consrnnt B as

B= 8nGpoR6 3c2

= 2 4:rrGPoH5R6 c2

3H5

(23.19)

Using the definition of the present-epoch deceleration pmruneter (23.10) we obtain

H2Rfi

(23.20)

B=2qo+Furthermore, using Equation (23.4) fork= I, we may write I

~

2 2 H ( 4:rrGPo ) H 2 JHl - I = -;} (2qo - I).

= -;}

Ro =

-iJo (2qo -

1)-

112

(23.21)

(23.22)

.

Substituting (2322) mto (23.20) we obtain

2qo

c

B:::: (2qo-1)3/2Ho·

(23.23)

Equation (23.18) may be rewritten in the form

!~ = B-R => (' cdt = cdt

Jo

R

{R /RdR.

v~·

Jo

(23.24)

Let us now introduce an angular parameter 1J as follows: R

=

Bsin2 ~ =

~ (1

- cos JJ),

(23.25)

with dR

= B sin~ cos ~d1].

(23.26)

Using Equations (23.25) and (2326) we may wnte

v~ B=Ji.dR = B

21J

sin 2,dTJ =

B

"2 (I

- cos TJ) dT).

(23.27)

Section 23.1

SolutlonsoftheFriedmannEquations

257

Using (23.25) and performing the elementary integration in (23.24) with (23.27), we obtain lhe two parameter equations R=

B

2

(1-cos1}),

B

ct="2(1}

sin"f})-

(2328)

The pm-.uneter Equations (23.28) define the scale radius R(t) by a pardllleter 1}. The function R(t) defined by (23.28) is a cycloid, and the scale radius is a cyclic function of the cosmic time. The universe starts expansion at the cosmic time t = 0 (1} = 0) with scale radius R = 0 and expands to a maximum size R = B at the cosmic time t = tL/2 (TJ = n ). Thereafter 0 at lhe cosmic time lhe universe contracts back to the scale radius R t = tL (1} 2n). The time tL in the closed model is called the lifespan of the present universe. It can be calculated from (23.28) with (23.23) for 1} = 2n, as follows:

=

=

(23.29) For example, the choice qo = 1 gives a lifespan of the universe equal to tL 2n/Ho. From Equations (23.22) and (23.23) we also note th
=

=

=

23.1.3 The Open Model (k

= -1)

For the open model with k = -1, Equation (23.4) g-ives Po < Pc. and from Equation (23.10) we find qo < 1/2. The Friedmann Equation (22.56) then becomes (2330)

where we again use the constant B given by (23.23). Equation (23.30) may be rewritten in the form

ldR

B+R

~di=~==>

['

lo

{'

{R

cdt= lo VB+R.dR.

(23-31)

Let us now introduce an angular parameter 1J as follows:

R=Bsinh 2 ~=~(cosh"f}-1),

(23.32)

Chapter 23 Nonstatlc Models of the Universe

255

with dR = Bsinh ~ cosh

~d1}­

(2333)

Using Equations (23.32) and (2333), we may write 2'7 B l/~ B+RdR = Bsinh 1,dT) = 2" (coshTJ -

l)d1}.

(2334)

Using (2332) and perforrmng elementary mtegrahons m (23.31) with (23.34), we obtain the two parameter equations

The parameter Equations (23.35) define the scale radius R(t) by a parameter '1- The function R(t) defined by (23.35) is growing indefinitely as t----.. oo (1} --+ oo). Like the flat Einstein-de Sitter solution, the open solution continues to expand forever. It should also be noted that the expansion of the universe in the open model is faster lhan lhat of the flat model because of the presence of the exponential functions in the parameter '1·

I23.2 I

Closed or Open Universe

The analysis of the three possible models for the expansion of the universe, presented in the previous chapter, does not resolve the question whether

the present universe is open or closed The answer to that question must be looked for in astronomical observations and estimates of the various parametersofthemodel. Thecurrentestimatesoflhepresent-epochHubble constant Ho and the critical density Pc are km

I

Ho=IOOa-s Mpc

Pc=2x10- 26ci2~.

O.Ssasl.

(23.36)

Estnnates of the i:resent-epochdeceleration parameterqo are mo re difficult to obtain. In order to study the deceleration par.uneter qo, we use the result for the Hubble constant H = k/R to calculate the second derivative of the scale radius with respect to cosmic time as follows:

R=HR

-o::::}

R=Hk+iIR=H2R+iIR.

(23.37)

Section 23.2 Closed or Open Universe

259

Substituting (23.37) into (23.7) we obtain I ii q(t)=--,-=HR

( ~+l iI ) H2

(23.38)

·

Usmg (23.38) we can calculate lhe present-epoch clece1erat:ion parameter qoas H(to) ) Qo=- ( H2(to) +1 .

(23.39)

The result (23.39) indicates that if we can measure if (to) and H(to) we can calculate the present-epoch deceleration parameter q0 . In order to measure if (to) we use lhefactthatas we lookdeeperinto space. we look farther back into time. For example, if we estimate the Hubble constant H(t) for objects 1 billion light years away, we are really estimating lhe HubbleconstantH(t) for lhe llllh-erse J billion years back in lhe cosmic time. The difficulty with such a method of determining lhe present-epoch deceleration parameterq0 is related to lhe difficulties of measuring lhe distances to lhe objects deep in space. In spite of these difficulties, some existing observational data suggest lhe value of lhe present-epoch deceleration parameter qo 1. As we have concluded before, for qo > 1/2, lhe universe is closed and such observational data suggest that the present universe is closed. However, using Equation (23.10) forqo = I, we obtain the following present-epoch matter (or rest-energy) density of lhe universe:

=

Po= 2pc

= 4 x 10- 26 a2~

(0.5

.s 11 .s J),

(23.40)

which is much l
260

Chapter 23 Nonstatic Models of the Universe

galaxies revolving around great spiral gaI
I23.3 I

Newtonian Cosmology

In the present section we study lhe evolution of an expanding universe wilhin the framework of lhe Newtonian theory of gravitation and compare lhe results with those obtained using lhe general theory of relativity. Let us imagine the large-scale matter-domillated Newtonian universe as an ideal fluid with negligible pressure moving according to the Newton laws of motion and gravity. Such fluid is often called the Ne"'1oninn dust. We now ccnsider Newtonian dust with a uniform density p(t) being in a state of uniform expansion. The only force acting on lhe dust particles is lhe force of gravity. The components of lhe three-dimensional position vector of an arbitrary dust particle at some cosmic time tare given by ~(t) = R(t)Aa

(a= I,2,3),

(23.41)

where A is a constam vector defined by lhe initial position of lhe dust particle, and R(t) is the scale radius of the uniform expansion. The first and second time derivatives of the coordinates (23.41) are given by

Section 23.3 Newtonian Cosmology

261

where H(t) = R/R is lhe Hubble constant of the Newtonian expansion. Let us now use lhe continuity Equation (1854) in the form

¥,+a. (pv")

¥, +i!'a.p +pa.if' = o.

~

(23.43)

Using (23.42) and lhe definition oflhe total time derivative of lhe density p. given by dp iJp iJp dxct P = dt =at+ a:xctdt'

(23.44)

we may rewrite (23.43) as follows: P+pHtJa.>!'=0==> P+3pH=O,

(23.45)

(23.46) Integrating Equation (23-46) we recover the result (22.53) obtaired using lhe Friedmann Equations, i.e., pR

3

= PoR~ =

Constant,

(23.47)

where Po and Ro are lhe matter density and the scale radius at the present epoch(t = to)- From theresult(23-47) we see thatin Newtonian cosmology lhe matter density also varies in time asR- 3_The isotropy and spherical symmetry of the universe require lhat any spherical volume evoh-e only under its own influence. If the observed spherical volume has radius r and mass M(r) = (4:rr/3)r3 p, the equation of motion ofa dust particle somewhere on its surface is given by lhe Newtonian Equation (17-44), as follows:

"i" = ~x" = - GM(r)x" ~ _ 4rrGpx". R

r3

3

(23.48)

As Equation (23.48) is valid for an arbitrary~, we obtain -·

4nGp

R = - --R. 3

(23.49)

Now using the result (23.47), Equation (23.49) becomes

R- _ 4nGP<JR~ -

3R2

-

(23.50)

262

Chapter 23 Nonstatic Models of the Universe

Multiplying both sides of Equation (23.50) by R, we obtain . -· 4nGPQR~ R RR+---Ri =U, 3

(23.51)

(!) =

~ di?.2 - 4nGPoR6 !!__. 2dt 3 dtR

Q_

(23.52)

Thus we may \\/rite

~

dt

(il' _

8nGP<JJl;l) 3R

= O.

(23.53)

Integrating Equation (23.53) we obtain

R.2 - 8:rrGPoR~ =

-kc2

3R

(23.54)

where kc2 is lhe integration constant. After some restructunng of Equation (23.54) we recover the second Friedmann equation (22.56), i.e.,

8:rrGPQR~

i?.2 +h?

~=-3-RJ"

(23.55)

Furthermore, usmg the result (23.54) rev.ritten as

1R2 +kc2

4nGP<JJl;l ~

2-R-

(23.56)

Equation (23.50) becomes

--

1R2 +kc2

R=----.

2

R

(23.57)

Dividing by R and restructuring Equation (23.57), we recover lhe first Friedmann equation (22.55), ie., (23.58) From the i:receding discussion we conclude that Newtonian cosmology gives the same dynamic equations for the scale factor R(t) of the expanding universe as the relativistic cosmology based on lhe Robertson-Walker metric. In Newtonian cosmology lhe pardllleter k is just an integration constant, whereas relativistic cosmology k is the curvature constant. In fact if k -::f=. 0 there is no loss of generality if we set k = ±1 in Newtonian cosmology eilher. The dynamic Equations (23.55) and (23.58) lead to exactly

263

Section 23.3 Newtonian Cosmology

=

the same three models of lhe expanding 11mverse with k -1, 0, 1. It should, however, be kept in mind lhat the Newtonian cosmology is just a nonrelativistic limit of lhe standard relativistic cosmological model and lhat it cannot account for a number of important physical features of the expanding universe. In particular it caIUlot handle lhe radiation-dominated early phases of lhe expanding universe and incorporate the pressure of the cosmic fluid in the proper way.

,,.. Chapter 24

Quantum Cosmology

The objective of the present chapter is to give a short introduction into lhe most profound speculations about the earliest moments of the i:resent universe. In the solutions of Friedmann equations, presented in lhe previous chapter, we have generally used the initial condition R(O) = 0. In effect such an initial condition means lhat in lhe earliest moments Of its evolution, the present universe may have been so compact that its size was comparable to the size of a single quantum particle. In such an early epoch lhe classical solutions of the Friedmann equations, derived in the previous chapter, are no longer adequate and we need a quantum theory of the very early universe, i.e., a quantum cosmology. Since quantum cosmology is a highly speculative theory in its early development phase, lhe presentation in this chapter will be limited to a few introductory tcpics. Furthermore, as quantum lheory is outside lhe scope of the present book, a few elementary quantum-mechanical results needed in lhe present chapter will be introduced without a detailed derivation from lhe first principles.

I24.1 I

Introduction

The nonstatic models of the expansion of the universe, discussed in lhe previous chapter, are based on a well-formulated gravitational field theory and available obseivational data. These models can be used to study the early moments of the universe down to the times t < 10-2 s. The subject

265

266

Chapter 24 Quantum Cosmology

of the present chapter is to investigate lhe so-called Planck epoch, ie., the times t :5 I0-43 s. The approach pursued in lhe i:resent chapter is to use lhe canonical quantization procedure to quantize lhe gravitational degrees of freedom and to derive lhe Klein-Gordon wave equation for the wave function of lhe universe governing both the matter fields and lhe space-time geometry_ Such a wave equation is generally known as lhe Wheeler-DeWittequatU»r. The wave function of lhe universe in this approach is a function of all possible three-dimensional geometries and matter field configurations, known as superspace. It should be noted that lhe ccncept of superspace used in quantum cosmology has nothing to do wilh lhe concept of supen;pace of lhe supersymmetric quantum theories. In order to reduce lhe problem to a manageable one, all but a finite number of degrees Of freedom are frozen out, leaving us with a finitedimensional supenipace known as lhe mini-superspace. In the present chapter we consider a simple mini-superspace model in which lhe only remaining degrees of freedom are lhe scale radius R(t) of a closed homogeneous and isotropic universe and a homogeneous massive scalar field¢. Furthermore all the degrees of freedom of the scalar field are also frozen out, and lhe only effect of the sealar field is to provide lhe vacuum energy density Pvac· Such a vacuum energy density contributes to a cosmological constant term defined by A = 8:rrGPvac· Using the mini-superspace model just described, we lhen calculate the semiclassical wave functions of the universe and lhe tunneling probability forlheuniversetomakelhetransitionfromR OtolhetransitionpointR = Ro, being lhe upper limit of the classically forbidden region 0 < R < Ro-

=

~ The Wheeler-DeWitt Equation Let us stan the derivation of the Wheeler-DeWitt equation by ret.-alling the Friedmann equations Of cos1nic dynamics, given by (22.44) and (22.45), i.e.,

R2 +kc2

8:nG

~=3P

R R.2 +1«:2

8:rrG 2R+~=-~P·

(24.1) (24.2)

In lhe very early phases of its evolution, the umverse was clearly nm matter dominated In fact, it is generally believed that in lhe Planck epoch it was not even radiation dominated, and lhe energy-momentum tensor was determined by the vacuum energy. In such a vacuum-dominated epoch

Sectmn 24.2 The Wheeler-OeWin Equation

267

the pressure of the extremely dense matter of lhe universe is given by p = -pc2 . Using .EQuation (22.54). we see lhat in the vacuum-dominated epoch we have

P=

0 ==> p

= Pvac = Constant.

(24-3)

Thus the Friedmann Equations (24.1) and (24.2) in lhe closed model (k =+I) become

R.2 +c2

8nG

A

-r=3A·a..·=3 ii

2R +

R2 +c'

- r = 8nGPvac =A.

(24.4) (24.5)

Substituting Equation (24.4) into (24.5) we obtain

R

R

A

2 R+3 =A=> R =

A

3-

(24.6)

Equation (24.6) can be written as follows: __

c2

(

R- RfiR =0

(3\

Ro= cV---f:.j.

(24.7)

The classical solution of Equation (24.7) is given by R(t)

= R0 cosh

{fa).

(24.8)

This solution describes a universe that was infinitely laige (R-----)- oo) in lhe infinite past (t -----)- -oo). Then it contracted to a minimum size of Ro at t = 0, after which itstartedexpanding again to an infinite size in lheinfinite future (t-----)- +oo). From Equation (24.8) we see lhat the classical solution has a forbidden range for 0 < R Ro and lhe tunneling probability of such a transition, we now construct lheaction integraloflhe universe in the Planck epoch as follows: IG = _ __c'__[dt£[R(t),R(t)j [ HdrdOd
(24.9)

268

Chapter 24 Ouanlum Cosmology

where Fgdrd()dq;

r 2dr

= R 3 ~ sin()d()dq;.

(24.10)

In Equation (24.9) we have used the result for the Ricci scalar (22.35) in lhe Robertson-Walker metric wilh k = + 1, i.e., Rj-

~ (~

1-cz

R+

R2+c2) R2

(24.11)

,

to conclude lhat lhe Lagrangian density£ [R(t),.k(t)] in lhe present model is space independent. Thus we can integrate out the spatial part of the action integral (24.9), which is just a three-dimensional spherically symmetric volume in lhe Robertson-Walker curved space-time metric (21.49), to obtain

1

1

1

,/=gdrd()drp=R3

0

l'

2

,.---------:; r dr

-rw

vl

1"

sin()d()

Cl

1"'

d
(24.12)

O

Integrating lhe elementary integmls over() and fP and introducing lhe new angular variable x as

r

= sinx =>dr = cosxdx.

(24.13)

the result (24.12) becomes

{ Fgdrd()dq;=4nR 3 {ir/Z sin2xdx =2n 2R 3.

Jo

J1r

(24.14)

Substituting (24.I 4) into (24.9) we obtain 4

1

la ~ - nc BG , dtR\tJC[R(t), R(t)].

(24.15)

Usin,glheresultforthe Ricci scalar (24.11) in lhe Robertson-Walker metric with k +1, we construct the Lagmngian density of the present model as follows:

=

.

6(il'-c' A) , 2

£[R(t),R(tJ]~1

c

~--+-

R

3

(24.16)

where we have eliminated lhe term including the second-0rder time deriv:ttive., which shonld not appear in lhe Lagrangian density, by reducing it to the total time differential. Substituting (24.16) into (24.15), we obtain 2

le = [dtL(t) = - J:nc [dt , 4G ,

(Rk

2 - Rc2 + !:_R 3), 3

(24-17)

Section 24.2 The Wheeler-DeWitt Equalion

269

where the Lagrangian L(t) is given by L(t)

3)

3:nc2(RR·2 -Re2 +JR A = -4(;

(24.18)

The Lagrange equation for the scale radius R(t) is then given by

~-~(~)=o.

(24.19)

Substituting (24.18) into (24.19), we obtain

2

3:rrc (·2 ·· "2) , -4(; R -c2 +AR2 -2RR-2R

(24.20)

(24.21) Thus lhe Lagrangian (24.18) gives lhe correct Equation (24.5) for lhe scale radius R(t). Let us now define lhe momentum conjugate to lhe scale radius R(t) as

PR=

aL

3nc2 •

aR_ = -wRR.

(24.22)

Using (24.22) we may express Rin terms of PR as follows;

k=-3!~1*·

(24.23)

Using Equations (24.22) and (24.23) we obtain lhe Hamiltonian of the present model as follows: . G p~ 3:rrc2 ( , A ') H=pRR-L=-J:rrc2fi+4G" Rc--3R .

('.24.24)

The Hamiltonian (24.24) is a classical Hamiltonian of the present minisuperspace model. The quantum-mechanical Hamiltonian operator ii is obtained using the canonical quantization prescription

PR~ iii~,

i

= .J=T.

(24.25)

The wave equation of lhe umverse in lhe present mini-superspace model, known as lhe Wheeler-DeWitt eqUlJtion, is lhen obtained fromlhe condition

270

Chapter 24 Quantum Cosmology

H = 0 as follows:

iJW(R)

=

[!!£_!:___ + Jnc2 (Rc2 - ~R3 )] ~(R) = 0 3n:c2R tJR 4G 3 • 1

[-n' a~'+

U(R)] 'i'(R)

~ 0,

(24.26)

(24.27)

with the potential U(R) defined by

The Wheeler-DeWitt equation written as in (24.27) has the familiar form of the one-dimensional Schrodinger equation for a particle with zero total energy moving in the potential U(R). The fonn of lhe potential (24.28) clearly indicates that there is a classically forbidden region for 0 < R < Ro and a classically allowed region for R > Ro. The point R =Ro is a turning point of the potential (24.28). Classically speaking and using the particle analogy, we conclude that a particle at R = 0 is stuck there and cannot travel to the region R > Ru. However, in quantum mechanics there is a finite probability that the particle can tunnel through the barrier and emeige at the classical turning point R = Ro. Thereafter it can evolve classically in the region R > Ro.

~ The Wave Function of the Universe In order to find lhe solutions oflhe Wheeler-DeWitt equation for the wave function of the universe W(R) and to calculate lhe tunneling probability of lhe universe through the potential barrier (24.28). we restructure Equation (24.TJ) as follows: d1 W dzl +Q2(z>W=O.

(24.29)

where z = RfRo is a dimensionless scale variable and the function Q 2
a= 3nc R~ 21iG

=

9rrc5 . 21iGA

(24.30)

Section 24.3 The Wave Function of the Universe

271

Using the Wentzet-Kramers-Brillouin (WKB) approximation, the wave function of the universe W(z) is given by W(z)

= {iNolQ(z)l-112 explw(z)I.

NolQ(z)l- 1 /2 cos(lw(z)I-~).

z< l z> I

(R
(R>Ro)

(2 .3l) 4

where No is the WKB normalization constant. which is not essential for the present analysis. The function w(z) is given by

a ( z2 - l )''' w(z)= [ i:Q(z)dz=J

(24.32)

For example, in lhe region where R »Ro the wave function (24.31) becomes

W(z)=Noa-•t2 ~cos(jz 3 --i)-

(24.33)

and it is a slowly falling periodic function. Let us now turn to the tunneling probability forthe universe to make a transition fromR = Oto lhetransition point R = Ro. The most general WKB result for the tunneling probability is given by

T~ [1+exr(z £1 Q(z)dz)T' ~ [i+exp(-¥)f

(24.34)

Substituting a from lhe result (2430) into (24.34), we finally obtain (24.35) It should be noted that in many quantum-mechanical problems lhe exponential in the expression foe the tunneling probability (24.34) is a large quantity. Jt is therefore customary in the litecature to use the approximate expression

However, since the gravitational action integral is a negative quantity, such an approximation is invalid in the present discussion and only the more general result (24.34) can be used. Even lhough lhe energy derEity Pvac and thereby the cosmological constant A of the vacuum--dOOiinated Planck epoch are very difficult to estimate, in order to get the idea of the ocders of magnitude of these quantities let us assume lhat the quantity a given

272

Chapte1 24 Quantum Cosmology

by (24.30) is of the order of unity. This gives the tunneling probability of the order of T ~ 2/3. Using the result (24.30). we obtain

9rrc 5 a= 21iGA ~ l

=)-

9nc 5 l A ,...., 21iG = 4.87 x 10+87 if.·

(24.37)

Now, u.~ing A= 8rrCP.,,ac. we have Pvac

= S:G,...., 2.9 X 10+96 ~.

(24.38)

The result (24.38) indicates an enormous energy density of the universe in the vacuum-dominated Planck epoch. Furthermore, using lhe result (24.7) we can estimate the radial distance of lhe turning point of lhe potential (24.28) as follows:

Ro=

cto = cA ,. .,

7.44 x 10-36 m.

(24.39)

The length scale of this order of magnitude is sometimes called the Planck length. The time scale to corresponding to this length scale is then given by to

=

A~

2.48 x 10-44 s.

(24.40)

and it is roughly to :::; 10-43 s, which is in agreement with our assertion in the introduction to the present chapter. A remarkable, and surprising, property of the wave function of the universe is thefactthatitis timeindependentanddependsonlyon thespacetirne geometry and the matter field content. A possible interpretation of the wave function W(R) is that it measures probabilistic correlations between lhe geometry and the matter field content. In this interpretation it is possible to use some function of the matter fields, e.g., the energy density of lhe matter fields, as a surrogate time variable. However, neither the role of time in the quantum cosmology nor lhe interpretation of the wave function of the universe is fully understood yet. We also note lhat just having the wave equation of the universe does not resolve lhe issue of the quantum evolution of the universe. To be more specific about the quantum evolution of the universe, we need information about lhe initial quantum state. There are cwrently several proposals for such an initial quantum state giving different physical predictions. If such predictions can be made to be sufficiently precise, lhen the present-day observations could be used, at least in principle, to test these different proposals for the initial quantum state of the early universe. The quantum cosmology efforts around the world today are very ambitious and lhe very limited presentation given in the present chapter only

Section 24.3 The Wave Function of the Universe

273

gives some flavor of this exciting subject. It is our hope that after mastering the material discussed in this chapter. an interested reader will be able to proceed to more advancedmonograph.~on the su~ect. Jtisalsoimportantto note tlrat despite a number of speculative efforts to create a self-consistent and empirically supported cosmological quantum lheory, a lot of work remains to fulfill that goal, and several fundamental questions still remain unresolved.

Bibliography

[1] M. Beny, Principles of Cosmology and Grm>itation. Adam Hilger, Bristol, 1989. [2] M. Carmeli, Classical Fields: General Relativity and GQlJ.ge Theory. Wiley, New York, 1982. [3] N. Dalarsson, "Phase-Integral Approach lO the Wave Equation of the Universe." Modern Physics Letters A Vol. 10, No. 33, 1995. [4] A. Einstein, "Die Grundlage der allgemeinen Relativitatslbeorie." Annalen der Physik Vol. 49, 1916. [5] J. Foster and J. D. Nightingale, A Shon Course in General Relativity. Longman Scientific and Technical, Harlow Essex, 1979. [6] E.W. Kolb and M. S. Turner, The Early Universe. Addison-Wesley, Reading, Massachusetts, I 990. [7] C. Moller, The Theory of Relativity. Oxford University Press, London, 1952. [8] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields. Addison-Wesley, Reading, Massachusetts, 1959. [9] W. Panofsky and M. Phillips, Oassical Electricity o.nd Magnetism. Addison-Wesley, Reading, Massachusetts, 1955. [JO] W. Rindler, Essential RelaJiJ>ity: Special, General o.nd Cosmological. Springer-Verlag, Berlin, 1977. [11] J_ Stewart, Advanced Genera!Relativity. Cambridge University Press, Cambridge, U.K., 1991. [12] N_ Straumann, General Relativity o.nd Relatnistic Astrophysics. Springer-Verlag, Berlin, 1981.

275

276

Bibliography

[131 R. C. Tolman, Relativity, Thermodynamics and Cosmology. Oarendon Press, Oxford, 1934. [141 R M. Wald, General Relatinty. The University of O:ticago Press, Chicago, 1984. [151 S. Weinberg, Cravnation and Cosmology. W11ey, New York, 1972.

Index A Absolutederlwtives,

ufvectors,69

...,__

uftensors,69--70

integral,90 ufelectroniagneticfield,151 Arc length,

ufcurves,48-9 element,48 Associated tensors,

vectocs.46-7 tensms..46-8

Continuity equation, ofelectriccharge,149 ufmass, 185 Coonfinates. ufsystems,6 Descartrs.18-19 octbogooat,51 physical, 52-3 spberical,19 Cosmic, backgrcund radiation. 225 dynamics,239--52 fluid,227 l:ime,234 Cosmological, principle,T27

redsluft,235-7

B

scale,225 Eliu:kholes, mini,222 region,206 supmnassive. 222

c Clui5mffetsymbols, ufthefirstkind,63 of the second kind, 63 Conse:wdun taw uf; enagy-nDillt:Iltllm, 130

angularmomentum, 131 electric charge, 149

mass,185

Guwrure, constant,233 Curvature tensor, alleru&ive expression, 102 covariant, IOI cyclicidenUry, IOI mixed, IOI

D Deceleration, paramru:r,237 Densily, charge, 148 critical(closure),254

2n

276 current,149 Lagrangian, 149

EE

eigenvectors,55

orthonormal,57 Einstein,

--deSiuersolufmn,255 field equations, 188-91 tensor,239--50 Electromagnetic, current vector, 147-9 field invariants, t45 field tensor, 137 Energy, momentum,126 restencrgy,127 totaienergy.126 ~'--ljI()(j1elj{ler!SOr.

uf~tterdistribution,187

&juatiunufmotiun,

""""'·

condition,141

ufthetheory, 141 Geodesic,

lines.89-95 equatio,n,89.92-5 Geometry. pseudo-.Euclidean,166 pseudo-Rlemannian, 166 Gravitational, constant, 173 field, 165-76 fieldpotentials, 166 potential, 171

H Hamiltonian, l'..-aperalDr, 79

oper.ttor.269 vaiiational principle, 90 Hubble,

ufafreep:micte.124

COil5tant,226

k:n'lpl.mll,216

law,225,226

Epoch, matter-domina1ed, 252 Plrurl.266 radiation-dominated, 266 vacwrrn-donfumk:d, '166-7 Events, intervalbetv."eell.112-16 horlzon,219

Indices, durmny indices, IO towerindices,5-6

upperindii..-'t'S..5-6 F Fluxut

eta.-"tricc:harge, 148 mass, 185 Friedmann, equations, 239. 250--2

Inertr.llsystems,

local. 167 ufreference,111 Interai...'liun, gravilationa1, 165 maximum speed u( 112

G Gauss, lheorem,85,86-7 ennature, 106

J:u..-Wian, oftransformation,33 Jacobitheorern.87

279 K

nonrelativistic limit, 173

universe,2ttl Kinetic,

Nonrebtivililiciimit, of:>Ctionintegral,125 ofkineticener,gy.128 of Lagrangian. 125 of momentum, 128 Newtunian,173

energy,128 energy-momentum tensor, 186 L Lagrangian, tx1uation,89-92 function,90 of charged particle, L16 Li.nearopemtocs. eigenv-.lluesof,56 maindin.'.Cb.unsof,58

u

._.,Th, commutator of, I 03 q:ieratt:rexpression, 103

secularequationof.56

Light,

cones,220 speedof, 112 Lorenu. condition, 141 force, 137

gauge,141 transformation, 116-19

Penhellon, advance, 207-L'i of a planet, 207

shtft,215 Potential, electrooiagl;eac,135

electricscalar,135

M

magneticvector,135 :r.latrix. adjunctmatrix,45

cofactorsof,45 columnmatrix,5

detemtlnantof.13 l:ramposedmatrix,45 unitmatrix,45 Metric. rnetnc,18 Scbwau:schild, 204, 205 Robertson-Walker, 225-37 Metric~

Euclidean,18 pseudo-Euclidean, 22 pseudo-Riemannian, 22 Riemannian,18-22

N Newmnian, dust,260 Newton law, 193--5

Q Quantum, cosmology, 265-73

theory,265

R Riccr.

scalar,105 tensor,104

Scale, radius,234 Se:llars, pseudoscalars, 42 scalar product, 18

280 Space-time. coordi.Dat.es, 114 geometry, 165 metric of, 165 Systems, first-ordersystem,6 ofreference,111 second-ordersystem,6 third-order system, 7 rubitrary-ordersystem, 7--S

T Tensors, tensorcapaciry,36 tensor density, 35 'fcamfonnaUon,

Index

expansionof.225 tmge-sc-.ik:structure,225 lifespanof.257 matter-dotmnated. 252 nonstatic model of, 239 radiation-dominated, 252 wave function of, 2711--3

v Vectors, axialvectors,42 curlvector,86 ofelectricfield,137

of rnagneUc induction, 137 polar vectors, 42

surfacevector,86

Gallilei,118 Iaw,23 ofcoordmates,23 ofeneqzy-momencum, 128--30

tan,gentunitvectors,49 uilltvectors.49

w

u Unit systems, ll-symbol,11 e-symboi, 12 Universe. darkrnatierin,259

Wheeler-De Witt, equation,266--70 World,

horizon,226 line,113 point,112