Eindhoven University of Technology

MASTER

A critical study on electrodynamics in non-inertial frames

Kerkenaar, R.W.P.

Award date:1974

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TEe H N I S C H E HOG ESC H 0 0 LEI N D H 0 V E N

A F DEL I N G D ERE L E K T ROT E C H N I E K

A critical study on electrodynamicsin non-inertial frames,doorR. Kerkenaa r.in de periode sept. 1972-juni 1974.

ET-7-1974

This study is performed infulfilment of the requirementsof the degree of Master ofScience (ir) at the EindhovenUniversity of Technology undersupervision of ir. C. Kooy.

Eindhoven, juni 1974.

TECHNISCHE HOGESCHOOL

EINDHOvu..J

STUDiEZilOllOTHEEK

ELEKT j-:OTl.':CritJL:I<

1

CONTENTS

List of referencesIntroductionAbstract

467

CHAPTER 1 CONCEPTS 8

464854

9-1011131313131414141616181819212222222428303032353535

tensors

3738414145

transformations

IntroductionConclusionsNotesMESH-SYS1'EMSDimensionalityLocation processMerit of a mesh-systemRelative character of co-ordinatesOrientation of a figureDistance as a functamental conceptQUADRATIC FORMSSpe~ial quadratic formsGeneral formIsome tryKinds of spaceGeodesicsSPACE-TUmThird fundamental hypothesisMeasurement of intervalRectangular co-ordinates and timeImmediate conciousness of timeFundamental velocityReal and pseudo-gravityKinds of space-timeTRANSFCRHATIONSBase-vectorsTensors~chematic connection between bases, vectors, transitionand fundamental tensorsSome important properties of vector componentsPhysical observables anct tensor quantitiesLorentz transformationsLocalization of tensors to observablesConnection between Lorentz transformations, tensorand localizationsGraphical representationsTime order

§ 1 •1 .11.21 .31 .41.51 .62.2.12.22.32.42.53.3.13.23.33.43.53.63.74.4.1L~. 2

4.3

4.44.54.64.74.8

4.94.10

Cl1AFTER 2

§ 1.2.12.2

MAXWELL EQUATIONS

IntrodutionConclusionsNote'3Geometrized unitsMaxwell-Einstein tensor equations for vacuumRelative tensors, generalized Kronecker symbol,

57

58595()606164

5.

6.16.26.3

permutation symbol and oriented tensorsDual tensorsMaxwell-Einstein tensor equation for macroscopic mediaScalar and three-vector potentialFour-vector potentialAnother form of Maxwell-Einstein tensor equations formacroscopic mediaLocalization properties of comoving framesLocal Maxwell equations for comoving framesLocal Maxwell equations for comoving frames expressedby local differential operatorsLocalization properties of synchronous framesLocal Maxwell equations for comoving synchronous frames

26466686971

727576

808384

CHAPTER 3 CONSTITUTIVE RELATIONS

868687888890939697

116

117117120

114

112

100100103107111

mediamediamediamedia

isotropicisotropicisotropicisotropicfor local

))

E,B2,13

IntroductionConclusionsNotesCONSTITUTIVE TENSORS OF LINEAR MEDIALocal tensorial descriptionPermittivity part of the constitutive four-tensorPermeability part of the constit~tive four-tensorConstitutive tensors of linear mediaCONSTITdTIVE rENsr HS OF LOCAL ISOTROPIC MEDIAcrNSTITUTIVE LOCAL THREE-VECTOR ~QUATIONS FOR LOCALISOTROPIC MEDIADerivation of D = function E,B) forDerivation of IT = function B,E) forDerivation of D function E,H) forDerivation of B = function ( H,E ) forConstitutive local three-vector equationsisotropic media moving rectilinearConstitutive three-vector equations for local isotropicmedia moving uniform and rectilinearConstitutive local three-vector equations for localisotropic media moving rectilinear with constant properacceleration 114Constitutive matrix for local em fields for local isotropicmediaCURHENT-FH~LD CONSTITUTIVE R,LATIONS FOR RECTILINEARMOVING MEDIADerivation of J = functionDerivation of J = function

§ 1 •1 • 1.1 .21 .31 .42..3.

3.113.123.213.223.31

3.32

3.33

3.4

4.

4.14.2

CHAPTER 4 POTEN'rI ALS 122

§ 1.2.

3.

IntroductionConclusionsNotesGeneral wave equation in terms of potentialsGeneral wave equation in terms of potentials forlocal isotropic mediaThree-vector wave equation as a special case of thegeneral wave equation for local isotropic media

123123123124125

132

CHAPTER 5 MEDIA WITH CONSTANT PROPER ACCEl.ERATHjN133

§ 1.1 .1

IntrodutionConclusionsNotesGEm'JETRYHyperbolic motion in flat space-time

13L~

1341351 j6

136

3

140141142

ll~4

1461 1-+6147

frame 1 C -'.JC

155155

laboratory

156

The metric matrix of medium coaccelerating frameInstantaneeous Lorent3 transformationLocalization of base four-vectors to coaccelerating fra~es

Graphical repre88n~Rtion of laboratory and coacceleratingmesh-systemsMAX'viELL EQUA'nONSTensors of the coacceleratinR frameLocal Maxwell ~quations of coaccelerating framesThree-vector local Maxwell equations of coacceler~ting

CONSfrIfrUrrrVE RELAfrIONSConstitutive tensors of the coaccelerating frameD = function ( E,B ), H = function ( B,E ) in theframe

§ 1.21.31. Ii

1.5

2.2.12.22.33.3.13.2

CHAPTEH 6

§ 1.1.11.21.31. 11? .3.

CHAPTEH 7

IntroductionConclusionsNotesGEOMEfrRYSteady rotational motion in flat space-timeThe metric matrix of the mediu~ corotating frameLocalization of laboratory frameInstantaneous Lorentz transformationsMAXWELL EQUATIONJCONSfrITUfrIVE RELrlrnONS

CURREi~T Dl~)l'RIBUTI()NS

159159160161161162164165

167

S 1.2.3.4.

IntroductionConclusionsNotesInertial current differential equationsCurrent distribution in a flat moving conductorAddendum to para~rHph 2Current distribucion in a rotating conducting cylinder

168168169170173179186

List of references

1 Becker R.Electromagnetic fields and interactionsvol. 1, Electromagnetic theory and relativityBlaisdell pub. com., New York, 1q64(LGGG 64-7001)

2 Eddington A.Space time and gravitationCambridge University Press, 1966 (1920)

3 Eddington A.The mathematical theory of relativityGambrid~e Urliversity Press, 1954

5 Einstein A.The meaning of relativityPrinceton University Press, Princeton, 1955

4 Einstein A.,Lorentz H., Weyl H., Minkowsky H.The principle of relativityDover, New York, 1952 (1q23)(lGGG A52-9845)

6 Kooy G.Electromar,netische krachten en bewegende mediaTechnische Hogeschool Eindhoven, 1973( lecture notes nr. 5.724.0)

7,8a Mo T.G.Theory of electrodynamics of media in non-inertial framesand applicationsCalifornia Institute of Technology, Pasadena, California, 1 0 69(technical report 48, thesis)

8 Mo T.G.Theory of electrodynamics of media in non-inertial framesand applicationsJorn. of Math. Physi~s, vol 11, nr. 8, aug. 1970

9 Rindler W.Hyperbolic motion in curved space-timePhys. Rev., vol. 199, nr. 6, sept. 1060

10 Hohrlich F.Classical charged particlesAddison-Wesley, Readjng, Massachusetts, 1q65(LGGG nr. 65-10932)

11 Sears F.W., Brehme R.~.

Introdution to the theory of relativityAddison-Wesley, Reading, Massachusetts, 1q68(LGGG nr. 68-19344)

12 Seidel J.Tensor rekeningTechnishe Hogeschool Eindhoven, 1969(lecture notes nr. 2j7)

13 Singh J.Great ideas and theories of modern cosmologyDover, New York, 1961Moderne kosmologieAula, Utrecht, 1967

14 Synge J.L.Talking about relativityNorth-Holland publ. com., London, 1970

15 Synge J.L., Schil~ A.Tensor calculus,University of Toronto Press, 1956

16 Synge J.L.Relativity, the special theoryNorth-Holland publ. com., Amsterdam, 1Q60

17 Synge J.L.Relativity, the general theoryNorth-Holland publ. com., Amsterdam, 1960

18 Zwikker C.Fysische materiaalkJndevol. 1Wetenschappelijke Uitgeverij, Amsterdam, 1966

19 Adler I.Nieuwe wiskundePrisma, Utrecht, 1966

20 Moon P., Spencer D.Field theory handbookSpringer Verlag, Heidelberg, 1961

21 Post E.J.Formal structure of electromagneticsNorth-Holland publ. com., Amsterdam, 1962

22 Tai C.T.Present vieuws on electrodynamics of moving mediaRadio Science, vol. 2, nr. 2, febr. 19(7(IEEE international convention, 1966)

~3 Gradshteyn I.S., Ryshik I.M.Tables of series, products, and inte~rals

Academic Press, New York,1965(LCCC nr. 65-290~7)

5

6

Introduction

This paper is in fact a collection of derivations of formulaeoccuring in a part of Mo's article 8 , and so it does not bringup anything new but concentrates on the concepts and methodsbehind the mentioned article. Though written with detailedargumentationit is not suitable for shallow reading but is intended as atechnical paper gliding those who want to make a serious studyof Mo's paper. That is the reason why most derivations arealso given so explicitely thus having two more advantages:1. derivations become easily clearer when reading thus avoiding

the use of pen and paper and2. the same is valid when checking correctness whileeventual mistakes

can be pointed out and traced e8sily.

For an electrotechnical student most concepts connected with thatarticle slch as fla t and cllrved space-time, special and p;eneralrelativity, localizations are unknown and in the first chapterwe shall give a brief introduction into these and other subjectsmos tly re ffering to Eddington [2] , [3], Eins tein [4], Sears ~11~ and }Synge [14] • rrhis in troduc tion is ra ther ma thema tical and di fficu"tiesconnected with the physical construction of inertial framesand transport of co-ordinates in a curved space-time are de­liberately dodged, by chosing examples in flat space-time (ch. 5).On the other hand concepts of metric and localization are stressedand these, certainly the last, are in my opinion usually treatedwith nonchalance.

The first part of this theoretical technical paper (ch.2,3,4)pays attention mainly to the behaviour of Maxwell laws andconnecting constitutive relations gouverning the electromagneticfields in (linear) media, when transformed from one co-ordinate(e.g. laboratory) frame into another (e.g. medium comoving)frame. Starting from the conviction that1. the familiar Maxwell equations (e.g. v.B=O,vx.E=-~B) are

valid in every inertial frame of which kind we usuallythink the laboratory frame to belong, and

2. the familiar simple constitutive equations for linearmedia (e.g. n=EE'~=fR) are valid in every inertial frame whichis inRtantaneously comoving with the medium (inertialmedium comoving frame) when accelerated sttins in themedium are neglected,

we show that mathematical symbols (tensors) can and will beconstructed such that transformations from one frame to an­other is quite simple.

From this we postdlate the "'eneral covadant !'1axwell-Einsteinte nsor eq:<a tions which hold good for every arbi trary frameincluding non-inertial ones. Thoup;h tensor efJuation arpearto be simple and mathem~tically ele~ant tensors as mathematicalsymbols have no direct physical significance and thereforewe derive from the tensor equations local Maxwell equations

7which combine the familiar field three-vectors (e.g.E,B)and an arbitrary chosen co-ordinate frame, usually non-inertial.Because of the non-inertialness we often arrive at str8ngeeffects in those local equations such as for instance (5.51a)~.B/O, which are due to the fact the c~osen co-ordinatesare not purely spatially nor as pure time.

A constitltive four-tensor, constructeJ by Mo for the firsttime, is also derived here and its importance lies in its tensorcharacter. ~e postulate its general covariance. Much effort isto derive from this tensor four t!'ree-vf'ctor constit,ltive relationsfor loca! isotropic media, partly as a check on the constr~~ted

four-tensor knowing the Same relations from Tai[2~ in the caseof rectilinear motions, and partly because these relations areno~ known for arbitrary motions. Interesting similaritiesappear between these four relations and they also show themixin,,; up of ,u.. and f. and also of two field vec tors. Restric tion torectilinear motion shows constitutive three-vector relationsare similar for all kinds of motion when the instantaneous velocityis substituted. A constitutive three-tens~is notfound connectingtrue currents and fields, so when deriving three-vector equationsexplicite use of a Lorentz transfor~ation has to be made.

In chapter four a somewhat academic subject is treated: the waveequation in terms of potentials in curved space-time.It showssome resemblance to our familiar wave equation, which is ofcourse a very special case of it ; a galge condjtion cares onlyse cond and null order deriva tives~ and the curva telre of space ­time considered is separated.

The behaviour of boundary conditions is not treated but it isobvious they behave much like constitutive relations.

The midpart of this paper (ch. 5) considers media with constantproper acceleration and is only intended as an example ofapplication of the topicsof the chapters 1-3, because of itssimpleness.

In the last part of the paper current distributions in a verysimplyfied model of a ball bearing motor consisting of1.a plane conducting~plate extending to infinity and fed by

constant current point source and2.a c0nducting cylinder extending to infinity and fed by a

constant current line source.The distribution in the first configuration is solved but forthe second only tIe differential equations are supplied yet.

ABSTHACT

In this report~a study of basic concepts and equations in relativisticelectrodynamics in non-inertial frames is given, to provide a soundbasis for theoretical research in technical systems, consisting ofparts in relative and non-uniform motion. To the important conceptof localization a great deal of attention is given.The general theoryis applied to the special cases of constant linear acceleration andsteady rotational motion.

ct. 1 -,

CHAPTER 1

CONCi:<::PTS

1. l·mSH-SYSTENS

1.1 Dimensionality1.2 Location process1.3 Merit of a mesh-system1.4 Relative character of co-ordinates1.5 Orientation of a figure1.6 Distance as a fundamental concept.

2. QUADRATIC FO~lS

2.1 Special quadratic forms2.2 General form2.3 Isometry2.4 Kinds of space2.5 Geodesics

3. SPACE-TIME

3.1 Third fundamental hypothesis3.2 Measurement of interval3.3 Redangular co-ordinates and time3.4 Immediate conciousness of time3.5 Fundamental velocity3.6 Real and pseudo-gravity3.7 Kinds of space-time

4. TRANSFORMATIONS

4.1 Base-vectors4.2 Tensors4.3 Schematic connection hetween bases, vectors, transition tensor

and fundamental tensors4.4 Some important properties of vector components4.5 Physical observables and tensor quantities4.6 Lorentz transformations4.7 Localization of tensors to observables4.8 Connection between Lorentz transformationu, tensor transformations

and localizations4.9 Graphical representations4.1CTime order

eY-t.1 -<.

Introduction

Basical concepts such as mesh-systems, quadraticforms, metricmatrices, space-time, and transformations are introduced in avery popular way. Emphasis is laid on quadratic forms anddirectly connected with these are the metric matrices whichplaya fundamental role of the first order becausa they completelydetermine the geometry of the considered space-time. The pre­eminance of the Minkowsky metric diag.(-1,1,1 ,1) is explainedand connected to the concept of an inertial frame.Localizationplays a second fundamental role and is connected with physicalobservability. The relation between Minkowsky metric, localization,Lorentz and tensor transfomations is supplied and appears tobe very use full •

The differeuce between real and pseudo gravity is demonstratedwith aid of a simple quotation of Sears but though being simpleit demonstrates the introdution of acceleration by technicalmeans does not affect the kind of space-time considered.Thisoften appears to be very confusing.

~aragraphs 4.1-4.5 give an indication of the small amount ofknowledge of tensor calculus which is required and also somespecial properties of tensors which appear to be usefull.

The last part of this chapter (4.9- ) is dedicated to graphicalrepresentations illustrating:1. the differences between contra and covariant components2. Lorentz and tensor trans-formations3. an elegant way of drawing a Minkowsky diagram.

9

vh 1-3

Conclusions

1. Special relativity does not only restrict itself tophenomena with constant velocity but is applicableto all phenomena in flat space-time thus extendingits validity to arbitrary motions considering instantaneousvelocities.

2. Intervals are fundamental 1 co-ordinates less. Clocks andscales always measure intervals in any situation.

3. Quadratic forms, metrics 1 localizations are fundamentalconcepts of which metric is the most important.

4. Only in Minkowsky frames we can measure physical quantitiesas we are accostomed.

5. It is possible and preferable to explore a n-dimensionalspace (-time) without considering higherdimensions.

6. The class of admissable co-ordinate systems overlayinga manifold is restricted by isometry.

7. Gravity acn be ca~sed by two sources: matter and acceleratedframes.

8. The Riemann-Christoffel tensor discriminates differentkinds of spaces (space-times).

9. Tensors are mostly not physical observable!

'0

10. Lorentz transformations connect only frames with Minkowskymetrices.Localizations transform frames with a Minkowsky metric intoframes with an arbitrary metric and vice versa.Tensor tranformations connect frames with arbitrary metrices.

11. When drawing a Minkowsky diagram in the familiar wayscale factors of both frames are different. We canavoid this.

cR.1- ~

Notes on chapter 1.

1. cf. Eddington[2] , p.772. cf. Eddington[3J, p.8-93. cf. Eddington[2J, p.874. geodetics: chap [1J, 2.55. cf. Eddington[3J, p.96. idem7. 40 other examples of orthogonal mesh-systems are given

by Moon and Spencer[20j8. cf. Eddington [2] , p. 799. idem, p.8010. idem11. cf. Seidel [12) , p.9,4012. idem13. cf. Eddington[2J, p.8014. Seidel [12J , p.41, gives the exaple of the isometry between

a helix surface and and the revolution figure of a chain line15. cf. Eddington[2] , p.8116. cf. Eddington[2] , p.S117. cf. Eddington[2] , p.8418. cf. Eddington[2] , p.8519. cf. Eddington[2] , p.8120. cf. Eddington[2] , p.89

cf. Eddington[3] , p.16, p.76: g's have to be constantscf. Eddington[~ , p.76, the vanishing of the Riemann­Christoffel tensor is a necessary and sufficient conditionfor flat space-timecf. Einstein[4], p.1/t1: same remark

21. cf. Eddington[~ , p.72: footnote22. cf. Eddington[~ , p.8323. cf. Synge & Schild [15J, p.37-4724. time and space are used in the Newtonian sense25. cf. Synge [14J , p.51 -5226. c f. Synge [14J, p. 5427. c f. Eddington [3J , p1128. idem29. quotation Eddington[3J, p.1230. quotation Eddington[3J, p.13-1631. because ds is a real measured quantity32. cf. Synge [16J , p.17-1 833. cf. Eddington[2], p.8334. cf. Mo[8J, p.259135. quotation Eddington[3] ,po 23-2436. cf. Synge (14J37. quotation Eddington~p.18-19

38. quotation Sears & BrehmeB~ p.184-18539. partial quotations from Eddington[~ , p.83-9240. Eddington~: note 5, see also paragraph 2.441. cf. MoC8~ p.259142. quotation Seidel~t p.843. summary of Seidel [12] , p.6-1144. cf. Einstein[4J, p.12245. cf. Zwikker [18J, p.10946. cf. Einstein[4], p.13347. cf. Seidel[12] , p.4748. cf. Seidel [12], p.13

cf, Einstein[4] , p.13 6

11

ch.1-s

49. see paragraph 2.450. cf. Mo~, p.259151. cf. Eddington~, p.1652. cf. Mo[S], p.259153. quotation Eddington[~ , p.17-1854. nonorthogonal space-time frames are meant55. cf. paragraph 3.6,56. Mo[8]57. cf.Mo[8], p.2591 , p.258958. cLseidel[12] , p.359. cf.Zwikker[181, p.1760. cf. Sears and Brehl!le~i.l, p .14 dX61. when covariant components are used : tan 0< = c3.T

the use of contravariant does not give a simple relation62. because we use geometrized units both scales have the same

units: m ; the scalefactor is the quotient of lenghtrepresenting the scale unit on the plot and scale unit

63. cf.Sears and Brehme111, p.1464. time dilatation and lenght contraction are represented

in both diagrams with dashed lines;comparision is easierin the last

65. interval is fundamental; Synge ~~ , p.81: succession ofevents

66. cf.Synge[14J, p.9167. cf.Synge[141 , p.136, p.176

cf.Sears and Brehme [ID , p.46-4768. cf.Synge[14] , p.137

12

§ 1.

§ 1. 1

§ 1.2.

§ 1.3

ch.1 - B

MESH-SYSTEMS

Dimensionality

To specify a point on a two-dimensional surf8ce it requirestwo independent numbers or co-ordinates. In three-dimensionalspace three numbers are required and in n-dimensional spacewe need n numbers. The number of numbers required to specifya point and the order of dimension of the space in which thepoint hRs to be specified are equivalent concepts. Suppose youknow by experience that the number of numbers needed is m,your conception or experience, then you can define the dimensionas m, your definition, or vice-versa.My aim is just to stressthe e'luivalency.

Location

Suppose you want to specify (locRte) a point A in a three­dimpnsional jelly. The procedure by which this locAtion iscarried out is very well known. Dividing the jelly into meshesby any three systems of two-dimensional slrfaces which crossone another and attaching consecutive numbers to the channelsbetween the surfaces, one nJmber from each system will specifya particular mesh. Every accuracy can be attained by makingthe subdivision sufficiently fine~

Note that the four following aspects are essential in thislocation process:

1. the way of dividing is arbitrary,2. the order of diBensionality of a dividing-system is

one lower than the system that has to be divided,3. the dividing systems have to be independent: they

cross each other,4. every accuracy can be attained.

Merit of a mesh-system

A well known example of the foregoing procedure is the systemof latitude and lon~itude meridians used on earth maps, and bytakin~ a good look at those maps all the essential featuresof the location process can be noticed. You can clearly seethat a mesh-system is somethin~ overlaid by an observer. onthe external world. It is a fictitious and arbitrary structureof great utility and comfort in describin~ phenomena of thatworld but has little to do with its intrinsic properties~This is also to be seen with the example of the jelly. Thejelly can be deformed as one likes but the location of pointA by a particular mesh-system remains undisturhed~

13

§ 1.L1

§ 1. 5

§ 1.8

e-1. 'I -7

Relative character of co-ordin3tes

Just suppling someone with a set of three numbers or co­ordinatps in order to locate point A in the jelly is notsufficient for him to point out A. You also have to tellhim what kind of mesh-system you have used. That is to say:you have to give him a clear description how you dividedthe jelly. So co-ordinates and mesh-system are totally inter­linked and should always be mentioned together. Puttin~ itin another way: co-ordinates are always relative to a particularmesh-system.

Orientation

Suppose you want to discuss the properties of a triangleformed by the cornerpoints A,B,C of which you have giventhe co-ordinates and a specification of the mesh-systemto which they belong. Now you are able to construct a tri­angle of geodetics4 with intersectingpoints A,B,C. If youare only interested in the fi~ure itself, thus in the con­figuration of points, the information so supplied is morethan complete because it specifies an arbitrary element,the orientation, which is irrelevant to the intrinsicpropeties of the considered figure and ought to be castaside from a description of those properties~

Distance as a fundamental concept

By stating the distances between the cornerpoints of thetriangle you can give a complete description of the f'gurewithout introducing superflous information like orientA-tionor anything else irrelevant to the intrinsic properties ofthe fig~re! The description by means of distances containsless information than the description by me~ns of locations.So, contrA-ry to the common ideas I state that distance isa more fundamental concepL than location; distance is afundamental concept and location is a computational resultorginating from the distances to other points. As Eddington[3J ,p 10, pu t sit:"Our ultimate analysis of space leads us not to a "here"and a "there" but to an extension such as that which relates"here" and "t here" It.

So space is not a lot of points close together; it is alot of distances interlocked. and accordin~ to the sameauthor our first fundamentai hypothesis has tolbe:

1~

0h.1-8 15

"Everything connected with loc8tion which enters into observationRlknowledp;e - everything we can know about the configuration ofpoints - is contained in a relation of extension between pairsof points".

This relation is called the line element or interval and itsmeaS'lre is denoted by "ds". Note that distance is a more~hsol~te thin~ tl·an location because the distance between twopoints can be stated without referring to any mesh-systemwhile location is always relative to a certain mesh-system.

A ",. ( h,3)

fig. 1: arbitrary mesh-system on a jelly surface

c~., -91G

§ 2. QUADRATIC FORMS

§ 2.1 Special quadratic forms

We now posses two methods of decribin~ configurations ofpoints namely the use of co-ordinates and the use of distance.'ilantinl'; to use both methods, the first because of its concisenessand the second for its absolute si~nificance, it is importantto find a formula that connects both. Great simDlification isobtained in these formulae if we content ourselves to veryshort distances e.g. distances within a mesh. By inte~rating

we can extend our results to lon~ distances. Some of theseformulae for short distances are very well known in ordinary7 v

geometry, e.IT.:for

rectangularpolarobliquespherical

co-ordinatesco-ordinatesco-ordinatesco-ordinates

The formula is fundamental we just define a rectangularmesh-system (x,y) hy making the division so that the co-ordinates(x, y) fi t the formula ds~::::: 1 dx.t -1- 1 dy'" and so on. Theformula is the only possihle test for,the kind of co-ordinates, the mesh-system, you are usin~.

In ~ome cases an observer adopts his mesh-system blindlyand lon~ afterwards discovers by accurate measures that dsdoes not fit for the ass~med formula and so concludes thathis mesh-system is not of the supposed nature~Consider forinstance the case of measurements on the earth's (spherical)surface and the mistaken adoption of a rectan~ular (flat)mesh-sys tern .

co-ordinatesco-ordinatesco-ordinatesco-ordinates

The use of special symbols for co-ordinates anticipates aknowledge which is really derived from the form of the formulae.In order not to ~ive away the Ge~ret pre~aturely it willbe be t ter to use the symbols(x"A,) in all cases. The four kindsof co-ordinates then give respectively the relations:

~~~ : ~ ~:;: d ~ ~:. c4;~s.t ::: 1 d.x 2 _ 2k: x,d.><- .. + 1 ~~/

.l. 1 J 1 2 .l.. .l.:s ~ l).)(, -1-- CC-::J x1 "'t

re eta ngu larpolarobliquespherical

If we have any mesh-system and want to know its nature, wemust make a number of measures of the lenght ds betweenad jace n t poin ts (xi' x.) and((:'<.,i" dx,),(xl+cL\\Iand te s t wh ic h f ormu lafits these measurements. If for example, we then find thatds~ R.lways equals dx,~i" x,ldx; ,we know tha t our mesh-systemis that in fig-ure ,X,and)<.:I. beinl'; the numbers usually denotedby the polar co-ordinates rande. The statement that polarco-ordinates are beinIT used is unnecessary hecause it addsnothinr; to our knowled~e whjch is net already contained inthe formula. It is merely a matter of ~jvinr; a name; but,of course, t~le name cellls to Olr minds a number of fClmiliqrproperties which otherwise not occur to us!

! ~. 1 -10

I

do; 1

IY--i--

X

dx

\7rectangular co-ordinates

r---

polar co-ordinates

5

oblique co-ordinates

[cJ1] ~ [-~ -~]

spherical co-ordinates

fig 2 special mesh-systems

c:l 1 -11

§ 2.2 Gener~l quadratic form

We now postulate as the ~enera1 form of the line elementthe followin~ quadratic function of the differences of co-

ordinates: d cl../!"" d d.,,'l. j3o( "( ,. ') 52. = :1 T d,x. cl.. X(3 , 5 / = 5 ; 0( J /- = 1,2.

and call it the second fundamental hypothesis. Written outexplicitely it states:" d d

d l. 11 d '" 12 I -+_:2.;I...tS ~ q )<" + 29 Oc,x,l X.z, 3 ><4

and all information ~out the mesh-system is supplied bythe coefficients go(P, which are usually called

'01. potentials,2.(contravariant) components of the first

fundamental tensor,'13.(contravariant) components of the metric

tensor ,u

and which can be determined by making physical meas'lrements.The hypothesis is induced by experience bu t Cddingt on [3] ,paragraph 97, gives a precise reason w~y the quadratic formshould have such a form.

§ 2.3 Is orne try

Althou~h the metric matrix(~11 9'2)\5~' g2:l

at first sight seems to be completely dependent on the arbitrarychoice of the mesh-system and thus seems to be wholly arbitrarythis is evidentely not so. Just let us take a good look atour examples a~ain : the first three relate mesh-systemsto a flat Euclidean surface as the fourth relates a mesh-system to a curved (spherical) surface. If the componentnof the metric tensor satisfy any of the first lhree formulaethen the surface is flat, if the fourth is satisfied thenthe surface is curved like a sphere.

{ z., )( 1-1

= + (2 K.) X<..)

"Z2.- Xl. +-1

(1K:)X 1

[ dz_, = d,x" + (,2 K )-'dx2..J dz =:: dx"-. + -lei

2-( 2 Ie ) X 1

1 t

= -V x/'+ X~= arcl:a..n- (X;l. X

1- 1 )

_ x1dxl +x~d.x.>..

:t~.~t1-~~~X~ 1- X~

{/,Y,tI

with

Ij~easily fol10';/8

so that

Call the co-ordinates in our four examples respectively:

Ac( • Yo(. Z Ii, U..: ; d, :: 1 , 2, ,then we can achieve the first line element out of the secondand out of the third by chan~ing to new parameters (co-ordjnates)according to :

==

§ Z.~

J.1-11

but it seems to be impossible to find two new parametersfor the fourth example so that the first line element c~n

be achieved.

Try how you will, you cannot draw a mesh-system on a flat(Euclidean) surface which agrees with the fourth formula~~

This is the well known problem mapmakers arrive at whenmappinr-; the earth on a flat sheet of paper instead of ona globe.

There is a third way of speakin~ about the some problem.Measure the distance between two adjacent points (the interval)on a flat sheet of paper and subsequently wrap the paperprecisely round a nphere. To achieve this you have to wrinklethe paper awfully Rnd so the distance measured on the sphericalsurface between those two points has changed in general.

Defintion.Two surfaces are isometric when the line elements are thesame or can be made the same by changinr-; to a sJitable mesh­system of the same dimensionality.

So the first three examples represent isometric surfaces(mesh-systems) but the fourth is clearly different~

Kinds of space

We now come to a point of far reaching importance. The generalquadratic form teaches us not only the character of thearbitrary chosen mesh-system but also classifies the natureof our two-dimensional surface on which the system is drawnand which is completely independent of that system.

So a genuinely two-dimensional beinr-; could not be cognisantof the difference between a plane and a roll; but a spherewould appear a different kind of space. The difference wouldbe reco'1nized hy measurement thus by t}ie formula for theline element~5

We see that there are different kinds of two-dimensionalspace betrayed by different kinds of metrical properties.They are naturally visualized as different surfaces inEuclidean space of three dimensions.

This picture is helpful in some way, bil t perhaps misleading'in others. The metrical relations on a plane are not alteredwhen the paper is rolled into a cylinder or a cone - themeasures being, of course, confined to the two-dimensionalworld represented by the paper and not allowed to take ashort cu t through "space". The formulae apply equally wellto a plane surface or a cylindrical surface or a conicalsurface; and so far as our pictJre draws a distintion betweena plane and a cylinder and a cone, it is misleading~There

is also a second disadvantage of this way of thinking.Later on we want to extrapolate our methods to a four­dimensional space-time and then we should have to embedour four-dimensional world in a ten-dimensional

19

Oh.1- 13

Euclidean space for an exact representation of the geometry.We may well ask whether there is merit in Euclidean geometrysufficient to justify going to such extremes'7The value of thepicture to us is that it enables us to describe importantproperties with common terms like "pucher" and "cu.rvature"instead of technical terms like "differential invariant".We have to be on our guard because analogies based on three­dimensional space do not always apply immediately to many­dimensional space. For instance: four-dimensional space withno CLlrva ture is not the same as flat space !18

So if we are two-dimensional (or four-dimensional) bein~s

we have to restrict ourselves to our two (or four) dimensionalworld and explore it by measurements in that world.

There are many kinds of mesh-systems and many kinds of two­dimensional spaces besides those illustrated in the fourexamples and we should like to discriminnte different kindsof space by the values of the components of the metric tensor.There is no characteristic visible to cursory inspectionwhich s~g~ests why the first three formulae should all b~long

to the same kind of space and the fourth to another one~Discrimination between different kinds of space is possible

by means 0!-R~e R~~:nn~chr~sl<-~f;l r~ntor:L~ If t 1_ ( ~ 1(t )( 1.2. a. ) . f'J dA\ \;J.<rJ exIT" YS:v j + yuo-Ho< 'J J lp.v Ho(q-1(1.26) {~1:= l~(~+re -w)

which is Cle~IY solely derived from the~omponents ofthe metric tensor alone and therefore belon~s to the classof fundamental tensors. If the ~fS of the different mesh­systems satisfy tensor enuations of the form

(1.3) -1«~jJ-vv - ,A,c:,u.vo-

whereA~~v~is the tensof specifin~ a certain kind of spacethen ybu are al10wed to say that those mesh-systems aredrawn in the same kind of space. Thus by meas1lring the g' sof an arbitrary mesh-system in a n-dimensional space youcan make a clear distintion between different kinds ofspace without taking a standpoint of higher dimensionality.

Going on with our examples we should of course like to seea test which the first three examples sat~sfy and the fourthdoes not. The condition for flat space is:

( 1 . -4 ) i=\ : p. '1 0-' - 0

which represents sixt~en mutually dependent second orderpartial differential equ.ations of which only one is independent~OThis is easily to be seen if you know the following symmetryproperties of the Rie~ann-Christoffel tensor:

( 1.5) A - A1. skew symmetry in the last two subscrip\s: 1· c.:,3r~-Al~. ';l'S2.cyclic symmetry in the subscripts: ~'1y-rK,/3JJ'd< T \;Z'j'Qi,/' (0

~riting out the condition in terms of the components of themetric tensor the relation hecomes, accordin~ to Eddin~ton [2.),

~_ ! _51.i._:r=-- ~~L 011 _ ~?3:<_~) +~ -( ~.9-3:11 - 1'1,,9 ~11 - _-3~ ~ ~11)= c'(1;"1\'3,, l; ~-'~x~J '/9 6X1 I dX2l~~ :<, '~5G;<''- 511Vj OX1

Every possible mesh-system in a flat two-dimensional s'lrfaceposse~a metric which has to satisfy this relation. You can~asily see the forth mesh-system does not.

20

,:41

~.1 - 1'" 2.1

It should be clearly understood that:

1. The only way of discoverin~ what kind of space beingdealt wit~ is from the values of the potentials, whichare determined practicRlly by measurement of intervals.

2. Different kinds of potentials (g's) do not necessarilyindicate different kinds of space.

3. All potentials which have the same Riemann-ChristoffelZ~tensor belong to the same space.

§ 2.5 Geodesics

In Euclidean three space we are used to define a straightline between two points as the shortest curve betwep~ thosepoints but difficuties in conception arrive when we~~urselveswhether there are straic;ht lines on a spherical surface.In order to avoid those difficulties we just modify our defintionslighty in such a way that it is applicable in all kindsof space.

DefintionA geodesic is a Clrve whose lenght has a stationary valuewith respect to arbitrary small variations of the curve,the end points bein~ held fixed~3

)o

o

This equation

(1. Gb~s ( ~ W )

a:; ~ (~)(1.6c)\~ d I I :S

QXi + J f) )\~$ _,~ L~;3J ~ cis S L.S a.rcle~ht

Physically a geod~sic is easily visualized by pulling astring thi~ht on a surface between two points. So geodesicsbelong to the intrinsic geometry of a surface because t~e

operation requires that yo'l do not get out of the s·lrface.

So a geodesic is a generalized straight line and the equationof a geodesic is independent of the used mesh-system andis consequently invariant -(in the sense that no particularmesh-system is mentiined when stating the equation).

(1.6Q) (; f. d..s = 0 "' 5 LS QrrL~n5h(.

Astates a simple variational process and res~lts in

)

A moving particle follows a geodesica geodesic null line and this is whytwo objects as more simpler means ofand clocks,(Eddington[],par 15).

OJ

DefintionA geodesicwhich u is

nul] line satiesfies thean arbitrary parameter:

J F ) d"," dAISLa( :3 J d";:~

I

followin~ equation in

o ::: q dA"'dx/;,J~':' d: d'~

Ll av-blrarjtrack and a tight pulseEddington thinks theseexploration than scales

§ 3.

§ 3.1

§ 3.2

~.1-15

SPACE-TIME

Third fundamental hypothesis

When describing physical phenomena we are used to "localize"them by three "space" co-ordinates and one "time" co-ordinateand such a set of four numbers is called an event:4 It iscovenient to have a name for the totality of all possibleevents and we might call it the "universe" but "space-time"is more commonly used.Synge makes one important reservationand that is that the hyphen is not to be removed exceptwith much discussion and precaution for we are not to findourselves talking about space and time separately as if weknew what those words mean~The essential thing is that weknow by experience that a set of four numbers is necessaryand sufficient to localize an event (phenomenom) in theuniverse and we postulate this experimental exrerience bywhat we call our third fundamental hypothesisf

"space-time is four-dimensional".

So an event is to space-time (relativity) what a point isto space (geometry).

By extrapolating the resllts of the fore goin~ paragraphsto our four-dimensional space-time we can

1. consider space-time continuum as a four-dimensionaldifferentiable manifold,

2. overlay space-time by arbitrary and fictitious mesh-systems,

3. take interval (line element) as a fundamental concept,4. postulate the same first and second fundamental hypothesis ,5. define isometry in the same way and6. discriminate between different kinds of space-time by

means of the Riemann-Christoffel tensor.

It should be clear to the reader that the metric tensorplays the same fundamental role in space-time as it playsin ordinary space and it is natural to conc~rate on itsform in order to make a clear distinction between ordinaryfour-dimensional Eu~lidean space and space-time.

Measurement of intervals

Whilst the form of the right hands ide of

(1.8) ds~ = :(PdAo(d")4 , (oCf) = O,1.Z,3

is that required my observation, the insertion of (ds)~onthe left, rather than some other funtion of (ds) is merelya convention. The quantityds is a mp.asure of the interval.It is necesAary to conAider cRrefully how measur&numbersare to be affixed to the different intervals occurin~ in naturein order to particularise the differences between intervals.The relation of extension between two events could be expressedaccording to many plans and to conform to the forege1ng

22

~.1- 16

formula a particular code of measure-numbers must be adopted:7

The comparision of two equal distances AB and CD with a measuringscal~ in two positions necessarily at different times we usuallycons~der as a test of equality of distance, but according to ourfir~t fundamental hy~othesis it is primarily a test of equalityof ~nterval or equal~ty of the same configuration of events andnot of equlity of events :8

In this experiment time is not involved; and we conclude that in spaceconsidered apart from time the test of equality of distance is equality ofinterval. There is thus a one-to-one correspondence of distances and intervals.We may therefore adopt the same measure-number for the interval as is ingeneral use for the distance, thus settling our plan of affixing measure­numbers to intervals. It follows that, when time is not involved, the intervalreduces to the distance.

1t is for this reason that the quadratic form (1. 8) is needed in order toagree with observation, for it is well known that in three dimensions thesquare of the distance between two neighbouring points is a quadraticfunction of their infinitesimal coordinate-differences-a result dependingultimately on the experiment,al law expressed by Euclid I, 47.

When time is involved other appliances are used for measuring intervals.If we have a mechanism capable of cyclic motion, its cycles will measureequal intervals provided the mechanism, its laws of behaviour. and all relevantsurrounding circumstances, remain precisely similar. For the phrase "preciselysimilar" llleans that no observable differences can be detected in the mechanismor its hehaviour; and that, as we have seen, requires that all correspondingintervals should be equal. In particular the interval between the eventsmarking the beginnin~ and end of the cycle is unaltered. Thus a clockprimarily measures equal intervals; it is only under more restricted conditionsthat it also measures the time-coordinate t.

In general any repetition of an operation under similar conditions, but fora different time, place, orientation and velocity (attendant circumstanceswhich have a relative but not an absolute significance-), tests equa.lity ofinterval.

It is obvious from common experience that intervals,which can bemeasured with a clock cannot be measured with a scale, and vice versa. Wehave thus two varieties of intervals, which are prov ided for in the formul"

I, since dB' may be positive or negative and the measure of the intervalwill accordingly be expressed by a real or an imaginary number. Theabbreviated phrase "imaginary interval" must not be. allowed to mislead;there is nothing imaginary in the corresponding relation; it is merely that inour arbitrary code an imaginary number is assigned as its measure-number.We might have adopted a different code, and have taken, for example, the.antilogarithm of ds' as the measure of the interval; in that case space­intervals would have received code-numbers from 1 to 00 , and time-intervals

numbers from 0 to 1. When we encounter V--=-l in our investigations, wemust remember that it has been introduced by our choice of measure-code,and must not think of it as occurring with some mystical significance in theexternal world. .

• They express rellltions to eventR which are not concerned in the teRt, e.g. to the sun lind

"tllrs.

" 2.9•

§ 3.3

vn. 1- 1/

Rectangular co-ordinates and time

" SupJ>08e that we have a small region of the world throughout which theg'8 can be treated 88 constants-. In that case the right-hand side of canbe broken up into the sum of four squares, admitting imaginary coefficientsif necessary. Thus writing

'!/I- U<:Ittl + a,xg + £laX, + a,x"'lh = bl Xl + b,a;. + b.x, +b,tt,; ef,c.,

so that d!/I = a l dXI + agdxg + a,dx. + a, dx,; etc.,we can choose the constlUlts aI, bh ••• so that (2'1) becomes

dsg = dyl' + dy,'+ dy.' + dy,' (4·1).

For, substituting ~or the dy's and comparing coefficients with C·1.9), v.e haveonly 10 equations ·to be l!latisfied by the 16 constants. There are thuA manyways of ma.king the reduction. Note, however, that the reduction to the sumof four squares of complete differentials is not in general possible for a largeregion, where the g's have to be treated as functions, not constants.

Consider all the events for which y, has some specified value. These willform a three-dimensional world. Since dy, is zero for every pair of theseevents, their mutual intervals are given by

ds' = dYI' + dy.' + dy.· (4·2).

But this is exactly like familiar space in which the interval (which we haveshown to be the same a.s the distance for space without time) is given by

ds' = dxl + dyg + dz' (4·3).

where x, y, z are rectangular coordinates.Hence a section of the world by y, = const. will appear to us as space, and

!/ll !/I, y. will appear to us as rectangular coordinates. The coordinate-frames'!h, y" Y., and x, y, z, are examples of the systems Sand S' of § 1, for whichthe intervals between corresponding pairs of mesh-corners are equal. Thetwo systems are therefore exactly alike observationally; and if one appearsto us to be a rectangular frame in space, so also must the other. One provisomust be noted; the coordinates YI, YI, y. for real events must be real, as inhmiliar space, otherwise the resemblance would be only formal.

Granting this proviso, we have reduced the general expression to

ds' = dx' + dy' + dz' + dy.' (4·4),

where x, y, z will be recognised by us as rectangular coordinates in space.Clearly y. must involve the time, otherwise our location of events by the fourcoordinates would be incomplete; but we must not too hastily identify itwith the time t.

~ --- - - - -

I suppose that the following would be generally accepted as a satisfactory(pre-relativity) definition of equal time-intervals :-if we have a mechanismcapable of cyclic motion, its cycles will measure equal durations of timeanywhere and anywhen, provided the mechanism, its laws of behaviour, andall outside influences remain precisely similar. To this the relativist wouldadd the condition that the mechanism (as a whole) must be at rest in thespace-time frame considered, because it is n0'Y known that a clock in motiongoes slow in comparison with a fixed clock. The non-relativist does not dis­agree in fact, though he takes a slightly different view j he regards the provisothat the mechanism must be at rest as already included in hiB enunciation,because for him motion involves progress through the aether, which (heconsiders) directly affects the behaviour of the clock, a.nd is one of those., outside influences" which have to be kept" precisely similar."

• It will be shown in § 36 that it is always possible to i~~~;;rorr;-th~-~~~rdi~~tes ;o-that thefirst derivatives of the g's vanish at a selected point. We shall suppose that this preJiminlU',ftransformation haa already been made, in order that the constanoy of the g's may be a validllpproxiulation through as large a region as possible round ,the seleoted poiut.

ct., - 1 S

this would have agreed with (4'6) in the only two cases yet discussed, viz.(1) when dt = 0, and (2) when dx, dy, dz = O. To show that this more generalform is inadmissible we must examine pairs of events which differ both intime and place.

In the preceding pre-relativity definition of t our cloeh had t.o remaillfltatiol1ary and were therefore of !w me for comparing time at different places.What did the pre-relativity phpicist mean by the diff"J'ence of time dtbetween two events at diffen'llt places? I do not think that we can attachallY meaniug to his hazy conception of what dt signified; but W~ know one

, Since then it is agreed that the mech-a:nlsm as a~hole 'Is to he at rest,and the moving parts return to the same positions after a complete cycle, weshall have for the two events marking the beginning and end of the cycle

dx, dy, dz = O.Accordingly (4'4,) gives for this case

dsI = dy,'.

We have seen in § 3 that the cycles of the mechanism in all cases correspondto equal intervals ds; hence they correspond to equal values of dy,. Bu~ bythe above definition of time they also correspond to equal lapses of time dt;hence we must have dy, proportional to dt, and we express this proportion­alit,y by writing

dy,= icdt (4·5).

where i = V- 1, and c is a constant. It is, of courtle, possible that c may bean imaginary number. but provisionally we shall suppose it real. 'l'hen (4'4)becomes

ds~ = dx' + dy~ + dzl - c~dt~ (4'6).

A further discussion is necessary before it is permissible to conclude that\4'6) is the most general possible form for ds2 in terms of ordinary space andtime coordinates. If we had reduced 1( 1.B) to the rather more general form

ds2 = dx2+dy2±flz~ - c2<!~~'}.crxdxdt - 2cf3dydt- 2c'Ydzd~ ~..(4·7),

_c:t. - o<c - fie -Ie[g o<pJ

- ol..e 1 0 0-

1-~c 0 0

- fc. 0 0 1

<ds) ~ 5~Pdxo(dx~ { 'f..,xl= tt. > X'/' z}- )

or two ways in which he was accustomed to determine it. One method whichhe used was that of transport of chronometers. Let us examine then what..happens when we move a clock from (x11 O. 0) at the time it to another place(let, 0, 0) at the time t,..

We have seen that the clock, whether at rest or in motion, provided itremains a precisely similar mechanism, records equal intenJala; hence thedifference of the clock-readings at the beginning and end of the journey willbe proportional to the integrated interval

fIB ds (4'81).

If the transport is made in the direct line (dy = 0, dz = 0), we shall haveaccording to (4'7)

- dsI == r;2d~ + 2cadxdt - da!-

{2a dx 1 (dte)'}= c'dt1 1 + - - - - -c dt c' dt •

Hence the difference of the clock-readings (4'81) is proportional to

f t. ( 2a.u U1)it. dt 1 + -0 - (j (4·82),

25

approximately

fig 3 wor1d1ine 1-2 over1 .by an arbitrary mesh-system

where 1.£= dx/dt, i.e the velocity of the clock. The integral will not in generalreduce to t 2 - t,.; so that the difference of time at the two places is not givencorrectly by the reading of the clock. Even when a =,0, the moving dockdoes not record correct time.

Now introduce the condition that the velocity u is very small, rememberingthat t~ - t l will then become very large. Neglecting u~/c', (4'82) becomes

f t. ( a dX)dt 1 + --t, c dt

a= (~ - tl ) + - (x~ - Xl)'

c

The clock, if moved sufficiently slowly, wiII record the correct time-differenceif, and only if, a = 0. Moving it in other directions, we must have, similarly,/3 = 0, 'Y = 0. Thus (4'6) is the most general formula for the interval, whenthe time at different places is compared by slow transport of clocks from oneplace to another.

I do not know how far the reader will be prepared to accept the conditionthat it must be possible to correlate the times at different places by movinga clock from one to the other with infinitesimal velocity. The methodemployed in accurate work is to send an electromagnetic signal from one tothe other, and we shall see in § 11 that this leads to the same !(lflllUlaC. Wecan scarcely consider that either of these methods of comparing time atdifferent places is an essential part of our primitive notion of time in thesame way that measurement at one pbcc by a cyclic mechanism is; therefore

they arebe~£regafdedMconverltiona.l.-LetTi:belinderstood, however, thatalthough the relativity theory has formulated the convention explicitly, theusage of the word time-diffm'ence for the quantity fixed by this convention isin accordance with the long established practice in experimental physics andastronomy.

Setting a = 0 in (4'82), we see that the accurate formula. for the clock­reading will be

ft'dt (1 - u'/ct)it,

== (1 - u';o')J: (t. - ~) (4'9)

for a uniform velocity 1£. Thus a clock travelling with finite velocity givestoo small a reading-the clock goes slow compared with the time-reckoningc(mv_e!1~~Eal_ly~d0.E~~d.~~ ._ __. _

lS

vYl. 1 - 20

----::-- ---_._-----------_ ..---------_._-

To sum up the results of this section, if we choose coordinates such thatthe gener&ol quadratic form reduces to

dst = dYl' + dYa' + dYaa+ dYl ,(4'95),

then Yl' y., Ya and y• ...c:l will represent ordinary rectangular coordinates andtime. If we choose coordinates for which

ds'= dy,' + dyt' + dYaK +dYl + 'Jady,dy. +2{3dYady. + 2'YdYady•...(4'96),

these coordinates also will agree with rectangular coordinates and time so faras the more primitive notions of time are concerned; but the reckoning by

11this formula of differences of time at different places will not agree with thereckoning adopted in physics and astronomy according to long established]practice. For this reason it would only introduce confusion to admit theseIcoordinates as a permissible space and time system. .

We who regard all coordinate-frames as equally fictitious structures haveno special interest in ruling out the more general form (4'96). It is not aquestion of ascribing greater significance to one frame than to another, butof discovering which frame corresponds to the space and time reckoninggenerally accepted and used in standard works such as the Nautical Almanac.

As far as §14 our work will be subject to the condition that we are dealingwith a region of the world in which the ris are constant, or approximatelyconstant. A region having this property is called jlat. The theory of thiscase is called the" special" theory of relativity; it was discussed by Einsteinin 1905-some ten years before the general theory. But it becomes muchsimpler when regarded as a r,;pecial case of the general theory, because it isno longer necessary to defend the conditions for its validity as being essentialproperties of space-time. For a given region these conditions lIlay hold, orthey may not. The special theory applies only if they hold j other Cl:l.Sl;S mustbe referred to the general theory. II ao

1.1

then itthat atreduces

If we suppose that the form of the interval is non singular:

ds~ = <jO<f dx-<d;<-f3 clet [ 50((3 J =+ 0

is known that we can always choose co-ordinates sosome specified event a non singular quadratic formto

<ds )~where the e's stand for +1 or -1. The numbers of + and ­signs occuring here are characteristic for the form andc~nnot be changed by changing co-ordinates. So there areonly five choices of signs to be considered:

a. + + + +b. + + + -c. + +d. + -e.

Since ds~is a real measured quantity, those space-timedirections in which a clock can move must satisfy

ds 4~a<,.B dX 0( dx ~ 31= > 0 .

the past B,

@the future A,the present P in

a. relativityb. Newtonian physics.

Q. 6

cJ..1-ZI

The regions of space-time adjacent to an event can be brokenup into a past,B, a present P, and a future A. The past andfuture contain the allowed directions for the world lines,and since these allowed directions are to satisfy

d5 :4= 30<13 d,x. 0(d~rd > 0

we may write

::

pas t and fu turepresentnull cone

ds~ >ds 2. <ds2.

ooo

past and fu turepresent

We must rule out (a.) because it gives no present and also(e.) because it gives no past and future. As for (c.) it gives

:' (ddX:): + cd x '); - (d,x..\): - (dx3).\ >0

: ( X) -t- (dx') (dx,z) - (dl)2. < 0

Here the present is four-dimensional as is required but itis possible to pass continuously from any event in past andfuture to any other without encountering the present.Thus(c.) must be ruled out just as (b.). The only possibility is

(1. 9)

d.

::::

=

§ 3.4

When the metric has the form

± [~ -i -L~ Jwe then say that the potentials have "Galilean" values

3) or

that the mesh-system belonging to this metric has a Minkowskianbase and its frame {l\.''''I}is called inertial or Minkowskian.We shall indicate a Minkowskian frame by brackets around thesuper or sUbscripts~

Immediate consciousness of time

" Our minds are immediately aware of a "flight of time" without the inter­vention of external senses. Presumably there are more or less cyclic processesoccurring in the brain, which play the part of a material clock, whose indica­tions the mind can read. The rough measures of duration made by the internaltime-sense are of little use for scientific purposes. and physics is accustomedto base time-reckoning on more precise external mechanisms. It is, however,de8il'3.ble to examine the relation of this more primitive notion of time to thescheme developed in physics.

Much confusion has arisen from a failure to realise that t.ime as currentlynsed in physics and astronomy devia.tes widply from the time recognisf:ld bythe. primitive time-sense. In fact the time of which we arc immediately con­scious is not in general physical time, but the morc fnD(hmental qnantitywhich we have called interval (confined, however, to timelike intervals).

--_..__._.~_..._~------_._---------.

~.1 -22.

Our time-sl'llse is not conccrued with erents outside our brains; it relatesonly to the linear chain of events along our own track through the world. Wemay learn from another obsNv('r similar information as to the time-successionof events along his track. Further we have inanimate observers-clocks­from which we may obtain similar information as to their local time-successions.The combinat,ion of these linear sllccessions along different tracks into a com­plete ordering of the events in relation to one another is a problem thatrequires careful ;malysis, and is not correctly solved by the haphazard intuitionsofpre-relativity physics. Recognising that both clocks and time-sense measureds between pairs of events along their respective tracks, we see that theproblem reduces to that which we have already been studying, viz. to passfrom a dcscription in terms of intervals between pairs of events to a description \in terms of coordinates.

The external events which we see appear to fall into our own localtime-succession; but in reality it is not the events themselves, but thesense-impressions to which they indirectly give rise, which take place in thetime-succession of our consciousness. The popular outlook does not troub~e todiscriminate between the external events thernsel ves and the events cons~iLuted

by their light-impressions on our brains; and hence events throughol'.t, theuniverse are crudely located in our private time-sequence. Through this con­fusion the idea has arisen that the instants of which we are conscious extendso as to include external events, and are world-wide; and the enduring universeis supposed to consist of a succession of instantaneous states. This crude viewwas disproved in 1675 by Romer's celebrated discussion of the eclipses ofJupiter's satellites; and we are no longer permitted to locate external eventsin the instant of our visual perception of them. The whole foundation of theidea of world-wide instants was destroyed 250 years ago, and it seems strangethat it should still survive in current physics. But, as so often happens, the

. theory was patched up although its original raison d'e'tre had vanished. Ob­sessed with the iuea that the external events had to be put somehow into theinstants of our private consciousness, the physicist succeeded in removingthe pressing difficulties by placing them not in the instant of visual perceptionbut in a suit,able preceding instant. Physics borrowed the idea of world-wideinstants from the rejl~cted theory, and construct.ed mathematical continuationsof the instants in the consciousness of the observer, making in this way time­partitions throughout the four-dimensional world. We need have no quarrelwith this very useful construction which gives physical time. We only insistthat its artificial nature should be recognised, and that the original demandfor a world-wide time arose through a mistake. We should probably havehad to invent universal time-partitions in any case in order to obt,ain a com­plete mesh-system; but it might have saved confusion if we had arrived at itas a deliberate invention instead of an inherited misconception. If it is foundthat physical time has properties which would ordinarily be regarded as con-traiytocofurnori sense, no surpnse lleed be felt; this highly technical constructof physics is not to be confilllnd,'u with the time of common sense. It is im­portant for us to discover the exact properties of physical time; but thoseproperties were put into it by t,he astronomers who invented it. " 3'5.

29

In view of the following quotations Synge has also the same conceptionof time:

p 102: "proper time is time in the deepest sence, co-ordinate timeis a second rate time ".

p 108: "time of an event is dependent on the chosen world line".p 130: "time is personal not universal" 36

ENb.§~

§ 3.5 The fundamental velocity

30

ds = 0, and hence

" Consider a. point moving along the x-axis whose velocity mea'lured by S'v', so that

, dx'v = dt' (6·1).

hen by (5'1) its velocity measured by S is

dx fJ (dx' - vdt')v-----dt - fJ (dt' - udx' Ie!)

v' -u= 1 'j. by (6'1) (u·2).-uv C"

I non-relati -ity kinematics we should have taken ib as axiomatic that

=v' - u.If two points move relatively to S' with equal velocities in opposite

fections + v' and - v', their velocities relati ve to S are

tl-u

Again it follows. from (5'2) that when

(~;~:)' (dy')2 (d~')2 = c'dt' + .dt' + dt' ,

(~lY + (~yy + (~;y = c'.

Thus when the resultant velocity relative to S' is c, the velocity relative teoS is also c, whatever the uirecl"ion. We see that the velocity c has a uniqu"

and very ret11:1.rkable pror(~rty.

Accordillg to the older views of absolute time thi:; result appears incredible.}loreover we have not yet shown that the formulae have practieal significanee,since c might be imaginary But experiment has revealed a real velocitywith this rCIll[l,rkable property, viz. 29£),860 km. per sec. We s:lall call this

t.he fllndnmental velocity.By good fortune there is an entity--light-which travels wit,h the funda-

mental velocity. It would be a mistake to snppos" that the cxi~tence of suchan entity is responsible t0r the prominence accorued to the fundan ,mtai velocityc in our scheme; but i~is helpful in rendering it mow directly acceo;sible to

('xperiment. 1157•

§ 3.S Real and pseudo-gravity

It is incorrect to suppose that gravitational phenomena are merelyt~e results of the choice of the mesh-system overlaying our space­tlme. There are subtle but essential differences between real andpseudo- gravitational effects.

ct.1 - 2~ 3\

" 1=0

0

0 0 0

0

LtI

i

!

Earth

f~!Ol11in

Earlh

fig A In general, in a nonuniform gravitational ficld, a frcely falling apple in afreely falling elevator move~ relative 10 the elevator.

Let us return to Einstein'selevator and imagine that we have made it so large that the falling reference frameextends O\er a large region of space, as in rL94 . The elevator is rigid in therelativistic sense; that is, in lhe rest frame of the elevator the distances betweenpoints on the walls, ceiling, and floor do not change. The ceiling, being furtherfrom the center of the earth than the floor is, would fall at a smaller accelerationthan docs the floor if ceiling and floor were not rigidly held together. The resultis that the enlire car falls at some inttrrmediate acceleration. However, an applencar the ceiling and not rigidly attached to the car accelerates relative to the earthat the lesser rate, and so relative to the car it will move toward the ceiling. Sim­ilarly, an apple free to move ncar the floor acceleraks toward the floor. An applenear the middle of lhe car stays there. Furthermore, sinee free objects acceleratetoward the center of the earth, apples ncar the walls of the elevator will gravitatetoward the center of the car as the car descends toward the earth. We use a largeelevator to exaggerate the nonuniform appearance of the gravitational field of theearth, but it is clear that this nonuniformity, although diflieult to detect, extendsto the smallest regions of space. It is impossible to invent realizable distributionsof matter so that the total gravitational field generated is exactly uniform. Itfollows th,lt real graVItational etlects ~··~illn()t 5c~-I-;tri--~Ty-~11;11ill,lleJ in a freelyfalling rigid coordinate frame. On the other hand, it is not possible to duplicatethe divergenl gravitatinnal field 0f the earth, or any other nonuniforl11 tkld, byaccelerating a rigid coordinate frame in a gravitation!ess region of the universe.

These slight differences in the nature of real and pseudo-grel\ ity would enablean observer, confined to a laboratory and unable to look at the outside worlddirectly, to decide whether he is at rest on the earth' or in an accelerated spaceship'. Nevertheless it is a remarkable feature of gravity that its effecb ,Ire virtuallyindistinguishable from those of pseudo-gravity. Einstein sUl11mariled these ideasin a statel1lent known as the e{jl/it:alellce prillciple:

All p!leIlOl/h'IW, !Jroceee!il1g i/l a certain way ill ,lie Jllclli!o-grat:il<ll iOllaljic!d (~f

l//I acce!eratee!jI"I1/11e.!J/'(}ccce! ill ,he .Iamc way ill ,lie c(!lIit:alclIl real grat:iwliona!ficle!. pl'Ot:idce!lhe rcgioll im'olt:ee! il .Il/wlI cllough so ,liat ,!Ie rcal,liele! is l'\I<'II­

t ially lIIlIjim11 .

._-~..- "-. · ...._--- .--~ ..-._....

,,309•

~.1-25 32

§ 3·7 Kinds of space-time

"

~" The question must now be put, Can every possible kind ofII space-time occur in an empty region in nature? Suppose we

give the ten potentials perfectly arbitrary values at every point;that will specify the geometry of some mathematically possiblespace-time. But could that kind of space-time actually occur­by any arrangement of the matter round the region?

The answer is that only certain kinds of space-time can occurin an empty region in nature. The law which determines whatkinds can occur is the law of gravitation.

It is indeed clear that, since we have reduced the theory offields of force to a theory of the geometry of thc world, if thereis any law governing fields of force (including the gravitationalfield), that law must bc of the nature of a restriction on thcpossible geometries of the world.

The choice of g's in any special problcm is thus arrived at bya three-fold sorting out: (1) many sets of values can be dismissedbecause they can never occur in nature, (2) others, while possible,do not rclate to the kind of space-time present in the problemconsidered, (3) of those which remain, one set of values relatesto the particular mesh-system that has been chosen. We havenow to find the law govcrning the first discrimination. What isthe criterion that decides what values of the g's give a kind ofspace-time possible in nature?

.~ _ .•__.,-_ ..~, ' - .-.----- .'-- - -- --,-'"-- ---~-.-.• . __., • __._._._... _ ... o__._._._, ~_.•~.~".•__

It can now be deduced that the space-time in whieh we live Picturing the space-time in the gravitational field round thlnot quite flat. If it were, a mesh-system could be drawn for 'earth as a pucker, we notice that we canllot locate the pllcke.

hich the g's have the Galilean values, and the geometry with Rt a point; it is "somewhere round" the point. At any spC'eiaspect to these partitions of space and time would be that point the pucker can be pressed ont lIat, alld the irregularit)seussed in Chapter III. For that geometry the geodesics, givillg runs off somewhere else. That is what the inhabitants of Jule~

,e natural tracks of particles, are straight lines. : Verne's projectile did; they flattencd out the pucker inside thlThus in flat space-time the law of motion is that (with projectile so that they could not detect any field of force therejtably chosen coordinates) every particle moves uniformly in i but this only made things worse somewhere else, and the)straight line except when it is disturbed by the impacts of,l would find an increased field of force (relative to them) on th~

her-partiCIes. Clearly this is-not true ofonr-\\:orf~f;-foreia1llllle;- other side of the earth.e planets do not move in straight lines although they do not What determines the existence of the pllckcr is not the value~

fier any impacts. It is true that if we confine attention to 8 of the g's at any point, or, what comcs to the sallie thing, thllall region like the interior of Jules Verne's projectile, all the field of force there. It is the way these values link on to thoselcks become straight lines for an appropriate observer, or, at other points-t.he gradient of the g's, and more particularlywe generally say, he detects no field of force. It needs a i the gradient of the gradient. Or, as has already been said, th~

'ge region to bring out the differences of geometry. That is I kind of space-time is fixed by differential equations.t surprising, because we cannot expect to tell whether a Thus, although a gravitational field of force is lIot all absolllt~

rfnce is tlat or curved unlcss we consider a reasonably large thing, and can be imitated or annulled at any point by allrtion of it. acceleration of the observer or a change of his mesh-system,According to Newtonian idcas, at a great distance from all neverthcless the presence of a heavy particle does modify thelttcr beyond the reach of any gravitation, particles would all , world around it in an ausolute way which canuot be imitatedwe uniformly in straight lines. Thus at a great distance from i artificially. Gravitational force is relative; but there is thismatter space-time tends to become perfcctly flat. This canrooreconiplex~Clll:tract-crofg~;~;itatio~~alil;flue~;;-e-;h~h-is

ly be checked by experiment to a certain degrec of accuracy, 'absolutc._ __ "asd there is some doubt as to ,vhether it is rigorously true. \Vc ~-~"--'~-----.

So we can define real gravitation as an!1llleave this afterthought to Chapter x, me:mwhile assumingextern field of force due to the kind ofth Newton that space-time far cnough away from everythingspace-time, that is completely specifiedflat. although near matter it is curved. It is this puckeringby the Riemann-Christoffel tensor, and

elf matter which accounts for its gravitational effects.pseudo-gravitation as an extern field ofJust as we picture diffcrent kinds of two-dimensional space'force due to the chosen mesh-system butdifferently curved surfaces in our ordinary space of threc-in the same kind of space.nensions, so we are now picturing different kinds of four-

nensional space-time as differently curved stlrfuces in aclidean space of five dimensions. This is a picture only •.e fifth dimension is neither space nor time nor anything that1 be perceived; so far as we know, it is nonsense. I should notieribe it as a mathematical fiction, bccause it is of no greatvantage in a mathematical treatment. It is even liable to;lead because it draws distinctions, like the distinction be­~en a plane and a roll, which have no meaning. It is, like, notion of a field of forc(' acting in space and time, merelyroduced to bolstcr up Euclidean geomet.ry, when EuclideanImetry has been found inappropriate. The real differcnceweenth_e__ \,~ri_ou~ __kinds of spaee-!ir~~_is_tha!..!~~l__~~~~Jcrent kinds of geometry, involving differcnt properties of theIt is no explanation to say that this is because the surfaces

differently curved in a rca I Euelidean space offive dinH'nsions.should naturally ask for au explanation why the space of

: dimensions is Euclidean; and presumably the' answer wouldbecause it is a plane in a real Euclidean space of six dimen­

IS, and so on ad infinitum.------------- -- ----A fifth dimension 8ulnecs for illustratini:( the property here eonsidCled;for an e. "let representation of the geometry of the wM.I<I, Euclidean BruCeen dimen8ions is required. We may well ask whether th~re is merit inlidea.n geometry sufficient to justify going to sneh extremeas.

vh. 1 - 2.8 33

In solving this problem Einstein had only two clues to guide This accollnt of our observational knowledge of nature showm. that there is no shape inherent in the absolute world, so tha(1) Since it is a question of whether the kind of space-time is when we insert a mesh-systl'm, it has no shapc initially, and l

)ssill]e, thc criterion must refer to those properties of the g's i rcctangular mesh-system is intrinsically no different from an~

hieh distinguish different kinds of space-time, not to those othcr mesh-system.hieh distinguish different kinds of mesh-system in thc same' Returning to our two clues, condition (1) makes an extralace-time. The formulae must therefore not be altered in ani ordinarily clean sweep of laws that might be suggested; amonjay, if we change the mesh-system. them Newton's law is swept away. The mode of rejection cal(2) We know that flat space-time can occur in nature (at be seen by an example; it will be sufficient to cousider tWI~eat distanccs from all gravitating matter). Hence the criterion dimensions. If in one mesh-system (x, y)

lust be satisfied by any values of the g's belonging to flat I els2 = glJ da;2 + 2g12 d.xdy + gndy2,

lace-time. "I and in another system (x', y')It is remarkable that these slender clues are sufficient to l~~cate"~J.:n"?~.~~9u~!l..,-.~ .l?a.r!~~tI!fl:r__la~~_Af~e~'Ya_!~~ __!!J.~J els

2= gl1'da;'2 + 2g12' d:c'dy' + g22'd!l'2,

uther test must be applied-whether the law is confirmed by, the same law must bc satisfied if the unaccented letters arbscrvation. ' throughout replaced by accented letters. Suppose the 18\The irrelevance of the mesh-system to the laws of nature is I glJ. =- ga2 i~ suggcsted. Change th~ mesh-system by, spacing ~h

:>metimes expressed in a slightly different way. There is one' y-ll1les t,?ce as far apart, that IS to say take y - lu, wltlype of observation which, we call scarcely doubt, must be: x' -= It. 'Ihcll1dependent of any possible circumstances of the observer ds

2"" gil d:.r

2 + 2g12 rl:rdy + g22 dy2, da;'2 4 dx'd' 4 d'8amely a complete eoincidencc in space and time. The track oC: "" gu + g12 Y + gu Y ,

particle through four-dimensional space-time is called its so that gu' "" 811 ' gu' = 4ga2 •

rorld-line. Now, the world-lines of two particles either intersect: And if gu is equal to g22' glJ' cannot be equal to gu'.r they do not intersect; the standpoint of the observer is 1I0t. After a few trials the reader will begin to bc surprised thnlvolved. In so far as our knowledge of nature is a knowlcdgcIany possible law could survive thc test. It seems so easy tlf intersections of world-lines, it is absolute knowledge inde- defeat any formula that is set up by a simple change of mesh,endent of the observer. If we examine the nature of our I syst.em. Certainly it is unlikely that anyone would hit on suclbservations, distinguishing what is actually seen from what is i a law by trial. But there are such laws, composed of exceedingl:lerely inferred, we find that, at least in all exact measuremcllts, I complicate'!. mathematical exp~ssiOl~~.. '~'~.:.the_o~r o~t~~~~Jur knowledge is primarily built up (>1' intersections of world- t called the "theory of tensors," and had already been workclnes of two or morc entities, that is to say their coincidences. I out as a branch of pure mathematics by niemann, Christoffc:'or example, an electrician states that he has observed a current I Ricci, Lcvi-Civita who, it may be presumed, never dreamt 0

f 5 milliamperes. This is his illfcrellce: his actual observution ! this physical application.ras a coincidence of the image of a wire ill his galvanometcr lOne law of this kind is the condition for flat space-timelith a division of a scale. A meteorologist finds that the tern- \ which is generally written in the simple, but not very illuminatin~erature of the air is 75°; his observation was the coincidence of formhe top of the mercury-thread with di"isiol1 75 on the scale of B: 0= 0 (4).is thermometer. It would be extremely clumsy to describe the The quantity on the left is called the Uiemann-Christoff(~sults of the simplcst physical experiment entirely ill tcrms of tensor, and it is written out in a less abbrcviated form in thoincidence. The absolute observation is, whether or not the Appendix~ It must be explained that the letters 11" v, a, I

oincidcnce exists, not when or where or under what cireum- indicate gaps, which are to be filled up by any of the numbertances the coincidence exists; unless we arc to resort to relative 1, 2, S, 4, choscn at pleasure. (When the exprcs!Jion is writte!nowledge, the place, time and uther cireumstan('es must ill out at length, the gaps are in the suffixes of the .1:'S and g's.l1eir turn be described by reference to other coincidences. But Filling the gaps in different ways, a large number of expressions, seems clear that if we could draw all the world-lines so as to Biu' B123' Bl32 • etc., are obtained. The equation (4) states thalOW all the intersections in their proper order, but otherwise all of these are zero. There are 4\ or 256, of these cxpression:rbitrary, this would contain a complete history uf t.he world, altogether, but many of them arc repetitions. Only 20 of th.nd nothing within reach of observation would be omitted. equations are really necessary; the others merely sny the sam_Let us draw such a picture, and imagine it embedded in a thing over again.

:lly. If we deform the jelly in any way, the intersections wiU It is clear that the law (4) is not the law of gravitation fOl:ill occur in the same order alon~ each world-line and no which we are seeking, because it is much too drastic. If it wercdditional intersections will be created. The deformed jelly will a law of nature, then only flat space-time ('ould exist in nature:present a history of the world, just as accurate as the one. nnd there would be no such thing as gravitation. It is tlot therigillalJy drawn; tllere-canbe"no-crIterion for distinguishing general condition, but a special case-when all attractitllrhieh is the best representation. matter is infinitely remote.Suppose now we introduce sllace and time-partitions, which nut in finding a general ('ondition. it mll.~· be a great help t(

Ie might do by drawing rectangular meshes ill both jellies. know a specinl case. Would it do to scIect n certain numbcr 01

\'e have now two ways of locating the world-lines and events the 20 equations to be satisfied generally, leaving the rest tc1 space and time, both on the same absolute footing. But be satisfied only in the special cnse? Unfortunately the equation!learly it. makes no diHerence in the result of the location whether hang together; and, unless we take them nIl, it is found thatre first deform the jelly allll then introduce regular meshes, or the condition is not independent of the mesh-system. But thererhether we introduce irregular meshes in the undeCormed jelly. happens to be one way of building up Ollt of the 20 conditioll!.nd so all mesh-systems are on the same footing. a less stringent set of conditions independent of the_~.c:s!l'

'stem. Let The foUowing conclusions can be stated.Gil ;; B~n +-1t~12 +11;13 +-.ll~---~- If B~.~= 0 (20 conditions)

ld, generally d . fi ._ B j space-time is flat. This is the state of the worl at an In mil

G". ,., B~.l + n:.~~B3,..V3 + B~·.:.:'_._.f.5:. .' distance from all matter and all forllls of encrgy.

len the conditions ! If G". = 0 (6 conditions)G,.. = 0 (5), i space-time is curved in the first degree. This is the state of tl,

ill satisfy our requirements for a general law of nature. world in an empty region-not containing matter, light cThis law is independent of the mesh-system, though this can electromagnetic fields, but in the neighbourhood of these formlly be proved by elaborate mathematical analysis. Evidently, of energy.hen all the B's vanish, equation (5) is satisfied; so, when flat If G,., 0 (1 condition)lllce-time occurs, this law of nature is not violated. Further I space-time is curved in the second degree. This is the state 0

is not so stringent as the condition for flatness, and admits : the world in a region not containing matter or elcctrons (boun~. the occurrence of a limited variety of non-Euclidean geome- I energy), but containing light or elcctromagnetic fields (fre,ies. Rejecting duplicates, it comprises 10 equations; but four Ii energy).. these can be derived from the other six, so that it gives' If G is not zero

" conditions, which happens to be the number required for a spacc-time is fully curved. This is the state of the world in I

w of gravitation*~ region containing continuous matter.The suggestion is thus reached that According to current physiC'll theory continuous matter doe:

G". = 0 not exist, so that strictly speaking the last case never arisesay be the general law of gravitation. Whether it is so or not Matter is built of electrons or other nuclei. The regions lyin!monty be settled by experiment. In particular, it must in between the electrons are not fully curved, whilst the region:'dinary cases reduce to something so near the Newtonian law, inside the electrons must be cut out of space-time altogetherlat the remarkable confirmation of the latter by observation l We ~~_n~!~~gin~~u_~~ly~~.~xpl~~.r:!n~_~heinsideof an electrolaccounted for. Further it is necessary to examine whether with moving particles, light-waves, or material clocks und

lere are any exceptional cases in which the difference between measuring-rods; hence, without further dcfinition, any geometryand Newton's law can be tested. \Vc shaU see that these of the interior, or Rny statement about space and time in the

sts are satisfied. interior, is meaningless. But in common life, and frequently inWhat would have been the position if this suggested law had physics, we are not concerned with this microscopic structure ofiled? \Ve might continue the search for other laws satisfying matter. We need to know, not the actual values of the g's atIe two conditions laid down; but these would certainly be far ia point, but their average values through a region, small fromore complicated mathematically. I believe too that they would, the ordinary standpoint but large compared with the molecular.t help much, because practically they would be indistinguish- structure of matter. In this macroscopic trcatment molecular.le from the simpler law here suggested-though this has not matter is replaccd by continuous matter, and uncurved spacc­:en demonstrated rigorously. The other alternative is that, time studded with holes is replaced by an equivalent fullyere is something causing force in natur~~.~_~omprised.i~-.!~:.curved space-time without holes.:ometrlcal schemehllherto considered, so that force is not It is natural that our senses should have developed facultiesIrely relative, and Ncwton's super-observer exists. for perceiving some of these intrinsic distinctions of the possiblePerhaps the best survey of the meaning of our theory can be states of the world around us. I prefer to think of matter and,tained from the standpoint of a ten-dimensional Euclidean energy, not as agents causing the degrees of curvature of thentinuum, in which space-time is conceived as a particular world, but as parts of our perceptions of the existence of theur-dimensional surface. It has to be remarked that in ten curvature.

mensions there are gradations intermediate between a flat It will be seen that the law of gravitation can be summed uprface and a fully curved surface, which we shall speak of as in the statement that in an empty region space-time can be.rved in the "first degree" or "second dcgree~" The dis- curved onlv in the first dcgree. .1 .~lction is something like that of curvcs in ordinary space, --._._~-~-~----_......----.~_._----._-----_.~---~-

:rich may be curved like a circle, or twisted like a helix; but the Conclusion:lalogyis not very close. The fuU "curvature" of a surface is a only those potentials which satisfy19le quantity caUed G, built up out of the various terms G..v in (1.11) G,.u.v -= 0

~ewh~t the same way as these are built up out of B~.". specify a possible space-time •

•' Isolate a region of empty space.time; and suppose that e:-erywhere outsidee region the potentials are known. It should then be pOSSIble by ~he law.of,vitation to determine the nature of spn.ce.time in the rec;ion. Ten dlff~rent\:ll

uations to"ether with the boundary·values would s\,ffice to determlDe the~ potentia~ throu~hout the region; but that would dete~~ine not lonly the~d of space· time but the mesh-system, whereas the partItions of t.le mesh­stem can be oontinued across the region in any arbit~ary w,ay. The fourtAl of p3.rtitions give a four·fold arhitrarIness; and to admit 01 tlus, the numberequations requued is reduccd to sa.

§ ~.

§ ~.1

cR.1-za

TRANSFORMATIONS

Base-vectors

After chosing a mesh-system in a certain problem we would liketo introduce base-vectors (e) as the tangent directions in apoint along the intersecting mesh-lines in that point:' So base­vectors are not defined by this but just introduced as a physicalconcept. Vectors are treated according to the ordinary rulesof the linear algebra.

35

fig 5 concept of base-vectors•

§ ~.2

So we think of a vector in a n-dimensional space as a setof n numbers to which we can attach a certain direction specifiedby those numbers while in general a vector is a "thing",e.g. a polynom, which obeys certain well defined mathematicalrules ~~These rules are combined in the mathematical stucture

of a vector space (Adler [18] ,p138)

Tensors

The reason for the use of tensors is clearly exposed by thefollowing quotation of Einstein [4] , p121:"

B. MATHEMATICAL AIDS TO THE FORMULATION OF

GENERALLY COVARIANT EQUATIONS

Ha.ving seen in the foregoing that the general p~stulate

of relativi_tl'J~a.ds to the_Ee~luirelll()~~ !l:a!111e_ equat_l()~l_~_of _physics shall be covariant in the face of any substitution ofthe co-ordinates Xl' •• :r4' we have to consider how suchgenerally covariant equations can be found. \Ve now turnto this purely mathelllllotical task, and we shall find that in itssolution a fundamental r61e is played by the invariant dsgiven in equation (3), which, borrowing from Gauss's theoryof surfaces, we have called the" linear element."

The fundamental idea of this general theory of covarillontsis the following :--Let certain things (" tensors") be ddinedwith respect to any system of co-ordinates by a number offuuctions of the co-ordinates, cillled the "components" ofthe tensor. There are then cl'rtain rules by which thesecomponents can he calclIia,ted for a new system of cn·()I:din-

- ate!), if they arc known for the original sy,.;tem of co-ordinates,and if the transformation c0unecting the two systems isknown. The things hereafter called tensors are further---_.._.." ..~ ..... ,,- _._'- ..- ---,._--~"------

c1. 1 - 29

characterized by the fact that the equations of transformationfor their components are linear and homogeneous. Accord­ingly, all the components in the new system vanish, if theyall vanish in the original system. If, therefore, a law ofnature is expressed by equating all the components of a tensorto zero, it is generally covariant. By examining the lawsof the formation of tensors, we acquire the means of formu­lating generally covariant Jaws.

DefintionA p-tensor is a funtion of. p vectors which is linear in eachof these vectors:'2.

For an exposition of tensor calculus there are many excellentbooks and we shall refer in particular to Seidel ~2] and Synge

36

Schematic connec tion between bases, vectors, cr'):lsi tion tensors and fundamental tensors.

arbi.tra.r ba..se tYtlns orms.

I'-'-'-j

iA- Ief> t

III

I

c..o vay:c.omponentst XQ{~

r­I

IIIIIIII r--

\\\

!I

cont rc.. vQ.r,ecm~nenb:.txo( j

co Vctr:

compc>n~nl:.s

{ X"I, jIIItIIIL._._._._

" "f!e - Q er·_·_"I,7_.J d.f '--'-'

iilIII

39

3.

§ ~.5

~, 1-.34

Physical observables and tensor quantities

Lj1

§ Lr.6

When we want to observe a tensor quantity physically we have tomeasure its components belonging to a local inertial (Minkowskian)co-ordinate frame, tx.'o/lj.{)C.Iol'} ,because only in inertial framesthe reckoning of differences in time at different places agreeswith the reconing adopted in physics according to long establishedpractice;;1 These tensor components are called "physical observables"in contrast with "tensor quantities" which are tJnsor components

with respect to any arbitrary, mostly noninertial, frame LxoC}, {xo<}'

So physical observables are special tensor quantities belongingto a local inertial frame and we shall indicate the differenceby brackets around the sub and superscripts of the physicalobservables;S°

Lorentz transformations

The Lorentz transformation is a teasor transformation whichtransforms one inertial co-ordinate frame into another andin this way the physical observables belonging to two differentinertial frames with each other.~

So frame{)(.'f'Jwith the metric matrix

[9 ] [ 1 -1 0 ] =: [~ (1. -1, - 1. - , )](i{)(f) == 0 -1 -1

is transformed by the Lorentz transformation A(V)_ ,according to't1

d (Y) 1\ (v) _ I t"jJ...)( 1 . 2 1) X _ ( IJ-) d. A /

into a frame \.Xl~l} with the ~etric matrix

[9,~X?)] = [~ -1 -1 _: ]

We shall now consider two Lorentz transformations of whichthe fir§t relates two inertial co-ordinate frames moving witha constant velocity with respect to each other and the secondrelates two inertial frames moving with a not necessarilyconstant velocity with respect to each other.

1. II Make the following transformation of coordinates

IlJ = fJ (x' - Itt'), Y = y', z = z', t = fJ (t' - ux'/o~)

fJ = (l-1JJjo~)-i,

where u is any real c2nstant not greater than c.We have by (5'1)

dJ,.!J. - c~dt~ = fJ~ {(dllJ' - udt')~ - c~ (dt' - udx'/o~)'}

= fJ' {(1 -~) dx" - (o~ - 11.:) dt'~}

= dx'~ - o'dt".,

......(5·1),

Hence from (40'6)_.__ ",__ .___~s. = dxt + dy~ +dz' - o~dt~ = d~'~ +~/~_~__~z" __ ~_~f'3 •••••. (.5'2).

cY1.1- 35

The accented and unaccented coordinates give the same formula for theinterval, so that the intervals between corresponding pairs of mesh-cornerswill be equal, and therefore in all observable respects they will be alike. We;shall recognise ai, y', z' as rectangular coordinates in space, and t/ as theassociated time. We have thus arrived at another possible way of reckoningspace and time-another fictitious space-time frame, equivalent in all itsproperties to the original one. For convenience we say that the first reckoningis that of an observer B and the second that of an observer B', both observersbeing at rest in their respective spaces·.

The constant u is easily interpreted. Since B is at rest in his own spacehis location is given by x = const. By (5'1) this becomes, in B"s coordinattls,x' - ut' = const.; that is to say, B is travelling in the x'-direction with velocity u.

Accordingly the constant u is interpreted as the velocity of B relative to B'.It does not follow immediately that the velocity of S' relative to B is

-u; but this can be proved by algebraical,solution of the equations (5'1) todetermine x', V', z', t/, We find

x' = S (x + 'ut), y' = y, z' = z, t' = /3 (t + llX/C2) ...... (5'3),

showing that an interchange of Sand B' merely reverses the sign of u.The' essential property of the foregoing transformation 'is that it leaves

the formula for ds2 unaltered (5'2), so that the coordinate-systems which itconnects are alike in their properties. Looking at the matter more generally,we have already noted that the reduction to the sum of four squares can bemalle in many ways, so that we can have

ds2= elf/!2 + dy/ + dyl + dy; = dy/' + dy..'2 + dY3'2 + dY4'2 (5'4).

• This is partly a matter of nomenclature, A sentient observer can force himself to "recollectthat he is moving" and so e.dopt a space in which he is Dot at rest; but he does not so readilyadopt the time which properly corresponds; unless he uses the space-time fr}lome in which he isat rest, he is likely to adopt a hybrid space-time which lea,ls to inconsistencies. There is DOambiguity if the "observer" is regarded as merely an involuntary llleasurin~ apparatus, which bythe principles of Ii 4 naturally partitions a spl~ce and time with re~pect to which it is at rest. " 53•

2,

oC+-Z.1- 35

§ L.. 7 L~c.:hoYV ~ ~ to ~~.\

lOC£\.lt::occt~ ~lc( tra.nsformo.b. cG()

COmoVl');l fra.rYle { xoe'~WLijI)

Qrbit~~ meL-.:c

""-'"'1. I - 1'-'"

§ ~.9

4

1 = OyCc

".~~ --f'---,-__,e1

/~ <f>/'

fig 7 two-dimensional reciproke bases

59

'"e,

co

cR.1- L,3 50

T

....

e--R.1- 4Li 51

'T

~C~/::;.5 I.CQ'n.O /> 0

lx

~

fig 9 : Lorentz diagramj

frames with equa units but different scale. eto s

x

cX,.1- 45 S2

~.fig 10 : Lorentz diagr I , ~ .

frames w~th e:Cal units and scale facto~.

53

...

x

L-n. 1- "1(

1

flg 12 : impossible world lines

~NL

eTa

~., - 4'1

rr y

"" ""f/

X

~ E~./.'"

--'-

FNL:BNL:FPo..?r

./

fig 13 :

ss

CTO :

Oh.1 -;30 5b

ch2..-1

CHAPTER 2

MAXWELL E(t,UATIONS

§ 1 •

2.12.2

2.3

3.

4.14.2

5.

6.16.26.3

Geometrized units

Maxwell-Einstein tensor equations for vacuumRela tive tensors, generalized Kronecker symbol, permu ta tionsymbol and oriented tensorsDual tensors

Maxwell-Einstein tensor equations for macroscopic media

Scalar and three-vector potentialFour-vector potential

Another form of Maxwell-Einstein tensor equations formacroscopic media

Localization properties of comoving framesLocal Maxwell equations for comoving framesLocal Maxwell equations for comoving frames expressedby local differential operators

7.1 Localization properties of synchronous frames7.2 Local Maxwell equations for comoving synchronous frames

~2.-2

Introduction

The familiar Maxwell three-vector equations are only valid forinertial frames and when we are dealing with moving matter weoften use medium comoving frRmes because of their conveniencefor certain problems (see for instance Mo 8 , chapter 5R ).These equations are modified when reffered to R non-inertialframe, usually a medium comoving frame, and the way in whichthis is carried out is revealed by the postulated covariantMaxwell-Einstein tensor equations. The induction of theseequations from the familiar three-vector equations is shownin the first part of this chapter (pflr. 1-5).

But there is one big disadvantage in the use of tensors ;tensors as mathematical symbols, in whose forms the physic~l

quantitlBs (E,TI,~,R'f;J) combine both the curvatures of1. the co-ordinate frame (pseudo-gravity) and2. soace-time ( real gravity) ,in order to enter the postulated simole covariant equations,have no direct physical significance.

We prefer dealing with local euuations because1. we wish to avoid unnecessary divergence" which may

occur to tensor quantities2. we wish to have the local physics of eventual interest

under direct consideration such that :a. we have a physical feeling ~bout the quantities

being handled andb. we are able to simplify equations with intuitive

symmetry argumem:.s when such physical symmetryhappens locally.

These local equations for general cOffioving frames will bederived in the second p1rt of this chapter (par. 6.1-7.~)

and shall be used in chapters 5 and/or 6.

Conclusions

S4

1. In our local equations (2.82),(2.86),(2.87), the modificationof the familiar Maxwell equations are shown.

2. The general covariant Maxwell-Eistein equations (2.20),(2.23a)are valid for every frame but cocern tensors,which we wish to avoid.

3. Paragraph shows a way to avoid the usually troublesomeChristoffel symbols in the Maxwell-Einstein tensor equations.

4. Flat space-time approximation of the modified Maxwellequations can be made when the space-time dependenceof gp." may be neglected.

5. Two misprints in Mo R , formula (4.7) are discoveredand indicated in paragraph 6.3, alinea 1 and5.

6. Duality rules (2.78) reduces work to the half.

7. Contra and covariant permutation symbols are not obtainedfrom one another by the process of raising and lowerin~

suffixes with tte metric tensor (2.32).

Notes on chapter 2

1. four-vector, one tensor and Lorentz tensor are equivalents2. cf. Synge & Schild[15] , p. 240-2503. cf. Mo[8], p. 2593, formula 4.2 ; symbols'18 ,used by the

author and Mo are to be interchanged4. not in geometrized units5. quotation Mo[8], p. 2592, 1 column6. cf. Mo [8J, p.2593, formula 4.57. idem 4.6 c,d,a and b8. quotation Mo[~ , p. 2593, 2 column9. cf. Post, p.65, footnote, [~'J10. cf. Synge & Schild 15 , p.54

§ 1, Wcm~z.ed-units

TAI3LE I. Conversion from geometrized to mks units* *

Quantity Geomctriled Dimension

mol

mol

11l

m'

m

m

m

mm

m

dimensionless

dimensionless

dimensionless

dimensionless

dimen~ionless

di mensionlcss

c'2.68 X 10 "E

B~G)I":

c= 0.811 X 1O-19B

= D I (~)1'2c2

t(1

= II !J.~ (G<0)';2e

= H !.(Q)1,2c3 EO

I= 1)- (, G)1:2

CZ u

I(T--

c<o3.768 X I02~

E*

B*

H*

<7*

i*

n*

'* = ,

r* = cr= 2.9':>7925 X 10'1

Gm* =-m

e'= 0.742 X 10- "m

q* =ql,(q)'"cw to

3.1J42 X 1O- t7qGk

T* Tc'1.14 X 1O-G7 T

F*G.

=-fc'

= 0.826 X 1O-"FG

p* = -- pc'G

f,* = -f,c'

f* = fieG

Pm * = ~; P

S* = S/k= 7.2435 X 1022 sec

Gh* = .-- h

e'2,61 X 10-70

V* = V,'cG

l.,,'· = ..- W

c'= 0.275 X 1O- s, IV

= i .1. (~;)1'2c3 tt'

= 1.014 X 10"i

J* = J! (G)L2cJ

fl.l

1(G)'"pall = PQ "

C" Ell

..._~--'-------------,._-_..._-_.-1(G' "

(T., c' ,JE ~,:(j)~'

kg 'sec-coul

caul/sec

(amp)

caul

m/seckg-m'

sec'(watt)

caul

coul

Ill'

m'

kg-m'I coul-sec'(joule ;'coul)

couP-sec

m'-kg

(weber 1m')coul

m'coul

(ne\\ tun caul)

sec-m

sec-nl"

kg/m'

sec

kg - m/sec'

OK

kg

kg-m' /scc'- OK(jou1c;oK)kg-m'

kg-m' /sec'(joule)I/sec

caul

msec

(newton)

kg/msec'

<7

IV

E

v

J

B

D

II

T

Pm

F

s

m

q

f

p

vol. electr. chge. density

mass density

>urLlce ekctr. ehge.density

volt

velocity

elcdri..: Ji'placement

m:Jgnctic intemity

elcctri..: lidJ

current

current demity

ll1;lgndic tlux density

Planck constant

conductivity

power

mass

frequency

pressure

entropy

temperature

force

lengthtime

energy

charge

dielectric constant coui'-sec'/kg-m'(farad/m) .*

<0

permeability kg-m /cou!'(henry/m)

dirnen,;onle5s

1-<0

.. From .\fa [1969]. In this table: m = meter. kg = kilogram, sec second, coul = coulomb; in geometrized unit, c ,~ 1J = I, k = 1; in mks units, G = 6.67 X 1O-1l and k = 1.3805 X 10-23• f = G Im~'i" ' G=6'1'''5'' N.,;;2f<1j-t2

EN

§ ~.1 M~-E~~ -h,,~ -e:vJ.a.bo.".~ ~ .ua~.f J

G1

\ .~,"-""t2)o t:"'~-,.~,..,...:\" .. :''-'-'/'''\.

'1(2)

- -I J

o"\, l";',( 1):J 0

-.::;--'" /2'dx.' J

r1 (3)

- _I?

J

4-- ,/'Y""'-d'A,.,.·~'l,A,,l:;:.Jl.r:'~.:,

-8Co.. ( 1 )

or.:l (2) _ ~ (,3~w ,_

.. ,

")\-~) I

!ic:

-'

\

~ ("'F~>I)

d,x. <-.7)

(2.'1)

__ , 't;1'- '(v} • i <. ' j

B0)

E -E 0Q) (1)

E.U,)

-BU)

B 0(V

o(2.l3 )

1 2. E.3o -E -E -

§ 2.2.

1.

vh.z- 8

cYt.2 - 9 65

.s.

2.3

1.

2.

~.2- 10

(2 .3,Sc... )

.3.

d,.2.-11

_ 1oBIT)

BIZ)

Bm

F - F'(1)() t3X1)

oF - F

(oU) (5XO)

F - F(oX~ (110)

_ B(1)

o_ E"l)

E(1)

- ~X3)+ ~3l(1)- ~O~)+ ~!>XO)

o-F + F

(oIU) \1)(0)

(:0)

- BE(3)

o_ E (1)

oE (1)

E (2)

E(.3)

.c~~ t4 ~u.lt<:2..~) . 'i

[ FrX~)] ~- E (1)

oB (.3)

_ B (2)

_ E (1)

_ B(.3)

oB (1)

_ E (.3)

B (~)

_ B (1)

o

,

------

§ 3.

cJ,.~ - 1'2

cJ!l.2- 13

~.... ~: ' ....."(2.1) ~ (2.2a.)

{V. B = 0 { V· D

V X E + ~ B = 0 Vx. H + ~ D'01 dT

1.

2..

.3.

-4.

s.

s

§ .4.2

c1.2.-15 7'

1

2

3.

ch.2-1G

2..

~ fU')J., JfW) 3

Jf-l2,lZ ~?l..3).3

f Dft"lllV f+V Mf(--J) v( 2. 6.,3 ) : = (-1 )

- Cl.z C.3 t C!3 Cz.

OJ.,S ...:. 0:.3~

cA.z- 18

§ 1:>.1

'75,

II 1'0 find the ph; ~ical tdrad basi,: C" I: and theappropriate way that the; arc carried allln); b; :0',(amovIng in {.\.~ f with velocit.\ U. \\ e first ,et

(2b2) (2.1'+ )

(2.b3)

since thc observer's proper-lime bp"e durll1l; d\" is£IT = (.1;00)) d\lI. i.e .. the local pwper-time directionfor {O} is the coordinate-time dIrection there. on];rescaled bv a factor of (~Oll)\' Since the loe:11 ~pati,11

basis vect;rs {S,!: must be orthogonal to ~{{J: and areorth~;~;:;~;Ti~ed~mong themselves for convenience,they arc only defined within a spatial rotation; thus,within (2.9),2U

{e lil } = O.N. {d, == e, - Cl;o)goo)eo~. (:~.15)• ~=~~3 ..

where the dt arc just the time-orthogonailled coon.11-

nate triads.II S

§ 6.2

7E;/

~.

1.

z.

J,.2.- 21 77

3.

~.2.- 2.2.

~(')(~\

(3)

(2.(;C<..) rt _D(1) _D(,,2.) _DU )

[GtrY-'ll>] 'V 0 _ HUI

\-I(l)

~ ::r = 0 _ H(1)

o

[:

cY,.?-13

1

o - Be') _ ~(") _ 15(.3)

o EUl _ E(.2)

o Eel)

o

2- ~'1 10k dua.ti:y ruJ~tav='

(278) (*F )rKV) dual FLf!v) , FJP")Lv)

BLl) • ElO

E(.L) __...--__.. _ ~ c..)..

dueJ C; <.j-LY.v)

• D lL)

- H(.')

END§S.2

§ 6.3

1.

~~w=i'fi:J::;:~II -----~-----------.. -------------

Ph\~iclil:, in \4,6al. the Inl\. Ot'!e)c"li curFellt lkn,it)J inlLl ch"lff:c density {I and the pre,cnce 0[' the curl­[ill' krill (,I' H cUlIlpen,ate t'e)r the fact th~lt the co()rdl­

n~lte dlwrgel1ce ot' D i~ not talcn purdy ~patjally;

,imilar remarks ~Ippl: te) (4,(Jl))++'6d), In fact if we

e\.press the c(wrdinale dit1'crential uperators by localdifferential ()pcraturs through ,Ix = dY"C" = drlzlc(2)'then (·~6a) and (4,(Jl)) becI)!nc, respectively,

j ,II [ .. II;Ii, 011 11,: oD I) Il']+ I -- ,,) ( ( "; ..------ - ----- -- J

\ ,-' j) ( ,axil oxo"'[( C')J,; ((1 ).1] DlZ)-t- -- ...., ((I) ....d)O )

l '

+ [, )' IIld: II '] II 0 1 - )\ -g 'I (' ("L' Iii,! II,) = (,+,ia

118

2.

d.0,.80)

~.2-2S 81

cv· ed.

82

83

(2,B~) e . L = e . = e (e). =(0) COH ' L

= eo ttl = 0

= 0,

G. '" "" " 00 01' ()o Oy J e + Cj.Je, == <e . e(.l)) = .q ey(L) - OlL.) 'JoCl)

MJ oJ_ 5°0 e, . e 0- 0 L' ) + - i (L) -

J J ~ ~)O.-h.e.-c~ eou) = 0

\\'I1"n tlte {\"~ h;l~ ~\nl'ItI\'I\\'lJ~ illl'llIl, I.l', go, "

0, .lii 1'",,', ""'Ii' ('IIlI, , ~lnd 1"'" I d1\I."h , Ill' c,ln rl"l,lil'tlllll' \(' that,:':llll '= I and dl'tinl' thl' l'pl'ral\\r~ 'iJ x alld'iJ. In 3-~p,ll'e dll'llrdln[' tll (,Ul) ;uch tlut

where 7."\ == c lJ ) [(_"jJ"k 'j 11' 111 'lddl'tl'Oll the' , l k ,..... (I) ,ll' • ~ ,

g", are 11l1t functions l)f time, (.f,~) \1 ill just reduce tothe ordinary .I-vector eqll;ltio1\~ in cun ilinear coordl­natt's.

'iJ . D c= p.

'iJ . B = 0,

a'iJxH,,=J+-D+(£·Dat ''iJ x E = - ~!! - 'I • Bat ' (,un

\1.,,,:=(

(n)

'VX ...f=d... ". :=

::::. (

:::::(

~3-1

CHAPTE.a 3

~U~STITUTIVE RELATIONS

85

§ 1. CONSTITU11Vi TbNSORS Or LINEAR MEDIA

1.1 Local teLsorial description1.2 Permil,tivit,y part of trle consticu.ti've t'our-tl:nsor1. j fermeabili ty part OJ tr,e consU tu.tive four-tensor1.4 Constitu.tive tensurs of linear m~dia

2. CONST1TUTIVE TENSORS OF LOCAL LINEAR ISOTR0P1C MeJIA

3. CONSl11Ul~VL LOCAL THREE-VECT~R EQUATIONS FOR LOCALISOTHOP,C IVkDIA

).11 Derivation of D function E 13 for local isotropic Inl;uia,5.12 lJerivatiun of H function 13,[ 10r 10cu.1 isotropic uledia,.21 DerivaLiun 01 IS lu.nction £,11 lor local isotrojJic meoiu).22 Derivation of B i'vtn<::tion ( H,E ) for local isotropic lliEld ia5.51 Constitutive local tnrel:- vector equations lor local isotropic

media llioving rectilinear3.32 Constitutive trlre8-vecLur equ.al-ions for local isotropic

media moving uniform ami rectilinear3.33 Constitutive local Lhree-vecLor equa t i ,jns for J.oca1 isotropic

IDeGla moving rectilinear with constant proper acceleration3.4 ConsLitl.i.tlve matrix for local em fields fur local isotropic

ilIeCI i a

4. CURRENT-FIELD CONSTITUTIVE R;:;LATIONS FOR RECTILINEARMOVING HEDIA

4.14.2

Derivation ofDerivation of

:f = func tionJ = function

E,i3 )E,B ) for local isotropic media

et.3- 2

Introduction

Realizing the familiar constitutive relations are only validwhen an observer is comoving with the medium, provided changesdue to accelerated strain are neglectable,we want to find outhow to modify these relations when observing a medium non-co­movingly. This can be done by constructing a constitutive fourtensor equation and postulating its covariance. Schematically:

familiar construction constitutive modified-------<..- • •

constitutive of tensors tensor equate constitutiverel~tions 1! rel~tions

comoving postulated laboratoryframe cov.ariant tensor eq frame

In this capter we confine most of our attention to constitutivethree-vector equations for local isotropic media with arbitraryand special moving modes. These equations can also be derivedwith out constructing a constitutive four-tensor ( by postulatingthe invariance of the Maxwell laws for all inertial frames),thus forming a nice check on our constructed four-tensor.

Conclusions

1. Constitutive relations for all kinds of moving modes,neglectingaccelerated stains, can be derived from the postulated

.. .... . ..constitutive tensor equation.

BE

• ITE13

ANTISYMMETIUCMATRIX

(u X " )+

EBEIfI:

SYNMETrncMATRIX=

For lineAr isotropic media consti tu tive three-vector equa tionswith respect to an inertial (laboratory) frame take thefollowing typical form :

DHIS13J

2.

.3. These constitutive three-vector equations take the sameform for all (rectilinear) moving modes when the instantaneousvelocity is substituted.

Notes on chapter 5.

equatiun

(24)(26)(27)(2b)(29)

(30)Ul)(35J,rirst

c.l. No[Sa] ,p.17-l9linearit~ in tlwb dOlliain is meant anti noL in fr~q~ency JOillainrespectivily parat,rapn 3.2 aLd 3.3skew s,;mmetry, ocw, allti symmetry ale equ.ivaleuLc. f. chapter 1, paraGraphteHsor characLer!dual. i t j

nownere deflneu bJ Mo[8], oac.L IVlo[ba] ,p.19, formula (32)idem p.l?l.u.ew p.l~

idem p.17ioem p.17idem p.ldidem p.1Sidem p.ldiaem p. H3iaem p.l9iaem p.20ideili p.20cf. IvIo[dJ,p.2~90, lorlJ.lula \1.2)cf. hO [8aJ ,p.21, L;rmula (37)we only use the noLational convention re~ular in tensor calculuscan also be derlved directly from Mo d ,p.2~99, form~la ( ~.12a)

cf. Tai[22] ,p.247, furmula (34),(35)lormulae lj.49)-(3.~4) artommittedcf. Mo[8], .·from IormUla (5.1) by differentiating to tcf. RindlerC~ ,p.208), form~la (le)cf, ZWikkerLl~ , p.110six-v~ctor ~ two-tensor

2d.29.

'j.

10.11.12.Ij.14.IJ.16.1 '( .18.1';1.20.21.22.25.24.25.26.27.

1­2.5.4.~.

6.( .

1.

1.1

~ ~~ ;vn<-J.4..

kx:cJ~~pt-o-n..1

cH1

1.

z.

3

5

<,..3.'5)

1 .2.

1.

c:h.3- 6

CO"Wlr~ rM-O-tbv l.3.1) w~i1 (.3.5) ~r'(.3.11) "i C (O)(l')(O)(J') = -L £. l'Xj')

2.

90

91

(.3.12) J(X,Ji) = -t ( ']O<-'j) + 9(-)<,1)) + -± (c:1c-><-'I) -<jc-Xl))

:Jek><-l) :== -± ( 5°(1) -+ j e- ><, j) ) ~o)/-(Xl) : = ±(5(7)-J (-Xl)) .

~~0(j tL r== o.q~'~ (AI) ~ {± (J~)<.l) + ~J 'f- '/) ) -+ 1- (~~(Xl) - <jex()\.'-/) }+

+ f -.t (JO~)l,,/) + Cj~(~-I)) + -± (5o",("() - JOX ()('-l)) ~

~o-n= , "-""""- .,;., , odd. "', ,

ge y- :::: t ( ~e)(.(Y-,!) -t- 9ex (><-,-j)) 'f, • '!CJt:= i ( gexO')/) - ~exe~'-7)) X 'I5oZ := i ( ~ O)/- (x II' l + 9ox ()(, -7) ) 'j x5; '. i l 90 x ('II - ~ox (><'71) '" '1w--d~~~V> vio -hro"", i tL au.~,J+--.t:..,'f'fi(.3.1~) ~5~ = tl (j()()') + 'j<-"}")) -+ ('j(X?/) + ")e-><"-7»\~t == l t (JU'j) + ~ (-'I'l') - (J (1\'-7) + J (-X '-7)) lJ2 = t (9()C.l) - 9 (-'1'/)) + (9(X~I) - 1 (-X'-7))

'.]7 :::: ~ (9(Xl) - ~ (-Xl)) - (J eXt;) - 9 (-X'-l)) =

7 ~"* (5("'7) - (x--- X)) - ( 'j- -1 )J2.3 ~J<-~ ~/~.~~ 1::0 ~4:: ~~ (~)

VlY.::>tead C~·'~~~ u...l.e- Cct/IL c~kuc1: Cl-

~'ldlan L, vrc:L:ao ~et ~ edt Vn.. ~ r J vn~ 'fIV),(d.f~<p..15) q I

Cf"v1; = *1 (C.yrrd. - ciFVd.) - (C. vcJ.;j~ - cfLd.t) I;:

= ~ (c"rt-< _(j-V)) - ( 0<--;3 )

3.1

3.2

c1.3- 8

(,3.11)

(.3.19)

1.

/Y'Yl. = 1 ;

;vn=2. ;

cJ....J - 10

95

96

II3. ELECTRO\lAG:"iETlC CO:"'STITCTlYE

TE:\SORS OF \IEOIA

Frl'l11 the eO\ananec postulate of rnacrosc"ric\1.1:\\\\.~11 equatIOns in media. we found"" that theCl)nsti,uti\r: tensor for a linear medllllTI \\jth 4­\'e!oeit\ \ \\ as

G,n = C,"'PF,p,

(3.28) C~'7Ii = p<,;(*rr'\*l·)J~,j + Ht"(<,,/lt,JI _ <, i1P r V)

- l'P(<,"v~ - <,'''[;')], (3.1)

whose physical meaning is revealed by its local form

G1u)(V) = CI/t1I")(71(PIFI7)(PI' (3.2)

Here, the local components of Kand E on a coordi­nate-transported physical tetrad {el~): in the framec(JIJIOl'ing lIith the lIil!diwll ha\ e the values

,,("(UI == 0. <,(0)(,) == O. K("w) == 0, K(i)ll,; == 0, ~

3

h«)lIO) == 1L hl';!li' (3.3)i 1

,,(,I()) and J\!rI(i) \\ith respect to {O~ in that frame havethe phy,jeal meaning

Dill = <,'''liIE(i). n(ll = K'l)ljlB(o'. (3.4)

With (3.1 ) (3AL the intnns'c physical propertIes ofthe linear medium enter covariantly into the [\1fl'rmJIISI11 F()r a "JCllLlm, C redllces to ['ill,p =1,')'1' I' s(' that G'l' = F"" in am frame and D = E and- 7.1 .

B = I ( I'l.r am nb.,cner.

ch.3- 13 97

2...

1

.3

~07f: Q .

.so (.3.34) lx_COVI\..U:> :(,3·3 C;) ~* u.) (Vf (~U.)~r = 6 ~~C- urLLcr

OI'nd lAJ~~ tL ~'Y\,u-hLcnL P~"I----~--~------------,

_ 1_ 1

- 1

l1""

~

0/.01 <r".l.f0< - ~ 0"""C1.3

~. ~

j3 -

_-1

(.3.::10) c.cV\I.. .be- L.0~~ ct..a :

(3'7) (~U)r""Y (-l4<().. )dfr = (b~ 8;-6a-6~) urul ++( u{E:,~ uj3 - LLv6~ uft) - ( lAf6}uo/. - u. -v9fu~):

_ 6;;; 1. + (u('6~uj3- u." D~~)-\ c{-j3)

LD~ch(3.38) '1. &~ 6; - 0 0' b~ =: 6~

(3.34) 2 uru..( =,Nottc.L tkJ. ~~ Z.3 /Y1O u.De..~~ oftL FFk oJ: ~M~QI\~~

J,.3 - 1599

4.

~.3 - 16 100

1. J-,.. bL rrr t1.- dvu:..."Lc-,,' tk n.Ja:l~(3.41') D'~) = I (K(1- u.(D)U(C») + E. u.(O)u(o»)5~ - cl<-£)UUJU(kl ~ EC.'('+

(.3.42.) D

=-B~ =: (K- E") [0

5 CA.~-L- tu.t~,5((2. ) 'i

[G; ~'YlJ =

(3.1)

.J

o - E'l _ Eti) _ EL3>o - 13i!'> B(i)

o _B(l)

o

{ ( K (D)) (0\ 5' i..= (1-u.. u- -rcU U.-)uk(0) (0)

{ (\(-E) U (0) E Lc: . ~ R (,w;)

- U'l) .. m.) u

~~toh~.

4t,e.d,

[

0= (m) (0)B ;= [B c"'] = (K- £,) u.

-.... • I.)

10Z

u.iii U (7, ju- u-'"' (I., _ C)

Ull.l ca>Uw (). %>m III

U U - LA U-u) U)

U1.3-19 103

1

(,3.55 )

H=C B

~

= -+ D E

a..

+

.5u..h~LtL\.t;~J( 2.. )

[ G tflVJ =

o - D(1) _ D(Z) -D(!)

_ H (3) H(2:)o - .>

o - H (1)

o

~·1o (3.1) )wt:cl CQ/l'L ev60 .-be. L0~ Q./.)

o - Effi - E'Zi - EO)

o -B(j} BUl

o - B(1)

o

(3.1)

3 ~--b-~ ceJ.cuJa-L~./

/'

Q. %~~t.-,.1gsa).c~~ ~i~_CVO' •• h

L.l LA CO> Eoki = - u CO) E..0J\~ - {ol E. t:k

h-c~ j LA Q~'

(..3.59)

b.

1 .3 2

2. ..3

3 l

J

~.3 - 2\ 105

c.

~.

+=

~o ~ c=L= _tLr1(3,~3) D H

E 8fa ·-E€. KJ< €

dt..3 -22

l~~v:'\(3.6;1)

(.3. bG )

106

..,;'

r

~;!>- 2.31°7

1

- €. ~ ~Ol. . I1 II t~J'11 +f~ ~trJu.tOlu _

lOI

1 /!J". tk"~ tL dvuAnb 1(.3. b8)

fE -1 0U (31 _ U <il-

- f~

U <Ii }.HlO- U (0) 01+fE Ll_UtTI

UlOJU(e-)

0

&q~~ J== ~d u<d.= ~/ WL~1L- B(3.42) D = As E. + Ba... \3(.3. 56 ) H _ C s B T DC<. E.so-(3. 1,9 ')

= ==-1= - ==-1-D - (AS - BQ.C Do..) E + Bo..C H .,

~ tlu, ~c--b.·crn h=:to~ ~~'Uy -e-X~ tM ~"O c0r:t,~ ~ aD vn (z18). ~I

2.

-'3

(3'70 )

ck.i- c: = ckt ( (K-c )

Ccx-«~

Q. (~£f

--t-

=

+;:;

~

-- 0

d. ( l \3K-£. I

1

>L (.3/°)

(3'71) eli: C _ d<_£(·[(.K-.).3_2UIUi (L)2. + UlutuJui !\-K )' 1 ~l<'-c k'-E:. 1<-E:. j

I(~- 2U('UL K.1( K-e::) + uiu .. ui uJ K( K-E:")2.

i< ( K- U'Ui( k-E)r ~ K (K-d (u'u,- ~,J

cl..-l: c 1- 0 O/>,J ~cnL c -1 .-<-xiJ;,

C-: D~~~P ~ ~l~~~

=

J,3-25

[) 12.. 1/ -2..( ~-£)

DZI \( -2-( -£)

\Ll~-L )(Jut-K)\<""£ K-f:

U\.I"~

u.3u -L2, I<-E:

~ C -I ~ ~o =)f~-k..:c...

u 2u2. - L\<-€

u..3u2..

.3' '\

U. U (Lfu~ - K )2.. )(_£

u.1 U2..

tko L-Lul cUe ob,k. "(llIU1- K )(u.I'U,'_ K. ) Lt'U (tiu.. -.K )

1(-(. 1<-(. 2.. L 1<-e:.

1 . (yt),l

- -~----

u.1u·J

-+- 0

u3

_u~lo U ) =

a

dl..3 - 28 110

1U,3 -l\.] [0 u~ -UL]o u., ).( -tK-E:)L Lt 0 ~ )

o (K-E:)u. u.,:- \'< 0:::::

1< ~ ol. l' , u1u.u'u J-UU-LlUl -E) lU.) .3 .L 4 :3- U'U -~U -u.'U U.l.U( K-q u.'u,,:- K .t .3 1

-U.z.U: - u.'U=

U.1U u\A,1 [, Ju ~U3] )(= U U 1( 1<_£)4 UOU - ULuc: I + u..tu ,

L ~- 0 Ll UJ. UU3( !<-£)ULu. -I< Lfu

1U.

3L,.l tillL

~ ~

8.

~ LUU ~~0V0~ "be un~c.h- L~cJ:L~

Cl.. (.3'78) 1 + I<-~ UOUo _ (K-oUlUt-I<-+ U."Uo(K'-E:) =<K-£)LlLU~-I~ (I<-C)U1u,:- K(uOu o + u.'U.:)

_ (1<-£) 1 - K-€. uiu. - K u°u.

l °-----c---€~----- = 1

€.. LllU' + Kuou uLU. + K u"ul" lEo

€. + UlU t (\(-E:) ( 1 +- K-e: UOu,,)(K-O u.LLAC-I<

== €. + LlIUc( I<-e:) ( E. ) :::;;E. LlIU· + KuOU 1

L c

::::: E ~ LlLU( T E: K uoU" -r LllU, (K _ u.l u,' E. 2. _ E.' U_o_U_o -----;-__

E.. u'u, + K uOu o 1 +..f..- .u'u.:'r( uOu"

L0e (~tc-t 1 ~ - 1 [ }- = --\-E( U"U" - LllU

(.3.&0) As - B.JC 6.J = I + ~+ & ulu,_k

L U"Uo1+~ uucK uOu/) I< uou

II

I I I

1 /J" tk, F"q~~ 1:k cIvu:~ ~(.3.81)

-'- u<.ii _B r{U(6I U

-fJ-E -1 }- @ I + ~tjJ H +

1 + fE utnu _1 +(£ u. liiu .,. LlCO}U

W _ 'II (0)u(O)u _ U. (O)u _

(0) (OJ

0u (i) ci)

lf: C - 1-u

+ i----L. I EtU u(O) Ut;i

1 + f£ u. Uti) 0

Ju(Oh -to) 0

L.. e .

3.

'+~ (.3.S L) 1)1C>U..b:(.3.8l) ~ 5 = c-1~L C-

1

D"" E

0;/ LwW- tl.-~x C-'Lo ~cry ~uonJ,/ (31.') ""oJ.-I:L~ti-1s~ -7 <yS).~ Ju,.c...,.t""",;, tL,,'~ CCP""r,l..U

W~th (3. c93) ~~c..~ ~.ban.c Q.nouA-d. -bL~ J

(3.8,3 ') (K-q U.\).L - K = (\<- t:)( 1- u.°uo ) - \( =

= - £. LLLU. - I< uOu = - I< u.°u (1 + E.. U'U,)l Cl 0\ _._

lAJ.e- ~1. )<..uvncn.JJ (.373) J(3.75 ) to. f t (3.a1): K UOU o

~!~d)~ 1 i (_\<1 + (K-<l[ LLiUJ

lJ) =K(_ Ku"uo) ( 1 + C u. U,' )

K LlOUo

~ -1 [ ~ 1\

o K UOuo

~.3 - 2B 11 2

3.31

1,

(.3.4~ Cl)

D = ic£: ,-,'0;'+ K(1-u.'C")) I T (K-n [:,;'2 ~(.3. S~Cl)

H

(3.81 a.)- 1

B - {~- Lf)r 1-+- j.J.f.. U. _ Uril

/ l1o,u. _(OJ

2.

A-

Ll -

( 1.1.&-1.31) ~ (.3. &4 ) CQ.K1.. k

~A1J (3.":;"c..) (3.SbQ.) ~:/'

113

(3.":;66)

[~0 0]_ [0 00]D = E y'2. ( 1 _ \.)"2. (e.()-1) o -1 E + ...1.(11f.) y2.AJ" 0 1 B0 (1.( l- U '2.(j) )? - 0

(~.s6b)

[~ - 1: (1-)""')('>..f 0 !] E0

o fH :::- 1 r2.(~ - LY'2.(j)+1) o . 0

JJ- i-1I y2(1_0'2.(;)

/

,

,

() .­o-.lL.".- (n2._ 1 )1\T ,1 - (;n.v)~

(.3.G8b I [1 0 0] ~o 0OJ-& 0 0 1 HD - C- O 0( 0 E

o 0 d.. o -1 0

(.3.81 b)

[ 00 J ~O 0 OJ- -r JLoo 1 E-B =r od..o Ho 0 0(- 0-1 0

c-R.3 - 30

I

11~

(;tJj ,-lJ.

~(" E

+-,

u t31

o

[

[

+ (f.l[-1u{ 0/ u:. -

1

Li)+ fJ.€. U. LJill

/ LJLOlU _/ (OJ

L lK-£)u't~~ du...bh.L.-~-,D-€-c:±:cn- (pLx-,u:ck f'.ecro--b.o-n..~n.o /Ynu.~ lU'-0-VY1J)<:V11c.c. LVL~ ~ b...l~ era.hrnv

l.3.1) G~J...V) _ C;u.x.-JXo(x./3) F:::J - I (d.y3)

r~-/2_:~ ~ ~a1o ('1) H) ~c.l:L-1' Vn ~V) cd ( E, B) vV1 ~/~occ.1 ~~.Lv\n~ co-vncrutn';)l . ~Q/>1~ •

'1/ ~ .--eJ::fL'o-n- CQftl ~~J~ un cJlq ~,e'ff1' 1] cLJ~'Zl.A(9o~,SO.~.L~1t ... rve.ck -e-1----LCrn, ~LD ~/1~ ~cL .1)t~ _q -tR.e-/Cv2-t~ tAl J::;, CcrvWlLfl~"CZ..

,-' - -+-,-,-,I

D

is

H

J...3-32I116

2 ~-tu:c-L~ ~~ to /~d~ ~-h-m.~cu.:cL 1('{3. l8b),p. ~hb) to

(3.Q1)

r~· = kU? ~tfJ~~~-~ ~Jl.EI [0 00] [1 0O]j

ll5 IJl 0 01.f- 0 O<o R

L 0 -1 0 / 0 C> 0( ...

...l.." /Yl.. = (€p.)~

I0( == , _1V1.

1 -((h.lr)"

J1 = (Irl.t- 1 ) A..r

1 _(fYlLT)2.

d"13 -33

-4.1

1

-1 (1) y-1 (1) J[ E (11Y v-. (-,) v-, <.3)

- (-') 0-(2.). E (~cr-. (.1.) . (,3) T- (3) _L3.1 E c.3)0-, (.l.) 0---, (3)

-1 (1) - I (1) ] [ Q (n]r ~ -1~(.1.) l...)

WI)) (:2.). (ij0-. (,3\ - ~ (a.) B

1.3) 0) c.3j0-. <.3) - v-. u) B

(1)

(j'. a)

U)V'(2)

-1r cr-0(1)

( 1)

- 0-:-(1)

- ~(')-1 0)-r cr-:- (1l

3.

(3.98)

(3.100)

(3.101)

(1)

rr- (3)(Ll

V . (.3)

1.3)cr- .(,3)

oo

o

)f 0

+ r<.t.r 0

~

J(1)

fA ;..:rcr- '(1)

0--(1)Y -1 '(1)

l o-(.1)

~1 '(1)

'V cr- UJ_I '(1)

~.2.

1

2

, E B' r r,. -h - ' - I= ~ .( , ); :en.../ l.oca.1.. lClO~ ~ct..L'Q..

~ i

[

0 0 J[8 (f~+(,DD- 0 0 -~. B(~o 1 0 f3~J

CHAP'FEH 4

POTENTIALS

§ 1 Gener8l wave equatjon in terms of potentials2 General wave equation in terms of potentials for

local isotropic media3 Three-vector wave equation as a special cade of

the general wave equation for local isotropic media.

I

122

Introduction

Starting from the covariant Maxwell tensor equations we derivein this short chapter a general wave equation and one valid forisotrcpic media both in curved space-time. The equation itselfwas already known from Mo[8l p.2594, but its derivation was notplblished.

The equation is expressed in terms of potentials and containsalso our familiar vector wave equation for isotropic media inflat space-time as should be expected.

Conclusions

1. I t is pre ferable to replace Mo' s condi tion W by L1.~A

2. To my opinion there is a misprint in Mo [8J, p.2594,formula (1+.10a)the fac tor K-€ has to be replaced by ( I<-e ) I E

Notes on chapter 4

I123

1 •2.3.

4.

cf. Mo[8J,p.2594, formula (4.9)idem (4.10)cf. Synge[17], p.15-19cf. Synge g. Schild [1'-)], p.89cf. Seidel, p.48

§ 1.

12~

1. >h- df-J~ :r;-bn t)occJ1y~~~ /"~ :1

(~.1) ( Cfv f Ad-;f ); Y =0 l I ft,a.;::, Q ()n~ -~ e-ta.na.c~ ~ L-0vU..be. cku.<.x.d jY)~.

END§ 1

§z

~.

ct4-6

Clc"mdq -1:0 sT<r [15J, F 10-4, t:L ICccc..:~ u, dz{~;.d -7(~,1b) I?~ := '1\ ~1V = 'Id ,

:=:0 (~, 15) CQ/n. ~ ..e.x.~cl a..o CL Kecct'~

(4.17) - K vrr- __ rA fK \( V (~,'b) _ c r~nf4'1C.1( =... v - J AJ oAK-V j J Ale.

= - Jr' jP Ie. K. 1<:>- = - l?-rr ._SubahLJq (-4.IS) ~ (-4'7) ~,to (L,.1-4) '-'-k~~<f=

(-43) j"K jfl\ Cf"",x A"f;v = \~ (At';v;v - Kr(Ar - Av;v Y)

/2u:c1/J\acl to..be- ~l~d.

(k20)

( ~.2.1 )

(~.n)

(~,23)

128

!S.

===0

G.

130

(2.51 )

(cry., J-l-- -IfL' v)/ '

o

1.. \ ) r A u..;v AJJ-,v~ ~~-5~<3JCCf">~CL"'L 9 \/~'I=O =) ! =M

-9 /( r = 0 ~

L.-I..-~ -eyte.o,"V C(.A ~ ~Cl.<.-<.r co-ndJ:<.b-n to

_ ( 1< Al-t,v _ d<'-E) 1 b~ Af"l'O),y + 0

o

cJ'CDAt:j a. M~~~l~'Q.N1-~--tru.'G [j.oZX(3)J= [cL~(1)-I)-')-I)J, U-J<.tlk' == .Ji"~ U ...K c6Jl{ 1'<.£L-0n..<.~ \ 4 . .32.) ~ ~

I

1(,L,t) _ -L ) 6 fA- n(o). - (1 E: to I J

. (d »0) rd. Ato )J... \....-{b) 1\ + L~)

fJ--I

J,S-1

CHAPTER 5

MEDIA WITH CONSTANT PROPER ACCELERATION

133

§ 1. GEOMETRY

1.1 Hyperbolic motion in flat space-time1.2 The metric matrix of medium coaccelerating frame1.3 Instantaneous Lorentz transformation1.4 Localization of base four-vectors to coaccelerating frames1.5 Graphical representation of laboratory and coaccelerating

mesh-systems

2. MAXWELL EQUATIONS

2.1 Tensors of the coaccelerating frame2.2 Local Maxwell eq~ations of coacceleratin~ frames2.3 Three-vector local Maxwell equations of coaccelerating frames

3. CONSTITUTIVE RELATIONS

3.1 Constitutive tensors of the coaccelerating frame3.2 D = function (E.B). H = funtion (B.E) in the laboratory frame

~.5.2

Introduction

This chapter is intended to serve as an example of the methodsto be used when dealing with moving media.

The equation of motion and the geometry, consisting of metric,localization and Lorentz matrices, of a medium subjected to aconstant proper acceleration in a flat space-time are derived.The concept of localization is made clear with a simple illustration.

ivhen for instance calculating propagation phenomena in moving mediait is often usefull to observe these from the point of vieuw ofa medium co-moving observer. In such a case the Maxwell lawsdeviate from their ordinary form which is shown. See also Mo[@, chapter 5A-B.

Neglecting changes due to accelerated strain we also derivesome constitutive relations both in the medium comoving frameand the laboratory frame.

Conclusions

1. Lorentz transformations and constitutive relations withrespect to the laboratory frame are similar to these formedia with constant velocity replacing constant velocityby instantaneous velocity.

2. Localizatior appears to be a rotation operating on co­ordinate axis.

3. Effects due to accelerated strain have to be stated expljcitelyin the simple comoving inertial local constitutive equations:e • g. C3 •3 ) DC" =t:'~'i)~j) + •••

4. Because of the assumed flatness of space-time no difficultiesarise from coordinate transport.

5. A misprint has been found in Mo[8l formula 5.11a

cJ,CS-3

Notes on chapter 5.

I

135

1. H.Minkowsky, Physik z. 10, p.104 (1909)M.Born, Ann.Physik 30, 1(1909),sec. 5

2. geometrized units3. cf. Rindler[9] , p.2083, formula (5)4. idem (6)5. idem ~8)6. quotation Rindler[q], p.2083, 2 n column7. quotation Mo(8], p.2~96

8. cf. Mo[8], p.2591, 1s column,formula(2.3)9. see paragraph 4.5, chapter 110. chapter 11'1. cf. Mo[8], p.2'596, formula (5.4)12. quotation Mo[8] , p. 25CJ6, formulae (5.6abc)13.14. no ordinary matrix multipication , see Mo 8 formula (5.6b)15. cf. Mo[8a], p.43-44, formulae (11a-c),(13a-b)16. cf. Synge [17] ,p.3, formula 817. cf. Mo[8],p.2598, fo~mulae (5.7-9)18. cf. Mo[8],p.2598, 2 n column19. quotation Mo[8], p.2598 , formula (5.11) with a corrected misprint20. idem (5.12)21. not necessarily with a Minkowskian metric!

136,

§ 1

r =

-l.. r.:wYvhca.-ba.

A _ ~1 (CO~J.L(G.i)- 1)a

Cl - CO"nQ~.t r~ a.c~tLnv~

1

§ 1. r

2.

<--------.------ --.J

~ C-.OJYu cvbo..be. ;U)M t±:em. ell:::> :

(5.1~) Q == ~(t0) ; ~)=~+r~(~-7r(7~-+1)~= r3~

0....\

c9-t.5 - 8

ro-=nmd A:",,~-L?i r «uu:>(5. 1;5 b) )...:r _ -;::::::=.0..:='1===:::::;-

V1 ;-(a..lt

4.

\-idently, lii.!;ht "lgnals emitted at the origin at or1'1 = 1 a never realh the receding particle. :\ ate also

any parI il'le P which moves uniformly :l1oll!!; thexis n~eets tlw particle Q performing hyperbolicion either not at all or t\\'ice, anc! thea with the sameti\-e ,peed on both occasions. Thi'i is obvious when-on,idcr that in the rest frame of P the plot of Q's1 b abo a rectangular hyperbola and that the plot)\ path is a straight line pcrpemlicular to the axis a..ilL' hyperbola. " S / I

_____----4C----e---1L----f.r--------------x.-

13~

5.1

5.2..

S-t~ r +L ~"'. 0;1 tL ~-kcr&.. y (131)

\ 131) c:1-l: ~ r ":1.'T'vue-- CQ.IY!- j1L~ }/ tfr., ~ oJ -tiu.. cl~cc.l ~L~Y p- ~

d ~..L I-t = (1 - ,v4) 1.. d T =

~~ ( 5".,51) cvrd w~ti, l"'~tJol.o=·h- w:lh c==-bd: CL q'-='d~) = (1 _ (ClI)2. ) t- d (Cll') = (1 + (a.\)2.ri d (~'T').

1 -+- (a.'T')2..

Abe<-t~ UJJl QA'd cd Gn~G.YAL [23], 2.01, r'S~) 1&1 ~r-x.-<-b.gLt~~ -tl. "",bJ!, e<n-d.:b"" / ( 'T = 0 • -l: = 0) ~-b, ~-J:o

. a..t = ~ (0.1 ):so-(5-10-.') T = -L~ (a.-t)

CL

L-Yll u..>cu:> -te:> h cL.ru:~./

..'::,~-LL.L'ij (5. I~ ) (j""'" .,;,to (5.151) en ",do, to -<»<pu= A'r - ,u-cb ,

p.2o.) IV" = a.T = ~ (a.t):c = -h-Cv1.l (Q.."b~1 + (c..l)].' ( ~a.(C\.t) )-to

QA -d 4 tk,,..J .,;, tL d..~,t'o.n i Y (1 ,~t) LU<-Cf-h '.(5,11) Y= 1 = C.O~"t( a..-t)

~ 1 +,v2. r

LUU Gu::t.c tL/bi: ~~tOrv +Jl~ to.be-- ~<xJ,/

_~ 1- _

cJ.s -8I

lLtO

o

(52)

~) • (5.31

() -I

o

-I

-I

oo

1"c.!.llIha{,

(\'III,IS.\ 111111,'1111 dll':kLltl"ll,'C I

as in Fig. 2 \\here origins have been adjusted ~o that at(= 0'; Ttheirrelati\,cvclocltyiszero.:\": ismediumcoacceleratmg in thc sense that ea,'h ptJint of fhed

- - .... .__ ._._ .. _.,_.., I

Let {.\:";: == ~Xo, Xi, Xi, X~: == \T, .\, L /: be a1:.lboratory inertial Mink,)\\~kian Crame. If a mediumis uniforr~)h linear accelerated, its COnHl\ ing frame:lI'- ~

can be described b) an {X/I} that

T = a I sinh (It. Y =)',

X=a1(cosha(-11+.\', Z=::, (5.1)

ThiS :.\,11: i~ jllst a c,)l1\enient \'r,II1lC dcscnbing thcmcdium mot;,m and I~ nClther s) nchrL)!Hlu, nor ,latlL',Hu\\c\cr. { is thc pn)pcr time of medium ('onlmins

obscr\ crs ;0;.'[ 7

2

(5.21) .

[

cJ.LZ -~hJ;' 0

_ thJi'-cl-1.t 0

o 0-1

o C 0

1.

1

QI\'1d d:c j\..£.cc.:1:.n.cx:al

(5.2,3) dx. l .,() = Nd.) dx.,l\)• (K)

2.

(1.'23')

(s.n)

"",bh -t:Lr~L ~7 .;.= (~r- ((~F+(~J'+~/j =(s.ld) , n~ (0 l-~ c..A\ - s,....f1 + 0 -t 0

1 =:: <~ .t1. ) _ U tOl~ _ ( u (i}- + u ttl a. -+ U (if)

§ 1 ..4

1

2

'-'1 l,. ~ - I I 1~3

3.

( 5.,2,q )

( 5.30)

D~~7'-";q (5..3l) to( S •.31q,,)

.3. A-.:. ~ COX1 h. c::k~ ~

(3 c.i sh 0

;J~ , [~-.st 0

~I[A/] = [All' (~~ [~ 1 0 ~ 00 ;;h 0 0 ~ .>

0 0 0 0 0 c:JvJ(5.32. )

7Vu:eh~~

~l~ Co ~+=-t I:L~ ~.

~~)-a-.L:d..1 ykn, (53') V>"'-~~COvnF~

1. A,~\~\ v; ~..um... r (5.z.4)

2.. L2.J e.j> Y ~~ foYn (5.26) ell 12cj;/~ = b~

.S f\'l~ A' S[1J e '00 = (0() f f

wU~ LJ.):Vf\W 1.o~,ZJ r U-ith Mo [&J. f~~)

~NTl q 1.~

(5.l9 )

ock'oo

r~ ~ 0 0] [;J,-'-ct °°1I' e' f1(5;!1a.) ~ cJ.. 0 o. 6 1 0 0 ;0::::

~'] lo() J - 0 0 1 Cl 0 0 1 0

o 001 0 00 1 j

~ ~ l~ ~ J

144I

1

/

/

I 'T' v

I 1// .~ 1ci.1'

" " / /dX

eo= <16i

!/~e;" (i>

1/1/ x./

C~)~

"

~~!~~~~ lines. of

~~nt'T"

?< ~~

1uc:~

~L~~'~ ~~ ~-f'~~klCcYta-S ~.al ~~

f ixFl = ~T',XIlA.)~tt jVnL-~ (Y'Y'C<-t-....:x \5.17)

I

[9fV] = [~ _~]~ aJa-i ~- -Gmv.. .

[~r] ~ [-:h ~~]w, CL~P~~'

0.-1

//

~.5- 1Lf 14G

9 2.

§ 2..1

a\.11 (' If I )Iy) , Iv) 1,1))~ = --,__ . \ i ,)(i') or )" = ("'I i,l f i,) > ]" = p + tanh lit) ,--,) ,) ,

a.'C' , I;,} cosh at(5,6:1.)

a\," axv , I") .' I' v F,,'I/31, '\' \'> f "I, ')r F '" = (' Cf'" = --- .----; (ii- I,,' 1,1 1#'ax' a.\' ' '

(5,6.:) " l~

(5 lib)

o

BIz)

l'osh (/ t

_Bll )ocosh (/ t

o

cosh (/ t

F'''' =

'lrnil.lrl) .

(j"'=[L .I),B ·1I111(~()b}1

th 0 0 1(+-thC~1 0 O. 111_ c~'010 w 1

o 0 1 I]) 1

1

-c:k' El1l - EW - t:h 13t3) - ELS) +thB(~l

C _ J;"Bts) d::.' BaJ ~

+ c~' BU) 0 - 1:1,(') r.... '

- Ji' BQ.) Bll) 0 i-'

( )

t1I1..J1> QlJo c u -E -t:.-E"

o 0 0 - i3»..al»1 0 • 0 _en)01- 0o

[Y] = [~'AK1(A)] = [e(~( J~ ['J'~]~J9) ~o

=

( ,(5Aob) , th[FfVJ~~l [er(~JY ~ ~~1

o

§ 2.2

~5- 15

(5.424.)

~'nt1lT D C41

.D(.S) +" ';t -t "3

(5.416)

H -HU)lt UlJ3

-et'H + H~, (1)'3

c1\ H - H(.\)'1 C1lU.

= 0

c!s- 16

Subl:LLq1:1(5.2.1) , 1 -~ 0 ( 5,2.'1 ) 1 0

-~ 1 [e !\~J~0 J-I

[~fVJ~ 1 evncl 1 ."

0 1 0 1

-l.. ._ (_ ).2.e,J,n t ')J '(l)' J

= ~Y1"J= cR, (~)

= c1 1 (,3)

:::=:1(1)

=:: e-t JT>'l== d /]0)

~.1

CAtS-7

( 1 H - 1 H ) = 1 (1) + 1 D(1)o H 0 n (.3)J,t (V,.3 '0

- a..ch (~)- J:Yh HC;3)JO + c.h (_~-I H ,... 1 H ) =: CJJ +~ DQ.) + 0 T\m+ Cl cA H +d, H 0 '-3))1 (1)13 'e G.. !.>'11. J.....J

Ctl ~)J 0 + cJ1..- ( dc-I H - 1 H ) ==: ch (3)+ cit DW + ' D(.])W J1 (1)J~ • 0 a. ./.Y1<1

1~ r==:r=~ >~~" ;1: <i>~ tl-.a-t tL M,,-~u->JLEw>~ ~Ll:01.- J21ua..~ cr- ~.c. ~~\:<.. , (*F)JV;v = 0 J G,rv;v =-J]Fl..Q/n-~ u..:vu:~ a.a

(5.~~) (~ F )fV,y + lyvD( \ Fj<>:= 0 ..

Ut 5 - 18 I :JU

d..s-19

~.~ b1

<-J,.-1):

- G (,)(~) i_ G; (1)(3)

>2,. , .3

H (.3) H W

r n ) -t- c...-t.h r 10 - 11q 'v q - -

==

151

en.;:; -20

fA- =:3/

END§ 2.2

tj. B - ;:;. .\lX E

V)<..E + Qx.. E +- ,v-K ~ ~{

C5 o S 1 0...)

(5.51 6)

{

(5.52 CL )

(S.5A- b ')

Q/lid (5.5~)

(5.5.3)

_~8 +)}-x.(QKB)

\7. D + :c;... 'l A H - t:r -t- ,u-'j'VX H + Cl A H +)J ><- ~t R - I J t- ~tD -IVX (ct:)\.. 0)

at.,_ ..:npb_tL)<cd CC>"Y'ttm"'7 er~ _- \/·1 - Ci·1 == ff + ;v.~1t ra:·~

CR.5-21

2.

o-a. H(31

a \-1(11

Ll) LI'J'). - a. r 1 Eo "kI.J

\7x' H

QX H

(5.56 ')

(£5')f..'A H) =t ")

( )1..1

( )ui

(lJ x (Cix 0))(1) =( )(~) =( lUI =.

o-t1 (+0. D(,2l)+tt (-a D<.3J

)

3.

3.1

chs- 2.2

(s.b1) J (5. b2), L-Ue. CQ.IY"L. ~o Fo-u<:. ~t

Q4 ~(l) = ~~. ~j ,~~aJ1-.6)<.0:) ~J

15-1

(5.65)- ",-L-V = Q.. .. e

d x.~

_ d- -

_ u.:)

e

Gh.s - 23 155

§ 3.1

Co?· '''I ,. ,c,'11 un" ': I' tdnll r I) f ~ tdilh "f r t. .tIl 1 £Ii .It ,if < dt ~. d: ,

),c,'h,ii l~11111 df ,'1'1' ,c,'h CIii ,(','11 '1":':' ,,:,'11 dl

' ",II , ,iI t ,I,

'l'l h I' 1C' i2 "Llnh ,; t l'" 1 '0 , ( (,,,1 \,']1 1,1\1'.." I :jl I .~ ,\ tallh dt (_I;. 1\ c,cl'h llt t ( t

19,

2.

(sl:A )

e:. (1XO) + tR £. (1)(1) €. (.lJ(O) + tJ, Eo (~(I)

sh E:. (1)(1) sh E(~X1)

e: (1)(~) E (~xa)

E:. (1)(3) E. (~lU.l

sh.E:(4l(~th..shE:(')('J e:.(O)(,1J+ tt. £,(1)(al

.sh' £ (IlU) .5h. e: (1)(;1)

sh e: (2.V.il e: liUl.a)

.sh e: (31<1) e:. au)

3.

1.

,

1S(;

.zo: ~ \.::,,) .

2

cRs -25.

C~q Q ~~ ,=t.eF,,- ~.~~ chv.e~kJ 1;/(3.8) E.. t\"Vl = E. 60 .) 1< ;:~Vl = 1< 6~

evncl P<-'-b,hLt~ ~ ~ (1;571) lAk C-OA" eke! -ll. ~cLrclvu:~)ocal cbdJ:.l-c1:lX. ~-Axchn. ero.lCm0 (3.1,6 b) e::vnd(!d.b)

~ ructJw=- ~q ~ob-, ~~"1

(5..lj ,AY" =; tevv-J., (a.:b ,r = co::J. (a.-t) ,yo- = ~J, (a..-b ,

(5'72.) [D (1') J= [~ t «_ClI(.o'-1 ,~ J{E((JJ+ yLi r~ I ~ K~e:ll. [I3 liJ]() 0 Y (e:- \<~) lo r L-I<-tE.. 0]

EN[)§3.2

<:-t.G - 1

CHAPTER 6

b'lElJlA 'd TH UN H'OHM HO'l'A'l'IUN

§ 1.

1.11.21.31.4

2.

3.

GEOM~'l'HY

Steady rotational motion in flat s~ace-time

The metric matrix of the medium corotating frameLocalization of laboratory frameInstantaneous Lorentz transformations

MAXWELL EQUATIONS

CONSTI'rUTIVE RELATIONS

vt. B- 2

Introduction

Geometry of frames, consisting of metric and localization andLorent8 matrices, relatea t~ the case of sGeady rotating mediain a flat space-time are derived.This can be used when consideringe.g. a ball bearing motor which was our initial aim.Maxwellequations and constitutive relations are not derived as originallyplanned because of time shortage and because we foresaw therewould be no need for them when considering our ball bearing motorfrom a laboratory fra~e.

Conelusion

159

When the angle speed is independent of the radius, .0. +.ocR}, thusfor small regions, geometry is ~nalogous to chapter 5 with theexception that one coordinate (~) has to be rescaled by the radius R .

ete - :3

notes on chapter 6

1. quotation Mo 8 • p.26022. isome try3. no difficulty with coordinate transport because of the

flatness of space-time

§ 1,

§ 1.1

Cl Consider a lab. cylindrical inertial frame {X':} =={T, R, <1>, Z:; then

(6.1) T = I, R = r, (I) = <P + O(r)1. Z =: (6.1)

carry {X") to a steady mediulll-corotating frame{x P } == {I, r, 4,,:: so that a fixed point (x') rotate~

with U == U(r) ahout the Z axis in the lah. 1\. Since Umust satisfy r~! < 1,4 it is impossihle to have U =

consl rotati()11 for lar,ee mcui~\' anu the most possihle

"rightlikc" continuous rotation U(r) should be

as

if

r .~ if), (6.2)

(63)

For example, if the proper centrifugal acceleration of arotating observer is to be proportional to r with pro­portionality constant U02, then

which satisfies (6.2) and can be taken as the relativisticanalog to "rigid" rotation, " 1

1.2

162.

§ 1.2.

+ tJLdr + df ~ JL = dd~(r) = - r( JLo }= _rJL~1/1 + (rJ'l.J

(b.n.) c.Q.N"\~ ~b-L:L-U ~te ~.10a.) 2. 01-b~ :

1 (dirt - (1(JR)2.-K.\d~)~- 1 (dZ).t) === cd-\:. /'( 1 - (rJ1.J)&) _ (dr)2( 1+crLfi.'f)' - (d'¢),~)-2 - (d-z( 1 -+

_(dt Xdr) 2. r4t ..f1.Jt' - (dt.Xdfp. r~JlJ _ <Jrxdf) 2. r"-t JLI

~~R~(,.J: ~7)1iL~fW'.L.

-- .... ...., - ...., .....->

16.13) [ 1 - (r..lL)a. - rtt Jl.JL' - r2.JL 0

l Jfv ] =- ~t JLJ~ - (1-+ (r bSl:~ - r.tt JLI 0 -_ r L J't" - r:tt JL' - r~ 0

0 1 30 0

~r.3t. J'L~(LM) 1 - (r-JLr2. - r~J'G 0

r 3 t JL-4 -(1 + (r:tt &1) ..-31:. JL3 0- -r:l.. JlJ + r~t J'L3 -r~ 00 0 0 1

~ iL conoiJ,;~ ,fa-- ~ ;vu--<-tLn. vYlQl-W IY'(Yt. l:l...//Y1~./

.+.

JLI - o

E~D§1.Z

~.6- 7

§ 1.3

RL0l~~c..

( I

~)

lLJ

~

~

(bIll [e~ClJ = 1 -y' [e 0(rJ = [..1"'.]- 1R- r 0, 0 ,

LA.le. cevn ~C~../eU d; c..eJv"t£C~ LUt"tJ, (b" Ib) •

~_ ~ ~t~

t~\~ R.i~ • -~

..Jt"YTlR-t..LC.

!'-"]- [: ~.]

END § 1.3

cJ,.6-8

§ 1..4~ L,..,.J,..~~.

§ 1.S

(.5.31)

~. 6 - 9

1

2..

3.

o o o ,1 c -J1 0

0 1 -JL4: 0-

0 0 0

0 0 0 1

0 .J\.-t-.Sl. 0

t' - -~t (j

~~ ·-1=

-r +r 0

a -10 Y

)..

o -l~ 0 -JG 0

o 1 - &t 0 (621)

o () 1 0 =o 0 0 1

~~ ~-b(5.31 a.) q ~ft

[.A] [ 1\ . (DC) • Ii A.A.] <1>.27.> [/\ • -:J 1e = e (6...:.!71 (~l

1 (II:) (Ie) 2.. <Cil:) ft - (leI' R-1

o ,\~l >. >..

(Ir.l~ () rJ'L 0 (~~ 0 -SL 0 tlQWo [-1 0 0 0 1 - J4-l:. 0 (6._~ If 0

=I rJL 0 1 0 0 0 R-1 0 ! rJl.o 0 0 y-l a 0 0 1 0

;J

END§1.~

§ 1.2.3.4.

CHAPTER 7

CURRENT DIS'l'RIBUTIONS

Inertial current differential equationsCurrent distribution in a flat moving conductorAddendum to paragraph 2Current distribution in a rotating conducting cylinder

Introduction

In this chapter we have deriv~d a set of three coupled partialdilferGJltial equations gouverning a stationary current distributionarid exprt:ssea iri any inertial illcdiu non-comoving (laboratory)coord ina te fra,l,e.

Applyinb tnese equations to two simple configurations :1. a uniformly liiOV ing flat. two-dimeGsional conuuctor fed by

a point source,2. a uniformly roLatlnc cylinder fea by a current line suurce,

we nave1. analytically solved tne current distribution in the fIrst

conliburatiun and2. given the decoupleu differential l~qu.ations in tne second cast:.

We aIm to discover and predict easJ measurable drag el'iects inorder to demonstrate in Which cases constitutive eg~ations asiound in cGapter 3 musL necessarily De applied. We su.spect theball.-bearirlb .llotor depeLJ.l1s on such aJ.J. effect and the two illentioneuconfi£..,urations are sHlIplyfiecL models of respectivily ring and ballof trie bearing. l'hese cases tnL.Ls lorm

1. examples of application 01 the foregoing chapLers and also2. all introQlttion to tne svuJy of the ball.-nearinG motor.

Conclltsions

L2.

3.

4.

Cl"orrelit draG bJ trle medi'um is possible.the quan ti ties (f-G"" A.T) rd..-1 ~ ( 1 _0<-1)

(v-speed of medium), character~ze tDe current orac in ~avinL

media.a cri tical value of tne ~cJpeeu appears to be

1lJ"c.:= ~ , I'"l = (Ej-l/£;po}"1:whid! is u.sually vel'J hiLh bu t can rea .... h 0 I'd inary values .ii

are Dig enough.large CUI'Jelit diCit.., efLects alJpear lor bi,~_ values of

notes on CHapter 7

1. z := XU), in tne furegoin6 chapters we always llsed x(l)as Lht. directionln which the medium moved

2. c1. Tai[22J, formula (30-31)3. idem 34- 554. i'urrru... lae" 7.66-'1.6':1) ommi Lted5. also tabula Led by Gradshte yn 123]2.53.5, nl'. 46. idem 57. formula (7.31) is ommitted8. H(x) Heaviside (unit step) function

(x) Dirac (impuls) function

1

2

ad c.

( :3 - , () [)Y.)?>

(::S.9 I )

l39 \ ).3

Q/I-d ~th a.id ~q.23) ~

t.. t ~-'·1) - 0:-1~ ('\]·1) - l1-d-1)Vdz 12: = fU"D)!o<-'d;c 1

cylJ.c """, h _,,~/('J ~y L<Wq(7-U:) v·l = 0

~_·Ir. ~ ,~-~c:-I:.O"L cnnlccl clHc"lWI"\-tcJ .Qc~~h.Cnv ~~)~ U.r-MLcrna.n..'-{ OUJV\.VYlt LA~ -I:JI:>b~&y /~CL~<.t<. D: ( <\

I 1\( a-~ 1) -(1-£' l''7d, J = ev'),1 ' 9, = rcr-",]d,-' I (P5)

,_vLJhxp,~,~roWt (1 10)~OA--d D-rrlca.bL b J ~~CVf\U.O.~ P=I ~ Idlll\ = () (nodraCl.e.~) END§11

§ 2,.

vh7-1 173I

1. 1

-----".."" ..../-­

./' /'

174I

1 . .3.

"7·3 9)

--;- I-(7"10) Jf\.I2:rrr fo ' --u---o

~ c LU1n.£.y,t cLL~butCrY<- ~ t~

( 7. '-J I )

2.1

2.2

2.21

/" ~k:d.uc'~(7-370..) q

/

2.22

e.-h·7- 10

~ra.rt ('0- c:ncL~CL~ Q/:) V'I (7" 3,7) :

1 :== r~ I' ' ~ ;== r ccal~ c crY),'..V"::>

An:::: QZt by ­h ,n

lI\:,t 0

2.32

11/.. I

17&I'

Q~ = 0 = b~

~'7-131 7 9

/

2.b

1.

1 .2 .

...3.

--------=-M--+-~fu_---- L

+++

-co..s-=I=-e,--t-+:;~~===')~~~~=c.=ao.e.~1Ii==~O_u..lc!.._C=O/Y">""--I_c:l;_'_ivr>-.i.nql-........_,~_l,,----,~I~frtlLO'l.._V_~LjYUtyu._ ~

o 0 0 0 lf1II • U II ~

: '~'I',+ a---"~~---" .E+ 1" I I ~

...-lC...x. ~~

~i lk+t+ji' [:~ (k+l.tj)~ (k_l-jt§J (k-[-J"r (ktl-jf' til I k+ ~-j)Cf ,I.l'j'·'~~ (I.L'jl<f :5

- - -- 11 6--'-' ...!J

[lJ ~~( l+jlf' [;,] (-2(L+j>cp) (-lj )-' [EJ(-2J'I'f (-2.l)-1 [~ {-ll~+ - - - 'iC,D- ['"

'oJ

r'[3 ( -2,l f' [§J (-llepl ('b·'[~]('l)'f I.><l-jf[~ (-"ljl)~- -t - - -os PuB p.j :tjr~Q)

....L,)

["l'i'I·'~;j"l,j'1 (1.(l-jlr1 eE] 1.ll"jl'f'

c:

[·lJ (+2jf' [-E] (2j)<f'~

- - +- ~B.c.... ~Tt-u:j

• (.L r' fE] ,l'f '-'if' [§J Hj'fl (-2jf' [~(-lj)<f [!]---+ 1 A.e .~ =

-'--' ._-, ' .. - -,~. Cf=O

[~] [~] (-2jf' tiJ (-2jlcr (.2jrl [_~] (2.p9'+ - + - C.......

c:: )-' [:E (2j9'l [~] [ ~] (lV' [-E (2J')Cf- ++ - '~B (2fr

-f-+ ~ A (2j r' [~El (2ifl £-2jl' [-;J (-ljCfI [!] [!Jl-B cr,5

[~] '-'i i' lEJ '-'j'fJ (-2.V' [~~ (-lJ'ICf [!J+ - - + D

<7-7 )

[ ~~ (k.l.jl' [.- ?-"::l<i] . _, 0 Iotl.', k.l1fl'-~.ll:::rj ,J ( k.l.jl"tl •~'-')H:j~

~(k-l-J) -( 1 - (-I) I.t4.~

,- (-I) +J -(1-(-1l J) ('-(-') ~Jl ('-<_I)~""J

0 Cl 0 0, , , , , ,-

~

[~- [ ~] [ ~] [!]iC,Dg-'.~

[ ~] [ ~JI

[~- [~].......vf A,B~c

[~]• ..J

[i] [I] -j [ ~](Jl

.5 B,C

~ 1

c:J.7- 1E 1 & ~..

~C [~] [§] [i ] [1]-_.

c ll]-- [1]-- [~ J [~ ].~

c [i] [-lJ.- [1]- [ ~ ].~

&...~lT1C

[!] [i] [l]~ [i]-~A1/1L.

- CII

[1]- lI] [i] [~J-~.... .-J 0

.' ......... " .

(7· 77)

1 M~j [0] I bl'i[ 0]I .tl+] [ 0]' k+l+j [ 0]- - -- ~

1 -H} ~ + 1-(-1) :~ T 1-H.) J t 1-(-1). -~

k+l+] .~. k-l.J +k+[-JO~-[+J 0

Oi O! 0: 0II '. II; II \ II

[fJ+ [-r]+ [i]+ [i]. ..-." --t\ ......... --l.

~E+.: I! Ii +~~--Jl.-J:-.J

t I i;i I ,~..:.!..k ..:.!

I : 2 .b+ -!--<4 S I- - +--+--- .3 .7

---+ S.B -

3.

1~4

\

\

5.

vh1-19 HlS

==

+

END § 7.3

§ -4 . Curownt cklnJ,LA-bon en Q rw~ ~t~ ey~I

tND§ 7~

ENDc:h.7/

END EvY-d~Q-0e/\I > 0 G - 0 G-)74 ./

./