剑桥大学——广义相对论手册(速成？).pdf

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1、1.MetricDierentialGeometryWeneedaformulationofphysicsvalidinarbitraryco-ordinatesystems.Physicalquanti-tiesmusthaveexistenceindependentofparticularco-ordinatesbeingused{hencemusttransformproperlyunderco-ordinatetransforms.Theyshouldberepresentedbytensors.1.1.T

2、ensors000aabaabConsidertheco-ordinatechangex!x(x)onspacetime,withinversex!x(x).Dene0a0@xap=aa@xa@xap=:00aa@x0ab0@x@xabbNotethatpp==,byusingthechainrule,where00aaaaa@x@x1ifa=bb=a0ifa6=bistheKroneckerdelta.000aaaUnderrepeatedco-ordinatechangex!x!x,wehavegrou

3、pproperty,usingthechainrule,00000aaapp=p:0aaaAcovarianttensorofnthrank,withcomponentsT.withrespecttoco-a:::a1naordinatesx,atapointPhastransformationlawaa1n000T!T=p


pT00a:::aa:::aaa:::a1n1naan12n100NotethatthegrouppropertyshowsthatthecomponentsTareuniquelyden

4、edwitha:::an1arespecttoanyco-ordinatesystemiftheyarexedinonesystemxthisprovidesawayofconstructingalltensors.a:::a1nAcontravarianttensorTtransformsas0000aaa:::aa:::aa:::a1n1n1n1nT!T=p


pTaa1nSimilarlyformixedtensorsforexample00bbabbT!T=ppT00aaabaItisimportantt

5、okeeptheorderoftheindicesthesame.Ascalarisatensorwithnoindices,invariantunderco-ordinatechange,forexamplethemassofaparticle.aAscalareld(x)isascalarfunctionforexamplepressure,orparticledensityina uid.MetricDierentialGeometry2bbAcovariantvectoreldv(x)isavecto

6、rfunctionofposition.Forexampleif(x)aisascalareldthen@v=:=aaa@xisacovariantvectoreld,sincea@@x@0v==:a00aaa@x@x@xAfurtherexampleisapressuregradientpina uid.aaContravariantvectors:supposeacurvex(),paramaterisedbyhasatangentadxaaofv=atthepointP,thenvisa

7、contravariantvector.Thefollowsbecaused00aaa0dxdxdxav==:addxdadxaOtherexamplesarethe4-velocityofanobserver,u=,whereispropertime.dExamplesofTensors(i)KroneckerDelta:000abbabbpp=pp=:000abaaaa2(ii)Metrictensorg,sincetheinvariantdscanbewrittenabab00@x@x2ababd

8、s=gdxdx=gdxdxabab00ab@x@x00ab00=gdxdxabab00whereg=gpp.00abababFurtherexampleswillbeprovidedinthesequelbythecurvatureandenergymomen-tumtensors.OperationsPreservingTensorPrope