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MetricDi erentialGeometryWeneedaformulationofphysicsvalidinarbitraryco ordinatesystems Physicalquanti tiesmusthaveexistenceindependentofparticularco ordinatesbeingused hencemusttransformproperlyunderco ordinatetransforms Theyshouldberepresentedbytensors TensorsConsidertheco ordinatechangexa xa xb onspacetime withinversexa xa xb De nepa a xa xa paa xa xa Notethatpa apba xa xa xb xa ba byusingthechainrule where ba ifa b ifa bistheKroneckerdelta Underrepeatedco ordinatechangexa xa xa wehavegroupproperty usingthechainrule pa apa a pa a Acovarianttensorofnthrank withcomponentsTa an withrespecttoco ordinatesxa atapointPhastransformationlawTa an Ta a a n pa a pana nTa anNotethatthegrouppropertyshowsthatthecomponentsTa a nareuniquelyde nedwithrespecttoanyco ordinatesystemiftheyare xedinonesystemxa thisprovidesawayofconstructingalltensors AcontravarianttensorTa antransformsasTa an Ta a n pa a pa nanTa anSimilarlyformixedtensorsforexampleTba Tb a paa pb bTbaItisimportanttokeeptheorderoftheindicesthesame Ascalarisatensorwithnoindices invariantunderco ordinatechange forexamplethemassofaparticle Ascalar eld xa isascalarfunctionforexamplepressure orparticledensityina uid MetricDi erentialGeometry Acovariantvector eldva xb isavectorfunctionofposition Forexampleif xb isascalar eldthenva a xa isacovariantvector eld sinceva xa xa xa xa Afurtherexampleisapressuregradientp aina uid Contravariantvectors supposeacurvexa paramaterisedby hasatangentofva dxad atthepointP thenvaisacontravariantvector Thefollowsbecauseva dxa d dxa dxadxad Otherexamplesarethe velocityofanobserver ua dxad where ispropertime ExamplesofTensors i KroneckerDelta paa pb b ba paa pb a b a ii Metrictensorgab sincetheinvariantds canbewrittends gabdxadxb gab xa xa dxa xb xb dxb ga b dxa dxb wherega b gabpaa pbb Furtherexampleswillbeprovidedinthesequelbythecurvatureandenergymomen tumtensors OperationsPreservingTensorProperty i Additione g Tab Wabistensor ii MultiplicationbyScalare g fTab iii OuterProductse g vaTbctransformsasva Tb c pa apbb pcc vaTbc iv Contractionofonecovariantwithonecontravariantindex ForexampleifTabcisatensor de nevc Taac transformingasvc Ta a c pa apba pcc Tabc bapcc Tabc pcc Taac pcc vc v Interchangeofindices forexampleTab atensor Tbawhichisalsoatensor Simi larlySymmetrisationandAnti symmetrisationT ab Tab Tba T ab Tab Tba theabovecanreadilybegeneralisedtomoreindices MetricDi erentialGeometry QuotientTheoremSupposeUa TabVbisavector forallvectorsVb Thenpaa Ua pa aTabVb andUa pa aUa Ta b Vb Ta b pbb VbSubtractingtheselasttwoyields Ta b pbb pa aTab Vb VbTa b pbb pa aTabMultiplyingbothsidesbypc byieldsTa b pbb pc b Ta b c b Ta c pc bpa aTab HenceTabisatensor InverseMetricTensorDe negab gba tobethematrixinverseofgab i e suchthatgacgcb ba NowforanyvectorVacande neavectorUa gabVb andwemaynotethatthereisaonetoonecorrespondancebetweenUbandVasincegabisnon singular hencewecanconstructallvectorsUbinthisway TheQuotienttheoremimpliesthatgabisatensor RaisingandLoweringofIndicesWecanusegabtoraiseanycovariantindex ForexampleTabgivesatensorTab gacTcb ifthe rstindexisraised Similarlycanusegabtoloweranyindex forexampleWabgivesWba gacWcb Theindexorderingmustbecarefullymantained Raisingandloweringareinverseoperations Onenormallyregardse g Tab Tba Tab Tabasdi erentversionsofthesameobject PartialderivativesoftensorsPartialderivativesoftensorsarenottensorsingeneral Forexamplesupposevaisavector eld Thenva xa xa va whence va xb xa xa xb va xa xa xb xb va xb paa pbb va xbingeneralTheonlyexceptionis aasmentionedearlier Lengths GeodesicsThesquaredmagnitudeofavectorvaorvaisde nedtobevava vavbgab vavbgab andisinvariantunderco ordinatetransformations ForexampleitcanbeevaluatedinaLocalInertialFramewheregab ab gab ab vais spacelikeifvava nullifvava timelikeifvava Asintheearliersection oflecturenotes ifvaisspacelikecan ndalorentztransformationintheLocalInertialFramemakingva vi Thenvava v v v jvj whichisthephysicallymeasured squaredmagnitudeofvainthatframe Ifvaistimelike canmakeva v thenvava c v AnglesbetweenvectorsSupposethatvaandwaarebothspacelike andthatonehasarrangedva vi wa wi insomeLocalInertialFrame Thentheangle betweenvaandwaisde nedbycos viwijvjjwj CartesianNotationinLIF gabvawb vcvc wdwd InvariantDe nition LengthsofcurvesIfxa descrivesaspacelikecurve whichisparameterisedby i e ifva dxad isspacelikealong thelengthof fromAtoBisZBAds ZBA gabdxad dxbd d Ifxa givesatimelikecurve i e va dxad istimelikealong thenthetimeelapsedalong fromAtoBis cZBAjdsj cZBA gabdxad dxbd d GeodesicsAgeodesic fromAtoBextremisesZBAjdsj ZBA gabdxad dxbd d ZBAL xa xa d Lengths Geodesics where xa dxad subjectto xedendpoints xa aretheco ordinatesofA andxa aretheco ordinatesofB Forexampleconsideraspacelikegeodesic
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本文标题：剑桥大学――广义相对论手册(速成？)
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