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1、Chapter2GeometryOurtouroftheoreticalphysicsbeginswithgeometry,andtherearetworeasonsforthis.Oneisthattheframeworkofspaceandtimeprovides,asitwere,thestageuponwhichphysicaleventsareplayedout,anditwillbehelpfultogainaclearideaofwhatthisstagelookslikebeforeintroducingthecast.Asamatteroffact,thegeometry
2、ofspaceandtimeitselfplaysanactiveroleinthosephysicalprocessesthatinvolvegravitation(andperhaps,accordingtosomespeculativetheories,inotherprocessesaswell).Thus,ourstudyofgeometrywillculminate,inchapter4,intheaccountofgravityofferedbyEinstein’sgeneraltheoryofrelativity.Theotherreasonforbeginningwith
3、geometryisthatthemathematicalnotionswedevelopwillreappearinlatercontexts.Toalargeextent,thespecialandgeneraltheoriesofrelativityare‘negative’theories.BythisImeanthattheyconsistmoreinrelaxingincorrect,thoughplausible,assumptionsthatweareinclinedtomakeaboutthenatureofspaceandtimethaninintroducingnew
4、ones.Iproposetoexplainhowthisworksinthefollowingway.Weshallstartbyintroducingaprototypeversionofspaceandtime,calleda‘differentiablemanifold’,whichpossessesabareminimumofgeometricalproperties—forexample,thenotionoflengthisnotyetmeaningful.(Actually,itmaybenecessarytoabandoneventheseminimalpropertie
5、sif,forexample,wewantageometrythatisfullycompatiblewithquantumtheoryandIshalltouchbrieflyonthisinchapter15.)Inordertoarriveatastructurethatmorecloselyresemblesspaceandtimeasweknowthem,wethenhavetoendowthemanifoldwithadditionalproperties,knownasan‘affineconnection’anda‘metric’.Twopointsthenemerge:firs
6、t,thecommon-sensenotionsofEuclideangeometrycorrespondtoveryspecialchoicesfortheseaffineandmetricproperties;second,otherpossiblechoicesleadtogeometricalstatesofaffairsthathaveanaturalinterpretationintermsofgravitationaleffects.Stretchingthepointslightly,itmaybesaidthat,merelybyavoidingunnecessaryass
7、umptions,weareabletoseegravitationassomethingentirelytobeexpected,ratherthanasaphenomenoninneedofexplanation.Tome,thisinsightintothewaysofnatureisimmenselysatisfying,andit6TheSpecialandGeneralTheoriesofRelativity
