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1、3CLASSICALANDSPECIALRELATIVITY3.1TheGalileantransformationEventsbelongtothephysicalworld—theyarenotabstractions.Weshall,nonetheless,introducetheideaofanidealeventthathasneitherextensionnorduration.Idealeventsmayberepresentedaspointsinaspace-timegeometry.Aneve
2、ntisdescribedbyafour-vectorE[t,x,y,z]wheretisthetime,andx,y,zarethespatialcoordinates,referredtoarbitrarilychosenorigins.LetaneventE[t,x],recordedbyanobserverOattheoriginofanx-axis,berecordedastheeventE´[t´,x´]byasecondobserverO´,movingatconstantspeedValongth
3、ex-axis.Wesupposethattheirclocksaresynchronizedatt=t´=0whentheycoincideatacommonorigin,x=x´=0.Attimet,wewritetheplausibleequationst´=tandx´=x–Vt,whereVtisthedistancetravelledbyO´inatimet.TheseequationscanbewrittenE´=GE(3.1)where10G=.–V1CLASSICALANDSPECIALRELA
4、TIVITY45GistheoperatoroftheGalileantransformation.Theinverseequationsaret=t´andx=x´+Vt´or–1E=GE´(3.2)–1whereGistheinverseGalileanoperator.(ItundoestheeffectofG).Ifwemultiplytandt´bytheconstantskandk´,respectively,wherekandk´havedimensionsofvelocitythenallterm
5、shavedimensionsoflength.222Inspace-space,wehavethePythagoreanformx+y=r(aninvariantunder22rotations).Wearethereforeledtoaskthequestion:is(kt)+xaninvariantunderGinspace-time?Directcalculationgives222222(kt)+x=(k´t´)+x´+2Vx´t´+Vt´22=(k´t´)+x´onlyifV=0!Wesee,ther
6、efore,thatGalileanspace-timedoesnotleavethesumofsquaresinvariant.Wenote,however,thekeyrôleplayedbyaccelerationinGalilean-Newtonianphysics:ThevelocitiesoftheeventsaccordingtoOandO´areobtainedbydifferentiatingx´=–Vt+xwithrespecttotime,givingv´=–V+v,(3.3)46CLASS
7、ICALANDSPECIALRELATIVITYaresultthatagreeswitheverydayobservations.Differentiatingv´withrespecttotimegivesdv´/dt´=a´=dv/dt=a(3.4)whereaanda´aretheaccelerationsinthetwoframesofreference.TheclassicalaccelerationisaninvariantundertheGalileantransformation.Ifthere
8、lationshipv´=v–Visusedtodescribethemotionofapulseoflight,movinginemptyspaceat8v=c@3x10m/s,itdoesnotfitthefacts.Forexample,ifVis0.5c,weexpecttoobtainv´=0.5c,whereas,itisfoundthatv´=c.Indee
