Sergiu Klainerman_ Introduction to General Relativity

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1、LECTURENOTESINGENERALRELATIVITY:Spring2009SergiuKlainermanDepartmentofMathematics,PrincetonUniversity,PrincetonNJ08544E-mailaddress:seri@math.princeton.eduCHAPTER1SpecialRelativity1.MinkowskiSpaceThen+1dimensionalMinkowskispace,whichwedenotebyRn+1,consistsofthemanifoldRn+1togetherw

2、ithaLorentzmetricmandadistinguishedsystemofcoordinatesx,=0;1;:::n,calledinertial,relativetowhichthemetrichasthediagonalformm=diag(1;1;:::;1).Wewrite,splittingthespacetimecoordinatesxintothetimecomponentx0=tandspacecomponentsx=xi;:::xn,ds2=m0dxdx=dt2+(dx1)2+(dx2)2+:::+(dxn)2:(1)We

3、usestandardgeometricconventionsofloweringandraisingindicesrelativetom,anditsinversem1=m,aswellastheusualsummationconventionoverrepeatedindices.Thecoordinatevectorelds@aredenotedby@.Thedual1forms@xaredx.Anarbitraryvectoreldcanbeexpressedasalinearcombinationofthecoordinatevectore

4、ldsX=X@withsmoothfunctionsX=X(x0;:::;xn).Anarbitrary1-formisalinearcombinationofthecoordinate1-formsA=Adx.Underachangeofcoordinatesx=x(x0)weobtain,@x@xds2=mdx0dx0;m0=m@x0@x0Twoinertialsystemsofcoordinatesareconnectedtoeachotherbytranslationsx=x0+x,Lorentztransformations,

5、(0)x0=x;m=m(2)andcombinationsofthetwox0=x+x.(0)Exercise.ShowthattheLorentztransformationsB(v)=B(v):R1+n!(0i)R1+n,with1

6、x,v=t+xandu0=t0x0,v0=t0+x0wehave,(1v)1=2u0=1u;v0=v;=(4)(1+v)1=2ShowthatB(v),jvj<1formsaoneparametergroupofdieomorphismsoftheMinkowskispaceandndtherelativisticlawofadditionofvelocities.341.SPECIALRELATIVITYDthe atcovariantderivativeofRn+1.Recallthatcovariantdierentiationas

7、so-ciatestoanytwosmoothvectoreldsX;YanothervectoreldDXYwhichveriesthefollowingrules.(1)GiventhreevectoreldsX;Y;Zandscalarfunctionsa;bwehaveDaX+bYZ=aDXZ+bDYZDX(fY)=X(f)Y+fDXY(2)ForanyvectoreldsX;Y;Z,Xm(Y;Z)=m(DXY;Z)+m(X;DYZ)Givenanarbitrary1-formA=AdxwehaveD!=@!.AvectorXissaidt

8、obetimelike,nullorspacelik