广义相对论导轮习题集Gerard't Hooft

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1、Problem1,Lorentztransformationsofelectricandmag-neticeldsWehavethat0000F=LLF;where,010B3B2iE1BBB30B1iE2CCF=:@B2B10iE3AiE2iE2iE30Notethatweusetheconventionfromthelecturenotes,inwhichx=(x;y;z;ict).(1)Consideranordinaryrotationaroundthez-axis.Showthatthemag-neticandelectric

2、eldstransformasordinaryvectorsundersucharotation.(2)Consideraboostalongthex-axiswithvelocityv.Showthatwehavethefollowingtransformation,mixingelectricandmagneticelds:(p=1=12,=v=c)000B1=B1;B2=(B2+E3);B3=(B3E2);000E1=E1;E2=(E2B3);E3=(E3+B2):RecallthataLorentzboostalongthex-axisisgiven

3、bythetransformation0114000441x=(x+ix)y=yz=zx=(xix):ConsidernowaninertialsystemIandtheLorentzboostedsystemI00!(i.e.InotrotatedrelativetoI),movingwithvelocityvrelativetoI.Theformulageneralizingtheaboveformulais:!!!v!!!v!000B=B+(v
B)(1)+E;v2c!!!v!!!v!000E=E+(v
E)(1)B:v2

4、c1(Youcantrytoprovethisformula,butitisnotpartoftheproblem)Assume!!thatE0andB0areconstantanddierentfromzero.!!(3)FindtheconditionthatE0andB0havetosatisfyinorderthatthere!!!existsavsuchthatB=0,andndthecorrespondingvexpressedin!!termsofE0andB0.Problem2'ConstantaccelerationpartI'oft

5、heenclosedexercises.Problem3,TheE•otv•osexperimentInthelectureitwasexplainedhowE•otv•osfoundawayoftestingtheEquiv-alencePrinciple(i.e.Mgrav=Minert)bycheckingthemisalignmentoftheforcesactingontwoobjects.ThegravitationalforceonthesurfaceoftheEarthis~rF~=GMLM;gNgravr3whilethecentrifugalforce

6、isF~=M!2~r(~!
~r)~!;!inert!2where!isEarth'sangularvelocity.Canyouderivemorecarefullythemea-surablemisalignmentbetweentheforcesF~(i)=F~(i)F~(i)g+!oftwoobjects:jF~(1)^F~(2)j=?jF~(1)jjF~(2)junderthe(justied)assumptionthatthegravitationalforceismuchstrongerthanthecentrifugalone?Problem4,C

7、onstantacceleration,partII(0)Recallyourresultsfromlastweek,wherewedidtherstpartofthisexercise2Nowconsideracoordinatesystemmoving"along"withtheacceleratedparticleandrigidinthesensethatdistancesmeasuredwithstandardrodsatrestinthesystemareconstantintime(which,in