Gaberdiel--Problem sets of General Relativity .pdf

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1、GeneralRelativity{HS09{Prof.M.GaberdielSheetIDue:weekofSeptember28Notation:Inthislectureweusenaturalunits,thatis,wearesettingc=G=1.Question1[AdditionLawforVelocities]:TheLorentztransformationlawmappinganintertialsystemintoonemovingwiththevelocityvrelativetotherstisoftheformt^=(tvx)x

2、^=(xvt)y^=yz^=z;pwhere=1=1v2.v1i)ShowthatthesuccessiveapplicationoftwosuchLorentztransformations,(x;t)!(^x;t^)v2and(^x;t^)!(~x;t~),whereonlythex-componentofv1andv2isnon-zero,isaLorentztransformation.IdentifythevelocityoftheresultingLorentztransformation.ii)Supposethatonlythex-compon

3、entofv1andthey-componentofv2arenon-zero.Showthatthesuccessiveapplicationoftwosuchboostscanbewrittenasthesuc-cessiveapplicationofaboost(withrespecttoarotatedcoordinatesystem),andarotation.Findthedirectionandtheabsolutevalueofthevelocityoftheresultingboost.Question2[LorentzTransformatio

4、nsforAcceleration]:SupposeacoordinatesystemK0ismovingwithavelocityvwithrespecttoanothercoordinatesystemK.AparticleinK0ismovingwithavelocityu0andissubjecttoaconstantaccelerationa0.DeterminetheLorentztransformationlawforaccelerationsbyshowingthatthecom-ponentsoftheaccelerationparallelan

5、dperpendiculartotherelativevelocityvinKaregivenby(1v2)3=2(1v2)0000ak=(1+v
u0)3ak;a?=(1+v
u0)3(a?+v(au)):Remark:Inthiscontext,velocityandaccelerationofaparticlemeanstherstandsecondderivativeofthetrajectoryoftheparticlewithrespecttothecoordinatetimeandnotitseigentime.Question3[CarP

6、aradox]:Acarentersa3mlonggaragewithaveryhighvelocitythatcorrespondsto=2.Thelengthofthecaratrestisknowntobe4m.Thegaragehastwogatesateachendofit,whichareopen.Intheperceptionofagatekeeperintherestframeofthegarage,thelengthofthecarisjust2m,soheconcludesthatittsperfectlyintothegarage.Howe

7、ver,intherestframeofthecar,thelengthofthegarageisjust1.5mandthereforethedriverofthecarwillconcludethatthecardoesnottinthegarage.Explainwhythedierentpointsofviewdonotcreateaparadox.Fromnowonsupposethatthegateattheendofthegarageisclosed.Fromthepointofviewofthegatekeeper,thecarhasabrak

8、ingdi