Specialrelativity,electrodynamics,andgeneral relativity:fromNewtontoEinsteinSecond EditionKogut
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Problems
advanced standard 12 XII Shashi Bhusan Tiwari Mc Graw Hill
Shashi Bhushan Tiwari
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ABOUTTHEBOOKCOVER
Thebookcoverpresentspicturesof fi veofthemostfamousphysicist mathematiciansofhistory.Ontheleft-handsidewe findCarlFriedrich Gauss(1777 1855)andabovehimBernhardRiemann(1826 66).On theright-handsidewehaveIsaacNewton(1642 1727)andabovehim JamesClerkMaxwell(1831 79).Andatthepinnacle,AlbertEinstein (1879 1955).The figuredepictstwoparalleldevelopments,mathematics ononesideandphysicsontheother,thatculminatedwiththefoundingof modernphysics:GaussandRiemanndevelopedthemathematicscritical fortheexpressionsofthephysicallawsdiscoveredbyNewtonandMaxwell, allofwhichledtothegrandsynthesis:Einstein’stheoriesofspecialand generalrelativity.Thisbookfocusesontheconceptsthesegreatscientists developed.Itshowswhichoneshavesurvivedthetestsoftime (locality interactionsoccuratspace timepoints,causality information travelsatthespeedoflight,andcovariance physicallawsshouldholdinall framesofreference)andwhichhavefallenovertime(actionatadistance, nonlocalconservationlaws).Withaminimumoftechnicaltrickerybuta largedoseoffundamentals,thebookdevelopsspecialrelativity,electromagnetism,andgeneralrelativityfromthesebasicprinciples.
PREFACE 1.SPECIALRELATIVITYWITHAMISSION
Thisbookisarevisionandextensionof “IntroductiontoRelativity.” That bookpresentsanapproachtorelativityinwhichthetheoryisdeveloped through “thoughtexperiments” thatillustratetheconceptsinsimple fashions.Forexample,toderivetimedilationandLorentzcontraction,a clockisconstructedoutofmirrorsandalightbeam,andwelearnhowit worksinitsrestframeandinaframeinwhichitmoveswithvelocity v Thisanalysisissimpleanddirectbecauselightbeamsandrodsare governedbythepostulatesofspecialrelativity:
1. Thelawsofphysicsarethesameinallinertialframesofreference.
2. Thereisacommon finitespeedlimitcinallinertialframes.
Throughoutthebooktherulesofnonrelativisticphysics(Newton’ s world)arecontrastedwithrelativisticphysics(Einstein’sworld).Itisthe secondpostulatethatdistinguishesNewtonianrulesfromEinstein’s.In Newton’sworld,velocitiesareunboundedbutinEinstein’sworld,the speedoflightcisanupperbound.Theexistenceofaspeedlimitcombined withPostulate1,thatallinertialframesareequivalent,producesrelativistic space time.
Thebookshowsthattimedilation,Lorentzcontraction,andthe relativityofsimultaneity,thatclockswhichareseparatedinspaceand synchronizedinoneframeare not synchronizedinaframeinrelative motion,areintimatelyrelated.
Thebookbuildsthesubjectfromthegroundup.ItusesMinkowski space timediagrams,whichshowthespatialandtemporalcoordinatesof twoframesinrelativemotion v.Timedilation,Lorenzcontraction,and therelativityofsimultaneitycanbeunderstoodthroughsimplepictures (cartoons,really),bothqualitativelyandquantitatively.Fromthisperspective, westudythetwinparadoxandseethatthereisnothingreallyparadoxical aboutit whenonetwinleavestheotherandtakesaroundtrip,shereturns youngerthanhersibling.Itisfuntoseehowitworksout!
Oncewehavemasteredkinematics themeasurementofspaceand timeintervals wemovetodynamics,thestudyofenergy,momentum, interactionsandequationsofmotion.InNewtonianmechanicsthereis momentumconservation,ontheonehand,andmassconservation,onthe
Newton’slawofaction reactionarethrownoverandarereplacedbylocal conservationlaws,whicharecompatiblewiththerelativisticnotionofcausality andthe finitenessofthespeedofpropagationofinformation.Inaddition,the integrityofparticlenumber,socrucialinNewton’sworld,isreplacedby particlecreationanddestruction, whichisforceduponusbytheenergy mass relationship,causalityandlocalityofrelativisticinteractionsofrelativity,etc.
Thusourintroductiontoelectrodynamicsisanalternativetothe traditionalapproachpopularizedbythetextbooksofJ.D.Jackson [1] and hismanyfollowers.Ourbookstartswithpostulates1and2ofspecial relativityandtheprimitivenotionthatchargesarethelocalsourcesof electric fields,anddevelopsthefundamentalsofelectromagnetism:unified magneticandelectric fields,theLorentzforcelaw,Maxwell’sequations, andthewaveequation.Aftermasteringthesefundamentals,thestudentis inapositiontodomorecomplexapplicationsofrelativityandelectrodynamics.Theapproachinthisbookissimpleandstraightforward.The derivationofMaxwell’sequations,havingcomefromsuchhumble beginningsinChapter1,shouldbeanepiphanyforthestudent.
2.GENERALRELATIVITY
Ourlasttopicinthisbookisthegeneraltheoryofrelativity,wherewe consideracceleratedreferenceframesinEinstein’sworld.Thekeyinsight hereisEinstein’sversionoftheequivalenceprinciple:thereisnolocal, physicalmeanstodistinguishauniformgravitational fieldfromanacceleratedreferenceframe.
Thisprinciplehasmanyinterestingformsandapplications.Supposeyou areonthesurfaceofEarthandwanttounderstandtheinfluenceofthe gravitational fieldonyourmeasuringsticksandclockscomparedtothoseof yourassistantwhoisatagreaterheightintheEmpireStateBuilding. Einsteinsuggeststhattheassistantjumpoutthewindow,becauseinafreely fallingframealleffectsofgravityareeliminatedandwehaveaperfectly inertialenvironmentwherespecialrelativityholdstoarbitraryprecision! Duringhisdescent,yourassistantcanmakemeasurementsofclocksand metersticks fixedatvariousheightsalongthebuildingandmeasurehow theiroperationdependsontheirgravitationalpotential.Wepursueideas likethisoneinthebooktoderivethegravitationalredshift,thefactthat clocksclosetostarsrunmoreslowlythanthosefarawayfromstars;the resolutionofthetwinparadoxasaprobleminacceleratingreference frames;andthebendingoflightbygravitational fields.
moderneraofparticleacceleratorswhereparticlevelocitiesclosetothe speedlimitwerepossible.
Anotherinfluentialbookis TheClassicalTheoryofFields byL.D.Landau andE.M.Lifshitz [3].Unlikemostbooksonelectromagnetism,which followthehistoricaldevelopmentofthe fieldandintroducerelativitytoward theend,LandauandLifshitzstartwithrelativityandbuildthetheoryof relativisticparticlesandlight.Theemphasisisonfundamentalprinciples ratherthanapplications.Unfortunately,thebookisaimedatadvanced studentsandmosteditionsare flushwithtypographicalerrors.Nonetheless, thebookisveryinspiringandhasuniqueinsightsintothesubject.Severalof themorechallengingproblemsinchapters7 10inthisvolumewere inspiredbyLandauandLifshitz.Thisbookalsointroducesgeneralrelativity andincludesderivationsoftheSchwarzschildmetric,gravitationalwaves,the bendingoflightraysinagravitational field,etc.Thediscussionsarebriefand powerful,theessenceofLandau’sbrilliantapproachtophysics.
TheexpositionbyN.D.Mermin [4] influencedseveraldiscussionsof theparadoxesofspecialrelativity.Thisbookisalsorecommendedtothe studentbecauseitshowsacondensedmatterphysicistlearningthesubject and findingacomfortlevelinitthroughthought-provokinganalysesthat avoidlengthyalgebraicdevelopments.ThehugebookbyJ.A.Wheeler andE.F.Taylor [5] titled SpacetimePhysics inspiredseveralofourdiscussions andproblemsets.Thisbook,aworkthatonlytheunique,creativesoulof JohnArchibaldWheelercouldproduce,isrecommendedforitsleisurely, interactive,thought-provokingcharacter.Finally,thebooksbyW.Rindler [6,7],apioneerinmoderngeneralrelativity,arealsorecommended.His book EssentialRelativity [7] isasolidintroductiontogeneralrelativity rootedintheeraofEinsteinandthepioneers.Afterthestudenthas masteredelectricityandmagnetismandLagrangianmechanics,heorshe couldtackle TheClassicalTheoryofFields byL.D.LandauandE.M.Lifshitz [3] and EssentialRelativity [7].
Thefutureofresearchinrelativityand fieldtheoryisbright.Hopefully thisbookwillsparksomeinterestinitsfuturepractitioners.
“Timetravelsindiversepaceswithdiversepersons,” Rosalind,fromAct3, Scene2, “AsYouLikeIt” byWilliamShakespeare.
“Mathematicsallowsyoutoexceedyourimagination,” LevLandau, Moscow,1952.
“Magic!”,student,anonymous,2018.
REFERENCES
[1]J.D.Jackson,ClassicalElectrodynamics,JohnWiley&Sons,NewYork,1962.
[2]A.P.French,SpecialRelativity,W.W.Norton,NewYork,1968.
[3]L.D.Landau,E.M.Lifshitz,TheClassicalTheoryofFields,PergamonPress,Oxford, 1962.
[4]N.D.Mermin,SpaceTimeinSpecialRelativity,WavelandPress,ProspectHeights,IL, 1968.
[5]E.F.Taylor,J.A.Wheeler,SpacetimePhysics,W.H.Freeman,NewYork,1992.
[6]W.Rindler,IntroductiontoSpecialRelativity,OxfordUniversityPress,Oxford,1991.
[7]W.Rindler,EssentialRelativity,Springer-Verlag,Berlin,1971.
Contents
1.1 Newton’sWorld:LawsandMeasurements1 1.2 Newton’sWorld:NoPlaceforMagnetism 8
1.1NEWTON’SWORLD:LAWSANDMEASUREMENTS
WhenyousetupaprobleminNewtonianmechanics,youchoosea referenceframe.Thismeansthatyousetupathree-dimensionalcoordinate system,forexample,soanypoint r canbelabeledwithan x measurement oflength,a y measurement,anda z measurement, r ¼ (x,y,z).Inaddition, youplaceclocksatconvenientpointsinthecoordinatesystemsoyoucan makemeasurementsandrecordwhenandwheretheyoccurred.
Newtonimaginedcarryingoutexperimentsonamasspoint m inthis coordinatesystem.Heimaginedthatthecoordinatesystemwasfarfrom anyexternalinfluences,andunderthoseconditionsheclaimed,onthebasis oftheexperimentsofGalileoandothers,thatthemasspointwouldmove inastraightlineataconstantvelocity.Newtonlabeledsuchaframeof reference “inertial.” ThisideaiscodifiedinNewton’ s firstlaw: Law1.Abodyinaninertialreferenceframeremainsatrestorinuniformmotionunlessactedonbyaforce.
Newtonandothersrealizedthattheremustbeawideclassofinertial frames.Ifwediscoveredoneframethatwasinertial,thenNewton arguedthatotherinertialreferenceframescouldbegeneratedby
1. Translation movethecoordinatesystemtoaneworiginandusethat system.
2. Rotation rotatethecoordinatesystemaboutsomeaxistoa fixed,new orientation.
3. Boosts consideraframemovingatvelocity v withrespecttothe first. Properties1and2arereferredtoastheuniformityandisotropyof space.Property3isreferredtoasGalileaninvariance:theGalileanboost
SpecialRelativity,Electrodynamics,andGeneralRelativity
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speedtowardbothclocks.Settheclockstozero,say,whentheyboth receivethesignal.Thetwoclocksaresynchronized,andwedidnoteven needtoknowthespeedofthesignalemittedfromthebeacon.Clearly,we couldusethismethodtosynchronizealltheclocksonthegrid.
Nowwearereadytodoexperimentsinvolvingspace timemeasurementsinthisframeofreference.Callthisframe S.Itconsistsofthegridworkofmeasuringrodsandclocks,allatrest,withrespecttoeachother. Now,supposewewanttocompareourexperimentalresultswiththose obtainedbyafriendofoursatrestinanotherframethatmovesatconstant velocity v ¼ (vx,0,0)withrespecttous.Accordingtoourpostulates,hisor hermeasurementsareasgoodasoursandallourphysicallawscanbe writteninhisorherframeofreference S0 withoutanychange(Fig.1.1).
Themeasuringrodsin S0 areidenticaltothosein S;theclocksin S0 are alsoidenticaltothosein S.Supposethattheoriginof S and S0 coincide whentheclockattheoriginin S readstime t ¼ 0andtheclockatthe originin S0 reads t0 ¼ 0.Nowforthecrucialquestion:Dotheothergrid markingsandclocksatthosemarkingsalsoagreeinthetwoframes S and S0 ? Therearetwodistinctphysicsissuestoconsiderhere.The firstisthe operationoftheclocksandrodsineachframe.FollowingNewton’ s principleofrelativity,thatallinertialreferenceframesareequivalent,allthe clocksandrodsin S workexactlythesameasthoseatrestin S0 .Weneed onlyknowthatbothframesareinertial therelativevelocitybetweenthe framesisphysicallyirrelevanttothedynamicswithineach.Thesecondissue isthephysicalmechanismbywhichwecantransmittheinformationinone gridofrodsandclockstotheotheratrelativevelocity v.InNewton’ s world,objectscanmovewithunboundedrelativevelocitiesrelativetoone another,asweshalldiscussfurtherin Eq.(1.2).Accordingly,signalsand informationcanbetransmittedatunboundedvelocities,essentially
Figure1.1 Twoinertialframesofreferenceinrelativemotion.
instantaneously.Therefore,thespatialgridworkandthetimesoneachclock in S0 canbeinstantaneouslybroadcasttothecorrespondingrodsandclocks intheframe S.Therefore,thegridworkofcoordinatesandtimesinboth framesmustbeidentical,eventhoughtheyareinrelativemotion.So,the lengthsofmeasuringrodsandtheratesofclocksareindependentoftheir relativevelocities.Wethereforeneedonlyonemeasuringrodandone clockatonepointinoneinertialframe,andweknowthepositionsand timesofallmeasuringrodsandclocksinanyotherinertialframe.Forthe purposesofthisbook,thisservesasthemeaningof “absolutespace” and “absolutetime” inNewtonianmechanics.Weneednothypothesizeabout thesenotionsfromaphilosophicalbasis,aswasdonehistorically,butcan deducethemfromthefactthatinNewtonianmechanicsthereisnospeed limit informationcanbetransmittedinstantaneously.
Nowconsidertherulebywhichtimesandpositionmeasurementsare comparedbetweenthetwoframes, S and S0 ,inrelativemotionin Newton’sworld,
where x, y, z,and t aremeasurementsin S and x 0 , y 0 , z 0 ,and t0 arethecorrespondingmeasurementsin S0 .Forexample,ifwemeasurethepositionof aparticlein S0 tobe x 0 , y 0 ,and z 0 ,attime t0 ,thenthecoordinatesofthis event(measurement)inframe S aregivenby Eq.(1.1).Theserelations, whicharesofamiliarand “obvious,” arecalledGalileantransformations. Notethatthe firstofthem, x ¼ x 0 þ vt,statesthatthepositionoftheparticleattime t consistsoftwopieces:(1)thedistance vt betweentheorigins ofthetwoframesattime t and(2)thedistance x 0 fromtheorigintothe particleintheframe S0 .Thisrulecontainsthenotion “absolute” space it usesthefactthatinaNewtonianworldthedistance x 0 intheframe S0 isalso measuredas x 0 intheframe S.Thelastequationin Eq.(1.1) isthestatement of “absolute” time.Notealsothatiftheparticlehasavelocity v 0 p with respecttotheframe S0 inthe x 0 direction,then x 0 ¼ v 0 p t 0 ¼ v 0 p t ; andits positioninframe S is x ¼ v 0 p þ v t ; soitsvelocityrelativetotheorigin offrame S is vp,
Thislawisthebasisofaconservationlaw:theconservationof momentum.Considertwointeractingparticles.Denotetheforceof particle1onparticle2 F12 andtheforce2on1 F21.Thethirdlawstates that,
Noticethatthisisavectorequation:theforcesareequalinmagnitudebut oppositeindirection.Wecanseethatthetotalmomentumofthesystem, p1 þ p2 isconstantintime,
Conservationlawsarefundamentaltoallsubfieldsofphysics.Newton’ s thirdlawisanessentialingredientinmechanics.Butitisimportanttonote somethingveryperplexingaboutthethirdlaw:itisa nonlocal conservation law.Theforcesofactionandreactioncancelasstatedin Eq.(1.7),butthe actionandreactionforcesacton different particles,whichcouldbeveryfar fromeachotheriftheforceactsoverlongdistancessuchaselectrostaticsor gravity. Eq.(1.7) makestheremarkablestatementthatateveryinstantof time t,particle1respondstothemotionofparticle2justsothesumoftheir momentaisaconstanteveniftheparticlesarethousandsofmetersapart. Thisispossibleinacausalworldonlyifinformationtravelsinstantaneously. ClearlythethirdlawwillnotsurviveinthisforminEinstein’sworldwhere informationcannottravelfasterthanc,thespeedoflight.Theimplications ofthefailureofthethirdlawwillbemoreprofoundthanwecanimagineat thispointinourstudies.
Themass m in Eq.(1.4) isclarifiedinpartbythethirdlaw,thelawof action reaction,whichimpliesinthecaseoftwointeractingparticlesthat
So,ifthe fi rstbodyischosentosetthescaleforinertia,inotherwords,if wede fi ne m 1 h 1,then m 2 isdeterminedfrom Eq.(1.8) .Thethirdlaw providesanoperationalmeaningtothe “ mass ” or “ inertia ” ofaparticle.
Arethelawsofdynamics,Newton’ssecondandthirdlaws,compatible withNewton’sprincipleofrelativity,thatallinertialreferenceframesare physicallyequivalent?Thekeyobservationisthataccelerationisa “Galilean invariant,” whichmeansthatanacceleration a isthesameinallinertial
referenceframes.Thisfollowsfrom Eq.(1.2).Differentiateitwithrespect totime,usethefactthat v isaconstantandlearnthat,
WelearnthatforcesareGalileaninvariants.Weareusingthefactthat massesareGalileaninvariantinNewton’sworldtomakethisobservation. ThispropertywasassumedinNewton’seraandwasverifiedexperimentally toreasonableprecisionatmodestvelocities.Weshallseethatmassesarenot invariantinEinstein’sworld,andtheimplicationsofthisfactwillproveto beverysignificant.
Wehavedealtwiththeseissuesveryexplicitlybecausetheywillhelpus appreciatespecialrelativity,wherethereisaspeedlimit,thespeedoflight. InalltherulesofNewtonianmechanics,therulesofhowthingswork,the secondandthirdlaws,donotdistinguishbetweeninertialframes.The dynamicsdosatisfyaprincipleofrelativity.Thedifferencebetweenthetwo theoriescomesfromthefactthatonehasaspeedlimitandtheotherdoes not.Thisaffectshowinformationissharedbetweenframes.Italsoaffects thedynamicswithineachframe Einstein’sformofNewton’ s “force equalsmasstimesacceleration” isdifferentbecauseitmustnotpermit velocitiesgreaterthanthespeedlimit.Buteachtheoryisconsistentwithin itsownrules.
1.2NEWTON’SWORLD:NOPLACEFORMAGNETISM
Itisinstructivetoreturntoourexampleofelectrostaticsandconsider Newtoniandynamicsinmoredetail.Thegoaltothisdiscussionisto demonstratethatifyouacceptNewton’slaws1 3andyouknow Coulomb’slaw,thenyoucanruleouttheveryexistenceofmagnetism!
Whatdowemeanbymagnetisminthiscontext?Weacceptthe experimentalfactthatelectricchargesproduceelectrostaticforces.Wewill writedownCoulomb’slawandwillmanipulateitandstudyit.Magnetic forcesare,bydefinition,thoseforcesproducedbyelectriccurrents.For example,ifathinwirecarriesacurrentI,thismeansthatelectronsare flowinginthewireatarateofICoulomb’spersecondpassinga fixedpoint alongthewire.Thewireisunchargedbutelectrons flowalongit.We knowfromourexperiencesinthelab,perhapsmakinganelectromagnet andbendingchargedparticletrajectorieswithit,thatthereisamagnetic fieldinthelabanditsstrengthisproportionaltothecurrentI.Inaddition,if
CHAPTER2
SpaceeTimeMeasurements
AccordingtoEinstein
Contents
2.1 AWorldWithaSpeedLimit11
2.2 MakingaClockWithMirrorsandLight13
2.3 LorentzContraction17
2.4 TheRelativityofSimultaneity19
2.5 TimeDilationRevisited22
2.6 LorentzContractionRevisited24 Problems27 Reference29
2.1AWORLDWITHASPEEDLIMIT
AccordingtoNewtonianmechanics,youcouldplaceachargedparticle (charge q,mass m)intoaconstantelectric field E andaccelerateittoan arbitraryvelocity, v(t) ¼ (qE/m)t.Inaddition,youcouldconsiderinertial framesinrelativemotionandtherelativevelocitycouldbearbitrarilylarge. Unfortunately,naturedoesnotallowsuchfreedom.Modernaccelerator experimentsindicatethatparticlescannotbeacceleratedbeyondauniversal speedlimit,whichisthespeedoflight, c ¼ 2.9979. $108 m/s.Lightis nature’sfastestrunner.Althoughhistoricallythefactthatthespeedoflight isthespeedlimitwasextraordinarilyimportant,itisnotimportantforthe logicaldevelopmentofthesubject.Justtheexistenceofaspeedlimitisall thatreallymatters.WedonotneedtoknowanythingaboutelectromagnetismtoderivetimedilationandLorentzcontractionandtherebyoverthrowNewton’svisionofabsolutespaceandtime.Butitiscrucialto understandthattheexistenceofaspeedlimitmustbecompatiblewith Postulate1ofrelativity,thatthelawsofphysicsarethesameinallinertial frames.Thismeansthatallinertialframesmust findthesameuniversalvalue forthespeedlimitthroughtheirownexperiments.
Inparticular,supposethatanexperimenteratrestinframe S0 shownin Fig.2.1 turnsona flashlightandpointsittotheright.Thoselightrayscan
SpecialRelativity,Electrodynamics,andGeneralRelativity
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Nowviewtheclock’soperationfromtheperspectiveofanobserverat restin S.Nowtheclockmovestotherightatvelocity v,andthelightray takesthepathinframe S asshownin Fig.2.3.Intheframe S,thelightray travelsalongthelinesegment ABandthenalong BAbacktomirrorA.The distancemirrorAin Fig.2.3 travelsbetweensendingandreceivingthelight rayis vDt.From Fig.2.3,thedistancethelightraytravelsis
Butthelightrayalsotravelsatthespeedlimit c inframe S accordingto Postulate2,so AB þ BA ¼ c Dt .(2.2)
Thisistheessentialpointinthisargument theuniversalspeedlimitenters here,andwehaveaclearqualitativedistinctionfromtherulesofNewtonian mechanics.Combining Eqs.(2.1)and(2.2),we find
whichallowsustosolvefor Dt,
Figure2.3 Fig.2.2 viewedfromframe S 0 .