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15TheSpecialTheoryofRelativity

151Theprincipleofrelativity

Forover200yearstheequationsofmotionenunciatedbyNewtonwerebelievedtodescribenaturecorrectly,andthefirsttimethatanerrorintheselawswasdiscovered,thewaytocorrectitwasalsodiscovered.BoththeerroranditscorrectionwerediscoveredbyEinsteinin1905.

NewtonsSecondLaw,whichwehaveexpressedbytheequation

wasstatedwiththetacitassumptionthat isaconstant,butwenowknowthatthisisnottrue,andthatthemassofabodyincreaseswithvelocity.InEinsteinscorrectedformula hasthevalue

wheretherestmass representsthemassofabodythatisnotmovingand isthespeedoflight,whichisabout km sec orabout mi sec .

Forthosewhowanttolearnjustenoughaboutitsotheycansolveproblems,thatisallthereistothetheoryofrelativityitjustchangesNewtonslawsbyintroducingacorrectionfactortothemass.Fromtheformulaitselfitiseasytoseethatthismassincreaseisverysmallinordinarycircumstances.Ifthevelocityisevenasgreatasthatofasatellite,whichgoesaroundtheearthat mi/sec,then

:puttingthisvalueintotheformulashowsthatthecorrectiontothemassisonlyonepartintwotothreebillion,whichisnearlyimpossibletoobserve.Actually,thecorrectnessoftheformulahasbeenamplyconfirmedbytheobservationofmanykindsofparticles,movingatspeedsranginguptopracticallythespeedoflight.However,becausetheeffectisordinarilysosmall,itseemsremarkablethatitwasdiscoveredtheoreticallybeforeitwasdiscoveredexperimentally.Empirically,atasufficientlyhighvelocity,theeffectisverylarge,butitwasnotdiscoveredthatway.Thereforeitisinterestingtoseehowalawthatinvolvedsodelicateamodification(atthetimewhenitwasfirstdiscovered)wasbroughttolightbyacombinationofexperimentsandphysicalreasoning.Contributionstothediscoveryweremadebyanumberofpeople,thefinalresultofwhoseworkwasEinsteinsdiscovery.

TherearereallytwoEinsteintheoriesofrelativity.ThischapterisconcernedwiththeSpecialTheoryofRelativity,whichdatesfrom1905.In1915Einsteinpublishedanadditionaltheory,calledtheGeneralTheoryofRelativity.ThislattertheorydealswiththeextensionoftheSpecialTheorytothecaseofthelawofgravitationweshallnotdiscusstheGeneralTheoryhere.

TheprincipleofrelativitywasfirststatedbyNewton,inoneofhiscorollariestothelawsofmotion:Themotionsofbodiesincludedinagivenspacearethesameamongthemselves,whetherthatspaceisatrestormovesuniformlyforwardinastraightline.Thismeans,forexample,thatifaspaceshipisdriftingalongatauniformspeed,allexperimentsperformedinthespaceshipandallthephenomenainthespaceshipwillappearthesameasiftheshipwerenotmoving,provided,ofcourse,thatonedoesnotlookoutside.Thatisthemeaningoftheprincipleofrelativity.Thisisasimpleenoughidea,andtheonlyquestioniswhetheritistruethatinallexperimentsperformedinsideamovingsystemthelawsofphysicswillappearthesameastheywouldifthesystemwerestandingstill.LetusfirstinvestigatewhetherNewtonslawsappearthesameinthemovingsystem.

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Fig.151.Twocoordinatesystemsinuniformrelativemotionalongtheir axes.

SupposethatMoeismovinginthe directionwithauniformvelocity ,andhemeasuresthepositionofacertainpoint,showninFig.151.Hedesignatesthe distanceofthepointinhiscoordinatesystemas .Joeisatrest,andmeasuresthepositionofthesamepoint,designatingits coordinateinhissystemas .Therelationshipofthecoordinatesinthetwosystemsisclearfromthediagram.Aftertime Moesoriginhasmovedadistance ,andifthetwosystemsoriginallycoincided,

IfwesubstitutethistransformationofcoordinatesintoNewtonslawswefindthattheselawstransformtothesamelawsintheprimedsystemthatis,thelawsofNewtonareofthesameforminamovingsystemasinastationarysystem,andthereforeitisimpossibletotell,bymakingmechanicalexperiments,whetherthesystemismovingornot.

Theprincipleofrelativityhasbeenusedinmechanicsforalongtime.Itwasemployedbyvariouspeople,inparticularHuygens,toobtaintherulesforthecollisionofbilliardballs,inmuchthesamewayasweuseditinChapter10todiscusstheconservationofmomentum.Inthe19thcenturyinterestinitwasheightenedastheresultofinvestigationsintothephenomenaofelectricity,magnetism,andlight.AlongseriesofcarefulstudiesofthesephenomenabymanypeopleculminatedinMaxwellsequationsoftheelectromagneticfield,whichdescribeelectricity,magnetism,andlightinoneuniformsystem.However,theMaxwellequationsdidnotseemtoobeytheprincipleofrelativity.Thatis,ifwetransformMaxwellsequationsbythesubstitutionofequations(15.2),theirformdoesnotremainthesametherefore,inamovingspaceshiptheelectricalandopticalphenomenashouldbedifferentfromthoseinastationaryship.Thusonecouldusetheseopticalphenomenatodeterminethespeedoftheshipinparticular,onecoulddeterminetheabsolutespeedoftheshipbymakingsuitableopticalorelectricalmeasurements.OneoftheconsequencesofMaxwellsequationsisthatifthereisadisturbanceinthefieldsuchthatlightisgenerated,theseelectromagneticwavesgooutinalldirectionsequallyandatthesamespeed ,or mi/sec.Anotherconsequenceoftheequationsisthatifthesourceofthedisturbanceismoving,thelightemittedgoesthroughspaceatthesamespeed .Thisisanalogoustothecaseofsound,thespeedofsoundwavesbeinglikewiseindependentofthemotionofthesource.

Thisindependenceofthemotionofthesource,inthecaseoflight,bringsupaninterestingproblem:

Supposeweareridinginacarthatisgoingataspeed ,andlightfromtherearisgoingpastthecarwithspeed .Differentiatingthefirstequationin(15.2)gives

whichmeansthataccordingtotheGalileantransformationtheapparentspeedofthepassinglight,aswemeasureitinthecar,shouldnotbe butshouldbe .Forinstance,ifthecarisgoing

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mi/sec,andthelightisgoing mi/sec,thenapparentlythelightgoingpastthecarshouldgo mi/sec.Inanycase,bymeasuringthespeedofthelightgoingpastthecar(iftheGalileantransformationiscorrectforlight),onecoulddeterminethespeedofthecar.Anumberofexperimentsbasedonthisgeneralideawereperformedtodeterminethevelocityoftheearth,buttheyallfailedtheygavenovelocityatall.Weshalldiscussoneoftheseexperimentsindetail,toshowexactlywhatwasdoneandwhatwasthemattersomethingwasthematter,ofcourse,somethingwaswrongwiththeequationsofphysics.Whatcoulditbe?

152TheLorentztransformation

Whenthefailureoftheequationsofphysicsintheabovecasecametolight,thefirstthoughtthatoccurredwasthatthetroublemustlieinthenewMaxwellequationsofelectrodynamics,whichwereonly20yearsoldatthetime.Itseemedalmostobviousthattheseequationsmustbewrong,sothethingtodowastochangetheminsuchawaythatundertheGalileantransformationtheprincipleofrelativitywouldbesatisfied.Whenthiswastried,thenewtermsthathadtobeputintotheequationsledtopredictionsofnewelectricalphenomenathatdidnotexistatallwhentestedexperimentally,sothisattempthadtobeabandoned.ThenitgraduallybecameapparentthatMaxwellslawsofelectrodynamicswerecorrect,andthetroublemustbesoughtelsewhere.

Inthemeantime,H.A.LorentznoticedaremarkableandcuriousthingwhenhemadethefollowingsubstitutionsintheMaxwellequations:

namely,Maxwellsequationsremaininthesameformwhenthistransformationisappliedtothem!Equations(15.3)areknownasaLorentztransformation.Einstein,followingasuggestionoriginallymadebyPoincar,thenproposedthatallthephysicallawsshouldbeofsuchakindthattheyremainunchangedunderaLorentztransformation.Inotherwords,weshouldchange,notthelawsofelectrodynamics,butthelawsofmechanics.HowshallwechangeNewtonslawssothattheywillremainunchangedbytheLorentztransformation?Ifthisgoalisset,wethenhavetorewriteNewtonsequationsinsuchawaythattheconditionswehaveimposedaresatisfied.Asitturnedout,theonlyrequirementisthatthemass inNewtonsequationsmustbereplacedbytheformshowninEq.(15.1).Whenthischangeismade,Newtonslawsandthelawsofelectrodynamicswillharmonize.ThenifweusetheLorentztransformationincomparingMoesmeasurementswithJoes,weshallneverbeabletodetectwhethereitherismoving,becausetheformofalltheequationswillbethesameinbothcoordinatesystems!

Itisinterestingtodiscusswhatitmeansthatwereplacetheoldtransformationbetweenthecoordinatesandtimewithanewone,becausetheoldone(Galilean)seemstobeselfevident,andthenewone(Lorentz)lookspeculiar.Wewishtoknowwhetheritislogicallyandexperimentallypossiblethatthenew,andnottheold,transformationcanbecorrect.Tofindthatout,itisnotenoughtostudythelawsofmechanicsbut,asEinsteindid,wetoomustanalyzeourideasofspaceandtimeinordertounderstandthistransformation.Weshallhavetodiscusstheseideasandtheirimplicationsformechanicsatsomelength,sowesayinadvancethattheeffortwillbejustified,sincetheresultsagreewithexperiment.

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153TheMichelsonMorleyexperiment

Asmentionedabove,attemptsweremadetodeterminetheabsolutevelocityoftheearththroughthehypotheticaletherthatwassupposedtopervadeallspace.ThemostfamousoftheseexperimentsisoneperformedbyMichelsonandMorleyin1887.Itwas18yearslaterbeforethenegativeresultsoftheexperimentwerefinallyexplained,byEinstein.

Fig.152.SchematicdiagramoftheMichelsonMorleyexperiment.

TheMichelsonMorleyexperimentwasperformedwithanapparatuslikethatshownschematicallyinFig.152.Thisapparatusisessentiallycomprisedofalightsource ,apartiallysilveredglassplate,andtwomirrors and ,allmountedonarigidbase.Themirrorsareplacedatequaldistancesfrom .Theplate splitsanoncomingbeamoflight,andthetworesultingbeamscontinuein

mutuallyperpendiculardirectionstothemirrors,wheretheyarereflectedbackto .Onarrivingbackat ,thetwobeamsarerecombinedastwosuperposedbeams, and .Ifthetimetakenforthelighttogofrom to andbackisthesameasthetimefrom to andback,theemergingbeams

and willbeinphaseandwillreinforceeachother,butifthetwotimesdifferslightly,thebeamswillbeslightlyoutofphaseandinterferencewillresult.Iftheapparatusisatrestintheether,thetimesshouldbepreciselyequal,butifitismovingtowardtherightwithavelocity ,thereshouldbeadifferenceinthetimes.Letusseewhy.

First,letuscalculatethetimerequiredforthelighttogofrom to andback.Letussaythatthetimeforlighttogofromplate tomirror is ,andthetimeforthereturnis .Now,whilethelightisonitswayfrom tothemirror,theapparatusmovesadistance ,sothelightmusttraverseadistance ,atthespeed .Wecanalsoexpressthisdistanceas ,sowehave

(Thisresultisalsoobviousfromthepointofviewthatthevelocityoflightrelativetotheapparatusis,sothetimeisthelength dividedby .)Inalikemanner,thetime canbecalculated.

Duringthistimetheplate advancesadistance ,sothereturndistanceofthelightis .Thenwehave

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Thenthetotaltimeis

Forconvenienceinlatercomparisonoftimeswewritethisas

Oursecondcalculationwillbeofthetime forthelighttogofrom tothemirror .Asbefore,duringtime themirror movestotherightadistance totheposition inthesametime,thelighttravelsadistance alongthehypotenuseofatriangle,whichis .Forthisrighttrianglewehave

or

fromwhichweget

Forthereturntripfrom thedistanceisthesame,ascanbeseenfromthesymmetryofthefigurethereforethereturntimeisalsothesame,andthetotaltimeis .Withalittlerearrangementoftheformwecanwrite

Wearenowabletocomparethetimestakenbythetwobeamsoflight.Inexpressions(15.4)and(15.5)thenumeratorsareidentical,andrepresentthetimethatwouldbetakeniftheapparatuswereatrest.Inthedenominators,theterm willbesmall,unless iscomparableinsizeto .Thedenominatorsrepresentthemodificationsinthetimescausedbythemotionoftheapparatus.Andbehold,thesemodificationsarenotthesamethetimetogoto andbackisalittlelessthanthetimeto andback,eventhoughthemirrorsareequidistantfrom ,andallwehavetodoistomeasurethatdifferencewithprecision.

Hereaminortechnicalpointarisessupposethetwolengths arenotexactlyequal?Infact,wesurelycannotmakethemexactlyequal.Inthatcasewesimplyturntheapparatus degrees,sothat

isinthelineofmotionand isperpendiculartothemotion.Anysmalldifferenceinlengththenbecomesunimportant,andwhatwelookforisashiftintheinterferencefringeswhenwerotatetheapparatus.

Incarryingouttheexperiment,MichelsonandMorleyorientedtheapparatussothattheline wasnearlyparalleltotheearthsmotioninitsorbit(atcertaintimesofthedayandnight).Thisorbitalspeedisabout milespersecond,andanyetherdriftshouldbeatleastthatmuchatsometimeofthedayornightandatsometimeduringtheyear.Theapparatuswasamplysensitivetoobservesuchaneffect,butnotimedifferencewasfoundthevelocityoftheearththroughtheethercouldnotbedetected.Theresultoftheexperimentwasnull.

TheresultoftheMichelsonMorleyexperimentwasverypuzzlingandmostdisturbing.Thefirst

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fruitfulideaforfindingawayoutoftheimpassecamefromLorentz.Hesuggestedthatmaterialbodiescontractwhentheyaremoving,andthatthisforeshorteningisonlyinthedirectionofthemotion,andalso,thatifthelengthis whenabodyisatrest,thenwhenitmoveswithspeedparalleltoitslength,thenewlength,whichwecall ( parallel),isgivenby

WhenthismodificationisappliedtotheMichelsonMorleyinterferometerapparatusthedistancefromto doesnotchange,butthedistancefrom to isshortenedto .Therefore

Eq.(15.5)isnotchanged,butthe ofEq.(15.4)mustbechangedinaccordancewithEq.(15.6).Whenthisisdoneweobtain

ComparingthisresultwithEq.(15.5),weseethat .Soiftheapparatusshrinksinthemannerjustdescribed,wehaveawayofunderstandingwhytheMichelsonMorleyexperimentgivesnoeffectatall.Althoughthecontractionhypothesissuccessfullyaccountedforthenegativeresultoftheexperiment,itwasopentotheobjectionthatitwasinventedfortheexpresspurposeofexplainingawaythedifficulty,andwastooartificial.However,inmanyotherexperimentstodiscoveranetherwind,similardifficultiesarose,untilitappearedthatnaturewasinaconspiracytothwartmanbyintroducingsomenewphenomenontoundoeveryphenomenonthathethoughtwouldpermitameasurementof .

Itwasultimatelyrecognized,asPoincarpointedout,thatacompleteconspiracyisitselfalawofnature!Poincarthenproposedthatthereissuchalawofnature,thatitisnotpossibletodiscoveranetherwindbyanyexperimentthatis,thereisnowaytodetermineanabsolutevelocity.

154Transformationoftime

Incheckingoutwhetherthecontractionideaisinharmonywiththefactsinotherexperiments,itturnsoutthateverythingiscorrectprovidedthatthetimesarealsomodified,inthemannerexpressedinthefourthequationoftheset(15.3).Thatisbecausethetime ,calculatedforthetripfrom to andback,isnotthesamewhencalculatedbyamanperformingtheexperimentinamovingspaceshipaswhencalculatedbyastationaryobserverwhoiswatchingthespaceship.Tothemanintheshipthetimeissimply ,buttotheotherobserveritis (Eq.15.5).Inotherwords,whentheoutsiderseesthemaninthespaceshiplightingacigar,alltheactionsappeartobeslowerthannormal,whiletothemaninside,everythingmovesatanormalrate.Sonotonlymustthelengthsshorten,butalsothetimemeasuringinstruments(clocks)mustapparentlyslowdown.Thatis,whentheclockinthespaceshiprecords secondelapsed,asseenbythemanintheship,itshows

secondtothemanoutside.

Thisslowingoftheclocksinamovingsystemisaverypeculiarphenomenon,andisworthanexplanation.Inordertounderstandthis,wehavetowatchthemachineryoftheclockandseewhathappenswhenitismoving.Sincethatisratherdifficult,weshalltakeaverysimplekindofclock.Theonewechooseisratherasillykindofclock,butitwillworkinprinciple:itisarod(meterstick)withamirrorateachend,andwhenwestartalightsignalbetweenthemirrors,thelightkeepsgoingupanddown,makingaclickeverytimeitcomesdown,likeastandardtickingclock.Webuildtwosuchclocks,withexactlythesamelengths,andsynchronizethembystartingthemtogetherthentheyagreealwaysthereafter,becausetheyarethesameinlength,andlightalwaystravelswithspeed .Wegiveoneoftheseclockstothemantotakealonginhisspaceship,andhemountstherodperpendiculartothedirectionofmotionoftheshipthenthelengthoftherodwillnotchange.How

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doweknowthatperpendicularlengthsdonotchange?Themencanagreetomakemarksoneachothers meterstickastheypasseachother.Bysymmetry,thetwomarksmustcomeatthesame and coordinates,sinceotherwise,whentheygettogethertocompareresults,onemarkwillbeaboveorbelowtheother,andsowecouldtellwhowasreallymoving.

Fig.153.(a)Alightclockatrestinthe system.(b)Thesameclock,movingthroughthesystem.(c)Illustrationofthediagonalpathtakenbythelightbeaminamovinglightclock.

Nowletusseewhathappenstothemovingclock.Beforethemantookitaboard,heagreedthatitwasanice,standardclock,andwhenhegoesalonginthespaceshiphewillnotseeanythingpeculiar.Ifhedid,hewouldknowhewasmovingifanythingatallchangedbecauseofthemotion,hecouldtellhewasmoving.Buttheprincipleofrelativitysaysthisisimpossibleinauniformlymovingsystem,sonothinghaschanged.Ontheotherhand,whentheexternalobserverlooksattheclockgoingby,heseesthatthelight,ingoingfrommirrortomirror,isreallytakingazigzagpath,sincetherodismovingsidewiseallthewhile.WehavealreadyanalyzedsuchazigzagmotioninconnectionwiththeMichelsonMorleyexperiment.Ifinagiventimetherodmovesforwardadistanceproportionalto inFig.153,thedistancethelighttravelsinthesametimeisproportionalto ,andtheverticaldistanceisthereforeproportionalto .

Thatis,ittakesalongertimeforlighttogofromendtoendinthemovingclockthaninthestationaryclock.Thereforetheapparenttimebetweenclicksislongerforthemovingclock,inthesameproportionasshowninthehypotenuseofthetriangle(thatisthesourceofthesquarerootexpressionsinourequations).Fromthefigureitisalsoapparentthatthegreater is,themoreslowlythemovingclockappearstorun.Notonlydoesthisparticularkindofclockrunmoreslowly,butifthetheoryofrelativityiscorrect,anyotherclock,operatingonanyprinciplewhatsoever,wouldalsoappeartorunslower,andinthesameproportionwecansaythiswithoutfurtheranalysis.Whyisthisso?

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Toanswertheabovequestion,supposewehadtwootherclocksmadeexactlyalikewithwheelsandgears,orperhapsbasedonradioactivedecay,orsomethingelse.Thenweadjusttheseclockssotheybothruninprecisesynchronismwithourfirstclocks.Whenlightgoesupandbackinthefirstclocksandannouncesitsarrivalwithaclick,thenewmodelsalsocompletesomesortofcycle,whichtheysimultaneouslyannouncebysomedoublycoincidentflash,orbong,orothersignal.Oneoftheseclocksistakenintothespaceship,alongwiththefirstkind.Perhapsthisclockwillnotrunslower,butwillcontinuetokeepthesametimeasitsstationarycounterpart,andthusdisagreewiththeothermovingclock.Ahno,ifthatshouldhappen,themanintheshipcouldusethismismatchbetweenhistwoclockstodeterminethespeedofhisship,whichwehavebeensupposingisimpossible.Weneednotknowanythingaboutthemachineryofthenewclockthatmightcausetheeffectwesimplyknowthatwhateverthereason,itwillappeartorunslow,justlikethefirstone.

Nowifallmovingclocksrunslower,ifnowayofmeasuringtimegivesanythingbutaslowerrate,weshalljusthavetosay,inacertainsense,thattimeitselfappearstobeslowerinaspaceship.Allthephenomenatherethemanspulserate,histhoughtprocesses,thetimehetakestolightacigar,howlongittakestogrowupandgetoldallthesethingsmustbesloweddowninthesameproportion,becausehecannottellheismoving.Thebiologistsandmedicalmensometimessayitisnotquitecertainthatthetimeittakesforacancertodevelopwillbelongerinaspaceship,butfromtheviewpointofamodernphysicistitisnearlycertainotherwiseonecouldusetherateofcancerdevelopmenttodeterminethespeedoftheship!

Averyinterestingexampleoftheslowingoftimewithmotionisfurnishedbymumesons(muons),whichareparticlesthatdisintegratespontaneouslyafteranaveragelifetimeof sec.Theycometotheearthincosmicrays,andcanalsobeproducedartificiallyinthelaboratory.Someofthemdisintegrateinmidair,buttheremainderdisintegrateonlyaftertheyencounterapieceofmaterialandstop.Itisclearthatinitsshortlifetimeamuoncannottravel,evenatthespeedoflight,muchmorethan meters.Butalthoughthemuonsarecreatedatthetopoftheatmosphere,some

kilometersup,yettheyareactuallyfoundinalaboratorydownhere,incosmicrays.Howcanthatbe?Theansweristhatdifferentmuonsmoveatvariousspeeds,someofwhichareveryclosetothespeedoflight.Whilefromtheirownpointofviewtheyliveonlyabout sec,fromourpointofviewtheyliveconsiderablylongerenoughlongerthattheymayreachtheearth.Thefactorbywhichthetimeisincreasedhasalreadybeengivenas .Theaveragelifehasbeenmeasuredquiteaccuratelyformuonsofdifferentvelocities,andthevaluesagreecloselywiththeformula.

Wedonotknowwhythemesondisintegratesorwhatitsmachineryis,butwedoknowitsbehaviorsatisfiestheprincipleofrelativity.Thatistheutilityoftheprincipleofrelativityitpermitsustomakepredictions,evenaboutthingsthatotherwisewedonotknowmuchabout.Forexample,beforewehaveanyideaatallaboutwhatmakesthemesondisintegrate,wecanstillpredictthatwhenitismovingatninetenthsofthespeedoflight,theapparentlengthoftimethatitlastsis

secandourpredictionworksthatisthegoodthingaboutit.

155TheLorentzcontraction

NowletusreturntotheLorentztransformation(15.3)andtrytogetabetterunderstandingoftherelationshipbetweenthe andthe coordinatesystems,whichweshallcallthe and systems,orJoeandMoesystems,respectively.WehavealreadynotedthatthefirstequationisbasedontheLorentzsuggestionofcontractionalongthe directionhowcanweprovethatacontractiontakesplace?IntheMichelsonMorleyexperiment,wenowappreciatethatthetransversearm cannotchangelength,bytheprincipleofrelativityyetthenullresultoftheexperimentdemandsthatthetimesmustbeequal.So,inorderfortheexperimenttogiveanullresult,thelongitudinalarm mustappearshorter,bythesquareroot .Whatdoesthiscontractionmean,intermsofmeasurementsmadebyJoeandMoe?SupposethatMoe,movingwith

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the systeminthe direction,ismeasuringthe coordinateofsomepointwithameterstick.Helaysthestickdown times,sohethinksthedistanceis meters.FromtheviewpointofJoeinthesystem,however,Moeisusingaforeshortenedruler,sotherealdistancemeasuredis

meters.Thenifthe systemhastravelledadistance awayfromthe system,the observerwouldsaythatthesamepoint,measuredinhiscoordinates,isatadistance

,or

whichisthefirstequationoftheLorentztransformation.

156Simultaneity

Inananalogousway,becauseofthedifferenceintimescales,thedenominatorexpressionisintroducedintothefourthequationoftheLorentztransformation.Themostinterestingterminthatequationisthe inthenumerator,becausethatisquitenewandunexpected.Nowwhatdoesthatmean?Ifwelookatthesituationcarefullyweseethateventsthatoccurattwoseparatedplacesatthesametime,asseenbyMoein ,donothappenatthesametimeasviewedbyJoein .Ifoneeventoccursatpoint attime andtheothereventat and (thesametime),wefindthatthetwocorrespondingtimes and differbyanamount

Thiscircumstanceiscalledfailureofsimultaneityatadistance,andtomaketheideaalittleclearerletusconsiderthefollowingexperiment.

Supposethatamanmovinginaspaceship(system )hasplacedaclockateachendoftheshipandisinterestedinmakingsurethatthetwoclocksareinsynchronism.Howcantheclocksbesynchronized?Therearemanyways.Oneway,involvingverylittlecalculation,wouldbefirsttolocateexactlythemidpointbetweentheclocks.Thenfromthisstationwesendoutalightsignalwhichwillgobothwaysatthesamespeedandwillarriveatbothclocks,clearly,atthesametime.Thissimultaneousarrivalofthesignalscanbeusedtosynchronizetheclocks.Letusthensupposethatthemanin synchronizeshisclocksbythisparticularmethod.Letusseewhetheranobserverinsystem wouldagreethatthetwoclocksaresynchronous.Themanin hasarighttobelievetheyare,becausehedoesnotknowthatheismoving.Butthemanin reasonsthatsincetheshipismovingforward,theclockinthefrontendwasrunningawayfromthelightsignal,hencethelighthadtogomorethanhalfwayinordertocatchuptherearclock,however,wasadvancingtomeetthelightsignal,sothisdistancewasshorter.Thereforethesignalreachedtherearclockfirst,althoughthemanin thoughtthatthesignalsarrivedsimultaneously.Wethusseethatwhenamaninaspaceshipthinksthetimesattwolocationsaresimultaneous,equalvaluesof inhiscoordinatesystemmustcorrespondtodifferentvaluesof intheothercoordinatesystem!

157Fourvectors

LetusseewhatelsewecandiscoverintheLorentztransformation.Itisinterestingtonotethatthetransformationbetweenthe sand sisanalogousinformtothetransformationofthe sand sthatwestudiedinChapter11forarotationofcoordinates.Wethenhad

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inwhichthenew mixestheold and ,andthenew alsomixestheold and similarly,intheLorentztransformationwefindanew whichisamixtureof and ,andanew whichisamixtureof and .SotheLorentztransformationisanalogoustoarotation,onlyitisarotationinspaceandtime,whichappearstobeastrangeconcept.Acheckoftheanalogytorotationcanbemadebycalculatingthequantity

Inthisequationthefirstthreetermsoneachsiderepresent,inthreedimensionalgeometry,thesquareofthedistancebetweenapointandtheorigin(surfaceofasphere)whichremainsunchanged(invariant)regardlessofrotationofthecoordinateaxes.Similarly,Eq.(15.9)showsthatthereisacertaincombinationwhichincludestime,thatisinvarianttoaLorentztransformation.Thus,theanalogytoarotationiscomplete,andisofsuchakindthatvectors,i.e.,quantitiesinvolvingcomponentswhichtransformthesamewayasthecoordinatesandtime,arealsousefulinconnectionwithrelativity.

Thuswecontemplateanextensionoftheideaofvectors,whichwehavesofarconsideredtohaveonlyspacecomponents,toincludeatimecomponent.Thatis,weexpectthattherewillbevectorswithfourcomponents,threeofwhicharelikethecomponentsofanordinaryvector,andwiththesewillbeassociatedafourthcomponent,whichistheanalogofthetimepart.

Thisconceptwillbeanalyzedfurtherinthenextchapters,whereweshallfindthatiftheideasoftheprecedingparagraphareappliedtomomentum,thetransformationgivesthreespacepartsthatarelikeordinarymomentumcomponents,andafourthcomponent,thetimepart,whichistheenergy.

158Relativisticdynamics

Wearenowreadytoinvestigate,moregenerally,whatformthelawsofmechanicstakeundertheLorentztransformation.[Wehavethusfarexplainedhowlengthandtimechange,butnothowwegetthemodifiedformulafor (Eq.15.1).Weshalldothisinthenextchapter.]ToseetheconsequencesofEinsteinsmodificationof forNewtonianmechanics,westartwiththeNewtonianlawthatforceistherateofchangeofmomentum,or

Momentumisstillgivenby ,butwhenweusethenew thisbecomes

ThisisEinsteinsmodificationofNewtonslaws.Underthismodification,ifactionandreactionarestillequal(whichtheymaynotbeindetail,butareinthelongrun),therewillbeconservationofmomentuminthesamewayasbefore,butthequantitythatisbeingconservedisnottheold withitsconstantmass,butinsteadisthequantityshownin(15.10),whichhasthemodifiedmass.Whenthischangeismadeintheformulaformomentum,conservationofmomentumstillworks.

Nowletusseehowmomentumvarieswithspeed.InNewtonianmechanicsitisproportionaltothespeedand,according(15.10),overaconsiderablerangeofspeed,butsmallcomparedwith ,itisnearlythesameinrelativisticmechanics,becausethesquarerootexpressiondiffersonlyslightlyfrom .Butwhen isalmostequalto ,thesquarerootexpressionapproacheszero,andthemomentumthereforegoestowardinfinity.

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Whathappensifaconstantforceactsonabodyforalongtime?InNewtonianmechanicsthebodykeepspickingupspeeduntilitgoesfasterthanlight.Butthisisimpossibleinrelativisticmechanics.Inrelativity,thebodykeepspickingup,notspeed,butmomentum,whichcancontinuallyincreasebecausethemassisincreasing.Afterawhilethereispracticallynoaccelerationinthesenseofachangeofvelocity,butthemomentumcontinuestoincrease.Ofcourse,wheneveraforceproducesverylittlechangeinthevelocityofabody,wesaythatthebodyhasagreatdealofinertia,andthatisexactlywhatourformulaforrelativisticmasssays(seeEq.15.10)itsaysthattheinertiaisverygreatwhen isnearlyasgreatas .Asanexampleofthiseffect,todeflectthehighspeedelectronsinthesynchrotronthatisusedhereatCaltech,weneedamagneticfieldthatis timesstrongerthanwouldbeexpectedonthebasisofNewtonslaws.Inotherwords,themassoftheelectronsinthesynchrotronis timesasgreatastheirnormalmass,andisasgreatasthatofaproton!Thatshouldbe times meansthat mustbe ,andthatmeansthat differsfrom byonepartin ,sotheelectronsaregettingprettyclosetothespeedoflight.Iftheelectronsandlightwerebothtostartfromthesynchrotron(estimatedas feetaway)andrushouttoBridgeLab,whichwouldarrivefirst?Thelight,ofcourse,becauselightalwaystravelsfaster.1Howmuchearlier?Thatistoohardtotellinstead,wetellbywhatdistancethelightisahead:itisabout ofaninch,or thethicknessofapieceofpaper!Whentheelectronsaregoingthatfasttheirmassesareenormous,buttheirspeedcannotexceedthespeedoflight.

Nowletuslookatsomefurtherconsequencesofrelativisticchangeofmass.Considerthemotionofthemoleculesinasmalltankofgas.Whenthegasisheated,thespeedofthemoleculesisincreased,andthereforethemassisalsoincreasedandthegasisheavier.Anapproximateformulatoexpresstheincreaseofmass,forthecasewhenthevelocityissmall,canbefoundbyexpanding

inapowerseries,usingthebinomialtheorem.Weget

Weseeclearlyfromtheformulathattheseriesconvergesrapidlywhen issmall,andthetermsafterthefirsttwoorthreearenegligible.Sowecanwrite

inwhichthesecondtermontherightexpressestheincreaseofmassduetomolecularvelocity.Whenthetemperatureincreasesthe increasesproportionately,sowecansaythattheincreaseinmassisproportionaltotheincreaseintemperature.Butsince isthekineticenergyintheoldfashionedNewtoniansense,wecanalsosaythattheincreaseinmassofallthisbodyofgasisequaltotheincreaseinkineticenergydividedby ,or .

159Equivalenceofmassandenergy

TheaboveobservationledEinsteintothesuggestionthatthemassofabodycanbeexpressedmoresimplythanbytheformula(15.1),ifwesaythatthemassisequaltothetotalenergycontentdividedby .IfEq.(15.11)ismultipliedby theresultis

Here,thetermontheleftexpressesthetotalenergyofabody,andwerecognizethelasttermastheordinarykineticenergy.Einsteininterpretedthelargeconstantterm, ,tobepartofthetotalenergyofthebody,anintrinsicenergyknownastherestenergy.

Letusfollowouttheconsequencesofassuming,withEinstein,thattheenergyofabodyalways

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equals .Asaninterestingresult,weshallfindtheformula(15.1)forthevariationofmasswithspeed,whichwehavemerelyassumeduptonow.Westartwiththebodyatrest,whenitsenergyis

.Thenweapplyaforcetothebody,whichstartsitmovingandgivesitkineticenergytherefore,sincetheenergyhasincreased,themasshasincreasedthisisimplicitintheoriginalassumption.Solongastheforcecontinues,theenergyandthemassbothcontinuetoincrease.Wehavealreadyseen(Chapter13)thattherateofchangeofenergywithtimeequalstheforcetimesthevelocity,or

Wealsohave(Chapter9,Eq.9.1)that .Whentheserelationsareputtogetherwiththedefinitionof ,Eq.(15.13)becomes

Wewishtosolvethisequationfor .Todothiswefirstusethemathematicaltrickofmultiplyingbothsidesby ,whichchangestheequationto

Weneedtogetridofthederivatives,whichcanbeaccomplishedbyintegratingbothsides.Thequantity canberecognizedasthetimederivativeof ,and isthetimederivativeof .So,Eq.(15.15)isthesameas

Ifthederivativesoftwoquantitiesareequal,thequantitiesthemselvesdifferatmostbyaconstant,say .Thispermitsustowrite

Weneedtodefinetheconstant moreexplicitly.SinceEq.(15.17)mustbetrueforallvelocities,wecanchooseaspecialcasewhere ,andsaythatinthiscasethemassis .SubstitutingthesevaluesintoEq.(15.17)gives

Wecannowusethisvalueof inEq.(15.17),whichbecomes

Dividingby andrearrangingtermsgives

fromwhichweget

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Thisistheformula(15.1),andisexactlywhatisnecessaryfortheagreementbetweenmassandenergyinEq.(15.12).

Ordinarilytheseenergychangesrepresentextremelyslightchangesinmass,becausemostofthetimewecannotgeneratemuchenergyfromagivenamountofmaterialbutinanatomicbombofexplosiveenergyequivalentto kilotonsofTNT,forexample,itcanbeshownthatthedirtaftertheexplosionislighterby gramthantheinitialmassofthereactingmaterial,becauseoftheenergythatwasreleased,i.e.,thereleasedenergyhadamassof gram,accordingtotherelationship

.Thistheoryofequivalenceofmassandenergyhasbeenbeautifullyverifiedbyexperimentsinwhichmatterisannihilatedconvertedtotallytoenergy:Anelectronandapositroncometogetheratrest,eachwitharestmass .Whentheycometogethertheydisintegrateandtwogammaraysemerge,eachwiththemeasuredenergyof .Thisexperimentfurnishesadirectdeterminationoftheenergyassociatedwiththeexistenceoftherestmassofaparticle.

1. Theelectronswouldactuallywintheraceversusvisiblelightbecauseoftheindexofrefractionofair.Agammaraywouldmakeoutbetter.

Copyright1963,2006,2013bytheCaliforniaInstituteofTechnology,MichaelA.Gottlieb,andRudolfPfeiffer

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