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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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TheFeynmanLecturesonPhysicsVol.ICh.15:TheSpecialTheoryofRelativity
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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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