getting the measure of einstein’s space and time

7/28/2019 GettingtheMeasure ofEinsteinsSpace andTime

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UNIVERSITY OF NEVADA, LAS VEGAS

GettingtheMeasure

ofEinsteinsSpaceandTime

An Introduction to Special RelativityLenZane8/1/2010

Thespaceandtimeintroducedin1905byAlbertEinsteinisexplainedbyexaminingaseriesof

simplethoughtorgedankenexperiments. Thedevelopmentmakesextensiveuseof

spacetimediagramstohelpreadersappreciatethefullextentofthechangesinour

understandingofspaceandtimepost1905.


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1

TableofContents

Introduction................................................................................................................................................ 4

ChapterOne: SpaceandTimeBefore1905............................................................................................ 6

SpeedandVelocity................................................................................................................................ 6

TheExperimentersareIntroduced...................................................................................................... 6

TheFirstSetofExperiments: EarthBased........................................................................................ 8

TheSecondSetofExperiments: BusBased..................................................................................... 11

BevandAnneCarefullyObservetheExperimentDoneontheBus............................................ 12

TheAdditionofVelocitiesFormula.................................................................................................. 15

APictureisWorthaThousandWords............................................................................................. 17

TheGalilean

Transformation

Equations

...........................................................................................

22

SummaryofChapterOne................................................................................................................... 26

ThePrincipleofRelativityandtheIsotropyofSpace................................................................. 26

SpaceandTimepre1905................................................................................................................ 26

ChapterTwo: TheSpeedofLight......................................................................................................... 27

ChapterThree: SpaceandTimeAfter1905......................................................................................... 30

ANewSetofExperimentsareProposed.......................................................................................... 30

NewStop

Watches

are

distributed

....................................................................................................

30

AnneandBevMeasuretheSpeedofLight...................................................................................... 30

TheSuperBusRollsintotheStory.................................................................................................... 33

AnneandBevObservetheLightFlashontheBus: I..................................................................... 35

AnneandBevObservetheLightFlashontheBus:II.................................................................... 39

TheAffectofMotiononSpace........................................................................................................... 46

TheAffectofMotiononTime............................................................................................................ 49

LightMovesthroughSpaceandTimeinaVeryStrangeWay.....................................................

52

ChapterFour: GeneralizingtheObservations.................................................................................... 54

DefiningtheNewProblem................................................................................................................. 55

FindingtheShrinkageFactor............................................................................................................. 56

TheTickingRateofMovingClocks.................................................................................................. 60


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QuantifyingtheDisagreementoverSynchronization.................................................................... 61

TestingtheTheory............................................................................................................................... 65

AQuestionaboutSynchronization................................................................................................... 67

GlobalPositioningSystem(GPS)....................................................................................................... 69

ChapterFive: TheRelationshipbetweenBusandEarthObservers.................................................. 70

CombiningSpacetimeGraphs........................................................................................................... 70

BevGeneralizesChucksSpacetimeDiagram................................................................................. 77

ASimpleExampleforDean............................................................................................................... 82

TheRevisedEquationfortheAdditionofVelocities..................................................................... 82

ChapterSix: ChuckSuggestsanExperimentandAnnehasaDream............................................ 87

ExperimentI

..........................................................................................................................................

88

ExperimentII........................................................................................................................................ 91

AnswersforExperimentI................................................................................................................... 95

AnswersforExperimentII.................................................................................................................. 98

AnnehasaDreamaboutaSuperluminalPigeon......................................................................... 102

ChapterSeven: TheBusandtheGarage........................................................................................... 106

AnnesProposal................................................................................................................................. 112

TheCollisionbetweenTheoryandReality....................................................................................

113

R.I.P.SuperBus.................................................................................................................................. 115

ChapterEight: TheSolarSystemandBeyond.................................................................................. 117

SynchronizingWatchesSeparatedbyLargeDistances................................................................ 118

ChapterNine: ChuckandDeanTraveltoAlphaCentauri............................................................. 120

May1,ThirteenYearsLater.............................................................................................................. 121

UsingtheLorentzEquations............................................................................................................ 127

LookingataMovingWatchTheDopplerEffect........................................................................

129

ChapterTen: SpaceTravel................................................................................................................... 133

TheEquivalencePrinciple................................................................................................................ 133

DesigningtheSpaceAdventure...................................................................................................... 136

TheResultsofChucksCalculations............................................................................................... 140


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HowdoesaRocketAccelerate?....................................................................................................... 145

APeekatGeneralRelativity............................................................................................................. 149

ChapterEleven: DoesitReallyHappen?........................................................................................... 152

TheFarAwayObserver(FAO)........................................................................................................ 152

ADynamicExplanationfortheBehaviorofWatchesandRulers.............................................. 156

AModelforaHydrogenAtom........................................................................................................ 157

ChapterTwelve: E=mc2...................................................................................................................... 160

AppendixA: Graphing......................................................................................................................... 167

AppendixB: ScientificNotation.......................................................................................................... 170

AppendixC: TheLightClock.............................................................................................................. 171

AppendixD:

The

Gravitational

Clock

Effect

....................................................................................

176


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4

Introduction

ThisbookwaswrittenaftermanyyearsofteachingSpecialandGeneralRelativityto

studentswithvariedbackgrounds. Thepresentationemphasizesspacetimediagrams

whichinmyexperiencehelpsstudentsvisualizespaceandtime. Thematerialisself

contained,allowingthebooktobeusedasatutorialforapersonwithlittlebackground

inbutsomefamiliaritywithalgebraandahealthycuriosityaboutSpecialRelativity.

Therearequestionsscatteredthroughoutthebooktoencouragethereadertotakesome

timetoreviewthematerialpresentedbeforemovingontonewmaterial. Thebookcan

alsobeusedasatextbookforacourseinspaceandtimefornonsciencestudents,the

audienceIprimarilyhadinmindwhenwritingit,orasanintroductorycoursefor

studentsplanningtostudysciencelaterintheirundergraduatecareers.

Chapter

One

establishes

a

simple

methodology

for

measuring

speeds

and

velocities

at

theusualnonrelativisticvaluesthatareencounteredineverydaylife. Inparticular,the

velocityofanobjectmovinginabusthatistravelingdowntheroadismeasuredby

peopleonthebusandsimultaneouslybyobserversstandingontheground. These

measurementsareusedtoprobetherulesthatgovernspaceandtime.

ChapterTwopresentsashortsummaryoflightsproperties. Forus,themost

importantpropertyislightsastonishinglylargevelocity.

ChapterThreereplacesthebusinChapterOnewithonethatcanmoveatrelativistic

speeds. Thisnewsuperbusisusedtoprobespaceandtimebydoingexperiments

analogoustothosedoneinChapterOne. Thesenewexperimentsforceour

experimenterstodrasticallyrevisetherulesgoverningspaceandtimethatemerged

fromChapterOne.

ChapterFourandFivegeneralizetheresultsoftheexperimentsdoneinChapterThree

culminatinginaderivationoftheLorentzTransformationequationsandtherelativistic

versionoftheadditionofvelocityequationfirstencounteredinChapterOne. Chapter

FourendswithasectiononhowaGPSverifiesthatmovingclocksrunslowandthatclocksfurtherfromthecenterofearthrunfast.

ChapterSixisprimarilyatutorialdesignedtogivereadersachancetoreviewallthe

earliermaterialbyusingspacetimegraphsandtheLorentzTransformationequationsto

analyzetwoimaginaryexperimentsdonewiththesuperbus.


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ChapterSevendescribesandanalyzesthewellknownPoleandBarnproblemand

attemptstogiveadefinitiveanswertothequestion,doesthepolefitinthebarnornot?

Inthischapter,thesuperbusreplacesthepoleandanewlyconstructedgaragetakes

theplaceofthebarn.

ChapterEightintroducestheastronomicaldistanceunit,thelightyear,andgivesa

briefdescriptionofourhomegalaxy,theMilkyWay.

ChapterNineanalyzesinsomedetailthefamousTwinParadox. Thisiconicparadox

ofSpecialRelativityariseswhenoneofthetwinstravelsawayfromearthinaspace

shipandreturnsyearslater. Uponreturning,itisdiscoveredthatthetwinthat

remainedonearthagedmorethanthetravelingtwin.

ChapterTenusesthelawsgoverningspaceandtimetostudythemotionofarocket

thatmoveswithconstantacceleration. Theacceleratingrocketisusedtoexaminethe

possibilitiesofhumantraveltoothergalaxies. ThelastsectionofChapterTenexplains

whyaclockfurtherfromthecenterofearthrunsfasterthananidenticalclosertothe

center.

ChapterElevengivestwoseparateargumentsdesignedtoshowthatrulersreallydo

shrinkandwatchesactuallyrunslowandtheseeffectsarenotjustillusoryor

theoretical.

ChapterTwelvegivesasimplederivationofEinsteinsfamousequation,E=mc2.

Thoughnotdirectlyrelatedtotheprimarythemeofthebook,itisdifficulttowritea

bookthatpurportstocoverSpecialRelativityandnotincludethatequation.


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ChapterOne: SpaceandTimeBefore1905

ThischapterexaminessomesimpleexperimentsthatestablishthespaceandtimeofGalileoand

Newton. Thisviewprevaileduntil1905whenAlbertEinsteinintroducedtheworldtoSpecial

Relativity. Itisalsothespaceandtimeofoureverydayliveswhererulersandwatchesarewell

behaved.

SpeedandVelocity

Theconceptsofspeedandvelocityarecolloquiallyunderstoodtobeadistancetraveleddivided

bytheelapsedtime. Inthischapter,velocitiesandspeedswillbemeasuredinfeet/secondorft/s.

Tophysicists,speedandvelocityarerelatedbutnotinterchangeableideas. Velocityincludesa

senseofdirectioninitsdefinition. Throughoutthisbook,anythingoranybodymovingfromleft

torightwillhaveapositivevelocity,forexample+10ft/s. Notethatavelocityof+10ft/sisthe

sameas10ft/s. Whileanyobjectorpersonmovingfromrighttoleftwillhaveanegativevelocity,forexample 10ft/sec,. Thespeedofanobjectisthesize,ormagnitude,ofitsvelocity.

Intheaboveexamples,bothobjectswouldhavethesamespeed,10ft/sec,althoughthosespeeds

wouldbeinoppositedirections.

Laterwewilllookverycarefullyathowspeedismeasured. Butfornow,itisclearthattofind

thevelocityorspeedofanobject,itisnecessarytomeasurebothadistancecoveredandthetime

ittooktotraversethatdistance. Rulersandstopwatchesaretheusualinstrumentsusedto

measuredistanceandtime. Imagineawarehousefullofexcellentlyfabricatedrulersandstop

watches.Alltherulersareidenticaltooneanotherandthesameistrueforthestopwatches.Theseinstrumentswillbehandedouttospeciallytrainedobserverswhowillcollectthedata

usedtodeterminethespeedorvelocityoftestobjectsinavarietyofcircumstances.Akeypoint

tokeepinmindisthattheexperimentsdescribedareallperfectlyreasonableanddoable,atleast

inprinciple,thoughsomemaybetechnologicallytoochallengingtobedonewithcurrently

availablerulers,stopwatches,andobservers.

TheExperimentersareIntroduced

Anne,Bev,Chuck,andDeanaregoodfriends,astuteobservers,andcuriousbynature.

Oneevening,afterwatchinganepisodeofStarTrek,theybegintalkingaboutspace

andtimeandSpecialRelativity,subjectsnoneofthemknowsverymuchabout. Finally

BevsuggeststhatinsteadofspeculatingaboutthemeaningofSpecialRelativity,they

oughttodosomecarefulexperimentstogetfirsthandknowledgeaboutspaceand


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7

time. Herthreefriendsquicklyagree,thinkingitcouldbeenlighteningtogetan

experimentallybasedunderstandingofthenatureofspaceandtime.

Anne,Bev,Chuck,andDeanarethemaincharactersinthisbook.AnneandBevarestationed

onearthduringalltheexperimentswithBevalwaysbeingtotherightofAnne. Conversely,ChuckandDeanarethetravelingpairofexperimenters. Theyrideinbusesandrocketcruisers

withDeanalwayssituatedtotherightofChuck.A&BandC&DareshorthandforAnneandBevandChuckandDean.Thesectionsinitalicsrepresenttheauthorinterjectinghimselfintothenarrative. Thehopeis

thattheseasideswilladdtoandnotdisruptthemainstory,theexperimentalprobingofspace

andtime.

Thenextmorning,brightandearly,ourfriendsmeettomapoutasetofexperiments.

TheyrecallthatSpecialRelativityhassomethingtodowithlightanditsvelocityand

thewayitisperceivedbydifferentobserversmovingwithrespecttooneanother.

Thoughtheyarenotsurewhatitisaboutlightthatissopeculiar,theydoknowthatthe

speedoflightisvery,verylarge. Annesuggeststhattheydosomesimpleexperiments

withsomethingthatmovesatapedestrianspeed;forexample,oneofthetrained

pigeonsthatshehasseeninthepark. Thesepigeonsallflyatexactly20ft/s.

Atthetimescientistsweregrapplingwiththeconceptsofspaceandtime,therewasgeneral

agreementthatlightwasawave. Oneoftheprinciplecharacteristicsofawaveisthatittravelsthroughsomemedium. Forexample;soundtravelsthroughairandothermaterialsubstances.

Thespeedofawaveisthespeedatwhichitmovesthroughthatmedium. Scientistsimagined

spacebeingpermeatedbyanetherealsubstance,thelumeniferousether,throughwhichlight

moved. Thespeedofapigeonisthespeedthatitmovesthroughair. Thisisanalogoustothe

wayawavemovesthroughamedium,ormorespecificallythewayscientistspicturedlight

movingthroughtheether. ThusAnneschoiceofapigeonastheobjecttostudywasreasoned

andnotfortuitous.

Deanvolunteerstoheadtotheparktosignupapairofpigeonsfortheirexperiments.

Afterheleaves,hisfriendsdecidetobreakthemselvesintotwoteamsAnneandBev

andChuckandDean. AnneandBevwillbetheteamthatmeasuresthingsfromthe

perspectiveofearthwhileC&Dwillrideinalaboratorythatmoveswithrespectto

earth. Chuckagreeswiththisplanandimmediatelyheadsdowntotheusedbuslotto


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findavehiclethatcouldbeappropriatelymodifiedtobecometherollinglaboratoryfor

himandDean.

Afterheleaves,A&Bgotothewarehousetopickupfouridenticalsetsofrulersand

stop

watches.

Later

that

day,

the

four

friends

reconvene

to

review

what

has

been

accomplishedsofarandwhatexperimentsoughttobedonefirst. A&Bdistribute

rulersandstopwatchestoC&D. Theydutifullytesttheequipmentandagreethat

theyareallidenticalandestablishthatthewatchesareaccuratetoonetenthofa

second,0.1second. Deanshowsoffthetwotrainedpigeonsthatvolunteeredtohelp

withtheexperiments. Thepigeonsdemonstratetheirskillbyflyingbackandforthin

tandematidenticalspeeds,eachmatchingtheotherflapforflap. FinallyChuckgivesa

tourofthebusthatwillactashisandDeansrollinglab. Hepointsoutthatithasan

excellentcruisecontrolthatwillensurethatthebusmaintainsaconstantspeedduring

anyexperiment. Bothteamsmeasurethelengthofthebusandagreethatitis100feet

long.

Inordertosimplifythediscussionsthatfollow,theworldofourfourexperimenterswillneedto

havesomepeculiar,butnotunreasonable,properties. Thefirstisthattheworldhasjustone

spacedimension. ThatisphysicstalkforthefactthatAnne,Bev,thepigeons,andthebuswith

ChuckandDeanaboard,canonlymovealongaline. Theycanmovetotheright,thepositive

direction,ortotheleft,thenegativedirection. Butmotionperpendiculartothatlineis

impossibleandmeaninglessfortheparticipantsinthevariousexperimentstobedescribedinthisandlaterchaptersbecausethereisnodirectionperpendiculartothelinetheymovealong! Fora

moreconcretepictureofaonedimensionalworld,imaginebeadsslidingonawire. Thebeads

canslidetoorfrobutitisimpossibleforthemtomoveoffthewire.

TheFirstSetofExperiments: EarthBased

Intheirfirstexperiment,A&Busetheirrulersandwatchestomeasurethespeedof

oneofthepigeons. Meanwhile,theotherpigeoniscomfortablyhousedonthebustobe

usedbyC&Dinlaterexperiments. A&Bdecidetohavetheirpigeonfly100feet,the

lengthofthebus,tomaketheirexperimentsmoredirectlycomparabletothoseC&D

willdo. C&DwatchasA&Bcarefullymeasureoff100feet. Anneholdsthepigeon

whileBevstationsherself100feettoAnnesright. TheplancallsforAnnetoreleasethe

pigeonwhilesimultaneouslystartingherwatch. Bevwillstartherwatchassoonas

Annereleasesthepigeonandbothwillstoptheirwatchesthemomentthepigeon

reachesBev.


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BeforeweletA&Bdotheaboveexperiment,letsmakesurethatthedescribedproceduremakes

sense. Thereisacertainreactiontimeinvolvedinstartingandstoppingawatch. SinceAnneis

startingherwatchatthesameinstantthatshereleasesthepigeon,wecanassumethatbothof

thoseactionstakeplaceatexactlythesameinstant.Afterall,thatistheleastwecanexpectfrom

welltrainedobservers! Ontheotherhand,Bevdoesnotknowexactlywhenthepigeonwillbe

released. Shewillstartherwatchafterashortdelaythatdependsonhowquicklyshecanreact

totheinitialmovementsofthepigeon. HumanreactiontimesforOlympicathletesarebetween

1and2tenthsofasecond. WhenBevcatchesthepigeon,sheinstantlystopsherwatch.Again

wecanassumethatsheisskilledenoughtodobothsimultaneously. ButnowitisAnne

watchingBevcatchthepigeonwhohastoreact,introducingsomeuncertaintyonherendofthe

experiment.

Bevrecords5.1secondsforthepigeonsflighttimewhileAnnegets4.9seconds. Being

goodscientiststheydecidetoaveragethetwotimes,concludingthatthepigeontook

5.0secondstofly100feetforaspeedof20feet/secondor20ft/s. Theyrepeatthis

experimentseveraltimes,andmeasure20ft/seachtimeforthespeedofthepigeon.

Justasadvertised,theirpigeonalwaysfliesat20ft/s. Notethatthepigeonsvelocityis

+20ft/sbecauseitfliesfromAnnetoBev,lefttoright.

Forthesecondexperiment,thepigeonfliesfromBevtoAnne,fromrighttoleft. When

theycalculatethespeedofthepigeon,theyget,unsurprisinglysinceitisawelltrained

pigeon,20

ft/s.

But

the

pigeons

velocity

isnow

20

ft/s.

After

this

experiment,

the

friendsdecidetobreakforlunch.

ThesmalldifferenceintimemeasuredbyBevandAnnewasexperimentalerrorintroduced

becauseofthefinitereactiontimesofourobservers.Inthefuture,wedontwanttoworryabout

whetherornotwelltrainedobserverscanrelease/catchapigeonwhilesimultaneously

starting/stoppingawatch. Nordowewanttoworryabouthumanreactiontimeswhich

needlesslycomplicatetheresultsofourexperiments. Consequentlypartofourdefinitionofa

welltrainedobserverwillincludetheabilitytoperformallthedutiesrequiredinagiven

experimentwithoutintroducinganyextraneouserrors.

Question1.1: Afterlunch,A&Bdecidetoredotheexperimentwiththepigeon. But

nowabreezeisblowingfromAnnetowardBevat10ft/s. Howdoesthewindaffectthe

resultsoftheexperimentwhenthepigeontravelsfromAnnetoBevandthenbackto

Anne? (Thinkaboutthisforabitbeforereadingon.)


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Whatdidyouconcludeabouttheflyingtimesforthepigeonwithabreezeblowingfromleftto

rightat10ft/sec? Thefirstthingtorecognizeisthatthespeedofthepigeonfoundduringthe

firstsetofexperimentswasthespeedofthepigeonwithrespecttotheairitwasflyingthrough.

Sincetheairwasnotmovingwithrespecttotheground,thepigeonsspeedwithrespecttothe

airwasthesameasthepigeonsspeedwithrespecttotheground. Thatisthereasonthepigeons

speedwasthesameflyingfromlefttorightasitwasflyingfromrighttoleft.

Butnowtheairismovingwithrespecttotheground. IneachsecondofflightfromAnnetoBev,

thepigeonadvanced20feetthroughtheairwhiletheairmoved10feetclosertoBev.

Consequentlythepigeonmoved30feetclosertoBevforeachseconditflew. Thepigeons

velocitywithrespecttothegroundwas+30ft/s. BevandAnnebothrecordedtheflighttimeas

3.3seconds.

Thereturntripwasmoredifficultforthepigeonbecauseitwasnowflyingintoaheadwind. Themeasuredflighttimeforthereturntripwas10seconds,makingthepigeonsvelocity 10ft/s.

Noticethatthetotaltimefortheroundtripinthe10ft/swindwas13.3secondscomparedto10

secondswhentherewasnowind.

Question1.2: Whatwouldbethetotaltimeforaroundtripwithawindblowing+15

ft/s?

Question1.3: Whatwouldhappentotheroundtriptimeifthewindwasblowingata

velocityof+20ft/sorlarger?

Rememberthatthepigeonsmotionthroughairisanalogoustothemotionofawavethrougha

medium. ScientistsatthedawnoftheTwentiethCenturyconsideredlightawave.

ExperimentsanalogoustothosedonebyA&Bwhenawindwasblowingweredoneby

scientistsonlight,withtheexpectationthatthespeedoflightwoulddependonthedirectionand

speedoftheetherwind. Theseexperimentsonlightwillbediscussedmoreinthenexttwo

chapters.

Theideathatapigeonflyinginstillairhasexactlythesamespeedregardlessofthedirectionflownisreallyastatementthatspaceisisotropic.Thatis,anyexperimentdonewithour

instrumentsalignedlefttorightoughttogiveexactlythesameresultifinsteadweorientedour

instrumentsrighttoleft. Thediscussionisrestrictedtoleft/rightbecausethosearetheonly

directionsavailabletoourexperimenters.Moregenerally,weexpecttheresultsofany

experimenttobeindependentoftheorientationoftheapparatusbecauseoftheisotropyofspace.


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Thisexpectationistakenbyscientistsasafundamentalpropertyofspace. Notethatthis

expectationisfalseiftheexperimentisdoneonawindydaybecausethewinddestroysthe

symmetrybetweenlefttorightandrighttoleft.Also,forthoseofuslivingonearth,motion

backandforthorfromsidetosideisisotropicbutgravity,likethewind,upsetstheisotropyof

spaceintheverticaldirection.

TheSecondSetofExperiments: BusBased

Aftersuccessfullymeasuringthespeedofthepigeonwithrespecttoearth,Anneand

BevsuggestthatitistimeforChuckandDeantodothesameexperimentintheir

rollinglaboratory. Thiswillgivethemachancetotesttheirexperimentalskill. C&D

getintheirbus,setthecruisecontrol,anddrivewithavelocityof+30ft/stowardA&B.

DeanisinthefrontofthebusandChuckintheback,exactly100feetseparatethem.

WhenDeanpeeksoutofthewindowheseestheonedimensionalearth,alongwithA&

B,movepastthebusfromrighttoleftwithavelocityof 30ft/s. Foraminute,Dean

forgetsthatheisinamovingbusandinsteadimaginesthattheearthismoving.

Itisimportanttonotethatthisbushasanincrediblygoodsuspensionsystem,travelsina

straightline,anditsspeedneverdeviatesfrom30ft/s. Therefore,withtheshadesdown,C&D

haveabsolutelynosenseofmovingandcouldlegitimatelythinkofthemselvesasstationary.

Ourearthlybiasmakesitdifficultinourheartofheartstoconsiderthebusstationarywhilethe

earthamblesbyat 30ft/s. Itisessentialthatweletgoofthisbias. Thealternativeistohave

BevandAnneinabusidenticaltotheoneusedbyChuckandDeanandthentohavethesetwo

busesmoverelativetooneanotherwithaspeedof30ft/s. Forexample,C&Dcanbetraveling

at+15ft/swhileA&Bdrovetowardthemat 15ft/s.A&Bwouldseetheotherbusmoving

withavelocityof+30ft/swhileC&Dsawtheotherbusmovingat 30ft/stowardthem.This

scenarioemphasizesthesymmetrybetweenA&BandC&Dandthereasonwhytheyareboth

perfectlyjustifiedinclaimingthattheyareonastationarybuswhiletheotherbusismoving.

Insteadofcreatingaperfectlysymmetricsituationandwastingfuelbyusingtwobusesinstead

ofone,you,thereader,willhavetoworktothwartyourbiasandbecomecomfortablewiththe

equivalencebetweenexperimentsdoneonthebusandthosedoneonearth.

ChuckandDeanfirstmeasurethetimeittookthepigeontoflyfromthebackofthebus

tothefront. Thentheycarefullyrepeatthemeasurementwiththepigeonflyingfrom

thefronttotheback. Inthefirstcase,thevelocityofthepigeonismeasuredtobe+20


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ft/swhileonthereturntripthevelocityis 20ft/s. Inbothcases,thespeedoftheir

pigeonisidenticaltothespeedofthepigeonusedbyA&B.

Embeddedinthediscussionofthesetwosetsofexperimentsisanimportantobservationthatwas

firstexplicitlystatedbyGalileoasthePrincipleof

Relativity. Theprinciplecodifiesour

experienceofflyingthroughsmoothaironajetliner. Ifyoufellasleepassoonasyouboardeda

plane,andwokesometimelater,youwouldbehardpressedtotellifyouwerestillsittingonthe

runwaywaitingforclearancetotakeofforcruisingat500mphat35,000feet. Infact,the

PrincipleofRelativitystatesthatthereisnoexperimentthatyoucoulddointheairplanethat

woulddifferentiatebetweenthetwostatesofmotion. BeforeEinstein,theprinciplewasmeantto

applytoanymechanicalexperimentandnottoexperimentsinvolvinglightormoregenerally

electromagneticphenomena. Wewilltakeitasawellestablishedexperimentaltruththatany

experimentdonebyA&BcanberepeatedbyC&Dontheirbuswithidenticalresults. The

pigeonexperimentisaparticularlysimpleaffirmationofthePrincipleofRelativity.

Moregenerally,thePrincipleofRelativitysaysthatallinertialreferenceframesarethesame.

Aninertialreferenceframeisacollectionofobserversmovingthroughspaceandtimewitha

constantvelocity.A&BandC&Dareobserversintwodistinctinertialreferenceframes.A&

BclaimthatC&Daremovingthroughtheirearthbasedreferenceframeat+30ft/swhileC&

DseeA&Bmovingthroughtheirbusbasedreferenceframeat 30ft/s.

BevandAnneCarefullyObservetheExperimentDoneontheBus

Beforegoingon,letsfinetunethedescriptionofouronedimensionalworld. Duringthe

experimentsdonewiththebus,itwasalwaysmovingfromlefttoright. Therefore,beforethe

startofeachexperimentthebusisfirstdriventosomestartingpointtotheleftofAnne,turned

around,andreadiedforitsnexttrippastA&B. Everyexperimentwillstartwhenoneofthe

earthobservers,AorB,isnexttooneofthebusobservers,CorD. Thestartofeachexperiment

willalsobemarkedbysomeparticulareventtakingplaceasthetwochosenobserverspassone

another,forexamplethereleaseofapigeon.

PleasedontaskhowAnneandChuckcanbeadjacenttooneanotherinaonedimensionalworldorhowthebuscanpassbyAnneandBevwithoutobliteratingthem. Ifyouneedaconcrete

picture,imaginethebuswithitsinhabitantsmovinginaonedimensionalworldparalleltothe

oneoccupiedbyA&B. Orgoingbacktothebeadandwireanalogy,A&BandC&Dare

beadsonparallelwires.


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A&BdecidetoaskC&Dtodotheirtwoexperimentsagainbutthistimetheywantto

determinehowfastthepigeononthebusflewwithrespecttothem,thatis,withrespect

toearth. Todothat,theyneedtomeasuretheearthdistancecoveredbythepigeonasit

fliesfromthereartothefrontofthebus. Thespeedofthepigeonisthatdistance

dividedbythetimeittookthepigeontoflyfromoneendofthebustotheother.

Fortheseexperiments,ChuckagreestoreleasethepigeonwhenheisadjacenttoAnne.

Atthatsameinstant,BevandDeanwillstarttheirwatches. Thatinstantalsosignalsthe

startoftheexperiment. Beforedoingtheexperiment,AnneandBevmulloverthe

questionofwherealongtheroadBevoughttostandsothatshewillbeadjacentto

Deanjustashecatchesthepigeon.

Question1.4: Beforereadingon,calculatehowfartotherightofAnneBevhastobe

standingtowitnessthearrivalofthepigeonatthefrontofthebus.

Bevobservesthatsincethebustravelsat30ft/sandthepigeontakes5secondstofly

fromthebacktothefrontofthebus,thebuswillmove150feetdowntheroadwhile

thepigeonisflying. Anneimmediatelyagreesandadds,Dean,inthefrontofthebus,

willbe100feetpastmewhenChuckreleasesthepigeon. Thereforehewillbe100plus

150feettomyrightwhenhecatchesthepigeon. Usingthisinformation,Bevstations

herselfdowntheroadatthe250footmarker. NowthatA&Bareready,Annegives

Deanthesignaltostartthebusrollingdowntheroadtowardthem.

ChuckreleasesthepigeonashepassesAnne. Bevstaresintentlyatthebusasitcomes

towardsherandseesDeancatchthepigeonjustasthebusdrivesby. Atthatinstantof

passing,sheandDeansimultaneouslystoptheirwatches. BevexcitedlywavestoAnne

thattheyhadcorrectlycalculatedtheplacesheneededtobestandingtowitnessDean

catchthepigeon.

Aftertheexperiment,thefourfriendscomparenotes. BevandDeanhavebothtimed

thepigeonsflightaslasting5seconds. BecauseBevisstandingataspot250feettothe

rightofAnne,thepigeonsvelocityis+50ft/swithrespecttoearth. C&Dseetheir

welltrainedpigeonflythelengthofthebus,100feet,withaspeedof20ft/s,justlike

expected.

Inthenextexperiment,thepigeonwillflyfromDeantoChuck,fromrighttoleft. The

planisforDeantoreleasethepigeontheinstantheisadjacenttoAnne. AgainA&B


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14

confertofigureoutwhereBevneedstostandinordertobeadjacenttoChuckwhenhe

catchesthepigeoninthebus.

Question1.5: WheredoesBevhavetobestandinginrelationtoAnnetobeatthe

correct

spot

to

observe

Chuck

catch

the

pigeon?

Bevpointsoutthatthebuswilltravel150feetduringthe5secondsittakesthepigeon

toflyfromDeantoChuckjustlikeinthepreviousexperiment. Anneagreesbutnow

thepigeonisflyingintheotherdirectionwhichmakesitharderforhertothinkabout.

Bevisalsohavingdifficultythinkingaboutthepigeonbutthenshesmilesandpoints

outthatthepigeonisnotreallyrelevant. Thepigeonfliesfromthefrontofthebusto

thebackofthebus,adistanceof100feet. Duringthattime,thebackofthebusmoves

150feet,thereforethepigeongetscaught,fromourperspective,50feettotherightof

theplaceitisreleased.

Annenodsslowly. Whenthepigeonisreleased,IwillbestandingnexttoDeanwhois

atthefrontofthebus. TherearofthebuswhereChuckissittingis100feettomyleft.

WhilethepigeonfliestowardChuck,therearofthebusmoves150feet. Therefore

Chuckwillbe50feettomyrightwhenhecatchesthepigeon. NowthatA&Bagree

onthespotwhereBevneedstobestanding,theyarereadytotesttheiranalysisagainst

theactualexperimentwiththebus.

Bevstationsherself50feettotherightofAnne. Thebuscomesrollingdowntheroad.DeanreleasesthepigeonashepassesAnne. ChuckandBevstarttheirwatches.Justas

expected,BevisadjacenttoChuckwhenhecatchesthepigeonandboththeirwatches

read5seconds. ButthistimeBevislocated50feettotherightofAnne,makingthe

velocityofthepigeonwithrespecttotheearth+10ft/s. Ofcourse,fromtheperspective

ofC&D,thepigeonsspeedisstill20ft/s.

Thefourfriendsreviewtheresultsoftheselasttwoexperiments. Aftersomecareful

thought,theyrecognizethattheresultsareconsistentandmakeperfectlygoodsense.

Thepigeonfliesinthestillairofthebuswhichismovingat30ft/s. Fromthe

perspectiveofA&B,abusmovingat+30ft/swithstationaryairinsideisanalogous

toasituationinwhichapigeonflieswhenawindblowsfromAnnetowardsBevata

speedof30ft/s.


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15

Whenthepigeonfliesfromthebacktothefrontofthebus,themovingbusactslikea

tailwind. Duringeachsecondofflight,itflies+20feetinthestillairofthebuswhile

thebusmoves+30feetalongtheroad. Therefore,fromtheperspectiveofA&B,the

pigeonflapsalongat+50feeteverysecond.

Ontheotherhand,forthereturnflight,themotionofthebusislikeaheadwind. Each

second,thepigeoncovers 20feetofbusdistancewhilethebuscovers+30feetofearth

distance. Thenetdistancepersecondtraveledbythepigeonflyingfromrighttoleftin

thebuswasonly+10feet.

Beforebreakingupfortheday,Chuckwondersaloudifthereissomegeneralprinciple

orrelationshipthatcouldexplaintheresultsofthetwoexperimentsdonewiththe

pigeonflyinginthebus. Bevsuggeststhattheysleeponitandmeetthenextmorning

tocomparenotes.

TheAdditionofVelocitiesFormula

Thenextday,AnneandBevareanxioustosharetheirthoughtswithChuckandDean.

C&DhaveahandfulofpaperstheywanttoshowA&B,buttheylettheirfriendshave

thefloorfirst. Annebeginsbywritingdownthefollowingtwoequations:

+50=+20+30 (1.1)

+10= 20+30. (1.2)

C&Dstareatherandtheequationswithblankexpressions. FinallyDean,whoisless

adeptatalgebrathanhisthreefriends,saysthatevenherecognizesthoseascorrect

equations,buthefailstoseetheirrelevancetoyesterdaysexperiments! Bev

impatientlyjumpsinandexplains,Inthefirstequation+50isthevelocityofthe

pigeonwithrespecttoearth,orVPEforshorthand,wherethePstandsforpigeonand

theEforearth. Ontheothersideoftheequationwehave+20,thevelocityofthepigeon

withrespecttothebus,orVPB,and+30,thevelocityofthebuswithrespecttoearth,or

VBE. Therefore,thatequation,intermsoftheshorthandnotation,canbewrittenas,

. (1.3)Chuckimmediatelyrecognizesequations1.1and1.2arespecialcasesofthemore

generalequation1.3. Theonlydifferenceisthatinequation1.2,thevelocityofthe


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16

pigeonwithrespecttothebus,VPB,is 20ft/sinsteadofthe+20ft/svalueithasin

equation1.1.

Exactly!saidBev.

AnnesummarizestheconclusionsthatsheandBevhavearrivedatbywritingdowna

moregeneralversionofequation1.3,

. (1.4)ThisequationgivestherelativevelocityofXwithrespecttoYifthevelocitiesofXwithrespect

toZandZwithrespecttoYareknown. Equation1.4succinctlyexplainstheresultsofthe

experimentsdonewiththepigeonandbus.

Theconnection

between

equations

1.4

and

1.3

requires

the

following

associations:

X

pigeon,

Yearth,andZbus. Laterinthischapter,itwillbeshownthatequation1.4,whichrelates

velocitiesinonereferenceframetothoseinanother,encapsulatesthepre1905conceptsofspace

andtime.

Useequation1.3or1.4toanswerthefollowingquestions.

Question1.6: Supposeageneticallymodifiedpigeonthatcanfly50ft/sisusedinthe

twoexperimentsdoneonthebus. AsthepigeonfliesfromChucktoDean,whatisthe

pigeonsvelocitywithrespecttoearth?

Question1.7: Whatisthisnewpigeonsvelocitywithrespecttoearthonthereturn

flightfromDeantoChuck?

Question1.8: WhatisthevelocityofXwithrespecttoX,VXX?

Strangequestion.Buttheanswerleadstoausefulidentity. ThevelocityofXwithrespecttoX

isalittlevague. Tobemorespecific,whatisthevelocityofthebuswithrespecttothebus? The

busisnotmovingwithrespecttothebussoitsvelocity=0. VXXisbydefinitionzero. ButVXX=

VXY+VYX=0whichleadsnaturallyto,

VXY= VYX.

Thisresulthasalreadybeenusedwhenitwaspointedoutthatifthebustravelswithavelocity

of+30ft/swithrespecttoearth,VBE,thenthebusridersseeearthtravelingpastthemwitha

velocityof30ft/s,VEB. Thisargumentreaffirmsthatintuitivelyreasonablenotion.


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17

APictureisWorthaThousandWords

ChuckandDeanareimpressedwiththeconclusionsreachedbyAnneandBev. Nowitistheir

turntosharetheiranalysisoftheexperimentswithA&B. C&Darenotasanalyticallyskilled

astheirfriends,sotheyworkedonawaytovisualizetheexperimentsdonewiththebusand

pigeon.

ChuckshowsA&Bhiseffortatdrawingapigeon,figure1.1. WhenA&Blookat

figure1.1,theybegintogiggle. ChucktellsA&BthatthepreviousnightDeanhad

burstoutlaughingwhenhelookedatthepigeon. Afterhestoppedlaughing,Deantold

Chuckthatthathissocalledpigeoncouldhavebeenahummingbirdorahoneybeefor

thatmatter. AtfirstChucksfeelingswerehurtbyhisfriendscomment,butthenhe

explainedtoA&B,thatDeansremarkledtoamajorbreakthroughinhiseffortsat

visualizingtheexperiments.

ChucktellsA&Bthattheactualexperimentwould

havebeenthesameifitwasdonewitha

hummingbirdorahoneybeeinsteadofapigeon.

Allthatwasneededwassomethingthatflewfrom

thebacktothefrontofthebus. Infact,thebus

couldhavebeenreplacedbyanRVorarailroadcar

movingat30ft/s.

BevandAnnebothappearabitperplexedbyChucksstatement. Chuckcontinuesby

pointingoutthatinanexperimentwithahoneybeeinabus,thehoneybeewouldbeso

muchsmallerthanthebusthatitcouldnothavebeendrawnwithanydetail. Instead,

thehoneybeewouldappearasameredotmovingfromoneendofthebustotheother.

FromtheperspectiveofC&Daspassengersonthebus,theexperimentisevenmoresimple.

Theyseeapigeonflyacrossthebuswhiletheyremainstationaryinsidethebus. Chuckstaysat

therearofthebusandDeanatthefront. Fromtheirbuscentricperspective,itisA&Bthatare

movingpastoutsidethebus. Therefore,fromtheirperspective,A&Bhadnothingtodowiththepigeonsflightinsidethebus.

Withthesesimplificationsinmind,ChuckshowsA&Bthefollowingsketches,figures

1.2a,1.2b,and1.2c:


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18

Chuck Dean

Figure1.2a

Thatismeontheleftwiththepigeon,thereddot,andDeanis100feettomyright.

Thebluelinerepresentsthebus. Noticethatthepigeonhasnotbeenreleasedyet. One

secondafterIreleasethepigeon,thenewsituationcanberepresentedbyananalogous

picture,figure1.2b:

Figure1.2b

Thepigeonisnow20feettomyright. Ofcourse,wehavenotmoved. Ourpositionsarefixedwithrespecttothebus. Twosecondsafterreleasingthepigeon,itflies40feet.

Afterfiveseconds,Deancatchesthepigeonatthefrontofthebus. Figure1.2c

representsthatsituation.

Figure1.2c

ThesesketcheshelpedDeanandIvisualizethekeyfivesecondsoftheexperiment. A

&Bnodandagreethatthosesketchesdoagoodjobofrepresentingtheflightofthe

pigeononthebus.

DeanthenexplainstoAnneandBevthatheandChuckwantedtoconsolidatetheseries

ofsketches,eachofwhichrepresentedaparticularinstantoftime,intoasinglepicture

showingthemotionofthepigeonthroughbothspaceandtime.

HedescribestoA&Bhowhehunchedoverhissketchpadforafewminuteswhile

mumblingaboutcapturingtheessence,nothingbuttheessenceoftheexperiment. He

finallydecidedthatthesmilingfacesinfigures1.2athrough1.2carecutebutnot

essential!

HeshowsA&Bfigure1.3,theconsolidateddiagramshowingtheflightofthepigeon

throughspaceandtime.


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19

Chuck Dean

Figure1.3

Annecomments,Whenyousaidessence,youwerentkidding. Iseefourdifferent

colorarrows;black,blue,red,andyellow. Iassumetheyeachstandforsomething

essential. Please,oneofyou,explainhowthispicturecapturestheflightofthepigeon

throughthebus.

DeandeferstoChuckwhobeginshisdescription:

ThediagramshowsthespaceoccupiedbymeandDean,inthiscasethebus,along

thehorizontaldirection. Thehorizontalblackarrowisthespaceaxis. Iamrepresented

bytheverticalbluearrowandDeanbytheyellowarrow. Weareseparatedby100feet

ofspace,thelengthofthebus. AlloftheimportantinformationaboutmeandDeanare

containedinthosetwolines. Theyshowwherewearewithrespecttothebusduring

thecriticalfivesecondsofthepigeonsflight. Consequentlywedecidedtocalltheblueline`ChucksWorldlineandtheyellowline`DeansWorldline. Thelinesarevertical

becauseneitherofusismovinginsidethebus. Forexample,ifmypositioninthebusis

labeledXbus=0thenDeanislocatedatXbus=100feet.

Theslantedredlinerepresentsthepigeonandisthepigeonsworldline. Whenthe

experimentbeganattimezero,T=0forshorthand,thepigeonwasattherearofthebus

withme. Thatisthepointwherethepigeonsworldlineintersectsmyworldline. When

Iletthepigeongo,thepigeonfliestowardDean. ThepigeonreachesDeanin5seconds

atthespotwheretheirworldlinesmeet;theplacewheretheredlinehitstheyellowline.

Justasspaceisrepresentedonthediagraminthehorizontaldirection,timeadvancesin

theupwarddirection. Thediagramsimultaneouslyshowsthespaceandtimelocation

ofme,Dean,andthepigeon.


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20

Chuck Dean

Figure1.4

Imaginethesituation2.5secondsafterthepigeonisreleased. IamatXbus=0andDean

isatXbus=100feet. Thepigeonflyingat20ft/sishalfwaybetweenusatXbus=50feet.

Thatinstantintimeiscapturedbythegreenhorizontallineintheabovefigure. That

lineshowstheexperimentalsituationexactly2.5secondsafterthereleaseofthepigeon.Theintersectionofthat2.5secondlinewithmyandDeansworldlinesshowsour

locationinthebusatthattime. Sincewearenotmoving,thatinformationisself

evident. Ontheotherhand,thepigeonsworldlineisslantedbecauseitismoving

throughthebus. The2.5secondlinehitstheredlineatauniqueplaceinthebus,Xbus=

50feet. Sothe`distancebetweenthehorizontalblackline,T=0,andanyother

horizontallinedrawnaboveit,representstheamountoftimethathaspassedsincethe

startoftheexperiment.

ChuckstopstalkingandlooksatAnneandBev. Bevspeaksfirst,SoifIunderstand

whatyouaresaying,thehorizontalblackarrowmarkspositionwithrespecttothebus.

YouareatXbus=0andDeanisatXbus=100feet. Butthatlineonlyrepresentsyour

locationatthestartoftheexperiment. Atthispoint,AnneremindsBevthatthe

experimentstartsatT=0;therefore,thehorizontalblacklineistheT=0lineandthe

horizontalgreenlineistheT=2.5secondline.

Bevcontinues,SoinFigure1.4,theforwardmarchoftimeisrepresentedbya

horizontalline

that

continuously

slides

upward.

She

grimaces

alittle

before

proceeding. Soatanyparticulartime,Deanisatthespotonthediagramthat

correspondstotheintersectionofa`timelinewithhisworldline. Smilingsheadds,I

canalmostpictureDeansmotionthroughtimeasasmilingfaceslidinguphis

worldline.


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21

Thesespaceandtimediagrams,orspacetimediagramsforshorthand,aregoingtoplaya

centralroleintheanalysisofspaceandtimeinlaterchapters. Taketimetobecomecomfortable

withtheideathatasinglespacetimediagramcapturestheflowofeventsthroughtime. Infigure

1.3,theblueline,Chucksworldline,representshismotionthroughspaceandtime. Heisnot

movingthroughthespaceofthebussincehestaysattherearbuthecannothelpbutmove

throughtime. DeansworldlineiscompletelyanalogoustoChucks. Ontheotherhand,the

pigeonismovingthroughbothspace,fromthereartothefrontofthebus,andtime,thefive

secondsittookthepigeontotraversethelengthofthebus. Sinceitisimpossibletostoptime,the

variousactorsdepictedonaspacetimediagramwillneverbecompletelystationary.Atthevery

least,theywillbemovingupthespacetimediagram,goingfromearliertolatertimes.

AnneandBevacknowledgethat,whilethespaceandtimediagramdrawnbyChuck

capturestheessenceoftheexperimentfromtheperspectiveofChuckandDean,they

sawasomewhatdifferentexperiment. Fromtheirperspective,thebus,withC&Dand

thepigeonridingalong,rollsdowntheroadat30ft/s. C&DarepreparedforA&Bs

commentandafterasecondortwoofsearching,showthemfigure1.5.

Anne ChuckBev Dean

Figure1.5

Chuckexplainsthat,asbefore,thehorizontalblackarrowrepresentsT=0,thestartof

theexperiment. Butitnolongershowslocationsonthebusbutinsteadshowswhere

AnneandBevarewithrespecttoearth. TheheavyblueandyellowlinesareAnneand

Bevsworldlines. Thoselinesareverticalbecausetheyarenotmovingwithrespectto

earth. Theworldlinesofthepigeon,Dean,andhimarethesameasbefore,red,yellow,

andblue. Thoseworldlinesareallslantedbecausetheyaremovingwithrespectto

earth. TheintersectionofeachworldlinewiththeT=0linegivesthelocationofthat


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22

participantintheexperimentwithrespecttoearthatT=0;Anne,Chuck,andthe

pigeonareatXearth=0,DeanisatXearth=100feet,andBeviswaitingatXearth=250feet.

Question: 1.9: ImaginedrawingtheT=1secondlineonfigure1.5. Atthatinstant,

what

are

the

Xearth

locations

of

Anne,

Chuck,

the

pigeon,

Dean,

and

Bev?

Question1.10: Makeasketchanalogoustofigure1.3thatshowstheworldlinesof

Anne,Chuck,thepigeon,Dean,andBevfromtheperspectiveofthebusobservers.

RememberC&Dseetheearthrollingbyat 30ft/sandthehorizontalblacklineon

thisspacetimediagramrepresentslocationswithrespecttothebus,Xbus,insteadof

Xearth.

ChuckasksAnneandBevwhattheythinkofthespacetimediagrams. A&B

immediatelyacknowledgethatthediagramsreallyhelptovisualizetheexperiment

withthebustravelingdowntheroadwhileapigeonfliesfromoneendofthebustothe

other. Withrelativevelocitiesandspacetimediagramstothinkabout,theydecideto

callitaday;averyproductivedayatthat.

Onthewalkhome,Annewondersaloudifthereisaconnectionbetweenthespacetime

diagraminfigure1.5andequation1.4, ,therelativevelocityequation.Bevshrugsindifferently,andcontinueswalkinghomeatabriskpace.

TheGalileanTransformationEquations

Thismaybeagoodplaceforreaderswhohavenotthoughtmuchaboutgraphsrecentlytoreview

AppendixA,aprimerongraphs. Theappendixhasashortdiscussionoftheequationy=mx+b

whichrepresentsastraightlinewithslopemandyinterceptb. Understandingtherolemandb

playwillbeusefulinlaterchaptersandbecomingcomfortablewithgraphswillpayimmediate

dividendsinthischapter.

Whentheyarrivehome,Annereproducesfigure1.5andstaresatitforawhile. She

pointsouttoBevthatthediagramisreallyagraphthoughnotthetypicalxversusy

graphseeninalgebraclasses. Annesays,Ifwethinkofthediagramasagraph,theblackhorizontallinethatChuckcallsT=0isalsotheXearthaxis. Bevnods,andAnne

continues,Myworldlineonfigure1.5istheXearth=0lineandalsothetimeaxisofa

graph. SoBev,Chucksspacetimediagramsareactuallyxversustgraphs.

Toemphasizethisnewperspective,Annemakesthefollowingsketch,figure1.6:


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23

Figure1.6

SheexplainstoBevthatthisisthestartofaspacetimegraph. Thehorizontallineisthe

xaxisandtheverticallineisthetaxis. Theaxesintersectattheoriginwherex=0andt

=0. Bev,knowingthatAnneisoffandrunning,triestoslowherdownbypointingout

thatthexaxisisalsothet=0lineandthatthetaxisisthesameasthex=0line. Of

course,saysAnne,butnoticethatthegraphrepresentsourentireonedimensional

universe!

Annecontinuesherexplanation,Whensomethinghappens,forexamplethepigeon

getscaughtbyDeanridingatthefrontofthebus,thatsomethingiscalledanevent.

Eventshappenataparticularspotandaparticulartime. Deancaughtthepigeonatthe

spacetimepointXearth=250feetandT=5seconds. Thateventhasauniquelocationon

thespacetimegraphgivenbytheintersectionoftheXearth=250footandT=5second

lines. Anyeventinthepastorfuturethathappenedorwillhappenisrepresentedbya

pointonagraphlikefigure1.6.

Bevstareswithrenewedinterestatthecosmicscopeofinnocentlookingfigure1.6.

AnnegivesBevalittletimetoreflectbeforeadding,Thexaxisinfigure1.6couldbe

eitherXearthorXbusdependingonwhothestationaryobservershappentobe. Oncethat

decisionismade,itispossibletoaddtheworldlinesofthevariousparticipantsinthe

experimentandtodeterminehowlargeasliceofspaceandtimeisnecessarytohave

thoseworldlinesfitonthegraph. Forexample,iffigure1.6isdrawnfromourperspective,thespaceslicehadtoincludethe250feetseparatinguswhilethetimeslice

required5secondsforthepigeontoflyacrossthebus. Ontheotherhand,only100feet

ofbusspacewasneededforthespacetimediagraminfigure1.3.

AnneaddstheworldlinesofChuckandDeantoherspacetimegraph,figure1.7,and

remindsBevthatChucksworldlinecorrespondstotheconstantXbus=0linewhile


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24

DeanstotheconstantXbus=100footline. Butthoseworldlinesareslantedonour

spacetimegraphbecausethebusismovingthemat+30ft/s.

Anne Chuck Dean

Figure1.7

Question

1.11:

Fill

in

the

following

table

showing

how

Xearth

for

Chuck

and

Dean

changewithtime.

Time 0s 1s 2s 3s 4s 5s

XearthforChuck

XearthforDean

ChucksworldlineisdescribedbytheequationXearth=30TfeetwhileDeansisgivenbyXearth=

100+30Tfeet. NotethatwhenT=0,ChuckisatXearth=0whileDeanisatXearth=100feetjust

asexpected. ButXearthforeachofthemincreasesby30feeteachsecond,thatincreaseisthe30T

termintheequationsfortheirworldlines. Theanswerstoquestion1.11oughttoagreewith

thoseequationswhenT=0,1,2,3,4,and5seconds.

Bevnowtakesovertheconversationbymakingthefollowingobservation; Every

locationinthebusreferenceframeisgivenbysomevalueXbuswhichcanbedrawnasa

constantXbuslineofourspacetimegraph,figure1.7. Onyoursketch,youdrewthe

particularconstantXbuslinesfor0and100feetandthoseslantedlinescrossedtheXearth

axis,theT=0line,at0and100feetrespectively. Asyoupointedoutearlier,anypoint

onourspacetimegraphcanbedescribedbytheuniquepairofvaluesXearthandT. For

exampleDeancaughtthepigeonatXearth=250feetandT=5seconds. Butthatpointis

alsouniquelydeterminedbytheintersectionoftheXbus=100feetandT=5second

lines.

Anneagrees,andwritesdownthemoregeneralequation,


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25

Xearth=Xbus+30T, (1.5)

whereXearthshowshowanyconstantbuslocation,Xbusmovesthroughtheearthframe.

The30inthatequationisjustthevelocityofthebuswithrespecttoearth. Equation1.5

connects

the

earth

based

coordinates,

Xearth

and

T,

to

the

bus

based

coordinates,

Xbus

and

T,foranyevent.

Question1.12: Apassengersittinginthemiddleofthebus,Xbus=50feet,seesthe

pigeonpassatT=2.5seconds. Whatistheearthcoordinate,Xearthforthatevent?

Question1.13: WhenDeancatchesthepigeon,Anneclapsherhandsinglee. Whereis

thebusobserverwhoisadjacenttoAnnewhensheclappedherhands? (Anneis

locatedatXearth=0andthepigeongetscaughtatT=5.)

ThebusislocatedbetweenXbus=0andXbus=100feetbutweneedtobeabletoimaginebus

observersoutsideofthatrangebecausesomeexperimentswillrequirebusobserverswhoarenot

actuallyonthebus! ThebusobserverwhoisadjacenttoAnnewhensheclappedisriding150

feetbehindthebusatXbus= 150feet!

Thebusispushingobserversalonginfrontwhohavebuscoordinatesgreaterthan100feetand

isdraggingothersbehindwithbuscoordinateslessthanzero.AsChuckbroughtupearlier,the

importantthingaboutthebusobserversisthefactthattheyaremovingat+30ft/swith

respecttoearthobservers. Insteadofridingona100footlongbus,wecouldhavehadthem

ridingonamilelongstretchofrailroadcars. Thatwouldhavegivenourmovingobserversa

longerbitofspaceonwhichtoarrangethemselves. Butsinceimaginationisaprerequisitefor

makinganysenseofSpecialRelativity,wewillsticktoour100footlongbuswithobservers

beingpushedorpulledalongasnecessary.

Anneisnowreadytothinkaboutthepigeonflyinginthebus. ThepigeonstartsatXbus

=0andflieswithavelocityofVPB,whereVPBfortheexperimentwedidwas+20ft/s.

ThelocationofthepigeonwithrespecttothebusisjustXbus=VPBT. Analogously,the

positionof

the

pigeon

with

respect

to

earth

isjust

Xearth

=VPE

T.

Rememberthepigeon

startedflyingatXearth=0. NowshereplacesXearthandXbusinequation1.5withVPETand

VPBTtoget,

VPET=VPBT+30T. (1.6)


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26

Dividingequation1.6byTgivesVPE=VPB+30. Butthe30inthatequationisjustthe

velocityofthebuswithrespecttoearth,VBE,

VPE=VPB+VBE. (1.7)

BevisdulyimpressedwithAnnesdemonstrationoftheconnectionbetweenthe

additionofvelocityequationandspacetimegraphs. ShegivesAnneawellearnedhug.

Nowtheyarebothreadytocallitanightthoughtheylookforwardtosharingtheir

newinsightsintospaceandtimewithChuckandDean.

InthegeneralcaseofabusmovingwithavelocityVBEwithrespecttoearth,equation1.6

becomes,

Xearth=Xbus+VBET. (1.7)

Thatequationconnectstheearthbasedcoordinates,XearthandT,ofanypointonspacetimegraph

withthebusbasedcoordinates,XbusandT,forthesamepoint.Equation1.7iscalledtheGalilean

Transformationequationbecauseittransformsearthbasedintobusbasedcoordinates.

Technicallytheextraequation,Tearth=Tbus=T,isneededtocompletetheGalileantransformation

ofcoordinates. Buttheuniformityoftimewassoembeddedinhumanconsciousnesspre1905,

thatincludingitseemedredundamt.

SummaryofChapterOne

Theimportantthingslearnedbyourfourfriendsarelistedbelow:

ThePrincipleofRelativityandtheIsotropyofSpace

Noexperimentcandifferentiatebetweenthelabonthebusmovingatconstantvelocityandthe

labattachedtoearth. Resultsofanexperimentareindependentoftheorientationofthe

apparatususedtodotheexperiment. Thispairofconclusionswastestedbymeasuringthe

velocityofthepigeonflyingfromlefttorightinstillairwithrespecttoearth,nodifference,and

comparingthoseresultstothevelocityofthepigeonflyinginthebusfrombacktofrontandthen

fromfronttoback. Thetwosetsofexperimentsgaveexactlythesameresults.

SpaceandTimepre1905

ThespaceandtimeofGalileo,Newton,andallphysicistspriorto1905isaccuratelysummarized

bytheexperimentsandconclusionsreachedbyAnne,Bev,Chuck,andDeaninthischapter.

TherelativevelocityequationderivedbyAnne,VPE=VPB+VBE,Chucksspacetimegraphs,and


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27

thetransformationequation,Xearth=Xbus+VBET,arethreeseparatebutequivalentwaysof

characterizingthespaceandtimeofthisera.

ChapterTwo: TheSpeedofLight

Thespeedoflightwasmeasuredinaseriesofexperimentsaround1850byHippolyteFizeau,a

Frenchphysicist. Thehistoryofeffortstomeasurethespeedoflightisworthpursuing,but

doingthatnowwouldbeadistraction. Theprimarythingtounderstandisthatthespeedof

lightisvery,verylarge. Lighttravelinginavacuumcovers300,000,000(3x108)metersina

secondorequivalently186,000(1.86x105)milesinasecond.AsDavidMerminpointsoutin

hisdelightfulbookItsAboutTime,thisspeedisverycloseto1footinananosecond(ns),abillionth(109)ofasecond. (Forthoseofuslivinginametricchallengedsociety,thatcoincidenceissofortuitous,thatIhavedecidedtousefeetinsteadofmetersasthestandardof

distanceinthisbook. Theactualvalueforthespeedoflightinfeetis0.98ft/ns.Amere

differenceof2%wasnotenoughtodetermefromusing1ft/nsforthespeedoflightthroughout

thisbook!)

Notethatsomeofthenumbersintheaboveparagraphwereslylywritteninscientificnotation.

Thoughitisnotnecessarytounderstandscientificnotation,ashortprimeronscientificnotation

isgiveninAppendixBattheendofthebook. Thisappendixoughttobehelpfultopeoplewho

arenotsofamiliarwiththisusefulwayofdealingwithlargeandsmallnumbers.

Aside: Itisveryusefultobeabletochangetheunitsusedtodescribethespeedoflight. For

example,tofindthespeedoflightintermsoffeet/second,startwiththespeedinmiles/second,

186,000miles/second. Usethefactthatonemileisequivalentto5280feet,1mile=5280feet.

Thismeansthattheratio,

1.

Anyexpressioncanbemultipliedbyonewithoutchangingit. Tochangethemilesin186,000miles/secondtofeet,multiplythatspeedby

. Themileunitscancelleavingfeetinits

stead. Nextmultiply186,000by5280toget982million. Thisshowsthat186,000miles/second

isequalto982millionft/swhichwasroundedoffto109ft/sor1ft/ns. InChapterFour,the

usefultrickofmultiplyinganexpressionby1willbeusedtosimplifyalgebraicexpressions.


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Thespeedoflightissolargethatundernormalcircumstancesthetimeittakeslighttotravel

fromheretothereisessentiallyzero. Forexample,theruleforestimatinghowfaryouare

fromalightningstrikestatesthatforeveryfivesecondsoftimebetweenseeingtheflashand

hearingtheresultingthundercorrespondstoaseparationofonemile. Ifyouhearthethunder10

secondsafterseeingthelightning,thebolthit2milesaway. Thisusefulrulecomesfromthefact

thatsoundtravelsatabout1000ft/sandamileis5280feet. Theassumptionisthattheflash

arrivesinstantaneouslysoforeachsecondyoucountoffbetweenseeingtheflashandhearingthe

thunder,thesoundtravels1000feet. Ofcourseifsoundtraveledfasterthanlight,youwould

hearthethunderfirstandseetheflashsecond!

Thetimeittakestheflashfromalightningstriketocoveronemileis1mile/186,000miles/sor

about5millionths(5x106)ofasecond. Obviously,oursensesaretotallyincapableofnoticing

timesthatsmall.

DuringthesameperiodthatFizeauandothersweredoingcarefulexperimentstoaccurately

measurethespeedoflight,ClerkMaxwell,anEnglishphysicistwascodifyingalltheassorted

phenomenainvolvingelectricityandmagnetismintoasetoffourequations,nowknownas

Maxwellsequations.Atthattime,noone,includingMaxwellexpectedtheretobeany

connectionbetweenlightandelectricandmagneticphenomena. Butaround1860,heshowed

thatelectricandmagneticfieldscantravelaslinkedwaves,electromagneticwaves,andthe

predictedvelocityofthesewaveswasgivenintermsoftwowellknownphysicalconstants,the

permittivity, 0,andpermeability, 0,whichhadnoapparentconnectiontolightoritsspeed.(Thepermittivityisaconstantthatshowsupintheequationusedtofindthemagnitudeofthe

electricforcebetweentwopointchargeswhilethepermeabilityisaconstantthatconnectsthe

currentinawirewiththemagnitudeoftheresultingmagneticfieldthatsurroundsthewire.)

ButwhenMaxwellcalculatedthespeedofhiselectromagneticwavesbyusingtheknown

valuesfor 0and 0hefoundthespeedeerilyclosetotheknownspeedoflightpromptinghimto

observe:

Theagreementoftheresultsseemstoshowthatlightandmagnetismareaffectionsofthesame

substance,andthatlightisanelectromagneticdisturbancepropagatedthroughthefield

accordingtoelectromagneticlaws.[1]Thehistoricalevolutionofourunderstandingofthenatureoflightisanothercuriousstorythat

wouldtakeusfarafieldfromourmaingoal. ButnotethatMaxwellspeaksaboutlightbeinga


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disturbancepropagatedthroughasubstance,whichisjustanotherwayofsayinglightisawave

travelingthroughamedium,thelumeniferousether.

Thequestionofwhetherlightisawaveoraparticlealsohasafascinatinghistory. Itturnsout

thatthequestionwasunanswerablebecausethebehavioroflightisparticlelikeundersomecircumstancesandwavelikeunderothers. Insteadoftryingtocategorizelightwithlabelslike

particleorwave,thinkoflightasthestuffthatbehaveslightlike. Inthisbook,pictureaflashof

lightassomethingproducedbyalaserbeingturnedonandoffveryquickly. Forexample,ifthe

laserisonfor1/10ananosecond,theflashis1/10ofafootlong,aboutaninch. Solaserflashes

inthisbookareshortenoughtobeconsideredobjectsthattravelinstraightlinesat1ft/ns.

Forexperimentsdoneoverdistancesof100feetorso,aflashoneinchlongisshortenoughto

qualifyasaparticleoflight.

Onelastpoint,in1983scientistsdecidedtodefinethespeedoflighttobeexactly299,792,458meters/second. Thischangedthespeedoflightfromanexperimentallydeterminedquantityto

onethathadafixedvalue. Previously,themeterwasdefinedasthedistancebetweentwoscratch

marksonabarinaParisvault. Thischangewasmadebecausescientistsareabletomeasure

timemuchmoreaccuratelythandistance. Withthisdefinitionforthespeedoflight,ameter

becamethedistancelighttraveledin1/299,792,458ofasecondinsteadofthedistanceonthat

Parisianbar! Thischangehasnoimpactontheconclusionsreachedbyourintrepidexplorersof

spaceandtimeinChapterThree,whentheyexperimentallydeterminethespeedoflight.

1. JamesClerkMaxwell,ADynamicalTheoryoftheElectromagneticField,PhilosophicalTransactionsoftheRoyalSocietyofLondon155,459512(1865).


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ChapterThree: SpaceandTimeAfter1905

ANewSetofExperimentsareProposed

Thespaceandtimeexplorerstookoffforafewdaystoenjoytheirtriumphover

GalileanspaceandtimeinChapterOne. Bytheirthirddayofchilling,Deanwasgettingrestlessandremindedthemthattheexperimentswiththepigeonandbuswere

justprecursorstothemorechallengingexperimentstheyneededtodowiththepigeon

beingreplacedbysomethingmovingatmuchlargerspeeds. Afterall,iftheywantto

understandthespaceandtimeofEinsteintheyneedanobjectmovingatornearthe

speedoflight.

Bevsuggeststhattheycuttothechaseanduseaflashoflightastheobjecttoreplace

thepigeon. Afterall,theywillbehardpressedtofindanythingelsethatcanapproach

thespeedofalightflash. Herfriendsagree.

NewStopWatchesaredistributed

ThestopwatchesusedinChapterOnewillbeoflittleusewhentryingtomeasurethespeedof

light. Rememberthatthosewatchesreadtimesto1/10thofasecond. In1/10thofsecondlight

travels30millionmetersor18,600milesor100millionfeet adistancelargerthan2/3the

circumferenceofearth.

NewstateoftheartstopwatchesaredistributedtoAnne,Bev,Chuck,andDean. Thesenew

watchesareaccurateto1/10thofananosecond(1010seconds). Inordertotakefulladvantageof

thesecuttingedgewatches,theywillneedreactionstimesof1/10thananosecondorbetter!

Althoughrealpeopleorstopwatchesforthatmattercannotbethisaccurate,itiseasyenoughto

imaginewelltrainedgeneticallyengineeredobserverswithextremelyaccuratestopwatches.

Thosepeoplewiththeaforementionedstopwatchesaregoingtobedoingexperimentsthrough

muchoftherestofthebook. TheexperimentstheyperformareintheparlanceofEinstein,

gedankenexperiments,orthoughtexperiments. Thoughtheexperimentscannotbedoneas

describedbecauseofhumanandtechnologicallimitations,theresultsoftheexperimentsare

completelyconsistentwiththecurrentscientificunderstandingofspaceandtime.

AnneandBevMeasuretheSpeedofLight

AnneandBevrequisitionalaserthatcanbeturnedonandoffin0.1ns. Thelightflash

producedis0.1footlong,oraboutoneinch. Theirplanistomeasurethespeedofthe

lightflashinmanneranalogoustothemethodusedtomeasurethespeedofthepigeon.


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ConsequentlyBevwalkstothe100feetmarkerontheroadwhileAnneremainsatthe

zerofootmark. BevstaresintentlyatAnneandisreadytostartherwatchassoonas

sheseesAnnepressthebuttononthelaserthatwillstarttheoneinchbitoflight

zoomingtowardher. Sheisstandingnexttoascreenthatwillflashtheinstantthelight

arrivessignalinghertostopherwatch.

AnnepressesthebuttonandhearsBevexclaim,RATS,Imustnothavebeen

concentratingintentlyenough. BevpullsherselftogetherandtellsAnnesheisready.

AnnelaunchesanotherlightflashandBevscreams,DOUBLERATS!

AnnewalksovertoBevtofindoutwhatiscausinghersuchconsternation. Bevtells

Anne,BothtimesthelighthitthescreenatthesameinstantthatIsawyoupressthe

button. IttookNOtimeforthelighttotravel100feet.

AnneremindsBevthatlighttravels1footinananosecond. Thereforeitoughttotake

light100nstocoverthedistancebetweenus. ShesaystoBev,Wehavebeenendowed

withextraordinaryreflexesandhavetheverybeststopwatchesavailablethatallowus

tomakemeasurementsaccurateto0.1ns. Thereisabsolutelynoreasonwhywecannot

dothissimpleexperiment!

Chuck,whohasbeenstandingbywatchingquietly,suddenlyyellsout,Experimenting

withlightisgoingtobetrickierthanwethought. Anne,Bev,andDean

simultaneouslylookatChuckwaitingforhimtoelaborate. Chuckishappytoobligehisfriends;BeviswatchingAnneandwaitingforhertopressthebuttonthatsends

thelightflashonitsway. ThereasonBevcanseeAnneisbecauselightistraveling

fromAnnetoBev. Ifitwerepitchblackoutside,Bevwouldnotevenbeabletosee

Anneletaloneseeherpressthebutton. TheimageofAnnestartingthelaserpulseon

itswaytravelstoBevatthespeedoflight,thesamespeedasthelightflash. Therefore

Bev`seesAnnepressthebuttonatthesameinstantthelighthitsthescreen.

Thefourfriendssitdownatapicnictableandmulloverthisdifficultyinexperimenting

withobjectsmovingatlightspeeds. Chuckpointsoutthatthesamesituationwas

takingplaceduringtheexperimentwiththepigeon. TheimageofAnnereleasingthe

pigeontook100nstoreachBev. Sounderthebestofcircumstances,Bevwasstarting

herstopwatch100nslate. Butthatlatestarthadnodiscerniblebearingonthetimeshe

measuredforthepigeontoflyfromAnnetoherbecause5seconds100nanoseconds=

4.9999999seconds. Thistimeappearedas5.0secondsontheiroldwatcheswhichwere


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onlyaccuratetotenthsofasecond. Butthiserrorwillbeaproblemanytimetheytry

toapplythismethodtoobjectsmovingatornearthespeedoflight. Chucksuggests

thatinallfutureexperimentseachofthemonlypayattentiontoreadingsontheirwatch

andtoeventsinthespaceimmediatelyadjacenttothemtoavoidthelighttraveltime

problem.

MeanwhileAnnehasbeenpacingaroundandmullingoverthetimingproblem. She

looksatherwatchandstartsandstopsitafewtimes. Finallyshecallsherfriendsover

tomakethesuggestionthatsavestheday. BevandIneedtostartourwatchesbefore

thestartoftheexperiment. Wecandothisbystandingsidebysidewhenwestart

them. Thiseliminatesthelighttraveltimeproblem.Afterstartingthem,weambleoff

toourstations100feetapartjustlikebeforeexceptnowwehaverunningwatchesthat

havebeensynchronized. AttheinstantIpressthebuttononthelaser,Isimultaneously

readthetimeonmywatch,TAnne. ThenBevrecordsthetimetheflasharrivesasTBev.

Afterwardswecalculatethetimedifference,TBevTAnne,whichisjustthetimeittookfor

theflashtotravel100feet. ChuckthinksAnnesideaisbrilliant.

Inthefuture,observersintheearthorbusframeswillrecordthetimethateventshappenat

theirfixedlocationinspace.A&BandC&Dwillsynchronizetheirwatchesusingthemethod

describedbyAnne. Beforethestartofanyexperiment,A&Bwillmeettosynchronizetheir

watches. C&DwillmeetonthebusasittravelstowardsA&Btosynchronizetheirwatches.

Alltheobserverswillhavewatchesthatarerunningbeforethestartofanyexperiment.Duringtheexperiment,eachobserverwillberesponsibleforrecordingthetimethateventshappeninhis

orherneighborhood.Aftertheexperimentisover,theselocalmeasurementswillbesharedto

formacomprehensiveviewoftheeventsthattookplace.

NowAnneandBevredotheexperimentwiththelightflashtravelingfromlefttoright.

Whentheycomparestopwatchtimes,theydiscoverthatTAnneTBev=100nsjustlike

expected. AnnegivesthelasertoBevandtheyrepeattheexperimentwiththelight

travelingfromrighttoleft. Bingo,againthelighttook100nstotravel100feet. Sothe

speedoflightwasthesamewhetherittraveledfromAnnetoBevorBevtoAnne.

Inthelate19thcentury,MichelsonandMorley,Americanphysicists,didverycareful

experimentstodeterminethespeedofearthwithrespecttothelumeniferousether. Thebasic

ideawasthatasearthorbitedthesun,itsvelocitywithrespecttotheetherwouldchange.

Thereforeatanygiventime,fromtheperspectiveofastationaryearth,therewouldbeanether


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windblowinginsomedirectionoverearthssurface. Consequently,MichelsonandMorley

expectedtomeasuredifferentspeedsforlightdependingonhowlightwastravelingwithrespect

totheetherwind. RememberA&Bdidanalogousexperimentswhentheymeasuredthespeedof

thepigeonflyingwithandagainstthewindthatblewfromAnnetowardBev.

MichelsonandMorley,throughaseriesofcarefulexperimentsstretchingoveryears,failedtosee

anyeffectonthespeedoflightduetothemotionofearththroughtheether. Fromtheirresult,it

appearedthatnatureconspiredtomaketheexistenceoftheetherimmunetoexperimental

verification.Asotherphysicistsinventedvariousschemestoexplainthisfailuretodetectthe

ether,Einsteindecidedonamoreradicalsolution. Hebanishedthelumeniferousethertothe

dustbinoffailedideas. Hisexplanationwasthatlightpropagatedthroughemptyspacewhich

madeitverydifferentandmuchstrangerthanalltheothersortsofwavesscientistshad

previouslyencountered,allofwhichtraveledthroughamaterialmedium.

TheSuperBusRollsintotheStory

ThebususedbyChuckandDeaninChapterOneiswoefullyinadequatetohelpduring

experimentswiththelaser. Ittakeslight100nstotraversethelengthofabus100feetlong.

Duringthetimeittakestheflashtomove100feet,thebus,movingat+30ft/swithrespectto

earth,wouldcover30ft/stimes100ns(107seconds)=30x107feet=3x106feet(3millionths

ofafoot!). Fromtheperspectiveoflight,abusmoving30ft/sisstationary!

Ifourintrepidexplorersofspaceandtimewanttostudythespeedoflightina

referenceframemovingwithrespecttoA&Bonearth,themovingframewillhaveto

haveaspeedcomparabletothespeedoflight. Consequently,ChuckandDeanordera

newsuperbuscapableofzippingsmoothlydowntheroadatconstantspeedsuptothespeedoflightor0.9ft/ns. Thisnewbusisexactly100feetlongjustliketheoriginalone. C&DborrowthelaserfromA&Bandtradeintheiroldstopwatchesfor

thenewmoreaccurateones.

Theoldrulersdidnothavetobeupgradedsincetheywereandarestillgoodenoughtomeasure

thelengthofthebusaccurately.

Question3.1: A&Baresurprisedonedaytoseearoguebuscomerumblingdownthe

roadtowardsthem. Theyseethebussoonenoughtobeabletosynchronizetheir

watchesandstationthemselvesalongtheroad. Thebuspassesthemandcontinues

downtheroadoutofsight.


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DescribeamethodthatA&Bcanusetomeasurethelengthandvelocityoftherogue

busasitspeedsthroughtheirreferenceframe. Thisquestionwillbeaddressedlaterin

thischapterinthesectionentitledTheAffectofMotiononSpace.

The

plan

for

the

bus

based

experiment

is

straightforward.

C

&

D

will

drive

down

the

roadatthespeedoflight,0.6ft/ns. Whilemovingataconstantvelocitywithrespecttoearth,ChuckandDeanwillmeasurethespeedoflightinsidethebus;firstforaflash

goingfromChucktoDeanandthenagainforaflashgoingintheoppositedirection,

fromthefronttotherearofthesuperbus. Theywillsynchronizetheirwatchesbythe

samemethodusedbyAnneandBev. A&Bsitdownandwatchasthebusheadsdown

theroad,turnsaround,andcomesrumblingbywithC&Dinsidemeasuringthe

velocityoflight. Innotimeflat,C&Darebacktosharetheresultsofthebusbased

measurements

of

the

speed

of

light.

ChucksummarizestheirresultsforA&B,Whenthelaserflashtraveledfromtherear

ofthebustothefront,Inotedthetime,TChuck,whenIpressedthebuttononthelaser.

ThenDeanwrotedownthetime,TDean,thattheflasharrivedatthefrontofthebus.

Usingthosetimes,wefoundthetransittimeforlight,TDeanTChuck=100ns. Thenwe

repeatedtheexperimentforaflashmovingintheoppositedirectionandgotan

identicaltimedifference,100ns. Soforlightmovingineitherdirection,thespeedof

lightwasthesameonthemovingbusasitwasonearth.

Nooneissurprisedbythisresultsinceitisconsistentwiththeresultstheygotusingthe

pairofpigeonsinChapterOne. Thespeedofthepigeonwasthesameonthebusasit

wasflyingbetweenA&Binthepark. Alsothespeeddidnotdependonwhetherit

wasflyinglefttorightorrighttoleft. Ourfourfriendsdecidetocallitquitsfortheday,

satisfiedthattheyaremakinggoodprogressinunderstandingspaceandtime. Before

headinghome,theyagreetomeetearlythenextmorningforanotherroundof

experiments.

ThefactthattheexperimentonthebuswiththelightflashreproducedtheresultsoftheexperimentdoneonearthisanotherconfirmationofthePrincipleofRelativity.Alsosincethe

speedoflightinbothcasesdidnotdependonthedirectionofthelightflash,theseexperiments

addcredencetothenotionthatspaceisisotropic.


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Thenextday,AnnewantsChuckandDeantoredotheexperimentwiththelaseronthe

bus. Butthistime,sheandBevwillalsocollectdataasthelightflashtravelsfromthe

reartothefrontofthemovingbus.

Chuck,

thinking

about

his

spacetime

graphs,

points

out

that

this

is

a

more

complicated

experimentbecauseitinvolvesobserversintwodifferentreferenceframes. Thetwo

experimentsdonethepreviousdayonlyinvolvedobserversinasinglereferenceframe:

firstA&BonearthandthenheandDeaninthebus.

Becauseofthisaddedcomplication,Chucksuggeststhattheycarefullygooverthe

detailsoftheexperimentbeforeactuallyhavingthebuszipbyat0.6ft/ns. Afterall,it

takesquiteabitoffueltogetthebusuptothatspeedsotheyoughttomakesurethey

getitrightthefirsttimetheytry.

AnneandBevObservetheLightFlashontheBus: I

Chuckwritesdownwhateachofthemwilldoduringtheexperiment. Whenheis

finished,heshareshislistwithhisfriends:

1. AnneandBevwillsynchronizetheirwatchesbeforethestartoftheexperiment.C&DwillsynchronizetheirwatchesonthebuswhileitzipstowardA&Bat0.6

ft/ns. Chuckwillbeattherearofthebus,Xbus=0andDeanwillbeatthefront,

Xbus=100feet.

2. TheexperimentwillbeginwhenAnne,standingatXearth=0,andChuckpassoneanother. Atthatpreciseinstant,ChucklaunchesthelightflashandheandAnne

recordthetimesontheirwatches. ThosetimesareTChuckandTAnne.

3. WhenthelightflashreachesDean,hewillbeadjacenttoBev. Theywillchecktheirrespectivewatchestonotethetimetheflasharrived. Thosetimeswillbe

recordedasTDeanandTBev.

Chucksoutlineoftheupcomingexperimentmeetswithgeneralagreement. Anne

headstotheXearth=0spotwhileC&Dboardthebusandbegintogoovertheircheck

list. Bevbeginstomoveoffdowntheroadbutcomestoaconfusedstop. Thenshe

yellsoutinavoiceloudenoughforeveryonetohear,WhereIamsupposedtostand

sothatIwillbenexttoDeanatthefrontofthebuswhenthelightflasharrives?


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Question3.2: UsingwhatourfriendslearnedaboutspaceandtimeinChapterOne,

howfartotherightofAnneshouldBevstandsothatsheisadjacenttoDeanwhenthe

lightflasharrives?

Chuck

reviews

his

pre

experiment

instructions

and

reluctantly

admits

that

a

crucial

bit

ofinformationwasmissing. Namely,Bevslocationsothatshewillbestandingatjust

therightspotontheroadtobeadjacenttoDeanjustasthelightflashreachesthefront

ofthebus.

AnnepointsoutthatsheandBevhavedonethissortofthingbeforewhenthey

observedthepigeonflyingfromoneendofthebustotheother. Shemumblesto

herself. AccordingtheexperimentsdonebyC&Dthepreviousday,thelightflash

willtake100nstocrossthebus. Duringthose100ns,thebusmoves60feetdownthe

road. Thereforetheflashmovesthelengthofthebusplusanextra60feet,foratotalof160feet. Bev,youneedtobestanding160feettomyright. Bevnodsinagreement.

Afterthiscarefulpreparation,ChuckandDeanboardthebus,drivedowntheroad,

turnaround,andheadtowardA&Breadytobegintheexperiment. Attheinstant

AnneisadjacenttoChuckthelaserflashbeginsitstriptothefrontofthebus. Sheand

Chuckbothrecordthetimesontheirwatches,TAnneandTChuck. Bev,waitinganxiously

160feetdowntheroad,ishorrifiedwhenthebuspassesherbeforethelightflashhas

gottentoDean! Dean,intentlywaitingfortheflashtoarrive,doesnotseethebuszip

byBev. Attheinstanttheflasharrives,helooksoutthewindowexpectingtoseeBev

butinsteadseesthe200footmarkeralongthesideoftheroad!

AnneandBevwaitforChuckandDeantoreturnsothefourofthemcantrytofigure

outwhatwentwrongwiththeexperiment. AssoonasC&Dstepoutofthebus,A&B

askthemwhatwenthappened. DeanlooksbewilderedbytheturnofeventsbutChuck

isreadytodefendtheirexperimentalskill. Chucksays,Fromtheperspectiveofthe

bus,everythingwentperfectly. Thelightflashtookexactly100nstogofromtheback

to

the

front

of

the

bus.

A&BshaketheirheadsanddisagreewithChucksrosyassessmentoftheexperiment.

EverythingcouldnothavegoneperfectlybecauseBev,whowasstandingexactlyatthe

160footmarkerontheroad,sawDeanpassbeforethelightflasharrived! Chucknow

admittedthattherewasonestrangebitofevidencethathehadnotmentioned,namely

thatDeanthoughthewasnexttothe200footroadmarkerwhenthelightflasharrived.


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Impossible,exclaimA&Bsimultaneously! Deandefensivelyasserts,IknowwhatI

saw. AndIalsoknowwhatIdidnotsee. InsteadofBev,Isawthe200footmarker.

Annepointsoutthatthereislittletobegainedbyarguing. Clearly,althoughthese

experiments

are

expensive

to

do,

this

one

has

to

be

repeated.

Chuck

suggests

they

reviewtheinformationgleanedfromtheirfirstexperimentbeforerepeatingit. This

makesgoodsensetoeveryone.

ThefailureofthissimpleexperimentwiththebusandlightflashtoconfirmtheGalileannature

ofspaceandtimemarksthebeginningofthedevelopmentofthespaceandtimeintroducedby

Einsteinin1905. Ourfourexperimentersareabouttoredotheexperimentmorecarefully. In

theprocesstheywillmakesomemeasurementsoftime. Thedatacollectedfromthissingle

simpleexperimentwillbeenoughtodevelopthecompletetheoryofSpecialRelativity.

Whilehisthreefriendsgrababitetoeat,Chuckdrawsthecarefulspacetimegraph,

figure3.1a,whichsummarizestheexperimentfromtheperspectiveofDeanandhim,

thebusriders. TheexperimentbeganatTbus=0whenhepressedthebuttonthatstarted

thelightflashzippingtowardDean,theblueworldlineatXbus=100feetonfigure3.1a.

Exactly100nslater,Deansscreenrecordedthearrivaloftheflash,theredworldline.

Deanisexactly100feettohisright. Everythingaboutfigure3.1amakesperfectsense.

NoticethatonthegraphdrawnbyChuck,

eachboxis20feetlongand20nanosecondshigh. Thismeansthatlighttravelingat1

ft/nsmoves20feet,oneboxtotheright,in20

ns,oneboxup.Allthespacetimediagrams

intheremainderofthebookwillhavespace

andtimescalesthatmaketheslopeofthe

worldlineofalightflasheither+1forlight

movingtowardtherightor 1forlight

movingtowardtheleft.

Oneotherpoint:theexperimentstartswhen

ChuckpassesAnne.Atthatinstant,their

watchesreadTChuckandTAnnerespectively.

Whentheexperimentisover,allthebus

observers,includingChuck,subtractTChuck


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38

fromwhatevertimestheyrecordedforeventsattheirlocationduringtheexperiment. This

adjustmentshiftsthebusstartingtimefortheexperimentfromTChucktozero.Analogously,all

theearthobserverssubtractTAnnefromtheirrecordedtimes. Theseadjustmentsintimemakethe

startingtimeforeachexperimentzeroaccordingtobothsetsofobservers. Thisallowsusto

continuecallingthestartingtimeforeveryexperimentT=0.

Chuck,ontheassumptionthatDean

didseethe200footroadmarker

outsidehiswindowjustastheflash

arrived,decidestoaddtheworldline

forthatroadmarkertothespacetime

graph. Fromtheperspectiveofthe

bus,thatmarkerhadavelocityof

0.6ft/ns. Soittraveled60feet,right

toleft,duringthe100nsittookthe

flashtocrossthebus. Sincethebus

is100feetlong,thatputthe200foot

roadmarker160feettotherightof

Chuck. Theworldlineforthemarker

isshownasthepurpledashedlineonfigure3.1b. Chuckrubshiseyes,frowns,and

mumblestohimself,Thisisveryperplexing.

Question3.3: Fromtheperspectiveofthebusriders,figure3.1b,howfarinfrontof

Chuckisthe200footroadmarkeratthestartoftheexperiment,Tbus=0?

Question3.4: AnneisadjacenttoChuckatthestartoftheexperiment. Atthatinstant,

fromherperspective,howfarinfrontofChuckisthe200footroadmarker?

Theanswerstothesequestions,thoughweird,areassimpleastheyseem!

Anne,Bev,andDeancomestrollingovertoChuck. Whatareyoustaringat?,asks

Dean. Heshowshisthreefriendsfigure3.1bandexplainsthatitcontainsallthedata

collectedfromtheexperiment. Chuckpointstotheworldlineoftheroadmarker,

Noticethatfrommyperspectiveonthebus,thatmarkeris160feetawayatthestartof

theexperiment. ButAnne,whowasrightnexttomewhentheexperimentbegan,

wouldinsistthatthe200footmarkeris200feetinfrontofher! Howcanthesame


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39

markerbe160feetfrommeand200feetfromher? Sincehewasaskingarhetorical

question,hedidnotexpectananswernordidhereceiveone!

FinallyDean,whoactuallysawthe200footroadmarkeroutsidehiswindowwhenthe

laser

flash

arrived

at

the

front

of

the

bus,

suggests

they

redo

the

experiment

with

Bev

200feetinfrontofAnne. A&Bagreewiththatsuggestion. Chuckpointstofigure

3.1b,andremindshisfriendsthatfromtheperspectiveofbusobservers,itsurelooks

likethe200footroadmarkerwasonly160infrontofhim. Theirnextexperimentought

todetermineonceandforallthedistancebetweenthe200footmarkerandheandAnne

whowereadjacentatthestartoftheexperiment.

Thefourofthemstareatthebusandwonderhowtogetabusobserver160feettothe

rightofChuckwhichisactually60feetinfrontofthebus! Deansays,Noproblem,I

canattacha60footboomwithaseattothefrontofthebus.

ChuckandDeanworkonaddingtheboomtothefrontofthebuswithaseatforthe

newobserver. Meanwhile,AnneandBev,scoutaroundforavolunteertorideinthat

seat. Edhappenstobepassingbywithnothingurgentonhisagendaandthinksitwill

beexhilaratingtobepushedalongat60%thespeedoflightinfrontofabus.

C&Dexplainhowthethreeofthemwillmeetinthebustosynchronizetheirwatches.

ChucktellsEd,Onceourthreewatchesarerunning,DeanandIwillstrolltoour

positionsatthefrontandrearofthebus. Meanwhile,youllcrawlalongtheboomtoreachyourseatatXbus=160feet.

Edstaresincredulouslyattheboom,looksatA&BforconfirmationthatChuckwas

serious,andsighs,Afterourwatchesarerunningproperly,Chuckwalkstotherearof

thebus,Deantothefront,andIcrawlalongthe60footboomtoreachmyseat! C&D

nodinagreement. Beingatrooper,heagreesbutwithlessenthusiasmandmore

trepidationthanhehadbefore.

Anne

and

Bev

Observe

the

Light

Flash

on

the

Bus:

II

Thenextmorning,thefiveofthemstartgettingreadyforthererunoftheprevious

daysexperiment. Chuck,Dean,andEdpracticesynchronizingtheirwatchesacouple

oftimessothatEdcangetusedtocrawling60feetalongtheboomtohisobservation

post. C&Dkeepremindinghimthatfromtheirperspective,thebusisstationaryandit

isearththatismovingat0.6ft/ns. Theirexplanationdoeslittletoeasehisanxiety.


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40

Chuckgathershisfriendstogetherandcarefullyreviewsthesequenceofeventsthat

willtranspireduringtheexperiment:

1. Beforetheexperimentbegins,A&Bwillsynchronizetheirwatches. Usingthesame

method,

C

&

D

&

E

will

synchronize

their

watches.

2. WhenAnneandChuckareadjacenttooneanother,Chuckwilllaunchthelaserflashandeachofthemwillrecordthetimesontheirrespectivewatches,TAnne

andTChuck.

3. EdandBevwillrecordthetimeontheirwatchesastheypass,TEdandTBev,1.ChuckremindsBevthatshehastwotimestorecord,hencetheextrasubscript.

4. Deanwillnotethetime,TDean,thatthelaserflashreachesthefrontofthebus.Andifthingsgoaccordingtoplan,BevwillbeadjacenttoDeanwhentheflash

arrives. ShewillnotethetimeforthateventasTBev,2.

WhenChuckisfinished,Annesays,Aswepassoneanother,eachofusoughttonote

thetimeontheotherpersonswatch. Thatwillactasachecktoinsurethateveryone

recordedtheirtimesaccurately. Chucklikesthatideaandaddsittohislist.

Chucksstepbysteprehearsaloftheexperimenthasgottenhisfriendsexcitedand

ready. ThethreebusobserverspileinandstartdrivingawayasA&Byell,bon

voyage. ThenA&Bgetdowntobusiness. Theysynchronizetheirwatches,takeup

theirstationsatXearth=0andXearth=200feet,andconcentrateonrecordingaccurate

timesfortheeventsthathap

getting the measure of einstein’s space and time

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