Class2:Einstein’sPostulate

Inthisclasswewillexploretheremarkableconsequencesthatfollowifthespeedoflightisaconstant,independentofthemotionofsource

andobserver

Class2:Einstein’sPostulate

Attheendofthissessionyoushouldbeableto…

• … describe Einstein’spostulate thatthespeedoflightisthesameineveryinertialreferenceframe

• … recognizetherelativityofsimultaneity:eventsthataresimultaneousinoneframe,arenotsimultaneousinothers

• … applytheconceptsoftimedilationandlengthcontraction,whichrelatetimeandspaceintervalsmeasuredindifferentinertialreferenceframes

• … recallthatthepropertimedifferencebetween2eventsismeasuredinaframewheretheyoccuratthesameplace

Thespeedoflight

Thespeedoflightisconstantinallinertialreferenceframes,independentofthemotionofthesourceand

observer

Newtonmeasureslightspeed𝑐

Torchmovinginwardatspeed𝑣,Newtonwouldmeasurelightspeed𝑣 + 𝑐

• Considerhowclassicalphysicswoulddescribetherelationbetweenspeedsindifferentframes…

• Thisideaisrejectedbyrelativity,whichbeginsinsteadwithEinstein’spostulate(1905):

Thespeedoflight

Thespeedoflightisconstantinallinertialreferenceframes,independentofthemotionofthesourceand

observer[NotethisintheWorkbook]

Einsteinmeasureslightspeed𝑐

Torchmovinginward,Einsteinstillfindsspeed𝑐

Einsteinmovinginward,hestillfindsspeed𝑐

• Asimpleidea,withsomepowerfulandalarmingconsequences!

Afewthoughtexperiments

• It’susefultoexploretheseideasbythoughtexperiments–imaginingsituationswhichmaynotbepreciselyreplicatedinalaboratory,butwhichconfrontthelogicofatheory*

https://www.cartoonstock.com/directory/t/thought_experiment.asp

*Andthereforeusuallygiveyouahead-ache!

Afewthoughtexperiments

• Wewilloftencomparethesethoughtexperimentsfromtheviewpointsofthestandardinertialframes,𝑆 and𝑆′,movingwithrelativespeed𝑣

• Thesereferenceframesareeachkittedoutwiththeusuallatticeofobserversandsynchronizedclocks

𝑆𝑆′

𝑣

Simultaneity

• Atrain(frame𝑆′)istravellingpastaplatform(frame𝑆),andalightbulbinthemiddleofthecarriageisswitchedon

• Thelightpulsespreadsout,andwe’llcomparewhenthelightfirsthitsthefront,andback,ofthecarriage

• Theseoccurrencesare2eventscalled𝐸' (front)and𝐸( (back)

FrontBack

𝑣

Simultaneity

• Wefirstconsiderthesituationasviewedfromthereferenceframeofthetrain (frame𝑆′)

• Thelightpulses,travellingatspeed𝑐, hitthefrontandbackofthecarriageatthesametime

• Theevents𝐸' and𝐸( aresimultaneous inframe𝑆′

(Trainisstationaryinthisframe)

• Nowconsiderthesituationasviewedfromtheplatform(frame𝑆),throughwhichthetrainismoving

• Thelightpulsesaretravellingatspeed𝑐,butthebackofthecarriagehasmovedcloser,andthefronthasmovedaway–sothelightreachesthebackfirst(𝐸( happensbefore𝐸')

Simultaneity

Emittedfromhere

Back Front

Simultaneity

Iftwoeventsaresimultaneousinoneframe,theyarenotsimultaneousinanotherframe

MinutePhysics:https://www.youtube.com/watch?v=SrNVsfkGW-0

• Considerthesetofsynchronizedclocksinframe𝑆′,whicharetickingsimultaneouslyasmeasuredinthatframe

• Thesetickscannotbesimultaneousinanotherframe– sotheclocksareunsynchronizedwhenmeasuredinframe𝑆 !!

Simultaneity

𝑣

TimeDilation

• Forournextexperiment,considerasimple“clock”inframe𝑆′ whichconsistsoflightbouncingbetweentwomirrors

• Eachtickoftheclockisacompleteoscillationofthelight,whichtakestime2𝐿/𝑐 inthisframe

• (Yes,thisisastrangewaytomakeaclock,butthefollowinglogicappliesequallywelltotheticksofyourwristwatch,orthebeatsofyourheart)

𝐿(Travellingatspeed𝑐)

TimeDilation

• Let’snowputtheclockinmotionatspeed𝑣 throughframe𝑆 – what’sthetimeperiod𝑡 inthisframe?

• Thelighthastotravelfurther,sotheperiodislonger

• We’llshowthat𝑡 = ./0× 2

2345/05�

𝐿 𝑣2.𝑐𝑡

𝑣𝑡

2.𝑐𝑡(Travelling

atspeed𝑐)

TimeDilation

Clockticksarefurtherapart,ifrecordedinaframethroughwhichtheclockismoving

[NotethisintheWorkbook]

https://www.kanopy.com/product/escaping-contradiction-simultaneity-relati

Propertime

• Weintroduceherethepropertimedifference,whichhasthesymbol∆𝜏 (𝜏 = theGreekletter“tau”)

• Apropertimedifferencebetween2eventsismeasuredinaframewheretheeventsoccuratthesameplace[NotethisintheWorkbook]

• Intheclockframe,thetickshappenatthesameplaceandareseparatedbyapropertimedifference– inthegroundframe,theseparationisalwayslonger:∆𝑡 = ∆𝜏/ 1 − 𝑣./𝑐.�

Clockmovingatspeed𝑣

Gammafactor

• Thequantity1/ 1 − 𝑣./𝑐.� occursthroughoutrelativityandhasthesymbol𝛾 (theGreekletter“gamma”)

𝛾 =1

1 − 𝑣.

𝑐.�

• 𝛾 isameasureof“howrelativistic”isthesituation

• 𝛾 = 1 for𝑣 = 0,𝛾 → ∞ as𝑣 → 𝑐

• 𝛾 isknownastheLorentzfactorhttps://en.wikipedia.org/wiki/Lorentz_factor

Lengthcontraction

• Forourfinalthoughtexperiment,considerarocketship(frame𝑆′)travellingwithspeed𝑣 between2pointswhichareseparatedbydistance𝐿 ontheground(frame𝑆)

• First,considerthesituationasviewedinthegroundframe𝑺:

• Accordingtogroundobservers,therocketshiptakestime∆𝑡@ABCDE = 𝐿/𝑣 totravelthisdistance

𝑣

𝐿

Lengthcontraction

• Howlongdoesthistriptakeintherocketframe𝑆′?

• Wecanusethetimedilationrelation:∆𝑡 = ∆F2345/05�

• Thestartandendofthetripoccuratthesameplaceintherocketframe,soareseparatedbyapropertimeinterval∆𝜏

• Inthegroundframe,thetimeintervalis∆𝑡@ABCDE = ∆𝑡 = /4

• So,timeinrocketframeis∆𝑡AB0GHI = ∆𝜏 = /4

1 − 𝑣./𝑐.�

𝑣 𝑣

Lengthcontraction

• Whatisthedistancetravelledaccordingtotherocket?

• Thegroundismovingpastthewindowatspeed𝑣,fortime∆𝑡AB0GHI =

/4

1 − 𝑣./𝑐.�

• Rocketpassengersmeasurethattheyhavemovedadistance𝐿AB0GHI = 𝑣∆𝑡AB0GHI = 𝐿 1 − 𝑣./𝑐.� = 𝐿/𝛾

• Thisdistanceislessthantheseparationontheground,𝐿 !!

𝑣 𝑣

Lengthcontraction

Thelengthofanobjectismeasuredtobeshorterinaframeinwhichitismoving

[NotethisintheWorkbook]

https://sciencenotes.org/moving-rulers-shorter-length-contraction-example-problem/

Triptothestars!

AspartofSwinburne’sPhysicsclassintheyear2118,yousuggestaratherambitious3rd yearGrandChallengeprojecttovisitthecloseststarProximaCentauri,located4.2lightyearsfromtheSun.

Yoursupervisor(whomayormaynotknowanyrelativity)complainsthatitwilltakeyouatleast4.2yearstogetthere,sinceyoucannottravelfasterthanlight,butyouonlyhave3monthstofinishtheproject!However,havingrecentlystudiedtimedilationyouknowbetter…

• HowfastwouldyouneedtotraveltoreachProxima Centauriin3months,accordingtoyourownwristwatch?

• Howcan your rocket ship travel 4.2light years in only 3months –doesn’t that involve faster-than-light travel??

• Is it possible tohand in your project back atSwinburne ontime??

Paradoxicalthinking

• Wehavelearntthat“movingclocksrunslow”.Considerclocks𝐴 and𝐵,atrestinframes𝑆 and𝑆′,respectively

• Wehaveanapparentparadox:bothclockscannotrunslow!

Clock𝐴 atrest,“Clock𝑩 runsslow”

Clock𝐴

Clock𝐵

Viewfrom𝑺

Clock𝐴

Clock𝐵

Viewfrom𝑺′

Clock𝐵 atrest,“Clock𝑨 runsslow”

Paradoxicalthinking

• Thisparadoxhasarisenbecausewehavenotbeenverypreciseindescribingthesituation.Let’sdobetter!

• Frame𝑆 observersthink:clockin𝑆′ hasonlyadvancedby45minsduring1hour– “movingclocksrunslow”

Situationat3:00inFrame𝑆 Situationat4:00inFrame𝑆

Frame𝑆

Frame𝑆′

Paradoxicalthinking

• Nowlet’spivottotheview-pointofframe𝑆′.Asmentionedbefore,frame𝑺′ observersthinkthattheframe𝑺 clocksareoutofsynchronization

• Frame𝑆′ observers:“frame𝑆 observersthinkourclocksarerunningslow,becausetheirsareoutofsynchronization!”

Situationat3:00inFrame𝑆′ Situationat3:45inFrame𝑆′

Frame𝑆

Frame𝑆′

Class2:Einstein’sPostulate

Attheendofthissessionyoushouldbeableto…

• … describe Einstein’spostulate thatthespeedoflightisthesameineveryinertialreferenceframe

• … recognizetherelativityofsimultaneity:eventsthataresimultaneousinoneframe,arenotsimultaneousinothers

• … applytheconceptsoftimedilationandlengthcontraction,whichrelatetimeandspaceintervalsmeasuredindifferentinertialreferenceframes

• … recallthatthepropertimedifferencebetween2eventsismeasuredinaframewheretheyoccuratthesameplace