In this class we will explore the remarkable consequences that...
Transcript of In this class we will explore the remarkable consequences that...
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