in this class we will explore the lorentz transformations...
TRANSCRIPT
Class3:LorentzTransformations
InthisclasswewillexploretheLorentztransformations,theequationswhichrelatethespace-timeco-ordinatesofaneventviewedin
differentreferenceframes
Class3:LorentzTransformations
Attheendofthissessionyoushouldbeableto…
• … befamiliarwiththeLorentztransformations,whichrelatethespace-timeco-ordinatesofaneventindifferentframes
• … applytheLorentztransformationstosuitableeventstoderivethephenomenaoftimedilationandlengthcontraction
• … understandthesignificanceofthespace-timeinterval𝑑𝑠#betweentwoeventsandtheideaofinvariants
• … applythelawofcombinationofvelocitiesinrelativity
Lorentztransformations
• Inthepreviousclass,weencounteredsomeoftheremarkablephenomenaofrelativity,suchastimedilationandlengthcontraction
• Wewillnowlearnhowtoderivetheseeffectsmoresystematically,usingtheLorentztransformations
H.Lorentz (1853-1928)https://www.nobelprize.org/nobel_prizes/physics/laureates/1902/lorentz-bio.html
Lorentztransformations
• Anevent isaphysicaloccurrenceatauniquepointinspaceandtime
• TheLorentztransformationsprovidethealgebraicrelationsforrelatingtheco-ordinatesofthisevent intwodifferentinertialreferenceframes,𝑆 and𝑆′
𝑆𝑆′
𝑣 Event
𝑥𝑥′
𝑦 𝑦′
Lorentztransformations
𝑥′ = 𝛾(𝑥 − 𝑣𝑡)
𝑡′ = 𝛾(𝑡 − 𝑣𝑥/𝑐#)
𝑦′ = 𝑦
𝑧′ = 𝑧
𝑥 = 𝛾(𝑥2 + 𝑣𝑡′)
𝑡 = 𝛾(𝑡2 + 𝑣𝑥′/𝑐#)
𝑦 = 𝑦′
𝑧 = 𝑧′
𝛾 =1
1 − 𝑣#/𝑐#�
Lorentztransformationsfrom𝑆 to𝑆′
Inversetransformationsfrom𝑆′ to𝑆
[NotethisintheWorkbook]
Lorentztransformations
• Thesetransformationsworkbecausetheypreservetheconstancyofthespeedoflight indifferentframes
• Consideraflashoflightemittedfromtheoriginof𝑆
𝑥
𝑦• Thepulseoflightexpandsinasphere
• Theco-ordinatesofthewavefront are𝑥# + 𝑦# + 𝑧# = (𝑐𝑡)#
𝛾# 𝑥2 + 𝑣𝑡′ # + 𝑦′# + 𝑧′# = 𝑐#𝛾#(𝑡2 + 𝑣𝑥2/𝑐#)#
𝑥# + 𝑦# + 𝑧# = (𝑐𝑡)#
Lorentztransformations
• Whataretheco-ordinatesofthewavefrontin𝑆′?
• SubstituteintheLorentztransformations:
𝛾#𝑥′# + 2𝛾#𝑥2𝑣𝑡2 + 𝛾#𝑣#𝑡′# + 𝑦′# + 𝑧′#= 𝛾#𝑐#𝑡′# + 2𝛾#𝑥2𝑣𝑡2 + 𝛾#𝑣#𝑥′#/𝑐#
𝛾#𝑥′# 1 − 𝑣#/𝑐# + 𝑦′# + 𝑧′# = 𝛾#𝑐#𝑡′# 1 − 𝑣#/𝑐#
𝒙′𝟐 + 𝒚′𝟐 + 𝒛′𝟐 = (𝒄𝒕′)𝟐= 1/𝛾# = 1/𝛾#
Lorentztransformations
• Thewavefrontalsoexpandsasasphere,withspeed𝑐,in𝑆′
• TheLorentztransformationsareconsistentwithEinstein’spostulateofspecialrelativity
𝑥
𝑦 𝑦′
𝑥′
𝑣𝑆 𝑆′
Lorentztransformations
𝑥′ = 𝛾(𝑥 − 𝑣𝑡)
𝒕′ = 𝜸(𝒕 − 𝒗𝒙/𝒄𝟐)
𝑦′ = 𝑦
𝑧′ = 𝑧
• IntheLorentztransformations– unlikeintheGalileantransformations– thereisnosuchthingasabsolutetime
𝑥′ = 𝑥 − 𝑣𝑡
𝒕′ = 𝒕
𝑦′ = 𝑦
𝑧′ = 𝑧
Lorentztransformation(correct)
Galileantransformation(incorrect)
Timedilationre-visited
• WeapplytheLorentztransformationsbyidentifyingeventsandtheirco-ordinates.Let’strysomeexamples.
• Consideraclockatrestattheoriginof𝑆′,whichticksattimes𝑡2 = 0 and𝑡2 = 𝜏.Whatisthetimedifferencebetweentheseticksasmeasuredin𝑆?
𝑥
𝑦 𝑦′
𝑥′
𝑣
𝑆 𝑆′
• Whatarethespacetime co-ordinatesofthetwoticksin𝑆2?
• UsetheLorentztransformationstotransformtheseco-ordinatesto𝑆
Lengthcontractionre-visited
• Consideraruleratrestin𝑆′,withendsat𝑥2 = 0 and𝑥2 = 𝐿
• Observersin𝑆 measurethelengthoftherulerbymarkingthepositionsofthetwoendssimultaneouslyin𝑺,attime𝑡.Whatisthelengthoftherulerasmeasuredin𝑆?
𝑥
𝑦 𝑦′
𝑥′
𝑣
𝑆 𝑆′
Space-timeinterval
• Considertwoevents,whichareseparatedinspace-timebyco-ordinatedifferences(∆𝑥, ∆𝑦, ∆𝑧, ∆𝑡) in𝑆
• Aspecialquantityisthespace-timeintervalbetweentheevents,whichhasthesymbol∆𝑠#
• Ifobserversin𝑆′ compute the space-timeintervalof the sameevents using their co-ordinatedifferences,(∆𝑥′, ∆𝑦′, ∆𝑧′, ∆𝑡′),theywillfindexactlythesameanswer![NoteintheWorkbook]
Space-timeinterval∆𝒔𝟐= −𝒄𝟐∆𝒕𝟐 + ∆𝒙𝟐 + ∆𝒚𝟐 + ∆𝒛𝟐
−𝑐#∆𝑡# + ∆𝑥# + ∆𝑦# + ∆𝑧#= −𝑐#∆𝑡′# + ∆𝑥′# + ∆𝑦′# + ∆𝑧′#
Space-timeinterval
• Let’sprovethatisthecasebyconsideringtwoticksofaclockatrestattheoriginof𝑆′,whoseeventswecomputedearlier
• Whatisthespace-timeintervalbetweentwoticksoftheclockasdeterminedby𝑆 and𝑆′?
𝑥
𝑦 𝑦′
𝑥′
𝑣
𝑆 𝑆′
Space-timeinterval
• Aquantitysuchas∆𝑠#,whichhasthesamevalueinallreferenceframes,iscalledaninvariant
• Quantitieswhichallobserversagreetobethesameareuseful!
• InEuclideangeometry,thedistancebetween2pointsisthesameifyourotateco-ordinates
https://www.mathsisfun.com/algebra/distance-2-points.html
Thespace-timeintervalinrelativity(sameinallinertialframes)isanalogoustoadistanceinEuclideangeometry(sameinallrotatedframes)
Space-timeinterval
• Thespace-timeinterval,∆𝑠# = −𝑐#∆𝑡# + ∆𝑥#,between 2events can be zero,negative or positive [ignoring 𝑦, 𝑧 forclarity]
• If∆𝑠# = 0,then the 2events can be linked by alight signal(∆𝑥 = 𝑐∆𝑡) in any frame
• If∆𝑠# < 0,then an observer can pass through them both(there is aframe with ∆𝑥 = 0)butthere is noframe in whichthe events occur simultaneously (∆𝑡 = 0)
• If∆𝑠# > 0,there is aframe in which the events occursimultaneously,butnonein which they occur atthe sameplace
• (We’ll discuss these points further in the next class)
Combinationofvelocities
• Inclassicalphysics,relativevelocitiesaddandsubtract
• Inrelativity,Einstein’spostulateshowsthisiswrong
10m/s
10m/s 20m/s
10m/s
𝑐 𝑐
Combinationofvelocities
• WecanusetheLorentztransformationstoderivetheexactformula.Firstconverttothedifferentials:
• Nowdividethesetworelationstofindthespeed𝑢K =LKLM
in𝑆,ofanobjectmovingwithspeed𝑢K2 =LKN
LMNin𝑆′:
• Doesthiswork?Yes– when𝒖𝒙2 = 𝒄,then 𝒖𝒙 = 𝒄
𝑥 = 𝛾 𝑥2 + 𝑣𝑡′ → 𝑑𝑥 = 𝛾 𝑑𝑥2 + 𝑣𝑑𝑡′
𝑡 = 𝛾 𝑡2 + 𝑣𝑥2/𝑐# → 𝑑𝑡 = 𝛾 𝑑𝑡2 + 𝑣𝑑𝑥2/𝑐#
𝑢K =𝑑𝑥𝑑𝑡 =
𝛾 𝑑𝑥2 + 𝑣𝑑𝑡′𝛾 𝑑𝑡2 + 𝑣𝑑𝑥2/𝑐# =
𝑑𝑥′𝑑𝑡′ + 𝑣
1 + 𝑣𝑐#
𝑑𝑥′𝑑𝑡′
=𝑢K2 + 𝑣
1 + 𝑢K2 𝑣𝑐#
Combinationofvelocities
• TheAmazon-spacedeliveryservice,arocketmovingatspeed𝑣 = 0.5𝑐,firesapackagetowardstheEarthatspeed𝑢2 = 0.75𝑐 intherocketframe.WhatisthepackagespeedasmeasuredbytheEarthobserver?
• Howdoesyouranswerchangeifthepackageisaccidentallyfiredintheoppositedirection,atthesamespeedintherocketframe?
Combinationofvelocities: 𝑢 =𝑢2 + 𝑣
1 + 𝑢2𝑣/𝑐# 𝑢′ =𝑢 − 𝑣
1 − 𝑢𝑣/𝑐#
[Exampleadaptedfromhttps://courses.lumenlearning.com/physics/chapter/28-4-relativistic-addition-of-velocities/]
Combinationofvelocities
• Howaboutvelocitiesintheperpendiculardirection?
• Although𝑦 = 𝑦′,𝑢c ≠ 𝑢c2 !!Carryingoutasimilarderivationtobefore…
• Dividingthesetworelations,
• The𝑦-velocity is altered due totimedilation
𝑦 = 𝑦′ → 𝑑𝑦 = 𝑑𝑦′𝑡 = 𝛾 𝑡2 + 𝑣𝑥2/𝑐# → 𝑑𝑡 = 𝛾 𝑑𝑡2 + 𝑣𝑑𝑥2/𝑐#
𝑢c =𝑑𝑦𝑑𝑡 =
𝑑𝑦′𝛾 𝑑𝑡2 + 𝑣𝑑𝑥2/𝑐# =
𝑑𝑦′𝑑𝑡′
𝛾 1 + 𝑣𝑐#
𝑑𝑥′𝑑𝑡′
=𝑢c2
𝛾 1 + 𝑢K2 𝑣𝑐#
Thesame,ornotthesame?
Whichofthefollowingquantitiesarenecessarilythesamewhenmeasuredintwoinertialframes,𝑆 and𝑆′?Whicharenot?
𝑆 𝑆′
𝑣
𝑥 𝑥′
𝑦 𝑦′
• Valueofthespeedoflightinavacuum
• Speedofanelectron
• Valueofthechargeonanelectron
• Kineticenergyofaproton
• Valueoftheelectricfieldatapoint
• Timebetweentwoevents
• Orderofelementsintheperiodictable
• Newton’sFirstLawofMotion
[ExamplefromTaylor&Wheeler,Spacetime Physics,SampleProblem3.1]
Class3:LorentzTransformations
Attheendofthissessionyoushouldbeableto…
• … befamiliarwiththeLorentztransformations,whichrelatethespace-timeco-ordinatesofaneventindifferentframes
• … applytheLorentztransformationstosuitableeventstoderivethephenomenaoftimedilationandlengthcontraction
• … understandthesignificanceofthespace-timeinterval𝑑𝑠#betweentwoeventsandtheideaofinvariants
• … applythelawofcombinationofvelocitiesinrelativity