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