Class1:SpecialRelativity

InthisclasswewillreviewsomeimportantconceptsinSpecialRelativity,thatwillhelpus

builduptotheGeneraltheory

Attheendofthissessionyoushould…

• … befamiliarwiththebasicingredientsofSpecialRelativity:reference/inertialframes,theLorentztransformations,Einstein’spostulateanditsconsequences

• … understandthesignificanceofthespace-timeinterval𝑑𝑠#betweentwoevents

• … beabletousespace-timediagramstodescribesomesimplevisualizationsofeventsinspace-time

• … understandhowthedefinitionsofclassicalmomentumandenergymustbemodifiedinSpecialRelativity

Class1:SpecialRelativity

Referenceframes

• Howdowedescribeoccurrencesinnature?

• Areferenceframeisaninformation-gatheringsystemconsistingofafieldofsynchronizedclocksonaco-ordinategrid

• Anevent isanoccurrencewithdefinitelocation/time

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_principles

Referenceframes

• Thelocationsandtimesofeventstogetherforma4Dco-ordinatesystemknownasspace-time.

https://xkcd.com/1524/

Inertialframes

• Somereferenceframesarespecial

• Inaninertialframe,afreely-movingbody(noexternalforces)moveswithconstantvelocity

• Inertialframesmoveuniformly withrespecttoeachother

https://en.wikipedia.org/wiki/Special_relativity

Inertialframes

• Someexamplesofnon-inertialframes!

https://www.thinglink.com/scene/877700809561735169https://www.pinterest.com.au/bestonamusement/beston-frisbee-rides-for-sale-or-fairground-pendul/

Principleofrelativity

• Lawsofnatureareidenticalinallinertialframes(mathematically,theyhavethesameformineachframe’sco-ordinatesystem)

• Thereisnosuchthingasabsolute rest/velocity

https://owlcation.com/stem/What-Were-Galileos-Contributions-to-Astronomy

(DuetoGalileo)

Principleofrelativity

• Example:Newton’sLaws�⃗� = 𝑚�⃗� areinvariantunderthetransformation�⃗�* 𝑡 = �⃗� 𝑡 + 𝑉./012302

• Nottruefornon-inertialframes,inwhichweexperienceforceswecannotaccountfor

https://xkcd.com/123/

Transformationsbetweenframes

• If(𝑡, 𝑥, 𝑦, 𝑧) aretheco-ordinatesofaneventinframe𝑆,whatareitsco-ordinates(𝑡*, 𝑥*, 𝑦*, 𝑧*) inframe𝑆′movingwithspeed𝑣 withrespectto𝑆?

• Galileantransformation:clocksin𝑆 and𝑆′ runatthesamerate(𝑡* = 𝑡),andvelocitiesadd(𝑢 = 𝑢* + 𝑣)

Theroleoflight

• AccordingtotheGalileantransformation,thespeedoflight𝑐 slowsdownto𝑐 − 𝑣 in𝑆′

• Thiscanbetestedexperimentallyandisfalse(e.g.Michelson-Morleyexperiment)

• TheGalileantransformationisinconsistentwithelectromagnetism

https://brooklyntomars.com/laser-beam-focus-352154e6acaf

Theroleoflight

• Inclassicalphysics,interactionsareinstantaneous(changeinonethinginstantaneouslyaffectsanother)

• Thisisnottrue– thereisafinitemaximumvelocityatwhichinteractionspropagate,whichturnsouttobethespeedoflight𝑐 = 3×10C m/s(large)

https://www.ihkplus.de/Das_Ende_der_Ketteninsolvenz.AxCMS

• Consequences:(1)bodiescannotmovefasterthan𝑐,(2)𝑐 isthesameinallframes,(3)causality

Theroleoflight

• Thefinitevelocityatwhichsignalscanpropagateisdeeplyconnectedtotheideaofcausality

https://en.wikipedia.org/wiki/Light_cone

Einstein’spostulate

• Einstein’spostulate(1905):thespeedoflightisthesameinallinertialframes (regardlessofthemotionofthesource/observer)

• TheGalileantransformationsarewrong!

https://www.patana.ac.th/secondary/science/anrophysics/relativity_option/commentary.html

• Howshouldwetransformtheco-ordinatesofevents?

• Imaginesendingoutalightsignalfromtheorigin:

InS:𝑥# + 𝑦# + 𝑧# = (𝑐𝑡)#

InS’:𝑥′# + 𝑦′# + 𝑧′# = (𝑐𝑡′)#

Lorentztransformations

Lorentztransformations

• TheseequationsaresatisfiedbytheLorentztransformations:

𝑥′ = 𝛾(𝑥 − 𝑣𝑡)

𝑡′ = 𝛾(𝑡 − 𝑣𝑥/𝑐#)

𝑦′ = 𝑦

𝑧′ = 𝑧

𝑥 = 𝛾(𝑥* + 𝑣𝑡′)

𝑡 = 𝛾(𝑡* + 𝑣𝑥′/𝑐#)

𝑦 = 𝑦′

𝑧 = 𝑧′

𝛾 =1

1 − 𝑣#/𝑐#�

Simultaneity

• TheLorentztransformationshavesomeremarkableconsequences

• First,considertwoeventsthataresimultaneousin𝑆(𝑡G = 𝑡#).BytheLorentztransformations:

• Theseeventsarenotsimultaneousin𝑺′!

• Thereisnosuchthingasabsolutetime,theorderofeventsdependsonthereferenceframe!

𝑡G* − 𝑡#* =𝛾𝑣𝑐# 𝑥# − 𝑥G

Lengthcontraction

• Considerastickatrestin𝑆′ withendsat(𝑥G* , 𝑥#* )

• Observersin𝑆 measureitslengthbyrecordingtheendpositions(𝑥G, 𝑥#) simultaneouslyin𝑆.Theyfind:

• Thelengthofamovingobject is contracted by 𝜸

𝑥# − 𝑥G = 𝑥#* − 𝑥G* /𝛾

https://www.slideshare.net/cscottthomas/ch-28-special-relativity-online-6897295

Timedilation

• Aclockattheoriginof𝑆′ ticksattimes(𝑡G* , 𝑡#* )

• Thespacingoftheseticksin𝑆 is:

• Amovingclockticksatlonger intervals

𝑡# − 𝑡G = 𝛾 𝑡#* − 𝑡G*

https://patriceayme.wordpress.com/2014/06/17/time-dilation/

Space-timeinterval

• TheLorentztransformationhasaspecialpropertyforhowthedifferenceinthespace-timeco-ordinatesoftwoeventstransformbetween𝑆 and𝑆′:

• Thiscombinationisthespace-timeinterval∆𝒔𝟐

• It is an invariant (has the samevalue in all frames)

−𝑐#∆𝑡# + ∆𝑥# + ∆𝑦# + ∆𝑧#= −𝑐#∆𝑡′# + ∆𝑥′# + ∆𝑦′# + ∆𝑧′#

Importanceofinvariants

• Invariants – quantitieswhichallobserversagreetobethesame– areuseful!

• InEuclideangeometry,thedistancebetween2pointsisthesameifyourotateco-ordinates

https://www.mathsisfun.com/algebra/distance-2-points.html

Thespace-timeintervalinrelativity(sameinallinertialframes)isanalogoustoadistanceinEuclideangeometry(sameinallrotatedframes)

Classifyingintervals

• Theinvariant space-timeinterval, ∆𝑠#= −𝑐#∆𝑡# + ∆𝑥# +∆𝑦# + ∆𝑧#,may be used toclassify pairs of events

• Events forwhich ∆𝑠#< 0 aretimelike (“timewins”)– aninertial observer can pass through them both.The timebetween the events in this frame is the proper time∆𝜏

• Eventsforwhich ∆𝑠#> 0 arespacelike – there’s an inertialframe in which they aresimultaneous.The separationbetween the events in this frame is the proper separation.

• Eventsforwhich ∆𝑠#= 0 arelightlike – theyoccuralongthesamelightray

Relativisticmomentum

• Conservationofmomentumcanbepreservedifwemodifytheconceptofthemassofaparticlesuchthatitdependsonvelocity

• We have introduced the rest mass𝒎𝟎

• As𝑣 → 0 werecoverclassicalmomentum𝑝 = 𝑚T𝑣

𝑚 𝑣 = 𝛾𝑚T =𝑚T

1 − 𝑣#

𝑐#�

Relativisticenergy

• Consideringtheworkdoneonaparticle∫ VWV2 𝑑𝑥�� as

theenergygained,werecoveramodifiedexpressionforrelativisticenergy

• Particles have arest energy𝒎𝟎𝒄𝟐

• As𝑣 → 0 wefind𝐸 = 𝑚T𝑐# +G#𝑚T𝑣#

• Mathematicallywefind:𝐸# − 𝑝#𝑐# = 𝑚T𝑐# #

𝐸 = 𝛾𝑚T𝑐# =𝑚T𝑐#

1 − 𝑣#

𝑐#�