class1 specialrelativity forbbcblake/class1... · 2018-04-29 · title:...
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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#�