in this class we will explore how space-time diagrams may...
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Class4:Space-timeDiagrams
Inthisclasswewillexplorehowspace-timediagramsmaybeusedtovisualizeeventsand
causalityinrelativity
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Class4:Space-timeDiagrams
Attheendofthissessionyoushouldbeableto…
• … createaspace-timediagramshowingeventsandworldlinesinagivenreferenceframe
• … determinegeometricallywhicheventsarecausallyconnected,viatheconceptofthelightcone
• … understandthateventsseparatedbyaconstantspace-timeintervalfromtheoriginmapoutahyperbola inspace-time
• … usespace-timediagramstorelateobservationsindifferentinertialframes,viatiltedco-ordinatesystems
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𝑐𝑡
Whatisaspace-timediagram?
• Aspace-timediagramisagraphshowingthepositionofobjects(events)inareferenceframe,asafunctionoftime
• Conventionally,space(𝑥)isrepresentedinthehorizontaldirection,andtime(𝑡)runsupwards
𝑥
Wehavescaledtimebyafactorof𝑐,soithasthe
samedimensionsasspace
Thisisthepathofalightraytravellinginthe𝑥-direction(i.e.,𝑥 = 𝑐𝑡),whichmakesanangleof
45° withtheaxis
Whatwouldbethepathofanobject
movingatspeed𝑣 < 𝑐?
Hereisanevent
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Whatisaspace-timediagram?
• Thepathofanobject(orlightray)movingthroughaspace-timediagramiscalledaworldlineofthatobject,andmaybethoughtofasachainofmanyevents
• Notethataworldlineinaspace-timediagrammaynotbethesameshapeasthepathofanobjectthroughspace
Rocketshipacceleratingin𝑥-direction(pathinspaceisstraight)
𝑐𝑡
𝑥
Rocketshipacceleratingin𝑥-direction(pathinspace-timediagramis
curved)
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Causality
• Inclassicalphysics,interactionsareinstantaneous(changeinonethinginstantaneouslyaffectsanother)
• Relativitytellsusthatthisisnottrue– thereisafinitemaximumspeedatwhichinteractionscanpropagate,whichturnsouttobethespeedoflight
• Thismaximumspeedleadstotheideaofcauseandeffect
https://www.ihkplus.de/Das_Ende_der_Ketteninsolvenz.AxCMS
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𝑐𝑡
Causality
• Consideranevent𝐸+ attheoriginofco-ordinates
• Consideringthatnothingcantravelfasterthan𝑐,whatregionofthediagramcontainseventswhich𝐸+ cancause?
• Whatregioncontainseventswhichcancause𝐸+?
𝑥𝐸+
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Causality
• Theseconsiderationsleadtotheconceptofalightconeinspace-timearoundanevent
https://en.wikipedia.org/wiki/Light_cone
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Causality
• Apairofeventscanbecausallyconnected,ornot,dependingontheirseparationinspace∆𝑥 andtime∆𝑡
• Tobecausallyconnected,werequire|∆𝑥| < 𝑐|∆𝑡|,orthespace-timeinterval∆𝒔𝟐= −𝒄𝟐∆𝒕𝟐 + ∆𝒙𝟐 isnegative
• Thespace-timeintervalisaninvariant– soif2eventsarecausallyconnectedinoneframe,theyareinallframes–i.e.relativitypreservescause-and-effect
• (Reminderfrompreviousclass:)If∆𝑠6< 0,thereisaframeinwhichtheeventsoccuratthesameplace(∆𝑥 = 0)butnoframeinwhichtheeventsoccuratthesametime(∆𝑡 = 0)
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Space-timegeometry
• Consideranevent𝐸6 whichtakesplaceatthesamespatiallocationin𝑆 as𝐸+,suchastwoticksofastationaryclock
• Howwouldtheseeventsappearinanotherframe𝑆′?WecanusetheLorentztransformationswith𝑥 = 0 todeducethat(𝑥;, 𝑡′) = (−𝛾𝑣𝑡, 𝛾𝑡),where𝑣 isthespeedof𝑆′
𝑐𝑡
𝑥𝐸+
𝐸6
𝑐𝑡′
𝑥′𝐸+
𝐸6
Positionsofeventsininertialframestravellingwithspeed𝑣
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• Whatmathematicalcurveisthepositionof𝐸6 tracingout?
• Recallthatthespace-timeintervalbetweentheevents,∆𝑠6= −𝑐6∆𝑡6 + ∆𝑥6,isaconstantineveryframe:“ 𝑐𝑡 6 − 𝑥6 = constant”isahyperbola
Locusof𝐸6 indifferentframes
𝐸+
Lightcone
𝐸6 alwaysremainswithinthefuturelightconeof𝐸+
Space-timegeometry
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Locusofpointsaconstantspace-timeinterval awayfromapointinspace-time(traceoutbychangingframe)
LocusofpointsaconstantdistanceawayfromapointinEuclideangeometry(traceoutbyrotation)
Space-timegeometry
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• Consider4eventswhichtakeplaceinframe𝑆 atspace-timeco-ordinates 𝑥, 𝑐𝑡 = 0,0 , 1,0 , 0,1 , (1,1)
• UsetheLorentztransformationstoplottheseeventsinaspace-timediagramfor𝑆′ withaxes(𝑥;, 𝑐𝑡′),where𝑣 = 0.6𝑐
𝑐𝑡
Changingframes
𝑥𝐸+
𝐸6
𝐸B
𝐸C
• 𝐸+ and𝐸6 couldrepresenttwoticksofaclocksittingattheoriginof𝑆
• 𝐸+ and𝐸B couldrepresentthemeasurementofthelengthofarulerin𝑆
• 𝐸+ and𝐸C couldbeconnectedbyalight-signal
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• Theeventsasmeasuredinframe𝑆′:
• (𝐸+, 𝐸B) and(𝐸6, 𝐸C) arenolongersimultaneousin𝑆′
• (𝐸+, 𝐸C) arestillconnectedbyalightsignalin𝑆′
𝑐𝑡
Changingframes
𝑥𝐸+
𝐸6
𝐸B
𝐸C
𝑐𝑡′
𝑥′𝐸+
𝐸6
𝐸B
𝐸C
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Changingframes
• Ratherthanchangethepositionsoftheevents,anicevisualizationapproachistotiltthespace-timeaxes!
• The𝑥′-axisisthelineof𝑡; = 0,socanberepresentedinan 𝑥, 𝑐𝑡diagrambytheline𝑐𝑡 = D
E𝑥 [using𝑡; = 𝛾 𝑡 − DG
EH]
• The𝑡′-axisisthelineof𝑥′ = 0,socanberepresentedinan 𝑥, 𝑐𝑡diagrambytheline𝑐𝑡 = E
D𝑥 [using𝑥; = 𝛾 𝑥 − 𝑣𝑡 ]
𝑐𝑡
𝑥𝐸+
𝐸6
𝐸B
𝐸C
𝑐𝑡′
𝑥′𝐸+
𝐸6
𝐸B
𝐸C𝑥′
𝑐𝑡′
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Changingframes
• Sothespace-timediagramof𝑆′ canbeoverlaidonthespace-timediagramof𝑆 asatiltedco-ordinatesystem!
𝑐𝑡 𝑐𝑡′
𝑥
𝑥′
Invariantspeedoflight(𝑥 = 𝑐𝑡,𝑥; = 𝑐𝑡′)
𝜃𝜃 tan 𝜃 =
𝑣𝑐
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Changingframes
• Thisprovidesuswithimmediatevisualizationsofthephenomenaoftimedilationandlengthcontraction
𝑐𝑡
𝑥
𝑐𝑡′
𝑥′Time
diffin𝑆′Time
diffin𝑆
𝑐𝑡
𝑥
𝑐𝑡′
𝑥′
𝑐𝑡 = 𝛾
𝑐𝑡; = 1𝑥; = 1
𝑥 = 1/𝛾Lengthin𝑆
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Faster-than-lighttravel
• Wecanalsouseourspace-timediagramstoshowthatfaster-than-lighttravelcontradictscause-and-effect!
𝑐𝑡
𝑥
TheMillenniumFalconleavestheDeathStaratspeed𝑣 =0.5𝑐 afteradaringraid
TheDeathStarfinallynotices,andfiresits“faster-than-light”weaponat𝑣 = 4𝑐
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Faster-than-lighttravel
• HowdotheseeventsappearfromtheMillenniumFalcon?
𝑐𝑡𝑐𝑡′
𝑥
𝑥′
The(𝑥;, 𝑐𝑡;) co-ordinatesystemistiltedwhenoverlaidon(𝑥, 𝑐𝑡)
Inthe𝑆′ frame,thefaster-than-lightweaponisfiredAFTERtheMillenniumFalconisdestroyed!!
Faster-than-lighttravelcontradictscause-and-effect!!
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Faster-than-lighttravel
• Whatotherissuesmakefaster-than-lighttraveldifficulttoreconcilewithstandardphysics?
13/8/18, 6(26 pmFaster-than-light neutrinos could be down to bad wiring - BBC News
Page 1 of 9https://www.bbc.com/news/science-environment-17139635
Science & Environment
Home Video World Asia UK Business Tech Science Stories
23 February 2012
The neutrinos are fired deep under the ItalianApennines to the Gran Sasso lab
Faster-than-light neutrinos could be downto bad wiringBy Jason PalmerScience and technology reporter, BBC News
What might have been the biggestphysics story of the past centurymay instead be down to a faultyconnection.
In September 2011, the Operaexperiment reported it had seenparticles called neutrinos evidentlytravelling faster than the speed oflight.
The team has now found twoproblems that may have affected their test in opposing ways: one in its timing gearand one in an optical fibre connection.
More tests from May will determine just how they affect measured speeds.
The Opera collaboration (an acronym for Oscillation Project with Emulsion-RackingApparatus) was initially started to study the tiny particles as they travelled through730km of rock between a particle accelerator at the European Organization forNuclear Research (Cern) in Switzerland and the Gran Sasso undergroundlaboratory in Italy.
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Class4:Space-timeDiagrams
Attheendofthissessionyoushouldbeableto…
• … createaspace-timediagramshowingeventsandworldlinesinagivenreferenceframe
• … determinegeometricallywhicheventsarecausallyconnected,viatheconceptofthelightcone
• … understandthateventsseparatedbyaconstantspace-timeintervalfromtheoriginmapoutahyperbola inspace-time
• … usespace-timediagramstorelateobservationsindifferentinertialframes,viatiltedco-ordinatesystems