Class4:Space-timeDiagrams

Inthisclasswewillexplorehowspace-timediagramsmaybeusedtovisualizeeventsand

causalityinrelativity

Class4:Space-timeDiagrams

Attheendofthissessionyoushouldbeableto…

• … createaspace-timediagramshowingeventsandworldlinesinagivenreferenceframe

• … determinegeometricallywhicheventsarecausallyconnected,viatheconceptofthelightcone

• … understandthateventsseparatedbyaconstantspace-timeintervalfromtheoriginmapoutahyperbola inspace-time

• … usespace-timediagramstorelateobservationsindifferentinertialframes,viatiltedco-ordinatesystems

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

Whatisaspace-timediagram?

• Thepathofanobject(orlightray)movingthroughaspace-timediagramiscalledaworldlineofthatobject,andmaybethoughtofasachainofmanyevents

• Notethataworldlineinaspace-timediagrammaynotbethesameshapeasthepathofanobjectthroughspace

Rocketshipacceleratingin𝑥-direction(pathinspaceisstraight)

𝑐𝑡

𝑥

Rocketshipacceleratingin𝑥-direction(pathinspace-timediagramis

curved)

Causality

• Inclassicalphysics,interactionsareinstantaneous(changeinonethinginstantaneouslyaffectsanother)

• Relativitytellsusthatthisisnottrue– thereisafinitemaximumspeedatwhichinteractionscanpropagate,whichturnsouttobethespeedoflight

• Thismaximumspeedleadstotheideaofcauseandeffect

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

𝑐𝑡

Causality

• Consideranevent𝐸+ attheoriginofco-ordinates

• Consideringthatnothingcantravelfasterthan𝑐,whatregionofthediagramcontainseventswhich𝐸+ cancause?

• Whatregioncontainseventswhichcancause𝐸+?

𝑥𝐸+

Causality

• Theseconsiderationsleadtotheconceptofalightconeinspace-timearoundanevent

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

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)

Space-timegeometry

• Consideranevent𝐸6 whichtakesplaceatthesamespatiallocationin𝑆 as𝐸+,suchastwoticksofastationaryclock

• Howwouldtheseeventsappearinanotherframe𝑆′?WecanusetheLorentztransformationswith𝑥 = 0 todeducethat(𝑥;, 𝑡′) = (−𝛾𝑣𝑡, 𝛾𝑡),where𝑣 isthespeedof𝑆′

𝑐𝑡

𝑥𝐸+

𝐸6

𝑐𝑡′

𝑥′𝐸+

𝐸6

Positionsofeventsininertialframestravellingwithspeed𝑣

• Whatmathematicalcurveisthepositionof𝐸6 tracingout?

• Recallthatthespace-timeintervalbetweentheevents,∆𝑠6= −𝑐6∆𝑡6 + ∆𝑥6,isaconstantineveryframe:“ 𝑐𝑡 6 − 𝑥6 = constant”isahyperbola

Locusof𝐸6 indifferentframes

𝐸+

Lightcone

𝐸6 alwaysremainswithinthefuturelightconeof𝐸+

Space-timegeometry

Locusofpointsaconstantspace-timeinterval awayfromapointinspace-time(traceoutbychangingframe)

LocusofpointsaconstantdistanceawayfromapointinEuclideangeometry(traceoutbyrotation)

Space-timegeometry

• 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

• Theeventsasmeasuredinframe𝑆′:

• (𝐸+, 𝐸B) and(𝐸6, 𝐸C) arenolongersimultaneousin𝑆′

• (𝐸+, 𝐸C) arestillconnectedbyalightsignalin𝑆′

𝑐𝑡

Changingframes

𝑥𝐸+

𝐸6

𝐸B

𝐸C

𝑐𝑡′

𝑥′𝐸+

𝐸6

𝐸B

𝐸C

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′

Changingframes

• Sothespace-timediagramof𝑆′ canbeoverlaidonthespace-timediagramof𝑆 asatiltedco-ordinatesystem!

𝑐𝑡 𝑐𝑡′

𝑥

𝑥′

Invariantspeedoflight(𝑥 = 𝑐𝑡,𝑥; = 𝑐𝑡′)

𝜃𝜃 tan 𝜃 =

𝑣𝑐

Changingframes

• Thisprovidesuswithimmediatevisualizationsofthephenomenaoftimedilationandlengthcontraction

𝑐𝑡

𝑥

𝑐𝑡′

𝑥′Time

diffin𝑆′Time

diffin𝑆

𝑐𝑡

𝑥

𝑐𝑡′

𝑥′

𝑐𝑡 = 𝛾

𝑐𝑡; = 1𝑥; = 1

𝑥 = 1/𝛾Lengthin𝑆

Faster-than-lighttravel

• Wecanalsouseourspace-timediagramstoshowthatfaster-than-lighttravelcontradictscause-and-effect!

𝑐𝑡

𝑥

TheMillenniumFalconleavestheDeathStaratspeed𝑣 =0.5𝑐 afteradaringraid

TheDeathStarfinallynotices,andfiresits“faster-than-light”weaponat𝑣 = 4𝑐

Faster-than-lighttravel

• HowdotheseeventsappearfromtheMillenniumFalcon?

𝑐𝑡𝑐𝑡′

𝑥

𝑥′

The(𝑥;, 𝑐𝑡;) co-ordinatesystemistiltedwhenoverlaidon(𝑥, 𝑐𝑡)

Inthe𝑆′ frame,thefaster-than-lightweaponisfiredAFTERtheMillenniumFalconisdestroyed!!

Faster-than-lighttravelcontradictscause-and-effect!!

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