The Lambda CDM-Model in Quantum Field Theory on Curved Spacetime and Dark Radiation
15b. Minkowski Spacetime 1.The Relativity of Simultaneity...
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15b. Minkowski Spacetime Principle of Relativity and Light Postulate entail: The speed of light c is the same in all inertial reference frames. • ' is moving at constant velocity with respect to O. t x ' t' x' θ θ • So : and ' must disagree on spatial and temporal measurements! light signal • and ' must measure same speed c for light signal. value of c for value of c for ' = = ∆ ∆ = ∆′ ∆′ 1 1. The Relativity of Simultaneity 2. Minkowski Spacetime 3. The Conventionality of Simultaneity 1. The Relativity of Simultaneity
Transcript of 15b. Minkowski Spacetime 1.The Relativity of Simultaneity...
15b.MinkowskiSpacetime
PrincipleofRelativityandLightPostulateentail: Thespeedoflightcisthesameinallinertialreferenceframes.
• 𝒪' ismovingatconstantvelocitywithrespecttoO.
t
x
𝒪 𝒪'
t'
x'
θ
θ
• So:𝒪 and𝒪'mustdisagreeonspatialandtemporalmeasurements!
lightsignal
• 𝒪 and𝒪'mustmeasuresamespeedc forlightsignal.
valueofcfor𝒪
valueofcfor𝒪'
=
𝑐 =∆𝑥∆𝑡
=∆𝑥′∆𝑡′
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1. TheRelativityofSimultaneity2. MinkowskiSpacetime3. TheConventionalityofSimultaneity1.TheRelativityofSimultaneity
15b.MinkowskiSpacetime
• 𝒪 and𝒪'makedifferentjudgementsofsimultaneity.
t
x
𝒪 𝒪'
t'
x'
• p andq aresimultaneousaccordingto𝒪'.
•
•
p
q
• p happensbeforeq accordingto𝒪.
RelativityofSimultaneityTherelationofsimultaneityisrelativetoinertialreferenceframes.
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PrincipleofRelativityandLightPostulateentail: Thespeedoflightcisthesameinallinertialreferenceframes.
1. TheRelativityofSimultaneity2. MinkowskiSpacetime3. TheConventionalityofSimultaneity1.TheRelativityofSimultaneity
Minkowskispacetime isa4-dimcollectionofpointssuchthatbetweenanytwopointsp,qwithcoordinates(t,x,y,x) and(t+Δt,x+Δx,y+Δy,z+Δz),thereisadefinitespacetimeintervalgivenby
• But:Includesthetimecoordinatedifference,too!Andit'snegative!
SpacetimeofSpecialRelativity=Minkowskispacetime
HermannMinkowski(1864-1909)
∆𝑠 = − 𝑐∆𝑡 $ + ∆𝑥 $ + ∆𝑦 $ + ∆𝑧 $
• SimilartoEuclideanspatial interval ∆𝑥 $ + ∆𝑦 $ + ∆𝑧 $.
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2.MinkowskiSpacetime
- Idea:Allinertialframesagreeonthespatiotemporal distanceΔs betweenanypointsp andq.
- ButtheydisagreeonhowΔs getssplitintoatemporal part andaspatialpart:Theydisagreeonmeasurementsoftimeandmeasurementsofspace.
t'
x'
x
t
•
•
p
q
Δs
• Allinertialframesagreeonthespatiotemporal distanceΔs betweenanytwopointsp andq.
• Theydisagreeonthetemporal distancebetweenp andq (timedilation)andonthespatial distance(lengthcontraction).
• TheydisagreeonhowtheysplitΔs intotemporalandspatialparts.
Δt
Δx
∆𝑠 = − 𝑐∆𝑡 $ + ∆𝑥 $Δt'
Δx' = − 𝑐∆𝑡′ $ + ∆𝑥′ $
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• Infinitesimally:ds2 =ημνdx μdx ν
TheMinkowskispacetimeintervalisencodedintheMinkowskimetricημν
• Absolutedistinction:AllinertialframesagreeonΔs,soallinertialframesagreeonwhichworldlinesaretimelike,lightlike,andspacelike!
ThreedifferenttypesofworldlineinMinkowskispacetime!
(Δs)2 = $",$%&
'
𝜂"$∆𝑥"∆𝑥$
=η00Δx0Δx0 + η01Δx0Δx1 +⋯+η33Δx3Δx3
=−(cΔt)2+ (Δx)2+ (Δy)2+ (Δz)2
Δx0= cΔt ,Δx1= Δx,Δx2= Δy , Δx3= Δz,
𝜂"$ =
−1 0 0 00 1 0 00 0 1 00 0 0 1
Threeformsof(Δs)2
(a)Timelike.(Δs)2< 0,or:
(b)Lightlike.(Δs)2=0,or:
(c)Spacelike:(Δs)2>0,or:
∆𝑥 ( + ∆𝑦 ( + ∆𝑧 (
∆𝑡< 𝑐
∆𝑥 ( + ∆𝑦 ( + ∆𝑧 (
∆𝑡= 𝑐
∆𝑥 ( + ∆𝑦 ( + ∆𝑧 (
∆𝑡> 𝑐
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•p
t
x
spacelikeworldline: (Δs)2 >0(objectswithspeeds> c)
timelikeworldline: (Δs)2 <0(objectswithspeeds< c)
Hence:TheMinkowskimetricdefinesalightcone atanypointp.
lightlikeworldline: (Δs)2 =0(objectswithspeeds= c)
Claim:Thedistinctionbetweenlightlike,timelike,andspacelikeworldlineswithrespecttoanypointp ismutuallyexclusive.
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physicalobjects:can'ttravelfasterthanorequaltoc
0 v< c
"tachyons"(?):can'ttravellessthanorequaltoc
v> cv= c
lightandEMwaves
ClaimA:Anobjecttravelingatv < cwithrespecttoaninertialframecannottravelatv ≥ cwithrespecttoanyotherinertialframe.ClaimB:Anobjecttravelingatv = cwithrespecttoaninertialframecannottravelatv > c orv < cwithrespecttoanyotherinertialframe.
ClaimC:Anobjecttravelingatv > cwithrespecttoaninertialframecannottravelatv ≤ cwithrespecttoanyotherinertialframe.
•p
Lightconeatp splitsspacetimeinto4 regions:1. Eventsinp'sforwardlightcone(futureofp).2. Eventsinp'sbackwardlightcone(pastofp).3. Eventsonp'slightcone.4. Eventsoutsidep'slightcone.
p'sfuture
p'spast
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Proof :• Given:v'B < c.• Suppose:vB ≥ c.- Now:S andS'measuresamespeedc forlightsignal(LightPostulate).- But:S' observeslightsignalovertakingB.- And:S observesB pacingorovertakinglightsignal.- So:S andS' areobservationallydistinct:ViolationofPrincipleofRelativity.
• Thus:Ifv'B < c,thenitcannotbethatvB ≥ c.
ClaimA:Anobjecttravelingatv < cwithrespecttoaninertialframecannottravelatv ≥ cwithrespecttoanyotherinertialframe.
v0 = speedofS'withrespecttoSv'B = speedofBwithrespecttoS'vB = speedofBwithrespecttoSc= speedoflightsignal
EarthS
c
RocketS'
v'B
v0
B
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1. Manyinertialframes;noneprivileged.2. Velocityisrelative.3. Accelerationisabsolute.4. Simultaneityisabsolute.
GalileanSpacetime
noprivilegedfamilyofstraights
privilegedfamilyofabsolutetimeslices
noabsolutetimeslices
noprivilegedfamilyofstraights
1. Manyinertialframes;noneprivileged.2. Velocityisrelative.3. Accelerationisabsolute.4. Simultaneityisrelative.5. Invariantlight-conestructureateach
point.
MinkowskiSpacetime
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Claim:GivenaneventA,thereisnoobjectivefactofthematterastowhatdistant eventsatrestwithrespecttoA aresimultaneouswithA.Thechoiceisamatterofconvention.
3.TheConventionalityofSimultaneity
Relativityofsimultaneity =Differentinertialframesjudgethesimultaneityofeventsindifferentways.(Entailedbythe2Postulates.)Conventionalityofsimultaneity =Withinasingle inertialframe,thesimultaneityofdistant eventsisnotfixedandcanbejudgedindifferentways.(Not entailedbythe2Postulates.)
HansReichenbach(1891-1953)
• Howcandistantclocksinthesameinertialframebesynchronized?- Einstein(1905):Uselightsignals.
• Howcanthesimultaneityofdistanteventsinthesameinertialframebeestablished?- Einstein(1905):Bysettingupsynchronizedclocksattheseevents.
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Aside:WhyEinstein'sfocusonclocksynchronization?
Answer:Clocksynchronizationwasonthecuttingedgeoftechnologyattheendofthe19thcentury:
• Railwaytechnology:Neededhighlyaccurate(synchronized)clocksfordependable,efficientservice.
• Electrificationofclocks:Tosynchronizeclocksto"railwaytime",sendelectricsignalsfromcentralclock.
• Galison(2003):Exampleofhowtechnologydrivestheoreticaladvances.
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• TosynchronizeClockB agivendistantfromClockA,
x
t
A B•
TA-emit
(1) EmitalightsignalfromA toB andrecordthetimeTA-emit onA.
• TB-reflect
(2) HaveB reflectthesignalbacktoA.RecordthetimeonB,TB-reflect.
•TA-return
(3) RecordtheA timeTA-returnwhenthelightsignalreturns.
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• Einstein'sStipulation:A andBmaybesaidtobeinsynchronyjustwhen
TB-reflect= T½≡ TA-emit+½(TA-return− TA-emit).
•T½
•
•
•
TA-emit
TB-reflect
TA-return
x
t
A B
- Assumption:Lighttravelsatthesamespeedc inalldirections.
StandardSimultaneityTheeventatTB-reflect issimultaneouswiththeeventatT½.
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•
•
TA-emit
TA-return
x
t
A B
Non-StandardSimultaneityTheeventatTB-reflect issimultaneouswiththeeventatTε.
•T½
- Assumption:Lighttravelsatthesamespeedc inalldirections.
- Assumption:Lightdoesnot necessarilytravelatthesamespeedc inalldirections.
• Reichenbach'sConventionalism:A andBmaybesaidtobeinsynchronyjustwhenTB-reflect= Tε≡ TA-emit+ ε(TA-return− TA-emit),forany valueofε,where0< ε < 1.
•Tε • TB-reflectStandardSimultaneityTheeventatTB-reflect issimultaneouswiththeeventatT½.
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• Einstein'sStipulation:A andBmaybesaidtobeinsynchronyjustwhen
TB-reflect= T½≡ TA-emit+½(TA-return− TA-emit).
Reichenbach'sClaim:(a) Tomeasuretheone-wayspeedoflight,weneedsynchronizedclocks.(b) Butwecanonlysynchronizeourclocksifwehavepriorknowledgeofdistant
simultaneity,whichrequirespriorknowledgeoftheone-wayspeedoflight.
• Who'sright:EinsteinorReichenbach?- Doeslighttravelatthesamespeedinalldirectionsornot?
•
•
TA-emit
TA-return
x
t
A B
Non-StandardSimultaneityTheeventatTB-reflect issimultaneouswiththeeventatTε.
•T½
•Tε • TB-reflectStandardSimultaneityTheeventatTB-reflect issimultaneouswiththeeventatT½.
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•
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TA-emit
TB-reflect
TA-return
x
t
A B
How can the "one-way" speed of light be measured?
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RealistResponse:• Agreethatthereisnoobservationaldifferencebetweenthestandardsimultaneityrelationandanynon-standardsimultaneityrelation.
• So:Ifempiricaladquacy(i.e.,agreementwithobservation)isthecriterionforhowonechoosesbetweencompetingtheories,thenthere'snoreasontopreferthestandardrelationtoanynon-standardrelation.
• But:Whythinkempiricaladquacyistheonlycriterionoftheorychoice?
- Supposesimplicity isacriterionoftheorychoice.
- However:Simplicityisahighlysubjectiveconcept...
Einstein
GeneralrelativityismuchmoresimplethanNewton'stheoryofgravity!
AverageJoe
???
- Then:Weshouldpreferthestandardsimultaneityrelation,sinceitassumeslighttravelsatthesamespeedinalldirections.
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- Supposeunifyingpower isacriterionoftheorychoice(i.e.,weshouldchoosethattheorythatfitsbetterwithothertheories).
- Then:Weshouldpreferthestandardsimultaneityrelation,sinceFriedman-Robertson-Walkerspacetimesingeneralrelativity(i.e.,"BigBang"spacetimes)areisotropicinawaythatsinglesoutthestandarddefinition.
- But:Adoptingsuchspacetimesasdescriptionsofouruniverserequiresmanyassumptions,oneofwhichjust isisotropy.
RealistResponse:• Agreethatthereisnoobservationaldifferencebetweenthestandardsimultaneityrelationandanynon-standardsimultaneityrelation.
• So:Ifempiricaladquacy(i.e.,agreementwithobservation)isthecriterionforhowonechoosesbetweencompetingtheories,thenthere'snoreasontopreferthestandardrelationtoanynon-standardrelation.
• But:Whythinkempiricaladquacyistheonlycriterionoftheorychoice?
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