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USTC,March18Cosmology,ImaginaryBlackHoles,ExtraDimensions1
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Cosmology,ImaginaryBlackHoles
andExtraDimensions
JohnWang@USTC-ICTSonMarch182005
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Outline
·QuickSummaryofResults
·StandardModel,Cosmology:ComparingTheoryversusExperiments
·ExtraDimensions,ImaginaryBlackHolesandStringFormation
Mostrelevantreferences
·C-MChen,P-MHo,I.NeupaneandJ.E.Wang,JHEP,
hep-th/0304177
·G.C.JonesandJ.E.Wang,PRDhep-th/0409070
·K.Hashimoto,P-MHoandJ.E.Wang,Phys.Rev.Lett.,
hep-th/0211090
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zoom in
our
universe
our
universe
USTC
PhysicsExtra
Dims
time T
iT
EE
Recentprojectsinclude-1)extradimensionscandrivecosmic
acceleration2)Imaginaryblackholesareconsistentandrelatedtode
Sitterspace3)cosmicstringnucleationcanbesmoothlydescribed
USTC,March18Cosmology,ImaginaryBlackHoles,ExtraDimensions4
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Inthe1980’sand1990’sthegrowingaccumulationoflaboratorydata
supportedthenowstandardmodelofgaugeinteractions
SU(3)×SU(2)×U(1)plusgravity.Allobservableandtestable
phenomenonseemedtofit.
USTC,March18Cosmology,ImaginaryBlackHoles,ExtraDimensions5
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ConceivablytheSMwouldcontinuetobeexplainallphysicsuptoat
least1013
GeVwherethegaugecouplingsappeartounify.Suchnew
highenergyphysicsrelatedtograndunifiedtheories,superstringsor
somethingelsemightneverbeseen.
log E (GeV)
20
40
20 1015 5
α
α
α α1
1
1
1
1
2
3−
−
−
−
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Inspiteofthesesuccesses,thereappearstobegrowingevidencefora
potentialREVOLUTIONinphysics!
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Thisnewmysteriousphysicshasbeennameddarkmatterand
darkenergy.
Wecanexplainearthboundphysicsinacceleratorsandlaboratories,
butunderstandingcosmologicalphenomenaappearstorequirea
changeinourbasicsconceptsofphysics!
USTC,March18Cosmology,ImaginaryBlackHoles,ExtraDimensions9
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Recentobservationshowasurprisingaccelerationtoouruniverseon
thelargestscales.Notonlyisouruniverseexpandingbutitis
undergoingacceleratedexpansion!Darkenergyisthetermusedto
describetheapparentcosmologicalacceleration.Cosmologistsobserve
thatthestarsinouruniversearenotonlymovingawayfromusbut
doingsoateverincreasingspeeds!
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Gravityisanattractiveforceandsoourexpectationisthattheremust
beanenergysourcetoaccelerateobjectsawayfromeachotherinour
universe.
Howthendoweexplainthisdarkenergy?
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StandardCosmology
Toattempttoanswerthisquestionletmebemoreconcreteand
discussthestatusofcosmologyandtheunderlyingobservational
assumptions.
Overlargescalescoveringmanygalaxies,itappearsthatouruniverse
isapproximatelyuniform.Thedensityofstars(andplanets)isroughly
thesameeverywhere.
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Asanapproximationletustakeourmodeloftheuniversetobe
exactlyuniform.Thisiswhatiscalleda
F(riedman)R(oberston)W(alker)universespecifiedinEinstein’s
languagebyametric
ds2
=−dt2
+1×
[
dr2
+r2(dθ
2+sin
2θdφ
2)]
(1)
ds2FRW=−dt
2+a
2(t)
[
dr2
1−kr2+r2(dθ
2+sin
2θdφ
2)]
(2)
S=
∫
d4xR(3)
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Astimepasses,theuniverseexpandsandhastwopossiblefates.Itcan
eitherexpandateverslowerratesorcollapseinabigcrunch.
big bang
big crunch
big bang
universe expands forever
FRWisnotourpresentuniverse!
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Einsteingravityalsoallowsfortheadditionofwhatiscalleda
cosmologicalconstantΛ.
S=
∫
d4x(R+Λ)(4)
Thistermcanbeaddedwithoutspoilingthetheoreticalconsistencyof
theequations.Physicallythistermcanariseduetothevacuumenergy
ofvariousfields,iezeropointenergy.
Apositivecosmologicalconstantcausesspacetimetoexpand,
counteractingtheusualgravitybetweenmasses.
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Ahomogeneousuniversewithapositivecosmologicalconstantis
calleddeSitterspace
ds2
=−dt2
+cosh2t[
dθ2
+sin2θdφ
2+cos
2θdψ
2]
(5)
Time
Recentaccelerationobservationscombinedwiththeapparent
uniformityofouruniverseshowouruniverseisapproximatelydeSitter.
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Theproblemisthatouruniverse’scurrentwouldbecosmological
constantisabout(10−3eV)
4whichisaverysmall.Howdoweexplain
suchasmallnumberwhenourscalesinphysicsaretypicallymuch
larger?Thisisthesocalledcosmologicalconstantproblem.
Onesuggestionisthattheaccelerationisnotcausedbyacosmological
constantbutbysomethingwhichonlyactssimilarlytoit.Our
accelerationmightbemoreeasilyexplainedinthismanner.
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Letusattackthesenewopenproblemsfromcosmology!
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Therehavebeenmanyproposalstoexplainwhatdarkenergymight
be.Almostalloftheseproposalsareessentiallyguessesinthatthey
introducenewphysicalinteractions(=fields).
·inflatons,phantommatter,tachyonmatter,ghosts
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Canweestablishwhatdarkenergyanddarkmatterarewithless
guessing?
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Oneapproachistotrytostartwiththestringtheorywhichbeganasa
theoryforstronginteractionsandevolvedtoatheoryencompassing
gaugeinteractions,gravityandquantummechanics!
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ExtraDimensions:
Asacandidatetheoryofouruniverse,canstringtheoryprovideus
withasensiblesourceofdarkenergy?
Toanswerthisquestionitisimportanttoknowwhatisdifferentor
newaboutstringtheory.Onerelevantfactaboutstringtheoryisthat
itrequires10dimensionsforspaceandtime!
Howdowedealwiththefactthatouruniversehasfourapparent
dimensions?
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Oneideaistodividethetendimensionsofstringtheoryintolargefour
dimensionsandsixsmalldimensions.
zoom in
our
universe
our
universe
USTC
PhysicsExtra
Dims
Despitetheirsmallsize,sincestringtheoryhasextradimensions,my
collaboratorsandIcheckedtoseeiftheyarerelevanttocosmology.
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Wefoundthatevensmalldimensionscanaccelerateourlarge
dimensions!ThisissurprisingsinceweearliersaidthatwithoutΛ
thereisnoacceleration.
Speciallychosentimedependentextradimensionsprovideenergyas
seenbythelargefourdimensions.
USTC
Physics
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WeexaminedthespacetimesolutionofEinsteingravity
R1,3
×H6(6)
ds2
=α2(t)
[
−a6(t)dt
2+a
2(t)dx
23
]
+α−2/3
(t)[
dψ2+sinh
2ψdΩ
25
]
(7)
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Dimensionallyreducethistofourdimensionstofindthespacetimeas
seenbythefourdimensionalobservers,us!
S=
∫
d4x[R−(∂φ)
2−15e
3φ/8]∼
∫
d4x[R−Λ(=15e
3φ/8)](8)
Thisisgravityandascalartheorywhichproducesaccelerationwhen
∂φ∼0.Toafourdimensionalobserver,extradimensionsactlikea
scalarfield.
Infactasopposedtomorephenomenologicalmodels,theinflatonhere
isnotarbitrarybutdeterminedbygravity!
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ImaginaryBlackHoles:
Blackholesarecrazy,interestingandhavebeenextremelyusefulin
betterunderstandingquantumgravity!Theyarefascinatingandtoday
Ihavethepleasuretodiscussaninterestingtwistontheusualstory
andwilldescribetheirapplicationinanewsetting.
ThepointIwishtogetacrossisthatthereisanaturalconceptof
imaginaryblackholeswhichisconsistent,hasusefulapplicationsand
afterthistalkIhopeyouwillagreethattheyarenotsostrange!
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BeforegoingintodetailsIwanttosummarizemymainpoint.
Imaginaryblackholesareblackholesplacedatimaginarycoordinate
valuesforexampleimaginarytime.Theyproducetimedependent
spacetimeswithinterestingproperties.
time T
iT
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Ablackholeisthespacetimecreatedbyapoint-likesourcewithmass.
Farawayfromthissourcethenwearemovearoundaswewantbutif
wepasstheblackholehorizonthenwearedoomed!
ThesimplestSchwarzschildblackholesolutionis
ds2BH=−(1−
2M
r)dt
2+
dr2
1−2Mr
+r2(dθ
2+sin
2θdφ
2)(9)
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Howeverletusstripawayallthesecomplexitiesofblackholeslike
theirnon-linearity,horizonstructure.Thisisacomplexityofthe
Einsteinaction.Wewanttojustdealwiththeessenceofablackhole
whichisitsmassivepoint-likeobjectsourcewhichwecanmove
aroundinspace.
x
y
z
mass
source
t=time
r=space
Notewehavespacetimenotspaceandthesourceexistsforalltime.
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Thefourrealcoordinatesofouruniversearet,x,y,z.Weliveinthe
“real”worldBUTwhataboutmovingsourcesincomplexcoordinates?
real coordinate T
imaginary
coordinate iT
Inotherwordswhilebeforewediscussedmovingtheblackholesource
inrealvalueofspacetime,nowwemovetheblackholesourceinto
complexvaluesofspacetime.Hencetheseareimaginaryblackholes.
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Tocreatearealsourceweaddingtwocomplexsources,
z+z∗
=(x+iy)+(x−iy)∈Rforallz∈C.
BHsourceδ(√
x2+y2+z2)
ImagBHsourceδ(√
x2+y2−(t−it0)2)+δ(√
x2+y2−(t+it0)2)
(10)
Taketheblackholesourceand1)rotateitsoitisnowextendedina
spatialdirectionandnottime2)moveitfromtheorigininpairs
it
z
t
it -it0 0
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ds2
=(1−2Mt
Σ)2[
(dx4)2
+Σ
4
(∆+(a2−M2)sinh2θ)3(−
dt2
∆+dθ
2)]
+∆sinh
2θ
(1−2MtΣ)2dφ
2
A=2aMtsinh
2θ
∆+a2sinh2θdφ
∆=t2−2Mt+a
2,Σ=t
2+a
2cosh
2θ
time
space
dS structure
dS structure
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ImaginaryblackholessourceafibereddeSitterwithoutrelyingona
cosmologicalconstant!Thisnewresultmayextendfieldtheory/string
theorydualitiestotimedependentsystems.
ds2
=((M2−a
2)cosh
2η+a
2sinh
2η)
2(
σ2(dx
4)2
Σ2+
+Σ
2+
(M2−a2)3(−dσ2/4σ
2+dη
2))
+(M
2−a
2)Σ
2+cosh
2ηsinh
2ηdφ
2
((M2−a2)cosh2η+a2sinh
2η)2,
A=2aMt
KH+dφ
a2+(M2−a2)coth2η
Σ+=(tKH+)
2+a
2.FiberingofdS2overtheηdirectionforη≥0.
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String/defectformation:Pointlikeobjectssuchasblackholes,or
spatiallyextendedobjectslikemassivestrings,canformintheearly
universeduetotheuniverse’ssmallsizeandhencehighenergies.
Insometheories,stringsariseasconfinedfluxtubes.Isthispossible
eveninstringtheorywherethefundamentalobjectsarestrings?YES!
Ifwehavedefects,howdoweseetheirformation?
GenericInitialConditions
T=0Sbraneshaveappeared
late timeremnants
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First,astringisaonedimensionalspatialobject.Toformastringwe
needtostartfromsomeotherobjectwhichissimilartoastring.In
otherwordsweneedthecorrectinitialconditions.
Asimplestartingpointforexampleisacylindricaltube.Ifthesizeof
thetubecollapsesthenwewillformastring.
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Startingfromstringtheory,wederivedanactiontodescribehowto
interpolatebetweentheinitialconditionsandthefinalstrings/defect.
Wealsofoundanalyticsolutionswhichdescribehowfluxtubesform
intostrings.
S=S0
∫
d3+1
x√
det(δij−∂it∂jt+Fij)(11)
Forexamplethisistheactionforacylinderwith3+1worldvolume
dimensions,locatedatpositionsxiandwithFijfieldstrengthie
electricandmagneticfields.
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Searchingforinterestingsolutionswefoundthoserepresentingtime
dependentcollapseoftubesofelectricfluxintothinlines.
EE
r=c
tE=1=criticalvalue(12)
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Theformationofsuchobjects,whichtypicallyarecalledtopological
defects,hasbeenstudiedbeforealthoughwithtwodifficulties.
Beforesolutionswereeithernumericalorcollapsedinfinitetimesothe
solutionscouldnotbetrustedatthecollapseddegeneratepoint.This
isthefirsttimeananalyticandnon-singularsolutionhasbeenfound!
Whatisperhapsmostamazingisthatwearedescribingtheformation
ofstringsinastringtheory!
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Summary
·Lotsofunexplainedphenomenon
·Extradimensionscanberelatedtocosmicacceleration
·Imaginaryblackholesmakesense!andhavedeSitterusestoo
·Newnon-singulartimedependentsolutionsdescribingstring
formation