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Cosmological Expansion and Its Effect on Small Systems

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Cosmological Expansion and Its Effect on Small Systems

    Cosmological Expansion and Its Effect on

    Small Systems

    Commun.Theor.Phys,(Beijing,China)46(2006)PP.987990

    ?InternationalAcademicPublishersVo1.46,No.6,December15,2006

    CosmologicalExpansionandItsEffectonSmallSystems

    B.MirzaandM.ZamaniNasab

    DepartmentofPhysics,IsfahanUniversityofTechnology(IUT),Isfahan(84156),Iran fReceivedJanuary23,2006;RevisedApril3,2006)

    AbstractInordertostudytheeffectoflargescalecosmologicalexpansiononsmallsystems,weassulneaFriedmann

    Robertson

    Wkertypecoordinatesysteminpresenceofanonzerocosmologicalconstantandderiveanon-static

    Reissner-Nrdstr6mmetric.Itisananalytic[unctionoflrforallvaluesexceptatr=0,whichissingular.Byde

    terrainintheequationofmotioninthismetricwecanestimatehowexpansionoftheuniversemayaffectPioneer~

    motion.Becausethemetricdoesnothaveanyeventhorizonandsohighpotentialregionsareaccessible,thismayhejp

    usjnbetterunderstandingAGNphenomenon.

    PACSnumbers:04.,

    Keywords:general

    04.20.Jb

    relativity,exactsolution

    1Introduction

    Theexpansionoftheuniverseiscurrentlyundergoing

    aperiodofacceleration,whichnowseemsinescapable:

itiscurrentlymeasuredfromthelightcurvesofsev-

    eralhundredtypeIasupernovae[Jandindependentlyin

    ferredfromobservationofthecosmicmicrowaveback

    groundfCMB)bytheWMAPsatellite[JandotherCMB experiments.L3_Oneofthepossibilitiesforsuchacceler

    atedbehaviouristheexistenceofacosmologicalcon- stant.Furthermore,acarefulanalysisoforbitaldatafr01"i1 pioneer10/11hasbeenreportedinRef.4],whichindi

    catestheexistenceofaveryweak,longrangeacceleration a=f8.741.33)×108cm/s2,directedtowardthesun. Itisintriguingthatthisaccelerationisconstantintime. Cosmicexpansionmightbeasourceforthisanomalous acceleration.Thesefactshasrisenupanewintereston nonstaticsolutionsofEinsteinequations. Historically.EinstienandStrausin1945publisheda paperonhowtwometric8,onestatic(1ikeSchwarzchild metric)andonetimedependent(1ikeRWmetric}can

    changedtogetheratsomefimits?[5JThisproblemhasbeen reconsideredbvdifferentauthorssincethen.lu_Some

    yearsbeforethese,in1933,McVittiefoundametricfor amaSspointintheexpandinguniverse.uItisjustthe

    SchwarzschildblackholeembeddedintheFriedmann- RobertsonWalkeruniverse.Later.in1993Kastorand Traschenfoundthemultiblackholesolutionintheback

    groundofdeSitteruniverse.zJwhichdescribesthedy

    namicalsystemofarbitrarynumberofextremeReissner——

    NordstrSmblackholesinthebackgroundofdeSitter universe.In1999ShiromizuandGenextendeditinto spinningversion.31In2000

    ;..

【】

    .

    Nayaketalbyusing

    Whittaker'sl5]andVaidya's16Jworksderivedasimpler

    caseoftheShwarzschildblackholeinthebackground ofEinsteinuniverse.whichtheycalledVESfVaidya- EinsteinSchwarzschild)spacetime.Inthispaper,byus. ingthemethoddevelopedinRef.i171wederiveanon

    staticsolutionforachargedsphericallysymmetricblack hole.

    Thispaperisorganizedasfollows.InSec.2weassume ageneralFriedmannRobertsonWalkertype(FRwtype)

    coordinatesystemandsolveEinstein'sfieldequationsfor amassiveparticlewithchargeQ.InSec.3,weseethat metricpropertieshaschangedandithasnosingularity exceptattheoriginr0.Becausethismetricisnon- singularforr?0andthereisnoeventhorizon,thehigh potentialregionsareaccessible.Thishugeamountofen

    ergycanbeconvertedtothermalorotherphysicalforms ofenergyandprovidingusanewsupermachineryIorhigh energyastrophysics.Thiscanmakedrasticchangeinour understandingoftheAGNphenomenon.InSec.4,we studythemovementofatestbodyinthismetricand determinetheeffectofexpansiononit.

    2Metric

    Toobtainanonstaticsphericallysymmetricsolution ofEinsteinequationsforanlassiveparticlewithcharge Q,weassumethegeneralformofFRWmetricas

    ds=B(r,t)dt.+0()[(r,t)dr.

    +r2(dO.+sinOde)].(1)

Usingthenon-vanishingcomponentsofga6,aftera

    straightforwardcalculationwearriveatthefollowingcom ponentsfortheEinsteintensor:

    V00=()+B+3()

    Gl1=2aSA+ahBA

    B2

    G

    A

    ) +(

    BB

    

    r2a2A上—ra2A2

    BAa2A1

    +一一_+

    r3r0ar2h.2r?

    .

    rB

    22一—2A2一—一百一—一十

    r2nar2n2r2a2A/~r2B .B22AB'4AB2.2AB

    r2a2A2r2BAr2B2

    +

    G33=sin2OG22

    ,

    whereprimesanddotsstand specttorandt.Theelectric Qis

    西:

    (2)

    (3)

(4)

    (5)

    (6)

    forthederivativeswithre

    potentialforapointcharge =Ao

    inwhichP=a(t)risthephysicalradialcoordinate.So

    thenonvanishingcomponentsofelectromagnetictensor

    are

    6=AbObA.,(8)

    F

    Q01

    0l一——,,

    rD02AB(),.=

    988B.MirzaandM.ZamaniNasabVbl_46 F=(),=1(),

    =

    iQ).'(10)

    (9dF),(11)

    To.=

    87ra2A(=((12)

    :=87tAB(.=sin20T2z.(13)

    AandBarerelatedtogethera.sfollows.Weknowthat

    VbF=47rj=0,(14)

    whichresultsin

    (+)(A+B)-o-(15)

    .()A_p2()A

    $calefactorsatisfiestheFriedmannequation,

    AssumingA=A(p),B=B(p)wehave

A=a(t)A,A=h(t)rA,(16)

    B=a(t)B,B=h(t)rB,(17)

    where()standsforthederivativesrespecttoP.Byusing Eqs.(15)^(17)onehas

    a-O,T?)(+)-o-(18)

    SinceAandBneedtobecomeoneatlargedistances,thus B=A_..(19)

    Einsteinfieldequationswithacosmologicalconstantare G.b+6=87r6.(20)

    UsingEqs.(1S),(iT),and(19),therr-fieldequationleads ((去一)一?AP2--Q2

    ()=()=

    sofromEq.(21)weobtain

    

    A

    3,

    3A+P

    =p

    p3+

    whereCistheconstantofintegration.DefiningD=pA,wecansolvetheequationabove,

    DJ.=

    [1+(C/p)+(Q./p.)(Ap./3)】士,/,『干_i-==-_=

    

    2Ap/3'

    (21)

    (22)

    (23)

    (24)

    ByevaluatingthepostNewtonianlimitsweobservethatmerelyD?

    isphysicallyacceptableandtheactualvalueof

Cis2M.Thuswehave

    B=Al=1[

    Thereforethefinalformofds2is d::f

    2l

    +2e2VZ~'~tfl

    +1-2M

    +.(25)

    +1-

    2M

    +

    Q2

    +1-2M

    +

    Q2

    ]-idr2+p2(dO2+sin20d.),(26) whichcanbewritteninaformsimilartoBoyer-Lindquistmetric,

    =

    dt2+e2p2dr2

    (dO2+sin20d]1

    Ap=

    [(p22Mp+Q.一会p4)+(27)

    Wemaysummarizethefunctionalbehavioroftheobtainedmetrica5follows.IfA=0,equation

    (27)reducesto

    well-knownReis8neNordstr6mmetric.ForM==0itchangestoFriedmann-Robertson-Walkermetricforde

    Sitterspace-time.WeshouldmentionthatthepresentednonstaticmetricandSchwarzschild-deSittermetric

    =

(++1-2M+Q2一会r.-1dr2+r2(dO2+sin20d(28)

    aretransformabletoeachotherbythefollowingcoordinatetransformation, p,+

    

    a-/

    IfM=Q:0thistransformationreducesto

    A(t3)fidt5

    1(2M/f)+(/.)(A./2)

    p_n(,f=t一言n(一会p.)

    (29)

    (30)

No.6CosmologicalExpansionandItsEffectonSmallSystems989

    3Singularity

    Equation(271easilyshowsthatAD=0obtainssingularitiesofourmetric.Butthisequationhasnorea1solution

    exceptatP0.Thisisaveryinterestingpropertybecauseitshowsthatifthereisanol1.zerocosmologicalconstantand

    smal1scalesactuallyparticipateincosmicexpansion,thereisnoeventhorizonsohighpotentialregionsareaccessible.

    Thishugeamountofpotentialenergycanbeconvertedtothermalorotherphysicalformsofenergyandprovidesus

    withanewsupermachinaryforhighenergyastrophysics.Thiscanmakeadrasticchangeinourunderstandingof

    theAGNphenomenon.IfwemakecoordinatetransformationslikeEq.(29)singularityappearsagain,butbecause

    weuseunchangedphysicalcoordinatesinallofcalculationsitseemsthatweproduceanartificialsingularitybythese

    transformations.Existenceofnakedsingularitiesandtheirinterestingproperties3xestillunderconsideration.

4EquationofMotionforaTestParticle

    OurnexttaskistoobtainandsolvethegeodesicequationsofafreelyfallingmaterialparticleinaproperFRW-type

    coordinatesystem.Inthestaticcase,thebasicequationswhichdeterminethegeodesicstructureofthemanifoldhave

    beenfullydeveloped.[18,19JWesolvetheprobleminthenon-staticcasebyusingtheequationsoffreefall

    +r=.(31)

    Usingthenon

    vanishingcomponentsofaffineconnectionobtainedfromEq.(1)resultsinfourequations.Sincethe

    fieldisisotropic,wemayconsidertheorbitofourparticletobeconfinedtotheequatorialplane,thatis0:7r/2.

    Then目一

    relevantequationimmediatelyissatisfiedandwemayforgetabout0asadynamicalvariable.Introducingthe

    previousvariablepandusingtheFriedmannequation,theaboveequationsarerewrittenas d2t

    

    dr2+

    +

    d2p

    dT2

    p

    (+2pA2+AP2A (+pA2+A2AA)()

    dtdp

    ()()+.(;

    d2t

    P

(P++P)()+()P().=.,

    d2~

    I

    2dpde-

    4-

    (pzdr):.…一一In一一I=IId7_2'pd7-dfd7_\, Thelastequationgives

    P.L,

    ,d7_

    whereJisconstantofintegration.SimplifyingEq.(32)andusingEq.(33)yields

    ()+dp

    (32)

    (33)

    (34)

    (35)

    (36)

    whereC1isconstantofintegration.Wecanreplacedp/dTby|pdt/dTadr/dTintheequationabo

    vethenif

    P--4o.oilehasdt/dT:1,dr/dr0.Thereforec1wouldb.efixedbyasymptoticbehaviorasC11.Eq

    uation

    (36)thenbecomes

    RewritingEq.(32)as

    d2t

    

    d,r2

    =

    A(?.PA).

    ()A*dtdpA3.2A/\ddt,~22dtdp

    +()2+]+A2-[dth2V~dtdp+()]:.,andusingthelineelementequationofamassiveparticleintermsofPv

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