Cirular Cosmi String Loop in Brans-Dike Theory
A.Barros y
, A. A. Sen z
,and C. Romero ?
y
Departamento deFsia,UniversidadeFederaldeRoraima
69310-270BoaVista,RR,Brazil
z
Harish-ChandraResearhInstitute
ChhatnagRoad,Jhusi,Allahabad, 211019,India
?
Departamento deFsia,UniversidadeFederaldaParaba
CaixaPostal5008, 58059-970Jo~aoPessoa, PB,Brazil
Reeivedon15Marh,2001
The gravitational eld of a stationary irular osmi string loop, externally supported against
ollapse,isinvestigatedintheontextofBrans-Diketheoryintheweakeldapproximationofthe
eldequations. Thesolutionisquasi-onformallyrelatedtotheorrespondingsolutioninEinstein's
GeneralRelativity(GR)andgoesovertotheorrespondingsolutioninGRwhentheBrans-Dike
parameter!beomesinnitelylarge.
I Introdution
Phasetransition in theearlyuniversemighthave
pro-dued sometopologialdefets [1℄. Amongstthese
de-fets,osmistrings,haveattratedalotofinterestfor
various reasons [2℄. For instane, they are apable of
produing observational eets suh as double images
ofquasarsandareonsideredtobepossibleandidates
as seedsforgalaxiesformation.
Insomegaugemodelsstringsdonothaveanyend,
andthusareeitherofinniteextentformorlosed
ir-ular loops. One of the most notable features of the
gravitationaleldofastraightinniteosmistringis
the presene of an \angular deit", in an otherwise
Minkowskianspaetime,havingamagnituderelatedto
the linear energydensity of thestring by the
equa-tion Æ = 8G. In fat, this angular deit plays a
keyrolefortheprodutionofdoubleimagesofquasars
[3℄.
The deit angle model is widely believed to be a
good approximation for desribing a spaetime
exte-riortothestringore. Frolov,Israeland Unruh(FIU)
[4℄ used this approximation for studying alosed
ir-ular osmi stringloopat amoment of time
symme-try. With the help ofthe initialvalue formulation[5℄,
theyproduedafamilyof momentarilystationary
ir-ular osmi strings,whih areregardedas thin loops
either atthetime offormationorattheturningpoint
of expansionand ollapse. An important assumption
in thisworkis thatallpointsontheirular stringbe
onial singularities with angular deit equalto that
ofainnitestraightstringof equallinearenergy
den-sity. Hughes, MManus, and Vandyk(HMV) [6℄
in-vestigated further the problem of angular deit of a
irularstring. Theyonsideredaweakeldstationary
solutionsof Einstein'seld equations for athin
iru-larstringandalsoestablishedthatradialstressshould
be introdued to support the stringloop against
pos-siblegravitationalollapse. Theydeterminedtheform
oftheradialstressfromthestressenergyonservation
relations. The main result of their study is that in
weakeldapproximationairularstringprodues,
lo-ally,alongthestringthesameangulardeitasdoes
astraightinnitelylongstringwiththesamelinear
en-ergydensity. Inanotherwork,MManus andVandyk
[7℄onsidered astringloopwitharotationoftheloop
whih providesthe neessaryentrifugal reation
par-tiallyorfullyinordertoavoidthepossiblegravitational
ollapse.
It turns outthat at suÆient high energysales it
seemslikelythatgravityisnotgivenbytheEinstein's
ation,butbeomesmodiedbythesuperstringterms.
Inthelowenergylimitofthisstringtheoryonereovers
Einstein'sgravityalongwithasalardilatoneldwhih
isnon-minimallyoupledto gravity [8℄. Ontheother
hand,salar tensortheories, suh as Brans-Dike
the-ory(BD) [9℄,havebeenonsiderablyrevivedin reent
years. ItwasshownbyLaandSteinhardt[10℄that
be-auseof theinterationoftheBDsalareldwiththe
Higgs type setor, the exponential ination in Guth's
model [11℄ouldbesloweddowntopowerlawoneand
thegraeful exit in theination isthus ompleted via
bubble nuleation. Although dilaton gravity and BD
theory arisefrom entirelydierentmotivations, it an
beshownthattheformerisaspeialaseofthelatter,
at least formally [12℄. Anothermotivation for
study-ing gravitationalproperties ofdefets in BDtheory is
that the onlydefets we anhope to observenoware
thoseformedafterorneartheendofination,andthe
formationofsuhsuperheavydefetsis relativelyeasy
toarrangein aBrans-Diketypetheory [13℄.
In the present work we have studied the
gravita-tional eld of a stationary irular osmi stringloop
in Brans-Diketheoryintheweakeldapproximation
oftheeldequations.
TheeldequationsintheBDtheoryarewrittenin
theform G =8 T + ! 2 ( ; ; 1 2 g ; ; ) + 1 ( ;; g 2); (1:1) 2= 8T
2!+3
; (1:2)
whereis thesalareld,! istheBDparameterand
T denotesthetraeoftheenergy-momentumtensorT
[9℄. Inthe weakeld approximationofBDtheory one
an assume g
= +h where jh
j << 1 and
(r)=
0
+(r)withj=
0
j<<1where 1=
0 =G 0 = (2!+3) (2!+4)
GinordertohaveaNewtonianlimitfortheBD
theory[9℄.
Ithas been shown reentlyby Barrosand Romero
[14℄,thatintheweakeldapproximationthesolutions
oftheBDequationsarerelatedtothesolutionsof
lin-earized equationsin GRwith the sameT
in the
fol-lowingway: ifg gr
(G;x)isaknownsolutionofthe
Ein-stein'sequationsin theweakeld approximationfora
given T
, then the BD solution orresponding to the
sameT
,in theweakeldapproximation,isgivenby
g bd
(x)=[1 G
0 (x)℄g gr (G 0
;x) (1:3)
where(x)mustsatisfy
2(x)= 8T
(2!+3)
; (1:4)
andGis replaedbyG
0
dened previously. Hene,to
get the spaetime for irular string loop one has to
solve the equation (1.4) with appropriate T
for the
irularosmistringloop.
Wehave taken theform of the energy momentum
tensorforthestringloopasproposed byHughesetal.
[6℄:
T t
t
= Æ(r a)Æ(z) (1:5a)
T
=kÆ(r a)Æ(z) (1:5b)
T r r = k r
(r a)Æ(z) (1:5)
in whih denotes the Heavisidestepfuntion. Here
weareonsideringaninnitelythinloopofstringwith
radiusa,lyinginthex-yplaneandenteredatthe
ori-gin. ForairularstringT
playsthesameroleasthat
of the longitudinal stress T z
z
for astraight stringand
T r
r
istheexternalradialstressrequiredforsupporting
the loop againstollapse and heneit is notloalized
onthestring.
With(1.5),equation(1.4)beomes
r 2
= 8
2!+3
[(k )Æ(r a)Æ(z)
+ k
r
(r a)Æ(z)℄ (1:6)
Writing=
1 +
2
,weanseparateequation(1.6)into
r 2 1 = 8
2!+3
[(k )Æ(r a)Æ(z)℄ (1:7a)
r 2 2 = 8
2!+3 [
k
r
(r a)Æ(z)℄ (1:7b)
Thesolutionsoftheequationsaboveanbefoundeasily
byusingtoroidaloordinates(;; )whiharerelated
toylindrialoordinates(;r;z)by[4℄
z=aN 2
sin (1:8a)
r=aN 2
sinh (01;j j) (1:8b)
where N 2
N 2
(; ) osh os . The surfaes
=
0
onstant are tori whose generating irles
have radii ash
0
and aoth
0
. The ring r = a,
z=0isnowgivenby=1.
In toroidal oordinates the solution of equation
(1.7a)nowbeomes[15℄
1 = 2 5 2 (k )
(2!+3)
N(; )
F(tanh (=2))
osh(=2)
(1:9a)
where F denotes omplete ellipti integral of the rst
kindandthesolutionfor
2 beomes[6℄ 2 = 4k
(2!+3)
N(; ) 1 X n=0 [H n ()P n 1=2 (osh) +G n ()Q n 1=2 (osh)℄" n
os(n ) (1:9b)
where H n () Z 0 dxN 1 (x;0)Q n 1=2
(oshx) (2:0a)
G n () Z 0 dxN 1 (x;0)P n 1=2
(oshx) (2:0b)
and P m n 1=2 and Q m n 1=2
are toroidal Legendre
fun-tions, "
n
2 Æ
0
n
, and also we have used the
from equation(1.3)thespaetime forastationary
ir-ularosmistringloopinBDtheoryintheweakeld
approximationisgivenby
ds 2
=[1 G
0 ( 1 + 2 )℄ds 2 HMV (G 0 ) (2:1) where 1 and 2
aregivenbyequations(1.9a)and(1.9b),
and ds 2
HMV (G
0
) is the metriobtained by Hughes et
al[6℄forirularstringloopin GRwithGreplaedby
G
0 .
To investigate the presene of onial singularities
on thering =1, weproeed as follows: rstof all,
for the salar eld solution in equation (1.9) we may
addanarbitraryonstantAsuhthat
=
1 +
2
+A (2:2)
For !1 oneannd the asymptotibehaviourof
1 and
2
fromequations(1.9)whih isgivenby[6℄
1 !
4(k )
(2!+3)
(2:3a)
2
!0 (2:3b)
Hene,wehave
1 G
0 =1+
4G
0
(k )
(2!+3) G
0
A (2:4)
Then, for the surfae t =onstant and = onstant
and using the toroidaloordinates one anobtain for
themetri(2.1)[6℄
ds 2
=[1+ 4G
0
(k )
(2!+3) G 0 A℄ [4a 2 e 2 e 4G 0
( k )+2b
(d 2
+d 2
)℄ (2:5)
wherebisthedimensionlessombinationofalladditive
onstantsappearinginthesolutions. Now,letusmake
thefollowingoordinatetransformation:
= (2:6a)
2ae
(+2G0(k ) b)
=r (2:6b)
Giventhatbisarstorderterminthisapproximation
[6℄wean write
1+ 4G
0
(k )
(2!+3) G
0 A=1
+ 4G
0 ( k)
(2!+3)
ln(r=2a) G
0
A (2:7)
By hoosing A = 4( k )
(2!+3) ln(r
0
=2a), where r
0
is a
on-stant,weget
ds 2 = 1+ 4G 0 ( k) ln(r=r 0 ) dr 2 (1 4G 0 ( k)) +r 2 d 2 (2:8) Deningdr 0 = dr p 1 4G 0 ( k )
, one anwrite theabove
equationas ds 2 = 1+ 4G 0 ( k)
(2!+3) ln(r 0 =r 0 ) h dr 0 2 +r 0 2 (1 4G 0 ( k))d 2 i (2:9)
Finally, if we put k = for \string"matter we are
ledto ds 2 = 1+ 8G 0
(2!+3) ln (r 0 =r 0 ) h dr 0 2
+(1 8G
0 )r 0 2 d 2 i (2:10)
Thisis exatlythe samemetriforastraightvauum
stringforsetiont=onstantandz=onstantwhih
wasearlierobtainedby Barrosand Romero[16℄ in BD
theoryin theweakeld approximation. Atthispoint,
itis worthwhilementioningthatthisspaetimehasan
angulardeit givenby
=8G 0 " 1 1
2!+3 ln r 0 r 0 !#
asmaybeseendiretlybyintegrating(2.10) arounda
irleofradiusr 0
.
Finally, itis wellknown that in the weak eld
ap-proximationwhen!!1theBDsolutiongoesoverto
theorresponding solutionin Einstein'sGR,although
thisisnotalwaystrueintheaseofexatsolutions[17℄.
Inouraseoneanhekthatfor!!1,both
1 and
2
beome zero and G
0
! G, so one anreover the
orrespondingGRsolutionforairular osmistring
obtainedbyHughesetal[6℄.
In onlusion, we have obtained the spaetime for
irular osmi string loop in BD theory in the weak
eldapproximation. Indoingso,wehavefollowedthe
method presribed by Barros and Romero[14℄. When
the loop is made of \string" matter we nd out that
at points near the string( ! 1)wereover the
re-sultpreviouslyobtainedbyBarrosandRomero[16℄for
astraight stati loal string in BD theory. The
solu-tiongoesoverto theorrespondingsolutionin GRin
thelimit ! ! 1. Hene We havegeneralised the
re-sultpreviuoslyobtainedbyHugheset.al [6℄forirular
osmistringin GR,inBrans-Diketheory.
Asthetrajetoriesofthelightrays,whiharegiven
by the nullgeodesis,the onlyhangeinvolved in BD
theory is the replaement of G by an new !
depen-dent\eetive"gravitationalonstantG
0 =(
(2!+3)
(2!+4) )G
andfor!tobeonsistentwithsolarsystemexperiment
andobservation,!500[18℄,thismeansthatphotons
travellinginthespaetimewillexperieneadereaseof
followsthatthedistortionoftheisotropyoftheCMBR
due to the gravitational eld of the irular osmi
string loops in BD theory may be alulated diretly
from the results obtainedin GR. A detail analysis of
thefullnonlinearBDeldequationswillertainlygive
moreinsight to the problem and for that a detail
nu-merial alulation should be done whih will be the
aimofourfuturestudy.
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