Physis in the Global Monopole Spaetime
E.R. Bezerra de Mello
Departamento deFsia-CCEN
UniversidadeFederaldaParaba,C.Postal 5.008,Brazil
E-mail: emellosia.ufpb.br
Reeivedon2February,2001
Inthispaperwereviewsomeimportantaspetsoftheglobalmonopolespaetimeandpresenthow
thismanifoldmodies,atlassialandquantumpointsofview,themovementofahargedpartile.
Theexpliitalulationsoftherenormalizedvauumexpetationvaluesoftheenergy-momentum
tensors, hT(x)iRen:, assoiated with massless bosoni and fermioni elds are also presented.
Moreovertheeetofthenonzerotemperatureinthispreviousformalismisanalyzed. Finally,we
brieypresentotherappliationsofthismanifoldinthetopologialinationandondensedmatter
system.
I Introdution
It iswellknownthat dierenttypesoftopologial
ob-jetsmayhavebeenformedduringUniverseexpansion,
suhasdomainwalls,osmistringsandmonopoles[1℄.
Thebasiideaisthatthesetopologialdefetsappeared
due tobreakdownofloalorglobalgaugesymmetries.
SupposethatwehavetheHiggseld a
(a=1;:::;N),
whosepotentialis
V()=
4 (
2
2
0 )
2
; 2
= a
a
; (1)
where
0
isthevauumexpetationvalueanda
ou-pling onstant. Thismodelgivesrisetodierenttypes
of topologialdefets: DomainwallforN =1,osmi
stringforN =2andmonopoles forN =3. Thesalar
mattereldplaystheroleofanorderparameterwhih
outsidethedefetaquiresanonvanishingvalue.
Inthisreviewwepresenttheeldequationsforthe
global monopole in setion II, using a spei model
oupled to the Einstein equations. In setion III, we
analyzethelassialandquantummotionofaharged
partileinthespaetimeofanidealizedpointlikeglobal
monopole, onsidering the indued eletrostati
self-interation. InsetionIV,wealulatetheGreen
fun-tionsassoiatewithmasslesssalarandfermionields
and obtain the renormalized vauum expetation
val-ues (VEV) of their respetive energy-momentum
ten-sors, hT
(x)i
Ren:
. The eet of the temperature in
the salar massless Green funtion and onsequently
itsinuene onthealulationofhT
iisevaluatedin
setion V. In setion VI, we briey present other
ap-pliations ofthisformalism. Finallyweleft forsetion
VII ouronludingremarks.
II Field Equation for a Global
Monopole
A.The Model
A global monopole is a heavy objet formed in
the phase transition of a system omposed by a
self-ouplingiso-salartriplet a
,whoseoriginalO(3)
sym-metryisspontaneouslybrokentoU(1)[2℄.
L= 1
2 g
(
a
)(
a
)
4 (
2
2
0 )
2
: (2)
Theeld ongurationdesribingamonopoleis
a
=
0 f(r)^x
a
; (3)
wherex a
x a
=r 2
. Themostgeneralstatimetritensor
withspherialsymmetryanbewrittenas
ds 2
= B(r)dt 2
+A(r)dr 2
+r 2
(d 2
+sin 2
d 2
): (4)
TheEuler-Lagrange equation forthe eld f(r)in the
metri(4)is:
(r 2
f 0
) 0
Ar 2
+ 1
2B
B
A
0
f 0
2f
r 2
2
0 f(f
2
1)=0; (5)
wheretheprimedenotesdierentiationwithrespetto
r. The energy-momentum tensor assoiated with the
mattereldisgivenby
T t
t
=
2
0
f 2
r 2
+ (f
0
) 2
2A +
4
2
0 (f
2
1) 2
;
T r
r
=
2
0
f 2
r 2
(f 0
) 2
2A +
4
2
0 (f
2
1) 2
; (6)
T
= T
=
2
0
(f 0
) 2
+
2
0 (f
2
1) 2
InatspaethemonopolehassizeÆ 1
p
0
. Itsmass
isM 0
p
.
Outside theoref 1and theenergy-momentum
tensoranbeapproximatedas
T t
t T
r
r
2
0
r 2
; T
=T
0: (7)
B. EinsteinEquations
Computing the Rii tensor for the metri tensor
(4)onends[3℄:
R
rr =
B 00
2B +
B 0
4B
A 0
A +
B 0
B
+ A
0
rA ;
R
= 1 r
2A
B 0
B A
0
A
1
A
; (8)
R
tt =
B"
2A B
0
4A
A 0
A +
B 0
B
+ B
0
rA :
WealsohaveR
=sin
2
R
:
Combiningonvenientlythese equationsweget
R
rr
2A +
R
tt
2B +
R
r 2
= A
0
rA 2
+ 1
r 2
1
Ar 2
= 1
2 R
r
r R
t
t
+R
; (9)
andonsequently
(r=A) 0
=1 r 2
1=2 R r
r R
t
t
+R
: (10)
With theaidoftheEinsteinequations
R
=8G
T
1
2 g
T
; (11)
weobtain
(r=A) 0
=1+8Gr 2
T t
t
: (12)
Tond the other omponent of the metri tensor we
write
R
rr
A +
R
tt
B =
1
rA
A 0
A +
B 0
B
= R r
r R
t
t
: (13)
Integratingwiththeondition
A(r)B(r)
jr=1
=1; (14)
weobtain
B(r)= 1
A(r) exp
8G Z
r
1 T
r
r T
t
t
rA(r)dr
: (15)
Thegeneralsolutionof(12)is
A(r) 1
=1 8G
2
0
2GM(r)=r ; (16)
whereM(r)isgivenby
M(r)=4 2
0 Z
r
0
(f 0
) 2
2A +
f 2
1
r 2
+
2
0
4 (f
2
1)
r 2
dr:
(17)
Beause T t
t T
r
r
outsidethe oreofthemonopole,in
thisregionwehave
B(r)=1=A(r): (18)
Also we an analyze the behavior of the funtion f.
Introduingnewsvariablesdenedas[4℄
x= p
0
r; =8G 2
0 ;
onendstheasymptotiexpansion
f(x)=1 1
x 2
3=2
x 4
+O(1=x 6
): (19)
Nowletusgobaktothemetritensor. Negletingthe
mass termand resalingthe t variable wean rewrite
themonopole metrias
ds 2
= dt 2
+ dr
2
2
+r 2
d 2
; (20)
where 2
=1 8G
2
0
=1 <1. Theabovemetri
tensor desribes an idealizedpointlike global monople
(G-M)defet.
Thespaetime desribed by (20)has thefollowing
interestingfeatures:
i)Itisnotat: thesalarurvatureR=R
=2
(1 2
)
r 2
:
ii)Thereis noNewtonianpotential: g
tt = 1.
iii) Thesurfae ==2has geometry ofaone, with
deitangleÆ=8 2
G 2
0 .
iv)Thesolidangleofasphereofunitradius is4 2
2
,
so smaller than 4 2
. There is a solid angle deit
Æ=32 2
G 2
0 .
v) Beause T
00
2
=r 2
, outside theglobal monopole
thetotalenergyislinearlydivergentatlargedistane:
E(r)4G 2
0 r:
Now after this brief review about the global
monopolespaetime, letus analyzesomelassialand
quantum eets produed by this geometry on the
movementofahargedpartile.
III Classial Eets
It iswell knownthat aharged/massivepartilewhen
plaedinthespaetimeofaosmistringbeomes
sub-jeted to a repulsive/attrative self-interation[5℄. A
similar phenomenon also our in the spaetime
pro-dued by a global monopole. These self-interations
aregivenby[6℄:
U = K
where r is the distane from the partile to the
monopoleand
K=q 2
S()=2>0; (22)
fortheeletrostatiase,or
K= Gm 2
S()=2<0; (23)
fortheinduedgravitationalone. Thenumerialfator
S() isafuntion oftheparameter
S()= 1
X
l=0 "
2l+1
p
2
+4l(l+1) 1
#
;
whihisniteandpositivefor<1.
These indued self-interations are onsequeneof
thedistortiononthepartile'seldausedbythe
ur-vatureand/orthenon-trivialtopologyofthespaetime.
Briey speaking these eets anbe explained by
alulatingtheNewtoniangravitationalself-interation
ofanarbitrarymassdistributionoutsidethemonopole
ore
U
G =
1
2 G
Z Z
drdr 0
m
(r)G(r;r 0
)
m (r
0
); (24)
where
m
isthemassdensity. Similarly,forthe
eletro-statiself-interationofanarbitraryhargedistribution
wehave
U
E =
1
2 G
Z Z
drdr 0
q
(r)G(r;r 0
)
q (r
0
); (25)
where now
q
isthehargedensity.
Foranisolatedpartileat somespei positionr,
weget
U
G =
1
2 Gm
2
G
R
(r;r ) (26)
and
U
E =
1
2 q
2
G
R
(r;r); (27)
whereG
R (r;r
0
)istherenormalizedGreenfuntion
de-nedas
G
R (r;r
0
)=G
(r;r
0
) G
H (r;r
0
); (28)
being G
(r;r
0
) the Green funtion dened in the
monopolespaesetion:
G
() (r;r
0
)= 1
r
> 1
X
l=0 2l+1
2
l +1
r
<
r
>
l
P
l
(os); (29)
where
l =
1
2 +
p
2
+4l(l+1)
2 .
TheHadamardfuntion,G
H (r;r
0
),assoiatedwith
thethree-dimensionalLaplae-Beltramioperatoris
G
H (r;r
0
)= 1
0
; (30)
being(r;r 0
) theone-halfofthe geodesidistane
be-tween the two points r and r 0
is the spae setion of
(20). Beauseweareinterested toevaluatethe
renor-malizedGreenfuntion intheoinidene limit,letus
take rst= 0
. Forthisase thethree-dimensional
lineelementin theG-M spaetimeis
d ~
l 2
= dr
2
2
;
whihgiveus
2(r;r 0
)= jr r
0
j
:
Soafter someintermediatealulationsweget
G
R (r;r )=
1
r
S() ; (31)
whihisniteforr>0.
The presene of this indued eletrostati
self-interationisrelevantin theanalysisofthemovement
ofan eletriharged partileplaedin the spaetime
ofaglobalmonopole. Inordertodothat wetakeinto
aounttheself-interationasthezerothomponentof
thefour-vetorA
. Thenextsubsetion isdevoted to
thisinvestigation.
A.ClassialAnalysis of the Motion
For the relativisti lassialanalysis of this
move-ment we shall use the Hamilton-Jaobi (H-J)
formal-ism. Aordingthisformalism,ourrstsetofequation
isgivenby[7℄
M dx
d +qA
=g
S
x
; (32)
being a parameter along the lassial trajetory of
thepartile.
Theseondsetofequationisgivenby
g
S
x
qA
S
x
qA
= M
2
; (33)
whereS isthefuntionalgivenby
S= Et+R (r)+()+L
Z
; (34)
with R (r) and () being unknown funtions and E
andL
Z
onstantsofthemovement.
Integrating (32) and (33) we anobtain the
equa-tionsofthetrajetoriesfordierentsituationsasshown
below. Deningu= 1
r
,weget,forthesurfae==2,
thefollowingsolutions:
u()= EK
L 2
K 2
+ p
(E 2
M 2
)L 2
+M 2
K 2
L 2
K 2
os
"
p
L 2
K 2
L
Z
(
0 )
#
;
forL 2
>K 2
,
u()= EK
K 2
L 2
+ p
(E 2
M 2
)L 2
+M 2
K 2
K 2
L 2
osh
"
p
K 2
L 2
L
Z
(
0 )
#
;
(36)
forL 2
<K 2
and
u()= M
2
2EK E
2
2
2
(
0 )
2
2K
; (37)
forL 2
=K 2
.
Fromtheaboveequationsispossibletoseethatthe
rst set of solutions presents bounded trajetory for
E 2
<M 2
and K<0,whih happensforthe
unphysi-alasewhen>1.
B. Quantum Analysis ofthe Motion
Fortheanalysis of thequantum motionweshould
usetherelativistiKlein-GordonequationorDiraone
ifthepartileis abosonorfermion,respetively.
Bosoni Case
Forthespin0asetheKlein-Gordonequation
writ-teninaovariantformreads
[2 iqA
p
g (
p
g) iq(
A
) 2iqA
q 2
A
A
R (x) M 2
℄(x)=0; (38)
with
2(x)= 1
p
g
p
gg
(x)
: (39)
whereg=det(g
). Intheaboveequationwealso
on-sidered the non-minimal oupling between the salar
eldwiththesalarurvatureR ofthemanifold.
NowapplyingthisformalismfortheG-Mspaetime
wehave
[
t 2+
2
r 2
r (r
2
r )
~
L 2
r 2
2i K
r
t +
K 2
r 2
M 2
r 2
℄(x)=0: (40)
Beauseourmetritensorisstatiandtheself-potential
istime-independent,weanadoptforthewavefuntion
theform
(x)=Cexp( iEt)R (r)Y
l;m
(;); (41)
where E is thepartile's self-energy. The solutionfor
the radialdierential equation are expressed in terms
ofhypergeometrifuntionsas
R (r)=r
l
e ik r
M( +1+i;2( +1); 2ikr); (42)
wheretheeetiveangularquantumnumberis
l =
1
2 +
p
2
+4[l(l+1)+ K 2
℄
2
with=2(1 2
)=2and
k= p
E 2
M 2
; = EK
2
k :
From the radial solution is possible to obtain the
phaseshiftÆ
l
,whihisthemostrelevantparameterin
thealulationofthesatteringamplitude.
Ifweareinlinedto onsiderthepossibilityofthis
systemtopresentboundstates,wehavetoassumethat
the parameter > 1, in whih ase K <0. So,
tak-ingE 2
<M 2
,wegetdisretevaluesfortheself-energy
givenby
E
n;l =M
2
(n+
l +1)
2
K 2
+ 2
(n+
l +1)
1=2
; (43)
withn=0;1;2:::
Fermioni Case
Consideringnowafermionipartile,wemusttake
theDiraequationin theovariantform
[i
(x)(
+iqA
(x)) M℄ (x)=0; (44)
where
(x)isthespinoeÆient. Usingan
appropri-atedtetradbasisfortheG-Mspaetimeweget[7℄:
[i (0)
t +i
(r)
r +
i
r
()
+
i
rsin
+i ( 1)
r
(r)
q (0)
A
0
M℄ (x)=0(45):
Alsousingthestandardproedureit ispossibleto
ob-tainthesetofsolutionfortheequationabove. Theyare
expressedin termsofhypergeometrifuntions. Again
from thesolution weanalulate the phaseshift,Æ
j .
Ifagainweadmittheunphysialsituationwhere>1,
it is possible to ndbound statesand theexpliit
ex-pressionforthedisreteself-energy
E
n;j =M
"
1+
K 2
(n+ p
(j+1=2) 2
K 2
) 2
#
1=2
:
IV Quantum Eets
Thenon-trivialtopologyoftheG-Mspaetimeimplies
thatthevauumexpetationvalue(VEV)ofthe
renor-malized energy-momentum tensor assoiated with an
arbitraryolletionofonformalmasslessquantumeld
mustbedierentfromzero[8℄.
TheexpliitalulationsforhT
(x)i
Ren
fora
mass-lesssalareld[9℄ and fermioni one[12℄ havebeen
de-veloped. Also,reently thermaleetshavebeen
on-sideredin thisontext[13℄.
Belowwepresentthemaineetsduetothis
spae-timein theVEVofsomequantum operators.
A. SalarCase
TheEulidean versionof theG-M metritensoris
givenby
ds 2
=d 2
+ dr
2
2
+r 2
d 2
: (47)
Themasslesssalareldpropagatorobeysthe
dieren-tialequationbelow
( 2+R )G
E (x;x
0
)= Æ
4
(x;x 0
)
p
g
; (48)
where wehaveinluded thenon-minimal oupling
be-tween the propagator with the geometry. The
Eu-lideanGreenfuntionanbeevaluatedusingtheheat
kernelapproah:
G
E (x;x
0
) = Z
1
0
dsexpf s( 2+R )g Æ
4
(x;x 0
)
p
g
= Z
1
0
dsK(x;x 0
;s): (49)
In order to obtain the heat kernel, K(x;x 0
;s), let us
solvetheeigenfuntionequationbelow
( 2+R )
(x)=
2
(x): (50)
Theompleteresultfortheequationaboveis:
(x)=
r
p
2r
exp( i!t)J
l (pr)Y
l;m
(;); (51)
with eigenvalues 2
= ! 2
+p 2
2
0 and
l =
1
p
(l+1=2) 2
+2(1 2
)( 1=8). Thesesolutions
obeytheompletenessrelation
X
(x)
(x)=
Æ 4
(x;x 0
)
p
g
: (52)
Nowweareinpositiontoobtaintheheat kernelby
K(x;x 0
;s) = Z
1
1 d!
Z
1
0 dp
X
l;m
(x)
(x
0
)exp( s 2
)
= exp
2
2
+r 2
+r 02
4 2
s
16(s) 3=2
(rr 0
) 1=2
1
X
l=0
(2l+1)I
l
rr 0
2 2
s
P
l
(os): (53)
d
Weaneasilyverifythatfor=1,
l
=l+1=2andin
thisaseitispossibletogetalosedexpressionforthe
heat kernel[14℄:
K(x;x 0
;s)= 1
16 2
s 2
exp
(x x 0
) 2
4s
: (54)
FinallyweanobtaintheEulideanGreenfuntionby
integrating(53). Ourresultis
G(x;x 0
) = 1
8 2
rr 0
1
X
l=0
(2l+1)
Q
l 1=2
2
2
+r 2
+r 02
2rr 0
P
l (os):
(55)
TheComputation ofh 2
(x)i
Ren
TheVEVofthesquareofthequantum eld
opera-torisgivenby
h 2
(x)i= lim
x 0
!x G(x;x
0
): (56)
Howevertheaboveexpressionisdivergent. Soin order
to obtain a nite and well dened value for h 2
(x)i,
we haveto renormalize it. We shall use thestandard
proedureasshownbelow:
h 2
(x)i
Ren = lim
0
[G(x;x 0
) G
H (x;x
0
whereG
H (x;x
0
)istheHadamardfuntion givenby[9℄
G
H (x;x
0
) = 1
16 2
2
(x;x 0
)
+ ( 1=6)R (x)ln
2
(x;x 0
)
2
; (58)
where is an arbitrary sale introdued in order to
avoidtheinfrareddivergene and(x;x 0
)theone-half
of the square of the geodesi distane between the
pointsxandx 0
. Beauseweareinterestedinthe
oin-idene limit, let us set rst 0
= in (57). Forthis
aseweget (x;x 0
)= (r r
0
) 2
2 2
. Ouraboveexpressions
reduethemselvesto
G(r;r 0
)= 1
8 2
rr 0
1
X
l=0
(2l+1)Q
l 1=2
r 2
+r 02
2rr 0
(59)
and
G
H (r;r
0
) = 1
16 2
4 2
(r r 0
) 2
+ 2
r 2
( 1=6)ln
2
(r r 0
) 2
4 2
: (60)
Beausethedependeneof
l
withlisnotsimpleone,it
isnotpossibletodevelopthesummationintheangular
quantumnumberlin (59)inanexatway;howeverif
we remember that the parameter 2
= 1 is lose
to the unity for realisti models 1
, it is possibleto
ex-pand
l
in powersof andonsequently toobtainan
approximate expressionfor (59). Soin lowestorder in
andusingappropriateintegralrepresentationforthe
Legendrefuntion weget:
G(r;r 0
) =
1
8 p
2 2
rr 0
Z
1
dt
1
p
osht osh
1
X
l=0
(2l+1)e
l t
; (61)
with osh=(r+r 0
) 2
=2rr 0
. Moreoverevaluatingthe
summationin luptotherstorderin,wehave:
S(t)= 1
X
l=0
(2l+1)e
l t
=e t=2
1+e t
(1 e t
) 2
1
2t
1 e 2t
(1 e t
) 2
+e t
: (62)
Also the logarithmi term in the Hadamard funtion
anbeexpressedintermofQ
0
(osh)=ln
r+r 0
r r 0
and
onsequentlyin integralform. So taking into aount
allthese onsiderationswefound, upto therstorder
in
h 2
(x)i
Ren
= lim
r 0
!r [G(r;r
0
) G
H (r;r
0
)℄
=
(p 2q)
8 p
2 2
r 2
( 1=6)
4 2
r 2
ln(r);
(63)
where pand q aretwonumbersgivenby theintegrals
below
p= Z
1
0 dt
e t=2
p
osht 1
1
3 +
t
sinht 1
1+e
t
(1 e t
) 2
; (64)
and
q= Z
1
0 dt
e t=2
p
osht 1
1
t(1+e t
)
1 e 2t
: (65)
Thesetwointegralsareniteandanbeevaluated
nu-merially. Theresultsarep= 0:39andq= 1:41[9℄.
2. Evaluation of hT
(x)i
Ren
InRef. [9℄thegeneralstrutureoftherenormalized
VEV of the energy-momentum tensor for a massless
salareld in theG-M spaetimeis presented. There,
this expression was obtained on basis of dimensional
arguments,symmetries,traeanomalyandtheexpliit
alulationoftheVEVofthesquareoftheeld
opera-tor. Theironlusionisthat theVEVforthis
energy-momentum tensorhasthefollowingstruture:
hT
(x)i
Ren =
1
16 2
r 4
A
(;)+B
(;)ln(r)
;
(66)
whereB
andA
arediagonaltensorsTheexpliit
ex-pressionforthelatteris
A
=diag(T+A r
r B
r
r ; A
r
r ; A
r
r
+1=2B r
r ;
A r
r
+1=2B r
r
) : (67)
For the partiular value = 1=6, T is given by the
trae anomaly. We have hT
(x)i
Ren
= T=16 2
r 4
=
=770 2
r 4
( 1 =2).
B. Fermioni Case
The spinor Feynman propagator is dened as
follows[10℄
iS
F (x;x
0
)=hT( (x) (x 0
))i : (68)
This propagator obeysthe followingdierential
equa-tion
(ir= M)S
F (x;x
0
)= 1
p
g Æ
4
(x;x 0
)I
(4)
; (69)
1
Infatforatypialgranduniedtheorytheparameter0isoforder10 16
Gev. So1 2
where g=det(g
). Thespinorovariantderivativeis
givenby
r==e
(a) (x)
(a)
(
+
) ; (70)
with the spin onnetion
given in terms of the
matrixandthebasistetrade
(a)
intheusualway:
=
1
4
(a)
(b)
e
(a) e
(b); :
IfabispinorD
F (x;x
0
)satisesthedierentialequation
2 M
2 1
4 R (x)
D
F (x;x
0
)= 1
p
g Æ
(4)
(x;x 0
);
(71)
where the generalized d'Alembertian operator is
ex-pressedby
2=g
r
r
=g
r
+
r
r
;
(72)
thenthespinorFeynmanpropagatormaybewrittenas
S
F (x;x
0
)=( ir=+M)D
F (x;x
0
): (73)
Nowafter thisbriefreview,letusspeializein the
al-ulation of the spin Feynman propagator in the G-M
spaetime.
Using appropriate basis tetrad for this spaetime,
theonlynon-vanishingspinonnetionsare[12℄:
2 =
1
2 h
(1)
(2)
os+ (2)
(3)
sin i
(74)
and
3 =
1
2 h
(1)
(2)
sin+ (1)
(3)
sinos
(2)
(3)
osos i
sin : (75)
We also adopt the following representation for the
-matrix
(0)
=
1 0
0 1
; (k )
=
0
k
k
0
: (76)
These matries obey the antiommutator relation
f (a)
; (b)
g= 2 (a)(b)
.
Aftersomeintermediatestepsweget 2
2 =
1
2
2
t +
2
r 2
r (r
2
r )
~
L 2
r 2
(1 ) 2
2r 2
1
r 2
~
~
L; (77)
where
~
=
~
0
0 ~
: (78)
The systemthat we shallonsider onsists ofa
mass-lessleft-handedfermionield. ForthisasetheDira
equationreduestoa22matrixdierentialequation
asshownbelow:
iD=
L
=0; (79)
where
D=
L =i
1
t
(r)
r 1
r
()
1
rsin
()
+
1
r
(r)
; (80)
with (u)
=~u,^ u^denotingtheusualunitvetoralong
thethree spatialdiretionsinspherialoordinates.
The Feynman two-omponent propagator obeys
nowtheequation
iD=
l S
F (x;x
0
)= 1
p
g Æ
(4)
(x;x 0
)I
(2)
; (81)
andanbegivenin termsofthebispinorG
F (x;x
0
)by
S
F (x;x
0
)=iD=
L G
F (x;x
0
); (82)
wherenowthisbispinorobeysthe2 2dierential
equa-tionbelow:
^
KG
F (x;x
0
)= 1
p
g Æ
(4)
(x;x 0
)I
(2)
; (83)
with
^
K= 1
2
2
t +
2
r 2
r r
2
r ~
L 2
r 2
1
r 2
1+~ ~
L
:
(84)
The VEV of the energy-momentum tensor an be
expressed in terms of the Eulidean Green funtion
G
E (x;x
0
) = iG
F (x;x
0
). Now we shall alulate
G
E (x;x
0
)byusing thesolutionoftheeigenvalue
equa-tionbelow
^
K
(x)= 2
(x); (85)
with 2
0,soweanwrite
G
E (x;x
0
)= X
(x)
+
(x
0
)
: (86)
Thenormalizedeigenfuntionanbewritten as
(k )
(x)=
r
p
2r e
iE
J
k (pr)
(k )
j;mj
; k=1;2; (87)
with
2
= E
2
2
+ 2
p 2
; (88)
and
1 =
l+1
1
2 ;
2 =
l
+
1
2
: (89)
Intheaboveequation (k )
j;m
j
arethespinorspherial
har-monieigenfuntionsoftheoperators ~
L 2
and ~
L[11℄.
2
Hereinthissetionweareusingthefollowingmetritensor: g =diag( 2
;1= 2
;r 2
;r 2
sin 2
TheexpressionforG
E (x;x
0
)anbegivenby
G
E (x;x
0
) = Z
1
1 dE
Z
1
0 dp
X
j;mj
(1)
(x)
(1)+
(x
0
)+ (2)
(x)
(2)+
(x
0
)
E 2
= 2
+ 2
p 2
:
(90)
Substituting (k )
(x)in theequationaboveweget
G
E (x;x
0
)= 1
2rr 0
X
j;mj h
Q
1 1=2 (u)C
(1)
j;m
j (;
0
)
+Q
2 1=2
(u)C (2)
j;mj (;
0
) i
; (91)
whereQ
(z)istheLegendrefuntion,u=1+( 4
2
+
r 2
)=2rr 0
andC (k )
j;mj (;
0
)= (k )
j;mj ()
(k )+
j;mj (
0
).
Againwean seethat for=1,
1 =
2
=l+1=2
andwegetalosedform forG
E (x;x
0
)
G
E (x;x
0
)= 1
8 2
1
(x;x 0
) I
(2)
; (92)
where2(x;x 0
)= 2
+(~r ~r 0
) 2
.
NowweanexpressthespinorGreenfuntionas
S
F (x;x
0
)=i
1
t
(r)
r +
1
r
(r)
~ ~
L
+
1
r
(r)
G
E
(x;x): (93)
1. VauumExpetation Value
Also,asinthesalarase, thegeneralstrutureof
the VEV of the energy-momentum tensor assoiated
with a massless fermioni eld in the G-M spaetime
hasthefollowingstruture:
hT
(x)i
Ren =
1
8 2
r 4
h
A
+B
ln
r
i
; (94)
where
A
=diag A 0
0
; T+A 0
0 +B
0
0 ;
T A
0
0
1=2B 0
0 ;T A
0
0
1=2B 0
0
(95)
and
B
=B
0
0
diag(1;1; 1; 1): (96)
Beause for this system T := r
4
Tra2
2 =
1 4
60 , our
problemofnding(94)onsiststoobtainthetwo
om-ponentsA 0
0 andB
0
0 only.
Usingthepoint-splittingapproah,theVEVofthe
respetiveenergy-momentumtensorforspinoreldhas
thefollowingform
hT
(x)i=1=4 lim
x 0
!x Tr[
(r
r
0
)
+ (r r
0
)℄S (x;x 0
): (97)
Forus it is neessaryonly to ompute hT 0
0
(x)i, sowe
have
hT 0
0 (x)i=
1
2
lim
x 0
!x Tr
2
G
E (x;x
0
): (98)
The above limit is divergent, so in order to obtain a
welldenedresultweshouldrenormalizeitsubtrating
fromtheEulideanGreenfuntion,G
E
,theHadamard
one, whih for this ase presents struture similar to
the salar ase. After a long alulation we found:
B 0
0 =
1 4
60
and A 0
0
a ompliated integral; however
wewritedownthislatteromponentforspeivalues
oftheparameter.
1)Forlargesolidangledeit(<<1),
A 0
0
1
60
ln+C
0 ; C
0
=0:0104:
2)Forsmallsolidangledeitorexess(j 1j<<1),
A 0
0 C
1
(1 ) ; C
1
=0:0773:
3)Forlargesolidangleexess(>>1),
A 0
0
C
3
4
; C
3
=0:0173:
V Thermal Eets
Nowwewouldliketopresenttheeetsofthenonzero
temperaturein the formalismofquantum eld theory
for a masslesssalar eld in the G-M spaetime. We
shalldeveloprstthealulationoftherespetive
Eu-lideanthermalGreen funtion. Oursubsequent
anal-ysis onerns in the alulation of the orretions to
theVEVofh 2
(x)i
andhT
(x)i
duetothenonzero
temperature[13℄.
A. The Eulidean ThermalGreen Funtion
Thethermal Green funtion, G
(x;x
0
), for a
mas-sivesalareld,non-minimallyoupledwiththe
geom-etryofthebakgroundspaetime,satisesthefollowing
onditions:
(i)Itobeysthedierentialequation
2 m
2
R
G
(x;x
0
)= Æ
(4)
(x;x 0
)
p
g
(99)
and
(ii) isperiodiin the Eulideantime , with period
givenby
= 1
B T
;
where
B
istheBoltzmanonstantandT theabsolute
BeausetheEulideanversionofourspaetimeisan
ultrastatione 3
,theEulideanthermalGreenfuntion
an be obtained by theShwinger-De Witt formalism
as follows:
G
(x;x
0
)= Z
1
0 dsK
(x;x
0
;s); (100)
where thethermalheatkernelan befatorizedas
K
(x;x
0
;s)=
3
i
4s i
2
4s !
K
1 (x;x
0
;s); (101)
being
3
thethirdthetaJaobifuntion
3 (zj!)=
1
X
n= 1
exp(in 2
!+2inz); (102)
with Im(!)>0. K
1 (x;x
0
;s)is thezerotemperature
heat kernelpreviouslydened byEq. (53)inthe
anal-ysisofthemasslesssalareld.
AftersomeintermediatestepswegettheEulidean
thermalGreenfuntionforasalarmasslesseldwhih
reads
G
(x;x
0
)= 1
8 2
rr 0
1
X
n= 1 1
X
l=0
(2l+1)P
l (os)
Q
l 1=2
2
( n)
2
+r 2
+r 02
2rr 0
; (103)
where we see that this Green funtion is a sum of a
zero temperature one plus some thermal orretion:
G
(x;x
0
)=G
1 (x;x
0
)+G
(x;x
0
).
Nowweareinpositiontoalulatethethermal
or-retionto theVEVofh 2
(x)i
andhT
(x)i
.
B.The Thermal Average h 2
(x)i
Thesingularontribution forh 2
(x)i
omesfrom
the zero-temperature term in the Eulidean thermal
Green funtion. So the proedure to renormalize this
thermalaverageisidentialtothepreviousonedened
inSe. 4:A:1:
h 2
(x)i
;Ren
= lim
x 0
!x [G
(x;x
0
) G
H (x;x
0
)℄
= lim
x 0
!x
G
1 (x;x
0
)+G
(x;x
0
) G
H (x;x
0
)
= h
2
(x)i
1;Ren
+ 1
4 2
r 2
1
X
n=1 X
l0
(2l+1)Q
l 1=2
(z
n );
(104)
d
where z
n =1+
2
n 2
2
2r 2
. Beause the rstterm in the
r.h.s. oftheaboveequationhasalreadybeendisussed
letusonentrateonthethermalorretion. Also,here
it isneessaryto alulatethis orretionexpandinga
series inpowersoftheparameter. After some
inter-mediate stepswegetthefollowingexpression:
h 2
(x)i
=
1
12 2
(1+)
p
2
8 2
r 2
S
1 (=r)
+
1
32 2
r 2
S
2
(=r); (105)
where
S
1
(=r)= X
n1 Z
1
n dt
1
p
osht osh
n t
sinh(t=2)
(106)
and
S
2
(=r)= X
n1 Z
1
n dt
1
p
osht osh
n t
sinh 3
(t=2) ;
(107)
where
n
=arosh
1+ n
2
2
2r 2
.
Unfortunately it is not possible to obtain expliit
resultsfor bothintegralsabovein termsof anyset of
elementaryfuntions. So, ifwewantto getsome
on-rete information about the behavior of the thermal
averageofh 2
(x)iweshouldproeedanumerial
eval-uationfor S
1 and S
2
for someintervalof the variable
! := =r. Speially weare interestedto know this
behaviorin thehightemperature,orgreatdistane
re-gion,i. e.,when!<<1. Byournumerialanalyses[13℄
3
An ultrastati spaetime admitsa globally dened oordinate system inwhih the omponentsof the metri tensor are
we foundthe following dependenes for both integrals
abovewhen!belongtotheinterval[0:01;0:1℄:
S
1 =
1
! a1
(108)
and
S
2 =
2
! a
2
; (109)
where
1
=14:26witha
1
=0:920:07and
2
=17:82
with a
2
=2:020:02. From these resultswean
in-fer theapproximate behaviorforh 2
(x)i
in thehigh
temperaturelimit:
h 2
(x)i
;Ren =
1
12 2
(1+1:68) 0:25
r
: (110)
Weanseethattheleadingtermisindependentonthe
non-minimal oupling andalso onthedistane from
thepointtotheglobalmonopole.
C. The Thermal Average hT
(x)i
.
The energy-momentum tensor assoiated to the
salareld in ageneralspaetimewithanon-minimal
ouplingisgivenby:
T
(x) = (1 2)r
(x)r
(x) 2(r
r
(x))(x)
+
2 1
2
g
(x)g
(x)r
(x)r
(x)
2 2
g
(x)R (x) 2
(x) G
(x)
2
(x) ;
(111)
d
where G
(x) andR (x) are,respetively,the Einstein
tensorandthesalarurvature.
The thermal average of the operator
energy-momentum tensor an be obtained using the thermal
Greenfuntion denedasusual
G
(x;x
0
)= Tr
e H
hT(x)(x 0
)i
Tre H
: (112)
Sothethermalaverageoftheenergy-momentumtensor
isgivenby
hT
(x)i
= lim
x 0
!x
[(1 2)r
r
0
G
(x;x
0
)
2r
r
G
(x;x
0
) G
G
(x;x
0
)
+
2 1
2
g
g
0
r
r
0
G
(x;x
0
)
2 2
g
R (x)G
(x;x
0
))
: (113)
We have already mentioned that the thermal Green
funtion assoiated with a salar eld in G-M
spae-timeanbewrittenasasumofazerotemperatureone
plus thermal orretions; so the above expression an
bewritten as
hT
(x)i
=hT
(x)i
1 +hT
(x)i
: (114)
Thegeneralstruture of theindependenttemperature
term in the r.h.s. of the aboveequation has already
been disussed previously in Se. 4:A:2. So, by this
reason,weshallonentrateonthepurelythermal
or-retion, hT
(x)i
. Forthesakeofsimpliityweshall
analyzeitszero-zeroomponentonly:
hT
00 i
= lim
x 0
!x
2
G
(x;x
0
)
+
2 1
2
g
0
(x;x 0
)r
r
0G
(x;x
0
)
+ 1
2
(1 4)R (x)G
(x;x
0
)
: (115)
The development of this equation is a long one. Our
nalresultis
hT
00 i
=
2
4 2
r 4
X
n1 X
l0
(2l+1) h
n Q
(2)
l 1=2 (z
n )
+
n Q
(1)
l 1=2
(z
n )+
l Q
l 1=2 (z
n )
i
;
(116)
where Q (k )
(z) are the assoiated Legendre funtions,
z
n =1+
n 2
2
2 ! 2
n =
1
2
n 2
! 2
+4
2(n 2
! 2
4) 1=2(n 2
! 2
+4)
; (117)
n =
1
n! 3
p
n 2
! 2
+4
2[(3n 2
! 2
2) 2
2℄
1=2[(3n 2
! 2
+2) 2
2℄
; (118)
l
= (2 1=2)+
l(l+1)
2
+
2
(1 4)
2
1
2
:
(119)
d
Also in this ase it is not possible to obtain some
onrete information about the behavior of hT
00 (x)i
with the temperature without to adopt the following
proedures: (i) Toexpandingaseries in powersof the
parameter = 1 2
, and (ii) to proeed the
sum-mationin theangularquantumnumberl. Afterdoing
that,someexpressionsobtainedallowustoproeedthe
summation in n. Unfortunatelywealso ouldnot
de-velop all the integrals that appears and onsequently
the summations in n; again, here we had to develop
some numerial evaluations for these termsin the
re-gionwhere!<<1. Ournal resultis:
hT
00 (x)i
=
2
30 4
+
1
24
6
1
r 2
2
+
1 ()
1
4
+
2 ()
1
r 2
2
+ 1
4 p
2 2
r 4
F(!)
; (120)
where
1 ()=
2
180
(11+52) (121)
and
2 ()=
1
768 64
2
+1725 832
: (122)
ThefuntionF(!)ontainsalltheintegralswhihwere
not possible to be expressed in terms of elementary
funtionsandonsequentlytoproeedthesummations
in n. Analyzing allthe termsin
F(!) numerially, in
thehightemperatureregimewehaveobtainedthatthe
leadingtermare, approximatelyproportionalto 1=! 4
,
followedbytermsproportionalto1=! 3
and1=! 2
.
As our nal onlusion about the thermal eets
in the renormalized VEV of the operators 2
(x) and
T
00
(x)weansaythat inthehightemperatureregime
wehave:
i)h 2
(x)i T
Ren =h
2
(x)i 0
Ren +T
2
f() (123)
and
ii)hT
00 (x)i
T
=hT
00 (x)i
0
+T 4
g() (124)
wheref andgarefuntionsofthedimensionless
param-eter =
B
rT. The analytiform forthese funtions
wereobtainedbyus, uptherstorderin.
VI Other Appliations
A.Topologial Ination
Topologial defets an be seeds for ination[15℄.
VilenkinandLindelaimedthattopologialdefets
ex-pandexponentiallyif
0
>O(m
P ).
In the partiular ase of a global monopole, 2
=
1 8G
2
0
beomesnegativeandthesolidangledeit
(Æ=4(8G 2
0
))biggerthan 4. Theroleofthe
ra-dial and time oordinates interhanges. The exterior
solutionorresponds to the G-M solutionis no longer
stati.
B.Gravitating Magneti Monopole
The ourreneof adeit angle is aonsequene
of the nontrivial topology of the bosoni eld
ong-urationoutsidethe themonopole. The non vanishing
gradients along nonradial diretions imply an energy
densitythat fallsoslowly:
T
00
a
;i
a
;i
2
r 2
:
(This result also points out that the total Newtonian
mass ontained within r is M(r) 4G 2
0
r. If we
taketheradialextentof aGalaxyto beR
G
'15Kp
andonsider atypialgranduniedtheory with
0 '
10 16
Gev, this mass turns out to beof order 10 69
Gev,
whihistentimesthetotalmassofatypialGalaxy.)
However onsidering a gravitating magneti
monopole,thegaugeovariantderivativeof thesalar
ledvanishesoutsidethemonopole:
(D
i )
a
=
i
a
e
ab A
b
Sothereisnodeitangle.
C. CondensedMatter
Theeetivegravityariseinmanyondensed
mat-tersystem. Thetypialexamplesaretherystalswith
disloations and dislinations linear defets. However
it appears that the superuid 3
He A provides the
most adequate analogies for relativisti models of the
eetivegravity.
Thequasipartilesin 3
He Aarehiralandmassless
fermions. Under spei irumstanesthese fermions
are`relativisti'withthespetrum[16℄
E 2
(k)+g ij
(k
i eA
i )( k
j eA
j
)=0: (125)
Here ~
Aisthedynamialvetorpotentialoftheindued
eletromagnetield, ~
A=k
F ^
land ^
listheunityvetor
inthemomentspae.
Themetritensorof theeetivespaewhih
gov-ernsthemotionofthefermionsisgivenby
g ij
=
2
? Æ
ij
l i
l j
2
k l
i
l j
; g 00
=1:
Here
? and
k =v
F (
? <<
k
)are thespeed of the
light propagating transverse to ^
l and along ^
l ,
respe-tively.
Forthespei asewhere ^
l=r,^ weget
ds 2
=dt 2
dr 2
a 2
r 2
d 2
;
where a 2
= 2
k =
2
?
>1. In this asewehavea metri
spaetimeanalogoustotheG-Monewithasolidangle
exess.
VII Conluding Remarks
Inthepresentpaperwehavereviewedsomeofthemost
importantaspetsabouttheglobalmonopolespaetime
anditsonsequenesonthethelassialandquantum
movement of a harged partile. Also we have
alu-lated the vauum polarization eets due to massless
salarandfermionieldsin thismanifold,andthe
ef-fetsofthenonzerotemperatureonthesequantities.
Besides themain resultsexhibited in thispaperas
mentioned above, the formalisms developed here an
beappliedtoothersub-areasofphysissinethe
ina-tionprobleminosmologytoondensedmattersystem.
Forallthesereasonswebelievethatthephysisinthe
global monople spaetimeis veryrih and deservesto
bestudied.
Aknowledgment
My aknowledgments to all my ollaborators (see
referenes([6℄), ([7℄), ([12℄) and([13℄)) whomadethis
work possible. This work was partially supported by
CNPq.
Referenes
[1℄ T.W.B.Kibble,J.Phys.A9,1387(1976).
A.Vilenkin,Phys.Rep.121,263(1985).
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