Salar Field Cosmology in Three-Dimensions
G.Oliveira-Neto
Departamento deFsia,Institutode Ci^eniasExatas,
UniversidadeFederaldeJuiz deFora,
36036-330, JuizdeFora, MG,Brazil
Reeivedon17Otober,2000
WestudyananalytialsolutiontotheEinstein'sequationsin2+1-dimensions. Thespae-timeis
dynamialandhasaline symmetry. Thematterontentis aminimally oupled,massless,salar
eld. Dependingonthevalue ofertain parameters,this solutionrepresentsthreedistint
spae-times. Therstoneisatspae-time. Then,wehaveabigbangmodelwithanegativeurvature
salar and a real salar eld. Thelast ase is a big bang model with event horizons where the
urvaturesalar vanishesandthesalareldhangesfromrealtopurelyimaginary.
I Introdution
Manyreentworks havefousedattentionontheissue
of general relativity in 2+1-dimensions [1℄. Most of
them dealwith twoimportantissues: blakholesand
osmology.
Themain motivation for the mostreentworks in
blak holephysisame from the importantdisovery
of the BTZ blak hole [2℄. They investigate dierent
lassialandquantumpropertiesoftheBTZandother
blakholesolutions[3℄,[4℄.
Onthe otherhand, the simpliity ofthe theory in
three-dimensions, when ompared with its version in
four-dimensions, is themain motivation for the works
in osmology. Most of them deal with quantum
as-petsofthetheory,inpartiulartheattempttoderive
awave-funtionfortheUniverse[4℄,[5℄,[6℄.
Inthepresentwork wewould liketo study an
an-alytial solution to the Einstein's equations in 2+
1-dimensions. The spae-time is dynamial and has a
linesymmetry. Thematterontentisrepresentedbya
minimally oupled,massless,salareld. Weshallsee
that thissolutiongivesrise,dependingonthevalueof
ertainparameters,tothreedierentspae-times. The
rst oneisat spae-time. Then, we haveabig bang
modelwithanegativeurvaturesalarandarealsalar
eld. Thelastaseisabigbangmodelwithevent
hori-zonswheretheurvaturesalarvanishesandthesalar
eldhangesfromrealtopurely imaginary.
InSe. II,westartby writingdownthe
appropri-ate Einstein'sequations. A solution(S) of this set of
equations hasalreadybeenfound, in dierent
irum-stanes Ref. [7℄. We shall use S suitably adapted to
ourproblem.
Sisomposedoffewfuntionsoftherelevant
oor-dinatesandfreeparameters(p
i
). InSe. III,weimpose
onditions uponS suh that it may be interpreted as
a physially aeptable solution. These onditions
re-strit the domains of the p
i
's, and divide S in three
distint typesof spae-times, depending onthe values
of the p
i
's. The spae-times are the ones mentioned
aboveand westudythemingreat details.
Finally, in Se. IV we summarize themain points
andresultsofthepaper.
II Einstein-salar equations
We shall start by writing down the ansatz for the
2+1-dimensional, dynamial, line symmetri,
spae-timemetrias,
ds 2
= 2e
(N(u;v)+U(u;v)=2)
dudv +e
2U(u;v)
dy 2
: (1)
where N(u;v) andU(u;v)are twoarbitraryfuntions
tobedeterminedbytheeldequations,(u;v)isapair
ofnulloordinatesvaryingintherange( 1;1),andy
isaoordinatetakingvaluesoverarealline: ( 1;1).
Thesalareld'willbeafuntion onlyofthetwo
nulloordinatesandtheexpressionforitsstress-energy
tensorT
isgivenby[8℄,
T
= ';
';
1
2 g
';
'
;
: (2)
where; denotespartialdierentiation.
Now,ombiningEqs. (1)and(2)wemayobtainthe
Einstein'sequations,whihintheunitsofRef. [8℄take
thefollowingform,
2U; uu (U; u ) 2 + 2N; u U; u = 2('; u ) 2 ; (3) 2U; vv (U; v ) 2 + 2N; v U; v = 2('; v ) 2 ; (4) 2N; uv +U; uv = 2'; u '; v ; (5) (e U ); uv
= 0: (6)
The equation of motion for the salar eld, in these
oordinates,is 2'; uv U; u '; v U; v '; u
= 0; (7)
Comparing the above set of non-linear,
seond-order, oupled, partial, dierential equations (3)-(7)
withtheonederivedinRef. [7℄,alledtypeA,wenotie
that theyare idential, although thesituation treated
hereisdierentfromtheonetreatedthere. Therefore,
weshalltakeasthesolutionof thesystemaboveEqs.
(3)-(7), the type A spae-times, obtainedin Ref. [7℄.
Theywillbeintroduedandstudiedingreatdetailsin
thenextsetion.
III Solutions
Fromourdisussionintheprevioussetion,wesuitably
adaptthetypeAspae-timesofRef. [7℄tothepresent
situation,andwritethefollowingsolutiontothesystem
Eqs. (3)-(7),
exp( U(u;v)) t 2
(u;v) = (1 u 2 v 2 ) 2 ; (8)
exp( N(u;v))W(u;v) = t 2Æ u v (1 v 2 ) 1=2 (1 u 2 ) 1=2 u v
+(1 v 2 ) 1=2 (1 u 2 ) 1=2 2 u (1 v 2 ) 1=2 u
+(1 v 2 ) 1=2 4aÆ + v (1 u 2 ) 1=2 v
+(1 u 2 ) 1=2 4aÆ (1 v 2 ) Æ 2 + (1 u 2 ) Æ 2 ; (9)
'(u;v) = 2alnt 2 + + ln 1 u (1 v 2 ) 1=2 v (1 u 2 ) 1=2
1+u (1 v 2 ) 1=2 +v (1 u 2 ) 1=2 + ln 1 u (1 v 2 ) 1=2 +v (1 u 2 ) 1=2
1+u (1 v 2 ) 1=2 v (1 u 2 ) 1=2 (10) d
where and 2 < and they are greater than or
equal to 1=2, Æ = 2a 2
+(3 1= 1=)=4, 2
=
(1 1=2)(1 1=2),Æ 2
+
=1 1=2, Æ 2
=1 1=2,
=(1=2)(Æ
+
Æ ),andaisaonstant,real,number.
Intermsof thenewquantities t(u;v)Eq. (8),and
W(u;v)Eq. (9),thelineelementEq. (1)beomes,
ds 2
= 2W(u;v)t 1=2
(u;v)dudv +t 2
(u;v)dy 2
; (11)
One maynotie from Eqs. (8)-(10), that for dierent
valuesof,anda,onehasdierentspae-times. Itis
also importanttonotie thatthe hoieof thelettert
inEq. (8),wasnotasual. Weshallbeabletointerpret
itasatimeoordinate.
Observing Eq. (11), we notie that these
spae-timeshaveasingularityatt=0. Itisaphysial
singu-larityasanbeseendiretly fromtheurvaturesalar
R ,andalsofrom Ref. [7℄.
Inorder to showthis resultwestart writingdown
theRiitensor that, in thepresentase, hasthe
fol-lowingexpression[9℄,
R = '; '; : (12)
Fromit,wemayomputeRstraightforwardlywiththe
aidofEqs. (8)-(11),nding,
R =
8v 2 1
u 2 1
(4a+Æ
+ +Æ )
2
Wt 5=2
: (13)
Finally, taking the limit t ! 0in R Eq. (13), we
otherphysialsingularityforthesespae-timesbeause
R iswelldenedoutsidet=0. Weanalsolearnthat
taking the limitt ! 1 of R Eq. (13), this quantity
goestozero.
The spae-times above will only be physially
a-eptableiftEq. (8)isareal, positivefuntion andW
Eq. (9)isarealfuntion. Therefore,onlyfewdistint
sets of values of , and a will be allowed. Eah of
them giving riseto a dierent type of spae-time. It
is importantto note thatsine t andW are funtions
of (u,v), the range of these oordinates shall also be
restrited.
Here we shall loose the ondition that the salar
eld ' Eq. (10), be real. We shall permit it to be
purely imaginary in some spae-time regions, whih
means that in those regions ' will be an example of
theso-alled`exotimatter'[10℄.
III.1 Spae-times
Westartnowthedeterminationof theallowed
val-uesof,andabyimposingthattbeareal,positive,
funtion. In order to simplify our study we shall
de-mandthat2and2beintegers,greaterthanorequal
to1.
ObservingthedenitionoftEq. (8),wenotiethat
if we permit 2 or 2 to be even, for positive t, we
would haveverylimiteddomains fortheoordinatesu
and v, respetively. Sine we would like to have the
biggestpossibledomainsfortheseoordinates,weshall
restritourattentionto odd,integer,valuesof2and
2. Thepositiveness oftwill alsoseletthephysially
aessiblespae-timevolume.
From the expression of W Eq. (9), we learn that
thethreedistintfuntionsofuandv,insidebrakets,
respetivelywithexponents2,4aÆ
+
and4aÆ ,maybe
negativeevenforpositivet. Sine,inordertointerpret
t as atime oordinate, W hasto be positive(see Eq.
(14)below), weshall eliminate these terms. Thebest
waytoaomplishthisisbysettingtheirsexponentsto
zero.
Withtheaidoftheformulaefor,Æ
+
andÆ ,given
justbelowEq. (10),weunderstandthattherearethree
distintmannerstosettheaboveexponentstozero: (a)
by hoosing a =0 and 2=2 =1, (b) by hoosing
2 = 2 = 1, and nally () by hoosing a = 0 and
either2or2equalto1. Asweshallseebelow,these
possibilitieswillproduethedistintsetsofspae-times
assoiatedtooursolution.
Now,wearein the position to identify t asatime
oordinate. ForpositiveW andt,wemayomputethe
norm of the quadri-vetor normal to surfaes of
on-stantt. Whihgives,
8u 2 1
v 2 1
1
Wt 1=2
; (14)
whih is always negative for 2 and 2, odd, integer,
numbers.
Gathering together all the onditions obtained
above,wemaygroupthesolutionssatisfyingthese
on-ditionsin threedistintsets.
III.1.1 Flatspae-time.
The rst is empty, at spae-time, obtained for
2 = 2 = 1 and a = 0. It has the following line
element,fromEqs. (8),(9)and(11),
ds 2
= 2dudv +t 2
dy 2
; (15)
where,t=1 u v.
Forpositivet,theoordinatesuand v willvary in
therange( 1,1),butunder theondition,
u +v < 1: (16)
Thesurfaet=0in thisaseisnotaphysial
sin-gularity,itisjustaoordinatesingularity.
III.1.2 Bigbang osmology without horizons.
Thesolutionsbelongingtothissethave2=2=1
and a 6= 0. Introduing these values in Eqs. (8), (9)
and(11),wemaywritethefollowingline element,
ds 2
= 2t 4a
2
dudv + t 2
dy 2
; (17)
where t =1 u v, u, v 2( 1;1) and satisfyEq.
(16).
ThesalareldEq. (10)isnow,
' = 2alnt 2
: (18)
Inthepresentase,wemayseefromEq. (18)that
thesalareldisalwaysreal,thereforethestress-energy
Eq. (2)isalwayspositive.
ThesalarofurvatureEq. (13),iswritten
R =
32a 2
t (4a
2
+2)
: (19)
From this expression is easy to see that t = 0 is still
aphysialsingularityforthesespae-times. Thesalar
eld Eq. (18), is also singular there. Therefore, we
may interpret this singularity as a big bang, for this
spae-time.
The dynamial nature of this spae-time may be
better appreiated ifwe re-writethe line element Eq.
(17)intermsoftandx=u v,
ds 2
= t
4a 2
( dt 2
+dx 2
)+ t 2
dy 2
where 1<x<1.
III.1.3 Bigbangosmologywitheventhorizons
Thelast set ofspae-times is determined when we
set a=0and either 2or 2 equalto 1. As amatter
ofdenition,and withoutloosing thegenerality,letus
hoose2 =1and 2=2n+1,where nis apositive
integer.
With the aid of Eqs. (8), (9) and (11), the line
elementofthesespae-timesis,
ds 2
= 2
t
1 u
( 2n
2n+1 )
dudv + t 2
dy 2
; (21)
where t =1 u v 2n+1
, u, v 2 ( 1;1)and satisfy
theonditionu+v 2n+1
<1.
Observing Eq. (21),weidentify besidesthe
singu-larity at t = 0, another one at u = 1. The seond
singularityisnotaphysialoneasanbeseendiretly
from REq. (13),whih forthepresentsituationis,
R = 4nv
2n
(1 u) (
2n
2n+1 )
t (
6n+2
2n+1 )
: (22)
Indeed, u= 1is an eventhorizon as weshall
demon-stratebelow.
Sine u = 1 is just a oordinate singularity, there
are newoordinates whih letEq. (21)regularatthis
event. On the other hand, in order to better
under-standthephysialeetofthehorizonandthe
dynam-ialnature ofthe spae-time, it is moreonvenientto
re-write the line element Eq. (21) in terms of, t and
x = 1+ u v 2n+1
. Whihgives,
ds 2
= 1
4n+2
4t
x 2
t 2
( 2n
2n+1 )
( dt 2
+dx 2
)+t 2
dy 2
:
(23)
Here,it islearthat wehavenotonly theu=0
hori-zon,whihinthenewoordinatesisthesurfaet=x,
but anotheroneatt= x. Intheoldoordinatesthis
isthesurfaev=0.
The basi property of event horizons, is the fat
that they isolate ertain spae-time regions from
an-other ones [8℄. This an be demonstratedfor the
sur-faes t = x, in the following way if we restrit our
attentiontothe(t;x)plane.
Let us start by alling setor I, the region in the
pastof thehorizons(0<t <x, 1<x<1). We
alsointroduethesetorII,astheregionto thefuture
of the horizons (x <t < 1, 1 <x <1). From
Eq. (23)weanseethatnullraysdesribe45 Æ
or135 Æ
straight lines in the (t;x) plane. Therefore, not even
thelightwillbeableto returnfromsetor IIto setor
I, oneit has entered it. Although t does nothange
therole of time with x after rossingthe horizons,as
weshall seefewimportantfats takeplaethere.
Thesalareld Eq. (10),may beobtained forthe
spae-timesbeingstudied,inthenewsetofoordinates.
'(t;x) = 1
2 r
2n
2n+1 ln
x + (x 2
t 2
) 1=2
x (x 2
t 2
) 1=2
: (24)
If one inspets the expression of '(t;x) Eq. (24),
onenoties that for t =0it diverges. Therefore, also
forthepresentspae-timeswemayinterprett=0asa
bigbangsingularity.
InsetorI, '(t;x) Eq. (24)isa realfuntion. On
the other hand, in setor II the salar eld is purely
imaginary. ItmeansthatinsetorII,thestress-energy
Eq. (2)maybenegativeandthesalareld willbean
exampleoftheso-alled`exotimatter'.
Observing R Eq. (22) we see that it vanishes in
bothhorizonsbuthasthesamesigninsetorsIandII.
Finally,Ifonewereinterestedin studyingquantum
eldtheory insetorII,theresultingtheorywouldbe
unitary. Thisis theasebeausethere isno
singulari-tiesthereand,fromEq. (14),thespae-timesunder
in-vestigationpossessaglobaltimelikeKillingvetoreld
[11℄.
IV Conlusions
Inthe present work we havestudied an analytial
so-lutionto theEinstein'sequationsin 2+1-dimensions.
Thespae-timewasdynamialand hadaline
symme-try. Thematterontentwasrepresentedbyaminimally
oupled,massless,salareld.
TheEinstein'sequationsforthissystemwere
iden-tialtoanotheronealreadysolvedintheliterature. We
haveusedtheknownsolution(S),andstudiedit.
We have imposed ertain onditions upon S suh
thatit ouldbeinterpreted as aphysially aeptable
solution. These onditions restrited the domains of
few freeparameters (p
i
) of S, and divided S in three
distinttypesof spae-times, dependingon thevalues
ofthep
i
's. Therstonewasatspae-time. Then,we
hadabigbangmodelwithanegativeurvaturesalar
and a real salar eld. The last ase wasa big bang
model with event horizonswhere theurvature salar
vanishesandthesalareldhangesfromrealtopurely
imaginary.
Aknowledgements
IamgratefultoA.Wangforsuggestivedisussions
I.D. Soares forhelpful disussionsand FAPEMIG for
theinvaluablenanialsupport.
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