Exact inhomogeneous Einstein-Maxwell-Dilaton cosmologies

10 Pages·0.18 MB·English

by  Stoytcho Yazadjiev

Exact inhomogeneous Einstein-Maxwell-Dilaton cosmologies
Stoytcho S. Yazadjiev
Department of Theoretical Physics, Faculty of Physics, Sofia University,
5 James Bourchier Boulevard, Sofia 1164, Bulgaria
E-mail: [email protected]
We present solution generating techniques which permit to construct exact inhomogeneous and
anisotropiccosmological solutionstoafour-dimensionallowenergylimitofstringtheorycontaining
non-minimally interacting electromagnetic and dilaton fields. Some explicit homogeneous and in-
homogeneous cosmological solutions are constructed. For example, inhomogeneous exact solutions
presentingGowdy -typeEMD universeare obtained. Theasymptoticbehaviourof thesolutions is
investigated. The asymptotic form of the metric near the initial singularity has a spatially varying
Kasner form. The character of the space-time singularities is discussed. The late evolution of the
solutions is described by a background homogeneous and anisotropic universe filled with weakly
1 interacting gravitational, dilatonic and electromagnetic waves.
0
0
2 PACS numbers: 98.80.Hw, 04.20.Jb, 04.50.+h, 11.25.Mj

n
a
J
I. INTRODUCTION In [4], Fienstein, Lazkoz, and Vazquez - Mozo have pre-

9 sented an algorithm for constructing exact solutions in

In the last decade string cosmologies attracted large stringcosmologyforheteroticandtype-IIBsuperstrings
3
v amountofinterest(seee.g.[1]andreferencestherein). In in four dimensions. They have also presented and dis-
6 the traditional approach the present (low energy) string cussed some properties of an inhomogeneous string cos-
5 cosmology is in fact the classical cosmology where gen- mology with S3 topology of the spatial sections. Clancy
1 eral relativity is generalized by including additional (in etal.[5]havederivedfamiliesofanisotropicandinhomo-
0 mostcases)masslessscalarfields. Ingeneral,oneexpects geneous string cosmologies containing nontrivial dilaton
1 that the inclusion of these extra matter degrees of free- andaxionfieldsbyapplyingtheglobalsymmetriesofthe
0
domwiththecorrespondingphysicalinterpretation,may string-effectiveactiontoageneralizedEistein-Rosenmet-
0
/ somehow resolve the long standing problems in cosmol- ric. Lazkoz[6]haspresentedanalgorithmforgenerating
h ogy. A recent interesting and promising development in families of inhomogeneous space-times with a massless
-t thisdirectionisthesocalledpre-big bangscenario[2]. In scalar field. New solutions to Einstein - massless scalar
p the framework of this scenario one assumes that the ini- field equations having single isometry, have been gener-
e
h tial state of the universe is characterizedby small string atedin[6]bybreakingthehomogeneityofmasslessscalar
: coupling and smallcurvature. This leads to aninflation- field G2-models along one direction.
v aryphaseforsufficiently homogeneousinitialconditions. It should be noted, however, that the inhomogeneous
i
X There are,however,no naturalreasonsfor the earlyuni- cosmologies in the framework of general relativity with
r verse not be inhomogeneous. At present it is not clear certain physical fields as a matter source have been in-
a how the large inhomogeneities may influence the pre-big vestigated long before the time of the string cosmology.
bang scenario. In a more general setting, the question ExactinhomogeneousvacuumEinsteincosmologieswith
of whether our universe (which looks now homogeneous S1 S1 S1, S2 S1 and S3 topology of spatial sec-
× × ×
and isotropic) may arise out of generic initial data still tions have been found by Gowdy [7] and studied after-
lacks a complete answer in both general relativistic and wards by Berger [8] and Misner [9]. Exact stiff perfect
string cosmology. fluid inhomogeneous cosmologies have been studied by
That is why the constructionand study of exact inho- Wainwright, Ince and Marshman [10]. Later Charach
mogeneous and anisotropic string-cosmologicalsolutions [11]andthenCharachandMalin[12](seealso[13])have
are still of great importance. foundandstudiedexactinhomogeneouscosmologicalso-
Theinhomogeneousstringcosmologieshavebeenstud- lutionstotheEinsteinequationswithanelectromagnetic
ied by a number of authors. In [3], Barrow and Kunze andminimally coupled scalarfield. These solutions have
have presented inhomogeneous and anisotropic cosmo- beeninterpretedasaninhomogeneousuniversefilledwith
logicalsolutions ofa low energystring theory containing gravitational,scalar and electromagnetic waves.
dilatonand axionfields when the space-time metric pos- Thepurposeofthispaperistopresentsufficientlygen-
sesses cylindrical symmetry . Their solutions describe eralsolution generating techniques which permit to con-
ever-expandinguniverseswithaninitialcurvaturesingu- struct exact inhomogeneous and anisotropic solutions to
larity. Theasymptoticformofthesolutionsneartheini- theequationsoflowenergystringtheorycontainingnon-
tial singularity has a spatially varying Kasner-like form. minimally coupled dilaton and electromagnetic fields -
1
the so called Einstein-Maxwell-dilaton (EMD) gravity. II. CONSTRUCTING SOLUTIONS WITH ONE
Examples of both homogeneous and inhomogeneous ex- KILLING VECTOR
actsolutionswillbealsopresentedandtheirasymptotics
will be investigated. We consider EMD-gravity described by the action
Besides the fact that the EMD cosmologies are inter-
esting in their own, there are at least two more moti-
vations for considering the EMD cosmologies. First, a = 1 ⋆R 2dϕ ⋆dϕ 2e−2ϕF ⋆F
long standing question in cosmology is the existence of A 16π Z − ∧ − ∧
(cid:0) (cid:1)
a primordial magnetic field. The string theory predic-
where ⋆ is Hodge dual with respect to the space-time
tion of the non-minimal coupling of the dilaton to the
metric g,Ris the Ricciscalarcurvature,ϕis the dilaton
Maxwellfield gives anopportunity to study the problem
field and F =dA is the Maxwell two form.
in a more general dynamical context. Recently, in the
Firstwewillexaminethe caseofspace-timeadmitting
homogeneous case this question has been addressed in
one space-like, hyper-surface orthogonal Killing vector
[14].
X. Physically this situation corresponds to a universe
Thesecondmotivationistoinvestigatethecharacterof
with homogeneity broken along two space directions. In
initialcosmologicalsingularitiesinthepresenceofmatter
thepresenceofaKillingvectortheBianchiidentitydF =
fieldsarisingfromthelowenergysuper-stringtheory. Ac-
0 and the Maxwell equations d⋆ e 2ϕF = 0 give rise to
cordingtotheBelinskii-Khalatnikov-Lifhsitz(BKL)con- −
the following local potentials Ψ and Ψ :
jecturethedynamicsofnearbyobserverswoulddecouple e m
near the singularity for different spatial points. A vac-
uum spatially homogeneous space-time singularity (IX
dΨ = i F , dΨ =e 2ϕi ⋆ F.
Bianchitype)isdescribedbyBKLasaninfinitesequence e − X m − X
ofKasnerepochs(”oscillatoryormixmasterbehaviour”)
Here i is the interior product of an arbitrary form
X
[15], [16]. BKL further speculated that a generic singu-
with respect to X.
larity should exhibit such a local oscillatory behaviour.
The hyper-surface orthogonal Killing vector X natu-
Interesting special cases are the so-called asymp-
rallydeterminesathreedimensionalspace-timesubman-
totic velocity-termdominated(AVTD) singularities[17].
ifold Σ with metric [21]
Their characteristic feature is the fact that the spatial
derivative terms in the field equations are negligible suf-
ficiently close to the singularities. Such singularities are P = g X X
described by only one Kasner epoch, not by an infinite N − ⊗
sequence of Kasner epochs i.e. an oscillatory behaviour where =g(X,X) is the norm of the Killing vector.
N
doesnotexist. Wethinkthatitisimportanttostudythe WenotethatthroughoutthetextwedenotetheKilling
behaviour of the singularities in the presence of matter fieldsandtheirnaturallycorrespondingone-formsbythe
fields coming form the super-string theory. In particu- same symbols.
lar, it is interesting to investigate the influence of non- The projection on Σ of the Einstein equations reads
minimalexponentialcouplingofthe dilatontothe U(1)-
field onthe characterof the singularities. As it has been ˆ =2dϕ dϕ+ 1 d d +
shownin[18],[19]the minimally coupled scalarfield can R ⊗ 2 2 N ⊗ N
N
suppress the oscillatory behaviour. 2
e 2ϕdΨ dΨ +e2ϕdΨ dΨ (1)
Very recently the influence of the exponential dilaton- − e⊗ e m⊗ m
N (cid:0) (cid:1)
electromagnetic coupling on the character of initial sin-
gularitieshasbeenstudiedbyNarita,ToriiandMaedain where ˆ is Ricci tensor with respect to the projection
R
[20]. These authors have consideredT3 Gowdy cosmolo- metric P. The Maxwell equations take the form
gies in Einstein-Maxwell-dilaton-axion system and have
shown using the Fushian algorithm that space-times in d† e−2ϕ −1dΨe =0 , d† e2ϕ −1dΨm =0. (2)
N N
general have asymptotic velocity-term dominated singu- (cid:0) (cid:1) (cid:0) (cid:1)
larities. Theirresultsmeanthattheexponentialcoupling Here d† =⋆d⋆ is the co-derivative operator.
The projection of the Einstein equations along to the
of the dilaton to the Maxwell field does not change the
nature of the singularity. It should be also noted that Killing vector gives
the exactsolutions foundin the presentworkleadto the
2 g(d ,d )=
same conclusion. N N − N N
Although, in the present paper we shall not consider 2 e−2ϕ e−2ϕg(dΨe,dΨe)+e2ϕg(dΨm,dΨm) . (3)
− N
theBKLconjectureindetails,webelievethatthepresent (cid:0) (cid:1)
work may serve as a good ground for further studying Finally, the vanishing of the twist = 1 ⋆ dX of the
T 2
of the above mentioned questions in the framework of Killing field leads to the constraint
Einstein - Maxwell-dilaton-axiongravity.
dΨ dΨ =0. (4)
e m
∧
2
Theequations(1),(2),(3)and(4)areequivalenttothe the nonlinear action of GL(2,R) gives
EMD - gravity equations in the presence of a spice-like,
hyper-surface orthogonalKilling vector.
Theconstraint(4)impliesthattheone-formsdΨm and =(detA)2 Nvde2ϕvd ,
dΨe are proportional i.e. there exists a function F such N c2Nvd+d2e2ϕvd
that dΨ = dΨ . Here we will consider the simplest
m e
F
case in which one of the potentials Ψ vanishes, say Ψ =
e
0. e2ϕ =c2 +d2e2ϕvd,
vd
N
In the case Ψ = 0 the full system dimensionally re-
e
duced equations can be obtained from the following ac-
tion
1 ac vd+bde2ϕvd
Ψ = N
AR = 161π Z √−detP (cid:18)R− 2Pij2DiNDjN m √2 c2Nvd+d2e2ϕvd
N where
PijD ϕD ϕ 2e2ϕ 1PijD Ψ D Ψ . (5)
i j − i m j m
− − N
(cid:1)
NowweintroducethefollowingsymmetricmatrixS a b
GL+(2,R) : ∈ A= .
(cid:18) c d(cid:19)
Itisworthnoting,however,thatsomeoftheGL(2,R)-
+2Ψ2 e2ϕ √2Ψ e2ϕ
N m m transformationsarepureelectromagneticgaugeorrescal-
S = .
ing of the solutions. Only the O(2,R)-transformations
√2Ψ e2ϕ e2ϕ
 m  lead to essentially new solutions.
Exact solutions to the EMD gravity equations can be
Making use of the matrix S, we may write the action
constructed from vacuum solutions by the method de-
(5) in the form
scribed in [22], too. In order to obtain solutions by
1 1 this method one assumes that the matrix S depends on
= √ detP PijTr(D SD S 1) .
i j − space-time coordinates through a harmonic potential Ω
AR 16π Z − (cid:18)R− 2 (cid:19)
(DiD Ω=0). Requiring then the constraint
i
(6)
1 dS dS 1
−
The action(6) has GL(2,R)as a groupof globalsym- Tr =1
−4 (cid:18)dΩ dΩ (cid:19)
metryforfixedprojectionmetricP. Explicitlythegroup
GL(2,R) acts as follows : to be satisfied, the EMD equations are reduced to the
vacuum Einstein equations
S ASAT ˆ =2D ΩD Ω,
→ ij i j
R
where A GL(2,R). DiDiΩ=0
∈
Note, that in the case when one of the electromag-
neticpotentialsvanishesthedimensionallyreducedEMD and a separated matrix equation
equations can be viewed as three-dimensional Einstein
gravity coupled to a nonlinear σ - model on the factor
d dS
GL(2,R)/O(2,R). S−1 =0.
dΩ(cid:18) dΩ(cid:19)
The GL(2,R) symmetry may be employed for gener-
ating new exact solutions from known ones. In particu-
Therefore, for every solution to the vacuum Einstein
lar, it will be very useful to employ the GL(2,R) sym-
equations, the solutions of the matrix equation give
metry to generate exact solutions with nontrivial elec-
classes of exact EMD gravity solutions (see for details
tromagnetic field from any given solution to the vac-
[22]).
uum field equations (i.e. pure Einstein equations) or
It shouldbe noted that the seedsolutions used for the
the dilaton-vacuum equations (Einstein equations plus
construction of exact EMD solutions via the above de-
a dilaton field).
scribed methods, must admit at least one Killing vector.
For example, starting with a dilaton-vacuum seed so-
The case with two commuting hyper-surface orthogonal
lution
Killingvectorswillbeconsideredinthefollowingsection.
0
vd
N
Svd = ,
0 e2ϕvd
 
3
1
III. CONSTRUCTING SOLUTIONS WITH TWO ∂ h=2∂ ϕ∂ ϕ +2∂ p∂ p+
z ξ z ξ z
KILLING VECTORS ξ
2e−2ϕ−2p∂ξΨe∂zΨe+ 2e2ϕ−2p∂ξΨm∂zΨm . (14)
A. General equations
Hereweshouldmakesomecomments. Wehavewritten
thesystemofpartialdifferentialequationsintermsofthe
Inthis sectionweconsiderthe EMDgravityequations
electromagneticpotentials Ψ and Ψ . These potentials
when the space-time admits two commuting space-like, e m
are derived with respect to the Killing vector X. In the
hyper-surface orthogonal Killing vectors X and Y. In
samewaywemayusetheelectromagneticpotentialswith
this case the metric can be written in the form
respect to the Killing vector Y denoted respectively as
ds2 =e2h−2p(dz2 dξ2)+e2pdx2+̺ e−2pdy2 (7) ΨYe and ΨYm. Moreover, we may also use a mixed pair
− of potentials, say Ψ and ΨY. In general, the transition
e e
where h, p and ̺ are unknown functions of z and ξ between different pairs of potentials may be performed
only. Therefore the Killing vectors are given by X = ∂ by the following formulas
∂x
Y = ∂ .
Th∂eyphysicalpropertiesofthemetric(7)dependonthe ∂ξΨYm =−ξe−2ϕ−2p ∂zΨe,
gradientofthe norm̺ ofthe two-formX Y . The case
∧
correspondingto∂ ̺beinggloballyspace-likeornullde-
µ
scribescylindricalandplanegravitationalwaves. Metrics ∂zΨYm =−ξe−2ϕ−2p ∂ξΨe,
where the sign of ∂ ̺∂µ̺ can change, describe colliding
µ
plane waves or cosmological models with space-like and
t∂im̺e-dleiksecrisbinegcuolsamritoileosg.icaMlemtroidcselswwitihthglsopbaaclel-ylikteimsein-lgikue- ∂zΨYe =−ξe2ϕ−2p ∂ξΨm,
µ
larities.
∂ ΨY = ξe2ϕ 2p ∂ Ψ .
Inthepresentpaperweshallconsideronlytheglobally ξ e − − z m
time-like case ̺=ξ2. It should be noted that the choice
For the reader’s convenience we have presented in the
̺ = ξ2 is dynamically consistent with the EMD gravity
Appendix the system (8) - (14) written in terms of the
equations because the Ricci tensor Rˆ satisfies mixed pair Ψ =ω and ΨY =χ .
e e
Letusgobackagaintothesystemofnonlinearpartial
g(Y,Y)Rˆ(X,X)+g(X,X)Rˆ(Y,Y)=0. differential equations (8) - (14). We have a complicated
systemofcouplednonlinearpartialdifferentialequations
In the presence of a second Killing vector the dimen- and it seems that finding all its solutions is a hopeless
sionalreductioncanbefurthercontinued. Withoutgoing task. Nevertheless, as we will see a large enough class of
intodetailswedirectlypresentthedimensionallyreduced solutions can be found.
equations
1
∂2p+ ∂ p ∂2p=e 2ϕ 2p (∂ Ψ )2 (∂ Ψ )2 B. Solution generating method
ξ ξ ξ − z − − z e − ξ e
(cid:0) (cid:1)
+ e2ϕ−2p (∂zΨm)2 (∂ξΨm)2 , (8) Here we consider the case in which one of the electro-
−
(cid:0) (cid:1) magnetic potentials vanishes. For definiteness we take
Ψ = 0. Let us introduce the new potentials u = p ϕ,
1 e
∂ξ2ϕ+ ξ∂ξϕ−∂z2ϕ=e−2ϕ−2p (∂zΨe)2−(∂ξΨe)2 φ=p+ϕ, ΨSm =√2Ψm and hS =2h. Then the sys−tem
(cid:0) (cid:1) (8)- (14)may be rewritteninterms ofthe new variables
e2ϕ−2p (∂zΨm)2 (∂ξΨm)2 , (9) as follows
− −
(cid:0) (cid:1)
∂ξΨe∂zΨm =∂zΨe∂ξΨm, (10) 1
∂2φ+ ∂ φ ∂2φ=0,
ξ ξ ξ − z
∂ ξe 2ϕ 2p∂ Ψ ∂ ξe 2ϕ 2p∂ Ψ =0, (11)
ξ − − ξ e z − − z e
−
(cid:0) (cid:1) (cid:0) (cid:1) 1
∂2u+ ∂ u ∂2u=e 2u (∂ ΨS)2 (∂ ΨS)2 ,
∂ξ ξe2ϕ−2p∂ξΨm ∂z ξe2ϕ−2p∂zΨm =0, (12) ξ ξ ξ − z − (cid:0) z m − ξ m (cid:1)
−
(cid:0) (cid:1) (cid:0) (cid:1) ∂ ξe 2u∂ ΨS ∂ ξe 2u∂ ΨS =0, (15)
ξ − ξ m − z − z m
1∂ h=(∂ ϕ)2+(∂ ϕ)2+(∂ p)2+(∂ p)2+ (cid:0) (cid:1) (cid:0) (cid:1)
ξ ξ z ξ z
ξ 1
∂ hS =(∂ φ)2+(∂ φ)2+(∂ pS)2+(∂ pS)2+
e−2ϕ−2p (∂ξΨe)2+(∂ξΨe)2 + ξ ξ ξ z ξ z
e 2ϕ 2p(cid:0)(∂ Ψ )2+(∂ Ψ )(cid:1)2 , (13) e 2u (∂ ΨS)2+(∂ ΨS)2 ,
− − ξ m ξ m − ξ m z m
(cid:0) (cid:1) (cid:0) (cid:1)
4
φ=φ +β ln(ξ), (18)
0 0
1
∂ hS =2∂ φ∂ φ+2∂ u∂ u+2e 2u∂ ΨS∂ ΨS.
ξ z ξ z ξ z − ξ m z m u=ln eφ21ξ1+α20 +e−2φ1ξ1−α20 , (19)
(cid:16) (cid:17)
It is not difficult to recognize that the system (15)
coincides with the corresponding one for the Einstein-
1 φ α
Maxwell gravity with a minimally coupled scalar field φ ΨSY = tanh( 1 + 0 ln(ξ)), (20)
e 2 2 2
for the metric
where φ , φ , α and β are arbitrary constants.
ds2 =e2hS−2u(dz2 dξ2)+e2udx2+e−2uξ2dy2. (16) The abo0ve s1eed0solutio0n (18) - (20) gives the follow-
−
ing homogeneous solution to the EMD equations. The
Here ΨSm is the corresponding non-vanishing electro- potential ΨYe is
magnetic potential in the case of a minimally coupled
scalar field.
In this way we have proven the following proposition ΨY = 1 tanh(φ1 + α0 ln(ξ)).
e 2√2 2 2
Proposition B.1 Let u, φ, hS and ΨS be a solution to
m
the Einstein-Maxwell equations with a minimally coupled The metric has the form
scalar field for the metric (16). Then p = 1(u + φ),
2
ϕ= 1(φ u), h= 1hS andΨ = 1 ΨS form asolution
2 − 2 m √2 m ds2 =A2(ξ)(dz2 dξ2)+B2(ξ)dx2+C2(ξ)dy2
to the equations of EMD gravity for the metric (7). −
where
Thisresultallowsustogenerateexactsolutionstothe
EMD gravity equations for the metric (7) directly from
knownsolutionsoftheEinstein-Maxwellequationswith A2(ξ)=eγ0−φ0ξα420+β02−β0 eφ21ξα20 +e−2φ1ξ−α20 ,
a minimally coupled scalar field. (cid:16) (cid:17)
Before going further we shall make some comments.
In the case when the space-time admits two space-like
Killing symmetries and one of the electromagnetic po- B2(ξ)=eφ0ξ1+β0 eφ21ξα20 +e−2φ1ξ−α20 ,
tentials is taken to vanish, it may be shown that the (cid:16) (cid:17)
equations governing the transverse part of the gravita-
tional field, dilaton and non-vanishing electromagnetic
potential are written in the compact matrix form C2(ξ)=e−φ0ξ1−β0 eφ21ξα20 +e−2φ1ξ−α20 −1.
(cid:16) (cid:17)
∂ξ ξS−1∂ξS −∂z ξS−1∂zS =0. (17) Here γ0 is a constant.
(cid:0) (cid:1) (cid:0) (cid:1) Theasymptoticbehaviouroftheexpansionfactorsand
The equation (17) is just the chiral equation. There
the dilaton field in the limit ξ 0 is as follows
arepowerfulsolitonictechniquesforsolvingthisequation →
(see [23]). Although, the study of cosmological soliton
solutions in EMD gravity seems to be very interesting, A2(ξ) ξα420+β02−β0−|α20|,
we will not consider them in the preset work. ∼
IV. EXAMPLES OF EXACT SOLUTIONS B2(ξ) ξ1+β0−|α20|,
∼
A. Homogeneous solutions
Homogeneoussolutionsareobtainedwhenwehavede- C2(ξ)∼ξ1−β0+|α20|,
pendence only on the ”time - coordinate” ξ. In this case
the differential constraint reduces to ∂ Ψ ∂ Ψ = 0.
ξ e ξ m
Thus we may choose one of the potentials to vanish. 1 1
ϕ (1 β α )ln(ξ).
0 0
Therefore we may apply the proposition from the pre- ∼−2 − − 2 | |
vioussectiontogeneratehomogeneoussolutionsstarting
The asymptotic form of the expansion factors and the
from the corresponding homogeneous solutions for the
dilaton field in the limit ξ 1 is
caseofaminimally coupledscalarfield. Inournotations ≫
thehomogeneoussolutionsforaminimallycoupledscalar
field are [13] A2(ξ) ξα420+β02−β0+|α20|,
∼
5
B. Inhomogeneous solutions
B2(ξ) ξ1+β0+|α20|, Here the proposition (B.1) will be applied again, this
∼
time to generate inhomogeneous exact solutions to the
EMDequations. Asaseedfamilyofsolutionswetakethe
C2(ξ)∼ξ1−β0−|α20|, Ccohsamroalcohgifeasmwiliythomf isnoilmutaiollnyscdouespclreidbisncgalianrhaonmdogeelencetoruos-
magnetic fields and with S1 S1 S1 topology of the
× ×
spatial sections. The Charach family of inhomogeneous
1 1 solutions and their asymptotics in the two limiting cases
ϕ (1 β + α )ln(ξ).
∼−2 − 0 2 | 0 | are presented in the Appendix.
Accordingtotheproposition(B.1)theinhomogeneous
Introducing the scynhroneous time dτ = A(ξ)dξ, the EMD metric is given by
lineelementinthelimitsξ 0andξ 1canbewritten
→ ≫
in the Kasner form
ds2 =e2(h−p)(dz2 dt2)+e2pdx2+ξ2e−2pdy2
−
ds2 dτ2+τ2p1dx2+τ2p2dy2+τ2p3dz2 where
∼−
where the Kasner exponents are defined by
1
2p=ln(2ξcosh( φ˜))+φ,
2
1+β 1 α
p1 = 1(1 α )02∓+(21| 0β|)2+ 3 , h= 1ln(ξ)+ln(2cosh(1φ˜))+
4 ∓| 0 | 2 − 0 2 2 2
1
F(φ˜ ,α ,A ,B ;ξ,z)+F(φ ,β ,C ,D ;ξ,z).
0 0 n n 0 0 n n
4
1 β 1 α
p = − 0± 2 | 0 | ,
2 1(1 α )2+(1 β )2+ 3 The dilaton field is
4 ∓| 0 | 2 − 0 2
1 1 1
1(1 α )2+(1 β )2 1 ϕ= φ ln 2ξcosh( φ˜) .
p = 4 ∓| 0 | 2 − 0 − 2 , 2 − 2 (cid:18) 2 (cid:19)
3 1(1 α )2+(1 β )2+ 3
4 ∓| 0 | 2 − 0 2
The inhomogeneous cosmological solutions introduce
asthesing” ”referstothelimitξ 0whilethesign characteristic length scales. In fact each normal mode
− →
”+” is for the limit ξ 1. has its own characteristic scale. The horizon distance in
≫
The dilaton filed is given by the ”z” direction is given by ds2 =0 and hence
x,y
|
ϕ σln(τ) ξ
∼ δz = dξ =ξ.
Z
where 0
In this way nξ can be viewed as the ratio of the hori-
zondistanceinthe ”z”directiontothe coordinatewave-
σ =−14(1∓|α10−|)2β+0∓(1212−α0β0)2+ 23 . lenTghtherλenair.ee.twnξo=limλδiznt.ing cases to be considered. The
firstcase iswhen the wavelengthis muchlargerthanthe
The parameters p1, p2 and p3 satisfy the Belinskii - horizon scale (nξ 1). The second case is when the
Khalatnikov relations wavelengthismuch≪lessthanthe horizonscale(nξ 1).
≫
In the first case (nξ 1) the asymptotic form of the
p +p +p =1, ≪
1 2 3 EMD metric is
p2+p2+p2 =1 2σ2.
1 2 3 −
ds2 =A2(ξ,z)(dz2 dξ2)+B2(ξ,z)dx2+C2(ξ,z)dy2
−
where
A2(ξ,z)=
eγ(z)ξ14α2(z)+β2(z)+β(z) e12φ˜∗(z)ξ21α(z)+e−12φ˜∗(z)ξ−12α(z) ,
(cid:16) (cid:17)
6
B2(ξ,z)=
eφ˜∗(z)ξ1+β(z) e12φ˜∗(z)ξ21α(z)+e−21φ˜∗(z)ξ−12α(z) , g =η +h
µν µν µν
(cid:16) (cid:17)
where
C2(ξ,z)=
e−φ˜∗(z)ξ1−β(z) e21φ˜∗(z)ξ12α(z)+e−21φ˜∗(z)ξ−12α(z) −1. η =diag(cid:16)−2ξβ02−β0e21Kξ,2ξ1+β0,
(cid:16) (cid:17) 1ξ1−β0,2ξβ02−β0e12Kξ
Inthe limitnξ 1the asymptoticformofthe dilaton 2 (cid:19)
≪
is
and
1
ϕ φ (z)+sign(α(z))φ˜ (z)
∼ 2(cid:16) ∗ ∗ (cid:17)−
1 1 β(z) 1 α(z) ln(ξ). h=diag 0, 2 ξ1+β0H(ξ,z), 2 ξ1−β0H(ξ,z),0 .
2(cid:18) − − 2 | |(cid:19) (cid:18) √ξ −√ξ (cid:19)
In order to discuss the cosmologicalsolutions near the When α0 =0 we have
6
singularity we have to consider homogeneous limit for ξ
approaching zero. Introducing the proper time, in the η =diag ξ14(1+α0)2+(β0−12)2−21e21Kξ,ξ1+12|α0|+β0,
homogeneous limit, by (cid:16)−
ξ1−21|α0|−β0,ξ14(1+α0)2+(β0−12)2−21e21Kξ ,
(cid:17)
τ = A(ξ)dξ,
Z
we find that the metric has the Kasner form h=diag 0,ξ1+21|α0|+β0(H(ξ,z)+ 1sign(α )H˜(ξ,z)),
(cid:18) √ξ 2 0
gµν ( 1,τ2p1,τ2p2,τ2p3). −ξ1−21√|αξ0|−β0(H(ξ,z)+ 21sign(α0)H˜(ξ,z)),0(cid:19).
∼ −
Here the Kasner indexes are spatially varying Theasymptoticformofthedilatoninthelimitnξ 1,
≫
for both α =0 and α =0, is given by
0 0
6
1+β(z) 1 α(z)
p (z)= ∓ 2 | | , 1 1
1 1(1 α(z) )2+(1 β(z))2+ 3 ϕ 1 β0+ α0 ln(ξ)+
4 ∓| | 2 − 2 ∼−2(cid:18) − 2 | |(cid:19)
1 1
H(ξ,z)+ sign(α )H˜(ξ,z) .
1 β(z) 1 α(z) √ξ (cid:18) 2 0 (cid:19)
p (z)= − ± 2 | | ,
2 1(1 α(z) )2+(1 β(z))2+ 3
4 ∓| | 2 − 2 The explicit form of the solutions and their asymp-
totics allow us to make some conclusions. The obtained
1(1 α(z) )2+(1 β(z))2 1 solutionsfallinthe categoryofasymptotic velocity-term
p (z)= 4 ∓| | 2 − − 2 dominated space-times. When the singularity is ap-
3 1(1 α(z) )2+(1 β(z))2+ 3
4 ∓| | 2 − 2 proached the spatial derivatives become negligible with
comparison to the time derivatives. Sufficiently close to
and satisfy the Belinski- Khalatnikov relations
the singularity the evolution at different spatial points
is decoupled and the metric is locally Kasner with spa-
tially dependent Kasner indices satisfying the Belinskii-
p (z)+p (z)+p (z)=1,
1 2 3
Khalatnikov relations.
The late evolution of the exact cosmological solu-
p2(z)+p2(z)+p2(z)=1 2σ2(z), tionsisdescribedbyahomogeneous,anisotropicuniverse
1 2 3 −
with gravitational, scalar (dilatonic) and electromag-
where
netic waves. The non-minimal dilaton-electromagnetic
1 β(z) 1α(z) exponentialcoupling influences mainly the homogeneous
σ = − ∓ 2 . background universe rather the scalar and electromag-
−1(1 α(z) )2+(1 β(z))2+ 3
4 ∓| | 2 − 2 netic waves on that background. The former may be
considered as minimally coupled scalar and electromag-
In the second case when nξ 1 the asymptotic form
≫ netic waves up to higher orders of 1.
of the EMD metric is as follows. ξ
Thefirstsub-caseisforα =0. ThentheEMDmetric
0
is given by
7
V. CONCLUSION ACKNOWLEDGMENTS
In this paper we have shown that it is possible to The author is deeply grateful to E. Alexandrova for
find exact inhomogeneous and anisotropic cosmologi- the support and encouragement. Without her help the
cal solutions of low energy string theory containing writingofthispapermostprobablywouldbeimpossible.
non-minimally interacting dilaton and Maxwell fields - TheauthorisalsogratefultoP.Fizievforthesupport
Einstein-Maxwell -dilaton gravity. First we have consid- and stimulating discussions.
eredspace-timesadmittingonehyper-surfaceorthogonal The author would particularly like to thank V. Rizov
Killing vector. It has been shown that in the case of for having read critically manuscript and for useful sug-
one vanishing electromagnetic potential the dimension- gestions.
ally reduced equations possess a groupof global symme-
try GL(2,R). We have described an algorithm for gen-
erating exact cosmological EMD solutions starting from
vacuum and dilaton-vacuum backgrounds by employing APPENDIX A: REDUCED EMD SYSTEM IN
the nonlinear actionof the globalsymmetry group. This TERMS OF THE POTENTIALS ΨE =ω AND
algorithm is especially useful for generating exact cos- ΨYE =χ
mological EMD solutions with only one Killing vector,
1
smtaetrhtiondgwwhitichhGpe1r-dmiliatstotno-vcoancusturmuctbaexckagctroEuMndD. sAolnuottiohnesr ∂ξ2p+ ξ∂ξp−∂z2p=e−2ϕ−2p (∂zω)2−(∂ξω)2
(cid:0) (cid:1)
startingfromsolutionsofthepureEinsteinequationshas + 1 e 2ϕ+2p (∂ χ)2 (∂ χ)2
been briefly discussed, too. ξ2 − ξ − z
(cid:0) (cid:1)
In the case when the space-time admits two commut-
ing, hyper-surface orthogonal Killing vectors we have
1
given a method which allows exact inhomogeneous cos- ∂2ϕ+ ∂ ϕ ∂2ϕ=e 2ϕ 2p (∂ ω)2 (∂ ω)2
mological solutions to the EMD equations to be gener- ξ ξ ξ − z − − z − ξ
(cid:0) (cid:1)
ated from the corresponding solutions of the Einstein- 1 e 2ϕ+2p (∂ χ)2 (∂ χ)2
Maxwellequationswithaminimallycoupledscalarfield. − ξ2 − ξ − z
(cid:0) (cid:1)
Usingthismethod,asaparticularcase,wehaveobtained
exact cosmologicalhomogeneous and inhomogeneous so-
lutions to the EMD equations. The initial evolution de- ∂ ω∂ χ=∂ ω∂ χ
ξ ξ z z
scribedbythesesolutionsisofaspatiallyvaryingKasner
form. The intermediate stage of evolution occurs when 1
∂2ω+ ∂ ω ∂2ω =
the characteristicscalesofthe inhomogeneitiesapproach ξ ξ ξ − z
the scale of the particle horizon. This stage is character-
2(∂ ϕ+∂ p)∂ ω 2(∂ ϕ+∂ p)∂ ω
ξ ξ ξ z z z
izedbystronglyinteractingnon-lineargravitational,dila- −
tonic and electromagnetic waves. The late evolution of
the cosmological solutions is described by a background 1
∂2χ ∂ χ ∂2χ=
homogeneousandanisotropicuniversefilled with weakly ξ − ξ ξ − z
interacting gravitational, dilatonic and electromagnetic
2(∂ ϕ ∂ p)∂ χ 2(∂ ϕ ∂ p)∂ χ
ξ ξ ξ z z z
waves. − − −
The cosmological solutions found in the present work
fall in the category of AVTD space-times. Near the sin- 1
∂ h=(∂ ϕ)2+(∂ ϕ)2+(∂ p)2+(∂ p)2+
gularity the dynamics at different spatial points decou- ξ ξ ξ z ξ z
ples and the metric has a spatially varying Kasner form. e−2ϕ−2p (∂ξω)2+(∂ξω)2 +
Therefore,thenon-minimalcouplingofthedilatontothe
1 (cid:0) (cid:1)
Maxwellfielddoesnotchangethenatureofthesingular- e 2ϕ+2p (∂ χ)2+(∂ χ)2
ity. The solutions are sufficiently generic and therefore ξ2 − ξ ξ
(cid:0) (cid:1)
weshouldexpectthattheT3space-timesinEMDgravity
have AVTD singularities in general.
1
Finallywebelievethatthepresentpaperwillbeasuffi- ∂zh=2∂ξϕ∂zϕ +2∂ξp∂zp+
ξ
cientlygoodbackgroundforstudyingsimilarproblemsin
the more-general case of Enstein-Maxwell-dilaton-axion 2e−2ϕ−2p∂ξω∂zω+ 2e−2ϕ+2p∂ξχ∂zχ
gravity.
8
APPENDIX B: CHARACH FAMILY OF EXACT and
SOLUTIONS
φ (z)=φ +
0
∗
For the reader’s convenience we present here the ∞ 2 n
C + D (γ+ln( )) cos(n(z z )),
Charach family of solutions written in our notations. n n n
(cid:18) π 2 (cid:19) −
The minimally coupled scalar filed is given by nX=1
φ=φ +β ln(ξ)+
0 0 φ˜ (z)=φ˜ +
0
∞ ∗
(CnJ0(nξ)+DnN0(nξ))cos(n(z zn)). ∞ 2 n
− A + B (γ+ln( )) cos(n(z z )).
nX=1 (cid:18) n π n 2 (cid:19) − n
nX=1
Here φ , β , C , D are arbitrary constants and J (.)
0 0 n n 0
Here γ is the Euler constant.
andN (.)arerespectivelytheBesselandNeumannfunc-
0
The asymptotic form of the Charach metric can be
tions.
written as
The metric function u and the non-vanishing electro-
magnetic potential ΨSY are correspondingly
e
ds2 =A2(ξ,z)(dz2 dξ2)+B2(ξ,z)dx2+C2(ξ,z)dy2
s − S S
1
u=ln 2ξcosh( φ˜) ,
where
(cid:18) 2 (cid:19)
A (ξ,z)=
S
ΨSeY =const+ 21tanh(21φ˜) eγ(z)ξ41α2(z)+β2(z)(cid:16)e21φ˜∗(z)ξ12α(z)+e−21φ˜∗(z)ξ−12α(z)(cid:17),
whereφ˜isanauxiliaryscalarfieldwhichisoftheform
φ˜=φ˜ +α ln(ξ)+ BS(ξ,z)=ξ e12φ˜∗(z)ξ21α(z)+e−21φ˜∗(z)ξ−12α(z) ,
0 0
(cid:16) (cid:17)
∞
(A J (nξ)+B N (nξ))cos(n(z z )).
n 0 n 0 n
−
nX=1 CS(ξ,z)= e21φ˜∗(z)ξ12α(z)+e−21φ˜∗(z)ξ−12α(z) −1.
The longitudinal part of the gravitationalfield is (cid:16) (cid:17)
The asymptotic behaviour of the scalar fields in the
1
hS =ln(ξ)+2ln(2cosh( φ˜))+ Charach solution in the high frequency regime (nξ 1)
2 ≫
is
1
F(φ˜ ,α ,A ,B ;ξ,z)+2F(φ ,β ,C ,D ;ξ,z)
0 0 n n 0 0 n n
2 1
φ φ0+α0lnξ+ξ−2H(ξ,z),
∼
where F is a solution to the system φ˜ φ˜0+β0lnξ+ξ−21H˜(ξ,z)
∼
1 1
∂ F = ((∂ φ)2+(∂ φ)2), where
ξ ξ z
ξ 2
1
∂ F =∂ φ∂ φ.
ξ z ξ z H˜(ξ,z)=Re ∞ (An iBn)e−i(zn−π4)ei(|n|ξ+nz),
For more details see [13]. n=X,=0 | |− | | 2π|n|
−∞6 p
In the limit ξ 0 the minimally coupled scalar field
→
and the auxiliary field behave as
H(ξ,z)=Re ∞ (C iD )e−i(zn−π4)ei(nξ+nz).
φ φ (z)+β(z)ln(ξ), n n | |
φ˜∼ φ˜∗ (z)+α(z)ln(ξ) n=−X∞,6=0 | |− | | p2π |n|
∼ ∗
The functions H(ξ,z), H˜(ξ,z) satisfy the D’Alembert
where
equations
2 ∞
α(z)=α0+ π Bncos(n(z−zn)), ∂ξ2H(ξ,z)−∂z2H(ξ,z)=0,
nX=1 ∂2H˜(ξ,z) ∂2H˜(ξ,z)=0.
2 ∞ ξ − z
β(z)=β + D cos(n(z z ))
0 π nX=1 n − n When α0 =0 the asymptotic form of the metric is
9
[13] M.Carmeli,Ch.Charach,S.Malin,Phys.Rep.76(1981)
[14] M. Giovannini, Phys.Rev. D 59, 123518-1 (1999)
gµν =ηµν +hµν [15] V. Belinskii, I. Khalatnikov, E. Lifhsitz, Adv. Phys. 31,
639 (1972)
where [16] Ch. Misner , Phys.Rev. Lett. 29 1071 (1969)
[17] D. Eardley, E. Liang, R. Sachs, J. Math. Phys. 13, 99
(1972); J. Isenberg, V. Moncrief, Ann. Phys. (NY) 199,
η =diag 4ξ2β02eKξ,4ξ2,1,4ξ2β02eKξ 84 (1990)
(cid:18)− 4 (cid:19) [18] V. Belinskii, I. Khalatnikov, Sov. Phys. JETP 32, 169
(1971)
and
[19] B. Berger, Phys.Rev. D 61, 023508-1 (1999)
[20] M. Narita, T. Torii, K.Maeda, E-print gr-qc/0003013
[21] R. Geroch, J. Math. Phys. 13, 394 (1972)
1
h=diag 0,4ξ2H(ξ,z)/4ξ, H2(ξ,z)/4ξ . [22] S. Yazadjiev, Int.J. Mod. Phys. D8, 635 (1999)
(cid:18) −4 (cid:19)
[23] V. Belinskii, V. Zakharov, JEPT 48, 985 (1978); JEPT
50, 1 (1980)
The constant K is given by
1 ∞
K = A2 +B2 +4(C2+D2) .
2π n n n n
nX=1(cid:0) (cid:1)
In the case when α = 0 the asymptotic form of the
0
6
metric is
η =diag ξ2β02+12|α0|+12|α0|2eKξ,ξ|α0|+2,
(cid:16)−
ξ−|α0|,ξ2β02+21|α0|+12|α0|2eKξ
(cid:17)
and
h=diag 0, ξ|α0|+2H(ξ,z)/ ξ,
(cid:16) ± p
ξ−|α0|H(ξ,z)/ ξ,0 .
∓ p (cid:17)
[1] J.Lidsey, D. Wands, E. J. Copeland Phys. Rep., Vol.
337, p.343 (2000)
[2] G. Veniziano, Phys. Lett. B 265, 287 (1991); M.
Gasperini,G Veneziano, Astropart. Phys. 1, 317 (1993)
[3] J. Barrow, K.Kunze,Phys. Rev D 56, 741 (1997)
[4] A.Feinstein,R.Lazkoz,M.Vazquez-Mozo,Phys.RevD
56, 5166 (1997)
[5] D.Clancy,A.Feinstein,J.Lidsey,R.Tavakol,Phys.Rev.
D60, 043503-1 (1999)
[6] R.Lazkoz, Phys. Rev D 60, 104008 (1999)
[7] R. Gowdy, Phys. Rev. Lett. 27, 827 (1971); Ann.
Phys.(N.Y.)83, 203 (1974)
[8] B. Berger, Ann. Phys.(N.Y.)83, 458 (1974); Phys. Rev.
D 11, 2770 (1975)
[9] Ch. Misner, Phys.Rev. D 8, 3271 (1973)
[10] J. Wainwright, W. Ince, J. Marshman, Gen. Rel. Grav.
Vol.10,259 (1979)
[11] Ch. Charach, Phys. Rev. D 19, 3516 (1979)
[12] Ch. Charach, S. Malin , Rev. D 21, 3284 (1980)
10