Exact inhomogeneous Einstein-Maxwell-dilaton cosmologies

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PHYSICAL REVIEW D, VOLUME 63, 063510

Exact inhomogeneous Einstein-Maxwell-dilaton cosmologies

Stoytcho S. Yazadjiev*Department of Theoretical Physics, Faculty of Physics, Sofia University, 5 James Bourchier Boulevard, Sofia 1164, Bulgar

~Received 13 November 2000; published 21 February 2001!

We present solution generating techniques which permit us to construct exact inhomogeneous and aniso-tropic cosmological solutions to a four-dimensional low-energy limit of string theory containing nonminimallyinteracting electromagnetic and dilaton fields. Some explicit homogeneous and inhomogeneous cosmologicalsolutions are constructed. For example, inhomogeneous exact solutions presenting a Gowdy-type EMD uni-verse are obtained. The asymptotic behavior of the solutions is investigated. The asymptotic form of the metricnear the initial singularity has a spatially varying Kasner form. The character of the space-time singularities isdiscussed. The late evolution of the solutions is described by a background homogeneous and anisotropicuniverse filled with weakly interacting gravitational, dilatonic, and electromagnetic waves.

DOI: 10.1103/PhysRevD.63.063510 PACS number~s!: 98.80.Hw, 04.20.Jb, 04.50.1h, 11.25.Mj

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I. INTRODUCTION

In the last decade string cosmologies attracted a laamount of interest~see, e.g., Ref.@1# and references therein!.In the traditional approach the present~low energy! stringcosmology is in fact classical cosmology where general rtivity is generalized by including additional~in most cases!massless scalar fields. In general, one expects that the ision of these extra matter degrees of freedom with theresponding physical interpretation may somehow resolvelong standing problems in cosmology. A recent interestand promising development in this direction is the so-calpre-big-bangscenario@2#. In the framework of this scenarioone assumes that the initial state of the universe is charaized by small string coupling and small curvature. This leato an inflationary phase for sufficiently homogeneous iniconditions. There are, however, no natural reasons forearly universe not to be inhomogeneous. At present it isclear how the large inhomogeneities may influence the pbig-bang scenario. In a more general setting, the questiowhether our universe~which looks now homogeneous anisotropic! may arise out of generic initial data still lackscomplete answer in both general relativistic and string cmology. This is why the construction and study of exainhomogeneous and anisotropic string-cosmological stions are still of great importance.

The inhomogeneous string cosmologies have been stuby a number of authors. In Ref.@3#, Barrow and Kunze havepresented inhomogeneous and anisotropic cosmologicalutions of a low-energy string theory containing dilaton aaxion fields when the space-time metric possesses cylindsymmetry. Their solutions describe ever-expanding uverses with an initial curvature singularity. The asymptoform of the solutions near the initial singularity has a sptially varying Kasner-like form. In Ref.@4#, Fienstein, Laz-koz, and Vazquez-Mozo have presented an algorithmconstructing exact solutions in string cosmology for heteroand type-IIB superstrings in four dimensions. They have a

*Email address: [email protected]

0556-2821/2001/63~6!/063510~9!/$15.00 63 0635

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presented and discussed some properties of an inhomneous string cosmology withS3 topology of the spatial sections. Clancyet al. @5# have derived families of anisotropiand inhomogeneous string cosmologies containing nontridilaton and axion fields by applying the global symmetriesthe string-effective action to a generalized Einstein-Rometric. Lazkoz@6# has presented an algorithm for generatifamilies of inhomogeneous space-times with a masslesslar field. New solutions to Einstein-massless scalar fiequations having single isometry, have been generateRef. @6# by breaking the homogeneity of massless scalar fiG2 models along one direction.

It should be noted, however, that the inhomogeneous cmologies in the framework of general relativity with certaphysical fields as a matter source have been investigatedbefore the time of the string cosmology. Exact inhomogneous vacuum Einstein cosmologies withS13S13S1, S2

3S1, andS3 topology of spatial sections have been foundGowdy @7# and studied afterwards by Berger@8# and Misner@9#. Exact stiff perfect fluid inhomogeneous cosmologihave been studied by Wainwright, Ince, and Marshman@10#.Later Charach@11# and then Charach and Malin@12# ~seealso Ref.@13#! have found and studied exact inhomogeneocosmological solutions to the Einstein equations withelectromagnetic and minimally coupled scalar field. Thesolutions have been interpreted as an inhomogeneousverse filled with gravitational, scalar and electromagnewaves.

The purpose of this paper is to present sufficiently genesolution generating techniques which permit us to constrexact inhomogeneous and anisotropic solutions to the etions of low-energy string theory containing nonminimalcoupled dilaton and electromagnetic fields—the so-caEinstein-Maxwell-dilaton~EMD! gravity. Examples of bothhomogeneous and inhomogeneous exact solutions wilalso presented and their asymptotics will be investigated

In addition to the fact that the EMD cosmologies areteresting in their own, there are at least two more motivatiofor considering the EMD cosmologies. First, a long standquestion in cosmology is the existence of a primordial mnetic field. The string theory prediction of the nonminim

©2001 The American Physical Society10-1

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STOYTCHO S. YAZADJIEV PHYSICAL REVIEW D63 063510

coupling of the dilaton to the Maxwell field gives an oppotunity to study the problem in a more general dynamicontext. Recently, in the homogeneous case this questionbeen addressed in Ref.@14#.

The second motivation is to investigate the characteinitial cosmological singularities in the presence of matfields arising from the low-energy superstring theory. Acording to the Belinskii-Khalatnikov-Lifhsitz~BKL ! conjec-ture the dynamics of nearby observers would decouple nthe singularity for different spatial points. A vacuum sptially homogeneous space-time singularity~Bianchi type IX!is described by BKL as an infinite sequence of Kasnerochs ~‘‘oscillatory or mixmaster behavior’’! @15,16#. BKLfurther speculated that a generic singularity should exhsuch a local oscillatory behavior.

Interesting special cases are the so-called asympvelocity-term dominated~AVTD ! singularities @17#. Theircharacteristic feature is the fact that the spatial derivaterms in the field equations are negligible sufficiently cloto the singularities. Such singularities are described by oone Kasner epoch, not by an infinite sequence of Kasepochs, i.e., an oscillatory behavior does not exist. We ththat it is important to study the behavior of the singularitin the presence of matter fields coming from the supersttheory. In particular, it is interesting to investigate the inflence of nonminimal exponential coupling of the dilatonthe U(1) field on the character of the singularities. As hbeen shown in Refs.@18,19# the minimally coupled scalafield can suppress the oscillatory behavior.

Very recently the influence of the exponential dilatoelectromagnetic coupling on the character of initial singulaties has been studied by Narita, Torii, and Maeda in R@20#. These authors have consideredT3 Gowdy cosmologiesin Einstein-Maxwell-dilaton-axion system and have shousing the Fushian algorithm that space-times in general hasymptotic velocity-term dominated singularities. Theirsults mean that the exponential coupling of the dilaton toMaxwell field does not change the nature of the singularIt should also be noted that the exact solutions found inpresent work lead to the same conclusion.

Although, in the present paper we shall not considerBKL conjecture in detail, we believe that the present womay serve as a good ground for further studying ofabovementioned questions in the framework of EinsteMaxwell-dilaton-axion gravity.

II. CONSTRUCTING SOLUTIONS WITH ONE KILLINGVECTOR

We consider EMD gravity described by the action

A51

16pE ~!R22dw`!dw22e22wF`!F !,

where! is Hodge dual with respect to the space-time meg, R is the Ricci scalar curvature,w is the dilaton field, andF5dA is the Maxwell two form.

First we will examine the case of space-time admittione spacelike hypersurface orthogonal Killing vectorX.

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Physically this situation corresponds to a universe withmogeneity broken along two space directions. In the prence of a Killing vector the Bianchi identitydF50 and theMaxwell equationsd!e22wF50 give rise to the followinglocal potentialsCe andCm :

dCe52 i XF, dCm5e22wi X!F.

Here i X is the interior product of an arbitrary form withrespect toX.

The hypersurface orthogonal Killing vectorX naturallydetermines a three-dimensional space-time submanifoldSwith the metric@21#

P5Ng2X^ X,

whereN5g(X,X) is the norm of the Killing vector. We notethat throughout the text we denote the Killing fields and thnaturally corresponding one-forms by the same symbols

The projection onS of the Einstein equations reads

R̂52dw ^ dw11

2N 2 dN^ dN12

N ~e22wdCe^ dCe

1e2wdCm^ dCm!, ~1!

whereR̂ is Ricci tensor with respect to the projection metrP. The Maxwell equations take the form

d†~e22wN 21dCe!50, d†~e2wN 21dCm!50. ~2!

Hered†5!d! is the coderivative operator.The projection of the Einstein equations along to the K

ing vector gives

NhN2g~dN,dN !522Ne22w@e22wg~dCe ,dCe!

1e2wg~dCm ,dCm!#. ~3!

Finally, the vanishing of the twistT5 12 !dX of the Killing

field leads to the constraint

dCe`dCm50. ~4!

Equations~1!, ~2!, ~3!, and ~4! are equivalent to the EMD-gravity equations in the presence of a spicelike hypersurforthogonal Killing vector.

The constraint~4! implies that the one-formsdCm anddCe are proportional, i.e., there exists a functionF such thatdCm5F dCe . Here we will consider the simplest casewhich one of the potentialsC vanishes, sayCe50. In thecaseCe50 the full system dimensionally reduced equatiocan be obtained from the following action:

AR51

16pE A2detPS R2Pi j

2N 2DiND jN2Pi j DiwD jw

22e2wN 21Pi j DiCmD jCmD . ~5!

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EXACT INHOMOGENEOUS EINSTEIN-MAXWELL- . . . PHYSICAL REVIEW D 63 063510

Now we introduce the following symmetric matrixSPGL1(2,R):

S5S N12Cm2 e2w A2Cme2w

A2Cme2w e2w D .

Making use of the matrixS, we may write the action~5! inthe form

AR51

16pE A2detPS R21

2Pi j Tr~DiSDjS

21! D . ~6!

The action~6! hasGL(2,R) as a group of global symmetrfor fixed projection metricP. Explicitly the groupGL(2,R)acts as follows:

S→ASAT,

whereAPGL(2,R).Note, that in the case when one of the electromagn

potentials vanishes the dimensionally reduced EMD eqtions can be viewed as three-dimensional Einstein gracoupled to a nonlinear s model on the factorGL(2,R)/O(2,R). The GL(2,R) symmetry may be employed for generating new exact solutions from known onIn particular, it will be very useful to employ theGL(2,R)symmetry to generate exact solutions with nontrivial electmagnetic field from any given solution to the vacuum fieequations ~i.e., pure Einstein equations! or the dilaton-vacuum equations~Einstein equations plus a dilaton field!.

For example, starting with a dilaton-vacuum seed solut

Svd5S Nvd 0

0 e2wvdD ,

the nonlinear action ofGL(2,R) gives

N5~detA!2Nvd e2wvd

c2Nvd1d2e2wvd,

e2w5c2Nvd1d2e2wvd,

Cm51

A2

acNvd1bde2wvd

c2Nvd1d2e2wvd,

where

A5S a b

c dD .

It is worth noting, however, that some of theGL(2,R)transformations are pure electromagnetic gauge or rescaof the solutions. Only theO(2,R) transformations lead toessentially new solutions.

Exact solutions to the EMD gravity equations can be cstructed from vacuum solutions by the method described

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Ref. @22#, too. In order to obtain solutions by this methoone assumes that the matrixS depends on space-time coodinates through a harmonic potentialV (DiDi V50). Re-quiring then the constraint

21

4TrS dS

dV

dS21

dV D51

be satisfied, the EMD equations are reduced to the vacuEinstein equations

R̂i j 52DiV D jV,

DiDiV50

and a separated matrix equation

d

dV S S21dS

dV D50 .

Therefore, for every solution to the vacuum Einstein eqtions, the solutions of the matrix equation give classesexact EMD gravity solutions~see for details Ref.@22#!.

It should be noted that the seed solutions used forconstruction of exact EMD solutions via the above describmethods, must admit at least one Killing vector. The cawith two commuting hypersurface orthogonal Killing vectowill be considered in the following section.

III. CONSTRUCTING SOLUTIONS WITH TWO KILLINGVECTORS

A. General equations

In this section we consider the EMD gravity equatiowhen the space-time admits two commuting spacelike hypsurface orthogonal Killing vectorsX andY. In this case themetric can be written in the form

ds25e2h22p~dz22dj2!1e2pdx21% e22pdy2, ~7!

whereh, p, and% are unknown functions ofz and j only.Therefore the Killing vectors are given byX5(]/]x) Y5(]/]y).

The physical properties of the metric~7! depend on thegradient of the norm% of the two-form X`Y. The casecorresponding to]m% being globally spacelike or null describes cylindrical and plane gravitational waves. Metrwhere the sign of]m% ]m% can change, describe collidinplane waves or cosmological models with spacelike atimelike singularities. Metrics with globally time-like]m%describe cosmological models with spacelike singularitie

In the present paper we shall consider only the globatimelike case%5j2. It should be noted that the choice%5j2 is dynamically consistent with the EMD gravity equtions because the Ricci tensorR̂ satisfies

g~Y,Y!R̂~X,X!1g~X,X!R̂~Y,Y!50.

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In the presence of a second Killing vector the dimensioreduction can be further continued. Without going into dtails we directly present the dimensionally reduced equati

]j2p1

1

j]jp2]z

2p5e22w22p@~]zCe!22~]jCe!

2#

1e2w22p@~]zCm!22~]jCm!2#, ~8!

]j2w1

1

j]jw2]z

2w5e22w22p@~]zCe!22~]jCe!

2#

2e2w22p@~]zCm!22~]jCm!2#,

~9!

]jCe]zCm5]zCe]jCm , ~10!

]j~je22w22p]jCe!2]z~je22w22p]zCe!50, ~11!

]j~je2w22p]jCm!2]z~je2w22p]zCm!50,~12!

1

j]jh5~]jw!21~]zw!21~]jp!21~]zp!2

1e22w22p@~]jCe!21~]zCe!

2#

1e22w22p@~]jCm!21~]zCm!2#, ~13!

1

j]zh52]jw]zw 12]jp]zp

12e22w22p]jCe]zCe

1 2e2w22p]jCm]zCm . ~14!

Here we should make some comments. We have writhe system of partial differential equations in terms of telectromagnetic potentialsCe andCm . These potentials arederived with respect to the Killing vectorX. In the same waywe may use the electromagnetic potentials with respect toKilling vector Y denoted, respectively, asCe

Y and CmY .

Moreover, we may also use a mixed pair of potentials,Ce andCe

Y . In general, the transition between different paof potentials may be performed by the following formulas

]jCmY52 j e22w22p ]zCe ,

]zCmY52 j e22w22p ]jCe ,

]zCeY52 j e2w22p ]jCm ,

]jCeY52 j e2w22p ]zCm .

For the reader’s convenience we have presented inAppendix the system~8!–~14! written in terms of the mixedpair Ce5v andCe

Y5x .Let us go back again to the system of nonlinear par

differential equations~8!–~14!. We have a complicated system of coupled nonlinear partial differential equations and

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seems that finding all its solutions is a hopeless task. Netheless, as we will see a large enough class of solutionsbe found.

B. Solution generating method

Here we consider the case in which one of the elecmagnetic potentials vanishes. For definiteness we takeCe50. Let us introduce the new potentialsu5p2w, f5p1w, Cm

S5A2Cm and hS52h. Then the system~8!–~14!may be rewritten in terms of the new variables as follows

]j2f1

1

j]jf2]z

2f50,

]j2u1

1

j]ju2]z

2u5e22u@~]zCmS !22~]jCm

S !2#,

]j~je22u]jCmS !2]z~je22u]zCm

S !50, ~15!

1

j]jh

S5~]jf!21~]zf!21~]ju!21~]zu!2

1e22u@~]jCmS !21~]zCm

S !2#,

1

j]zh

S52]jf]zf12]ju]zu12e22u]jCmS]zCm

S .

It is not difficult to recognize that the system~15! coin-cides with the corresponding one for the Einstein-Maxwgravity with a minimally coupled scalar fieldf for the met-ric

ds25e2hS22u~dz22dj2!1e2udx21e22uj2dy2. ~16!

HereCmS is the corresponding nonvanishing electromagne

potential in the case of a minimally coupled scalar field.In this way we have proven the following proposition.

Proposition B.1. Letu, f, hS, andCmS be a solution to the

Einstein-Maxwell equations with a minimally coupled scafield for the metric~16!. Then p5 1

2 (u1f), w5 12 (f2u),

h5 12 hS and Cm5(1/A2)Cm

S form a solution to the equations of EMD gravity for the metric~7!.

This result allows us to generate exact solutions toEMD gravity equations for the metric~7! directly fromknown solutions of the Einstein–Maxwell equations withminimally coupled scalar field. Before going further we shmake some comments. In the case when the space-timemits two spacelike Killing symmetries and one of the eletromagnetic potentials is taken to vanish, it may be shothat the equations governing the transverse part of the grtational field, dilaton and nonvanishing electromagnetic ptential are written in the compact matrix form

]j~jS21]jS!2]z~jS21]zS!50. ~17!

Equation~17! is just the chiral equation. There are powerful solitonic techniques for solving this equation~see Ref.

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@23#!. Although, the study of cosmological soliton solutioin EMD gravity seems to be very interesting, we will nconsider them in the present work.

IV. EXAMPLES OF EXACT SOLUTIONS

A. Homogeneous solutions

Homogeneous solutions are obtained when we havependence only on the ‘‘time coordinate’’j. In this case thedifferential constraint reduces to]jCe]jCm50. Thus wemay choose one of the potentials to vanish. Thereforemay apply the proposition from the previous section to gerate homogeneous solutions starting from the corresponhomogeneous solutions for the case of a minimally coupscalar field. In our notations the homogeneous solutions fminimally coupled scalar field are@13#

f5f01b0 ln~j!, ~18!

u5 ln~ef1/2j11a0/21e2f1/2j12a0/2!,~19!

CeSY5

1

2tanhS f1

21

a0

2ln~j! D , ~20!

wheref0 , f1 , a0, andb0 are arbitrary constants.The above seed solution~18!–~20! gives the following

homogeneous solution to the EMD equations. The potenCe

Y is

CeY5

1

2A2tanhS f1

21

a0

2ln~j! D .

The metric has the form

ds25A2~j!~dz22dj2!1B2~j!dx21C2~j!dy2,

where

A2~j!5eg02f0ja02/41b0

22b0~ef1/2ja0/21e2f1/2j2a0/2 !,

B2~j!5ef0j11b0~ef1/2ja0/21e2f1/2j2 a0/2!,

C2~j!5e2f0j12b0~ef1/2ja0/21e2f1/2j2 a0/2!21.

Hereg0 is a constant.The asymptotic behavior of the expansion factors and

dilaton field in the limitj→0 is as follows:

A2~j!;ja02/41b0

22b02ua0u/2,

B2~j!;j11b02ua0u/2,

C2~j!;j12b01ua0u/2,

w;21

2 S 12b021

2ua0u D ln~j!.

The asymptotic form of the expansion factors and thelaton field in the limitj@1 is

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A2~j!;ja02/41b0

22b01ua0u/2,

B2~j!;j11b01ua0u/2,

C2~j!;j12b02ua0u/2,

w;21

2 S 12b011

2ua0u D ln~j!.

Introducing the scynhroneous timedt5A(j)dj, the lineelement in the limitsj→0 andj@1 can be written in theKasner form

ds2;2dt21t2p1dx21t2p2dy21t2p3dz2,

where the Kasner exponents are defined by

p15

11b071

2ua0u

1

4~17ua0u!21S 1

22b0D 2

13

2

,

p25

12b061

2ua0u

1

4~17ua0u!21S 1

22b0D 2

13

2

,

p35

1

4~17ua0u!21S 1

22b0D 2

21

2

1

4~17ua0u!21S 1

22b0D 2

13

2

,

and the minus sign refers to the limitj→0 while the plussign is for the limitj@1.

The dilaton field is given by

w;s ln~t!,

where

s52

12b071

2a0

1

4~17ua0u!21S 1

22b0D 2

13

2

.

The parametersp1 , p2, and p3 satisfy the Belinskii-Khalatnikov relations

p11p21p351,

p121p2

21p325122s2 .

B. Inhomogeneous solutions

Here proposition B.1 will be applied again, this timegenerate inhomogeneous exact solutions to the EMD eqtions. As a seed family of solutions we take the Charafamily of solutions describing inhomogeneous cosmolog

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with minimally coupled scalar and electromagnetic fields awith S13S13S1 topology of the spatial sections. ThCharach family of inhomogeneous solutions and theirymptotics in the two limiting cases are presented in the Apendix.

According to proposition B.1 the inhomogeneous EMmetric is given by

ds25e2(h2p)~dz22dt2!1e2pdx21j2e22pdy2,

where

2p5 lnF2j coshS 1

2f̃ D G1f,

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2ln~j!1 lnF2 coshS 1

2f̃ D G1

1

4F~f̃0 ,a0 ,An ,Bn ;j,z!

1F~f0 ,b0 ,Cn ,Dn ;j,z!.

The dilaton field is

w51

2f2

1

2lnF2j coshS 1

2f̃ D G .

The inhomogeneous cosmological solutions introducharacteristic length scales. In fact each normal mode haown characteristic scale. The horizon distance in the ‘z’’direction is given byds2ux,y50 and hence

dz5E0

j

dj5j.

In this way nj can be viewed as the ratio of the horizodistance in the ‘‘z’’ direction to the coordinate wavelengtln , i.e., nj5dz/ln .

There are two limiting cases to be considered. The ficase is when the wavelength is much larger than the horscale (nj!1). The second case is when the wavelengthmuch less than the horizon scale (nj@1).

In the first case (nj!1) the asymptotic form of the EMDmetric is

ds25A2~j,z!~dz22dj2!1B2~j,z!dx21C2~j,z!dy2,

where

A2~j,z!5eg(z)2f* ~z!j (1/4)a2(z)1b2(z)2b(z)~e(1/2)f̃

*(z)j (1/2)a(z)

1e2(1/2)f̃*

(z)j2 ~1/2! a(z)!,

B2~j,z!5ef̃*

(z)j11b(z)~e(1/2)f̃*

(z)j (1/2)a(z)

1e2(1/2)f̃*

(z)j2(1/2)a(z)!,

C2~j,z!5e2f̃*

(z)j12b(z)~e(1/2)f̃*

(z)j (1/2)a(z)

1e2(1/2)f̃*

(z)j2(1/2)a(z)!21.

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In the limit nj!1 the asymptotic form of the dilaton is

w;1

2$f* ~z!1sgn@a~z!#f̃* ~z!%

21

2 S 12b~z!21

2ua~z!u D ln~j!.

In order to discuss the cosmological solutions near thegularity we have to consider homogeneous limit forj ap-proaching zero. Introducing the proper time, in the homoneous limit, by

t5E A~j!dj,

we find that the metric has the Kasner form

gmn;~21,t2p1,t2p2,t2p3!.

Here the Kasner indexes are spatially varying

p1~z!5

11b~z!71

2ua~z!u

1

4@17ua~z!u#21F1

22b~z!G2

13

2

,

p2~z!5

12b~z!61

2ua~z!u

1

4@17ua~z!u#21F1

22b~z!G2

13

2

,

p3~z!5

1

4@17ua~z!u#21F1

22b~z!G2

21

2

1

4@17ua~z!u#21F1

22b~z!G2

13

2

,

and satisfy the Belinski-Khalatnikov relations

p1~z!1p2~z!1p3~z!51,

p12~z!1p2

2~z!1p32~z!5122s2~z!,

where

s52

12b~z!71

2a~z!

1

4@17ua~z!u#21F1

22b~z!G2

13

2

.

In the second case whennj@1 the asymptotic form of theEMD metric is as follows.

The first subcase is fora050. Then the EMD metric isgiven by

gmn5hmn1hmn ,

where

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h5diagS 22jb022b0e~1/2!Kj,2j11b0,

1

2j12b0,

2jb022b0e~1/2!KjD

and

h5diagS 0,2

Ajj11b0H~j,z!,2

2

Ajj12b0H~j,z!,0D .

Whena0Þ0 we have

h5diag~2j (1/4)(11a0)21(b021/2)221/2e(1/2)Kj,

j11(1/2)ua0u1b0,j12(1/2)ua0u2b0,

j1/4(11a0)21(b021/2)221/2e(1/2)Kj),

h5diagS 0,j11(1/2)ua0u1b0

AjFH~j,z!1

1

2sgn~a0!H̃~j,z!G ,

2j12(1/2)ua0u2b0

AjFH~j,z!1

1

2sgn~a0!H̃~j,z!G ,0D .

The asymptotic form of the dilaton in the limitnj@1, forboth a050 anda0Þ0, is given by

w;21

2 S 12b011

2ua0u D ln~j!

11

AjS H~j,z!1

1

2sgn~a0!H̃~j,z! D .

The explicit form of the solutions and their asymptotiallow us to make some conclusions. The obtained solutifall in the category of asymptotic velocity-term dominatspace-times. When the singularity is approached the spderivatives become negligible with comparison to the tiderivatives. Sufficiently close to the singularity the evolutiat different spatial points is decoupled and the metric iscally Kasner with spatially dependent Kasner indices satising the Belinskii-Khalatnikov relations.

The late evolution of the exact cosmological solutionsdescribed by a homogeneous, anisotropic universe wgravitational, scalar~dilatonic!, and electromagnetic waveThe nonminimal dilaton-electromagnetic exponential copling influences mainly the homogeneous background uverse rather the scalar and electromagnetic waves onbackground. The former may be considered as minimcoupled scalar and electromagnetic waves up to higherders of 1/j.

V. CONCLUSION

In this paper we have shown that it is possible to fiexact inhomogeneous and anisotropic cosmological solut

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sth

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ns

of low-energy string theory containing nonminimally inteacting dilaton and Maxwell fields—Einstein-Maxweldilaton gravity. First we have considered space-times admting one hypersurface orthogonal Killing vector. It has beshown that in the case of one vanishing electromagnetictential the dimensionally reduced equations possess a gof global symmetryGL(2,R). We have described an algorithm for generating exact cosmological EMD solutiostarting from vacuum and dilaton-vacuum backgroundsemploying the nonlinear action of the global symmegroup. This algorithm is especially useful for generating eact cosmological EMD solutions with only one Killing vector, starting withG1-dilaton-vacuum background. Anothemethod which permits us to construct exact EMD solutiostarting from solutions of the pure Einstein equations hbeen briefly discussed, too.

In the case when the space-time admits two commuhypersurface orthogonal Killing vectors we have givenmethod which allows exact inhomogeneous cosmologicallutions to the EMD equations to be generated from the cresponding solutions of the Einstein-Maxwell equations wa minimally coupled scalar field. Using this method, asparticular case, we have obtained exact cosmological hogeneous and inhomogeneous solutions to the EMD eqtions. The initial evolution described by these solutions isa spatially varying Kasner form. The intermediate stageevolution occurs when the characteristic scales of the inmogeneities approach the scale of the particle horizon. Tstage is characterized by strongly interacting nonlinear grtational, dilatonic and electromagnetic waves. The late elution of the cosmological solutions is described by a baground homogeneous and anisotropic universe filled wweakly interacting gravitational, dilatonic and electromanetic waves.

The cosmological solutions found in the present work fin the category of AVTD space-times. Near the singularthe dynamics at different spatial points decouples andmetric has a spatially varying Kasner form. Therefore,nonminimal coupling of the dilaton to the Maxwell field doenot change the nature of the singularity. The solutionssufficiently generic and therefore we should expect thatT3 space-times in EMD gravity have AVTD singularitiesgeneral. Finally we believe that the present paper will bsufficiently good background for studying similar problemin the more-general case of Einstein-Maxwell-dilaton-axigravity.

ACKNOWLEDGMENTS

The author is deeply grateful to E. Alexandrova for tsupport and encouragement. Without her help the writingthis paper most probably would have been impossible. Tauthor is also grateful to P. Fiziev for the support and stimlating discussions. The author would particularly likethank V. Rizov for having critically read the manuscript anfor useful suggestions.

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APPENDIX A: REDUCED EMD SYSTEM IN TERMS OF THE POTENTIALS CeÄv AND CeYÄx

]j2p1

1

j]jp2]z

2p5e22w22p@~]zv!22~]jv!2#11

j2 e22w12p@~]jx!22~]zx!2#,

]j2w1

1

j]jw2]z

2w5e22w22p@~]zv!22~]jv!2#21

j2 e22w12p@~]jx!22~]zx!2#,

]jv ]jx5]zv ]zx,

]j2v1

1

j]jv2]z

2v52~]jw1]jp!]jv22~]zw1]zp!]zv,

]j2x2

1

j]jx2]z

2x52~]jw2]jp!]jx22~]zw2]zp!]zx,

1

j]jh5~]jw!21~]zw!21~]jp!21~]zp!21e22w22p@~]jv!21~]zv!2#1

1

j2e22w12p@~]jx!21~]zx!2#,

1

j]zh52]jw]zw 12]jp]zp12e22w22p]jv]zv1 2e22w12p]jx]zx.

ray

du

d

APPENDIX B: CHARACH FAMILY OF EXACTSOLUTIONS

For the reader’s convenience we present here the Chafamily of solutions written in our notations. The minimallcoupled scalar field is given by

f5f01b0 ln~j!1 (n51

`

@CnJ0~nj!1DnN0~nj!#

3cos@n~z2zn!#.

Here f0 , b0 , Cn , Dn are arbitrary constants anJ0(•••) andN0(•••) are, respectively, the Bessel and Nemann functions. The metric functionu and the nonvanishingelectromagnetic potentialCe

SY are correspondingly

u5 lnF2j coshS 1

2f̃ D G ,

CeSY5const1

1

2tanhS 1

2f̃ D ,

wheref̃ is an auxiliary scalar field which is of the form

f̃5f̃01a0 ln~j!1 (n51

`

@AnJ0~nj!1BnN0~nj!#

3cos@n~z2zn!#.

The longitudinal part of the gravitational field is

hS5 ln~j!12 lnF2 coshS 1

2f̃ D G1

1

2F~f̃0 ,a0 ,An ,Bn ;j,z!

12F~f0 ,b0 ,Cn ,Dn ;j,z!,

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ch

-

whereF is a solution to the system

1

j]jF5

1

2@~]jf!21~]zf!2#,

1

j]zF5]jf]zf.

For more details see Ref.@13#.In the limit j→0 the minimally coupled scalar field an

the auxiliary field behave as

f;f* ~z!1b~z!ln~j!,

f̃;f̃* ~z!1a~z!ln~j!,

where

a~z!5a012

p (n51

`

Bn cos@n~z2zn!#,

b~z!5b012

p (n51

`

Dn cos@n~z2zn!#,

and

f* ~z!5f01 (n51

` H Cn12

pDnFg1 lnS n

2D G J cos@n~z2zn!#,

f̃* ~z!5f̃01 (n51

` H An12

pBnFg1 lnS n

2D G J cos@n~z2zn!#.

Hereg is the Euler constant.

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te

he

-

s


The asymptotic form of the Charach metric can be writas

ds25As2~j,z!~dz22dj2!1BS

2~j,z!dx21CS2~j,z!dy2,

where

AS~j,z!5eg(z)j (1/4)a2(z)1b2(z)~e(1/2)f̃*

(z)j (1/2)a(z)

1e2(1/2)f̃*

(z)j212 a(z)!,

BS~j,z!5j~e(1/2)f̃*

(z)j (1/2)a(z)1e2(1/2)f̃*

(z)j2(1/2)a(z)!,

CS~j,z!5~e(1/2)f̃*

(z)j (1/2)a(z)1e2(1/2)f̃*

(z)j2(1/2)a(z)!21.

The asymptotic behavior of the scalar fields in tCharach solution in the high frequency regime (nj@1) is

f;f01a0 ln j1j21/2H~j,z!,

f̃;f̃01b0 ln j1j21/2H̃~j,z!,

where

H̃~j,z!5Re (n52`,Þ0

`

~Aunu2 iB unu!e2 i (zn2p/4)

A2punuei (unuj1nz),

H~j,z!5Re (n52`,Þ0

`

~Cunu2 iD unu!e2 i (zn2p/4)

A2punuei (unuj1nz).

The functionsH(j,z), H̃(j,z) satisfy the D’Alembert equations

. D

ev

tiv

06351

n ]j2H~j,z!2]z

2H~j,z!50,

]j2H̃~j,z!2]z

2H̃~j,z!50.

Whena050 the asymptotic form of the metric is

gmn5hmn1hmn ,

where

h5diagS 24j2b02eKj,4j2,

1

4,4j2b0

2eKjD

and

h5diagS 0,4j2H~j,z!/4j,21

4H2~j,z!/4j D .

The constantK is given by

K51

2p (n51

`

@An21Bn

214~Cn21Dn

2!#.

In the case whena0Þ0 the asymptotic form of the metric i

h5diag~2j2b021(1/2)ua0u1(1/2)ua0u2eKj,j ua0u12,

j2ua0u,j2b021(1/2)ua0u1(1/2)ua0u2eKj)

and

h5diag~0,6j ua0u12H~j,z!/Aj,7j2ua0uH~j,z!/Aj,0!.

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Exact inhomogeneous Einstein-Maxwell-dilaton cosmologies