Table Of ContentSolution generating in scalar-tensor theories with a massless scalar field and stiff
perfect fluid as a source
Stoytcho S. Yazadjiev
Department of Theoretical Physics, Faculty of Physics, Sofia University,
5 James Bourchier Boulevard, 1164 Sofia, Bulgaria
E-mail: [email protected]
We present a method for generating solutions in some scalar-tensor theories with a minimally
coupled massless scalar field or irrotational stiff perfect fluid as a source. The method is based on
the group of symmetries of thedilaton-matter sector in the Einstein frame. In thecase of Barker’s
theory the dilaton-matter sector possesses SU(2) group of symmetries. In the case of Brans-Dicke
and the theory with ”conformal coupling”, the dilaton- matter sector has SL(2,R) as a group of
symmetries. We describe an explicit algorithm for generating exact scalar-tensor solutions from
2
solutions of Einstein-minimally-coupled-scalar-field equations by employing the nonlinear action of
0
the symmetry group of the dilaton-matter sector. In the general case, when the Einstein frame
0
dilaton-matter sector may not possess nontrivial symmetries we also present a solution generating
2
technique which allows us to construct exact scalar-tensor solutions starting with the solutions
n of Einstein-minimally-coupled-scalar-field equations. As an illustration of the general techniques,
a
examples of explicit exact solutions are constructed. In particular, we construct inhomogeneous
J
cosmological scalar-tensor solutions whose curvature invariants are everywhere regular in space-
0
time. A generalization of themethod for scalar-tensor-Maxwell gravity is outlined.
2
2
v PACS numbers: 04.50.+h, 04.20.Jb,98.80.Hw
1
0
0 I. INTRODUCTION exact solutions are useful as comparisons for numerical
8
andapproximatesolutionsandaschecksofthecomputer
0
1 Scalar-tensor theories of gravity are considered as the codes.
0 most natural generalizationsof generalrelativity [1]- [6]. Solving of scalar-tensortheories equations in the pres-
/ Inthesetheoriesgravityismediatednotonlybythemet- ence of a source is a difficult task due to their com-
c
q ric of space-time but also by a scalar field (the so called plexity in the general case. In fact, scalar-tensor grav-
- gravitational scalar). Scalar-tensor theories contain ar- ity equations are much more complicated than Einstein
gr bitrary functions of the scalar field that determine the equations. In the so called Einstein frame the source-
: gravitational ”constant” as a dynamical variable. From lessscalar-tensorequationsarereducedtoEinsteinequa-
v a theoretical point of view it should be noted that spe- tions with a minimally coupled scalar field. In this case
i
X cificscalar-tensortheoriesarisenaturallyasalow-energy much progress has been achieved in finding exact homo-
r limit of string theory. geneousandinhomogeneouscosomologicalsolutions[28]-
a In the weak field limit scalar-tensor theories differ [37]. Little has been done in solving scalar-tensor equa-
slightlyfromgeneralrelativity. Inthestrongfieldregime, tions with a source. The known solutions are perfect
however,thepredictionsofscalar-tensortheoriesmaydif- fluid homogeneous cosmological solutions depending on
fer drastically from those of general relativity as it was the time coordinate only (see Refs. [12]- [20] and ref-
shown in Refs. [7] and [8]. erences therein). In Ref. [12], Barrow investigated a
Scalar-tensor theories have also attracted much in- method which enables exact solutions to be found for
terest in cosmology (see Refs. [9]- [27] and references vacuumandradiationdominatedFriedmann-Robertson-
therein). Walker(FRW)universesofallcurvaturesinscalar-tensor
Theprogressintheunderstandingofscalar-tensorthe- theories with an arbitrary form of the coupling function
ories of gravity is closely connected with finding and ω(Φ). Particular classes of solutions were presented for
investigation of exact solutions. A theoretical discus- specific choices of ω(Φ), including Brans-Dicke, Barker
sionsofmanyaspectsofthe earlyuniverse,gravitational and Bekenstein theories. Barrow and Mimoso [13] pre-
waves, gravitational collapse and the structure of the sentedamethodforderivingexactsolutionsforflatFRW
compact objects within the framework of scalar-tensor cosmological models with a perfect fluid satisfying the
theories (as in general relativity) necessitates the use of equation of state p = (γ 1)ρ where γ is constant with
−
exact solutions. Besides the theoretical motivation for 0 γ 2, in scalar-tensor theories with an arbitrary
≤ ≤
constructionofexactsolutions,thereisalsoamoreprac- form for the coupling function ω(Φ). A number of ex-
tical one. With the advent of numerical calculations, plicit solutions for inflationary universes and p = 0 uni-
1
verseswereobtained. ExactFRWcosmologicalsolutions considerasimple source. Onthe otherhand, indifferent
in general scalar-tensor theories for stiff perfect fluid or contexts,thescalarfield(differentfromthegravitational
radiation were derived by Mimoso and Wands in Ref. scalar) plays an important role in modern physics: the
[14]. Homogeneousbutanisotropiccosmologiesinscalar- scalar field has been proposed as a candidate for gravi-
tensor theories of gravity were examined by Mimiso and tationallensing [38],[39]and for darkmatter atgalaxies
Wands in Ref. [15]. The authors presented a method for scales[40],aswellasatcosmologicalscales[41],[42],[43].
deriving solutions for any isotropic perfect fluid with a Examples of different kinds of explicit exact solutions
barotropic equation of state in a spacially flat cosmol- are also considered.
ogy. For stiff fluid or radiationor in vacuumthe authors
were able to obtain solutions in a number of anisotropic
Bianchi and Kantowski-Sachs metrics. Extending the II. SCALAR-TENSOR THEORIES WITH A
earlierworkof[11],[12]and[13],BarrowandParsons[16] MINIMALLY COUPLED SCALAR FIELD AS A
provided a detailed analysis of FRW universes in a wide SOURCE AND SYMMETRIES OF DILATON -
range of scalar-tensor theories of gravity. The authors MATTER SECTOR
constructed a range of exact solutions for open, closed
and flat isotropic universes containing matter with an Scalar-tensor theories are described by the following
equation of state p (1/3)ρ and in vacuum. The early action in the Jordan frame:
≤
andlate-timebehavioursofthesolutionswereexamined,
1
too. InRef.[17],BillyardandColeydiscussedtheformal S = d4x g˜ F(Φ)R˜ Z(Φ)g˜µν∂ Φ∂ Φ
µ ν
equivalences between Kaluza-Klein gravity, Brans-Dicke 16πG − −
theoryandgeneralrelativitycoupledtoamasslessscalar ∗ Z p (cid:16) 2U(Φ))+Sm[Ψm;g˜µν]. (1)
−
field. Using the formal equivalences the authors showed
that exact solutions obtained in one theory will corre- Here, G is the bare gravitational constant, R˜ is the
spondtoanalogoussolutionsintheothertwotheories. A Ricciscala∗rcurvaturewithrespecttothespace-timemet-
phase-space analysis of the FRW models in Brans-Dicke ric g˜µν. The dynamics of the scalar field Φ depends
theorywasperformedbyKolitchandEardley[18]. Their on the functions F(Φ), Z(Φ) and U(Φ). In order for
analysiswasimprovedonbyHoldenandWands[19]who the gravitonsto carry positive energy the function F(Φ)
presented all FRW models in a single phase plane. Par- must be positive. The action of matter depends on the
ticular attention was focused on the early and late-time material fields Ψm and the space-time metric g˜µν but
behaviour of the solutions and on whether inflation oc- does not involve the scalar field Φ in order for the weak
curs. The qualitative properties of scalar-tensortheories equivalence principle to be satisfied. It should be men-
of gravity were also studied by Coley in Ref. [20]. The tioned that the most used parametrization in the litera-
author presented exact solutions which are analogues of ture is Brans-Dickeone,correspondingto F(Φ)=Φ and
the generalrelativistic Jacobsstiff perfect fluid solutions Z(Φ)=ω(Φ)/Φ.
and vacuum plane wave solutions which act as past and It is much clearer to analyze the equations in the so-
future attractors in the class of spatially homogeneous calledEinsteinframe. Letusintroducethenewvariables
models in Brans-Dicke theory. gµν and ϕ, and define
It should be noted the the methods developed in [12],
[13], [14], and [15] are solution generating methods only gµν =F(Φ)g˜µν ,
for homogeneous cosmological solutions and are not ap- dϕ 2 3 dln(F(Φ)) 2 Z(Φ)
plicable to more general cases. = + , (2)
dΦ 4 dΦ 2F(Φ)
The purpose of this paper is to present a general (cid:18) (cid:19) (cid:18) (cid:19)
method for generating exact solutions to the gravity (ϕ)=F−1/2(Φ) ,
A
equations with a minimally coupled massless scalar field 2V(ϕ)=U(Φ)F 2(Φ) .
−
(MCSF) and irrotational stiff perfect fluid within the
framework of some scalar-tensortheories whose Einstein From now on we will refer to gµν and ϕ as Einstein
frame dilaton-matter sector has nontrivial symmetries. frame metric and dilaton field correspondingly.
In the general case, when the Einstein frame dilaton- In the Einstein frame the action (1) takes the form
matter sector may not possess nontrivial symmetries
1
we also present a general solution generating technique S = d4x√ g(R 2gµν∂ ϕ∂ ϕ 4V(ϕ))
µ ν
which allows us to construct exact scalar-tensor solu- 16πG − − −
∗ Z
tions starting with the solutions of Einstein-minimally- +S [Ψ ; 2(ϕ)g ] (3)
m m µν
coupled-scalar-field(EMCSF).Thistechniqueisbasedon A
the geodesics of the Riemannian metric associated with where R is the Ricci scalar curvature with respect to
the dilaton-matter sector. the Einstein frame metric g .
µν
The motivations to consider a MCSF as a source are We consider scalar-tensor theories with a minimally
the following. Inview ofthe complexityof the equations coupled (massless) scalar filed (MCSF), σ, as a matter
ofthe scalar-tensorgravity,it isnaturalasa firststep to source. A minimally coupled massless scalar field also
2
corresponds to an irrotational stiff perfect fluid with en- which corresponds to the functions F(Φ)=Φ and
ergy density
(1+3K) 3Ka2Φ
8πG ρ˜=8πG p˜= g˜µν∂µσ∂νσ (4) Z(Φ)= 2KΦ(a2−Φ 1)
∗ ∗ − −
and velocity field
wherea>0andbarearbitraryconstants. Sinceeq.(2)
define ϕ up to a constant we may put b=0. In the case
u˜ =∂ σ/ g˜µν∂ σ∂ σ (5)
µ µ µ ν
− K =1 and a=1 we obtain the Barker’s theory [44].
provided g˜µν∂ σ∂ σ <p0 [36], [37]. For negative Gauss curvature (K < 0) the isome-
µ ν
The Jordan frame action for the scalar field is try group is SL(2,R). The scalar-tensor theories whose
dilaton-matter sector has negative curvature are charac-
1
S = d4x g˜( 2g˜µν∂ σ∂ σ). (6) terized by
m µ ν
16πG − −
∗ Z p
In what follows we will consider only the special form (ϕ)=aexp( K ϕ)+bexp( K ϕ)
A | | − | |
of the potential U(Φ): U(Φ)=2ΛF2(Φ) or V(ϕ)=Λ=
corresponding to thepfunctions F(Φ)=Φpand
const.
The full Einstein frame action then is
(1 3 K )+12 K abΦ
Z(Φ)= − | | | |
1 2 K Φ(1 4abΦ)
S = d4x√ g(R Λ+ DM) (7) | | −
16πG − − L
∗ Z where a > 0 and b 0 are arbitrary constants. In
≥
where the case b = 0 we obtain the Brans-Dicke theory with
K =1/(3+2ω) . When ab=0 the function (ϕ) can
= 2gµν∂ ϕ∂ ϕ 2 2(ϕ)gµν∂ σ∂ σ (8) | | ∗ 6 A
LDM − µ ν − A µ ν be presented in the form
is the dilaton-matter sector of the theory Lagrangian.
Remarkably, in the case of some specific scalar-tensor
(ϕ)=Ωcosh( K ϕ+c)
theories the dilaton matter-sector (8) of the theory pos- A | |
sesses hidden symmetries which allow us to generate ex- where Ω = 2√ab and c is apconstant. Without loss of
actsolutions. Inordertounveilthese symmetrieswe de-
generality we may put c = 0. In the special case Ω = 1
fine a two-dimensional abstract Riemannian space with
and K = 1/3 we obtain the theory with ”conformal
a metric | |
coupling”. This theoryis alsodescribedby the functions
dl2 =dϕ2+ 2(ϕ)dσ2. (9) F(Φ)=1− 16Φ2 and Z(Φ)=1.
A The scalar-tensor theories possessing flat (K = 0)
Solution generation techniques consist in finding in- dilaton-matter sector have an isometry group Iso(R2)
variant transformations of the dilaton-matter sector of and are described by
†
the Lagrangian in the action (7). This is equivalent to
finding the isometry group of the metric (9). In two (ϕ)=1+aϕ
A
dimensions the isometry group of (9) can be either G
1 corresponding to the functions F(Φ)=Φ and
or G . Clearly our metric possesses G isometry corre-
3 1
sponding to the Killing vector ∂/∂σ. From a physical 1 3a2Φ
point of view, however, this symmetry is not interest- Z(Φ)= −
2a2Φ2
ing because it generates just a shift of the scalar field σ:
σ σ+const. Themetric(9)hasG groupofisometries where a=0 is an arbitrary constant.
3
→ 6
onlywhentheGausscurvatureK isconstant. Reversely, It is worth noting that he scalar-tensor theories with
the constant curvature condition imposes a differential the same group of isometries of the dilaton-matter sec-
equation for the function (ϕ): tor may have rather different behaviour from a physi-
A
cal point of view. As for example we may consider the
d2 (ϕ)
K =−A−1(ϕ) dAϕ2 . (10) Ba(ϕrk)er’sctohse(√orKyϕw)itwhhAer(eϕK) ∼=c1o.sF(ϕor)tahnedBaartkheero’srythweoitrhy
A ∼ 6
the effective gravitational ”constant” is a real constant
Inthisway,solvingthediferentialequation(10)weob-
G G (to first order of the weak filed limit) while
tain the scalar-tensor theories whose dilaton-matter sec- foerfaf t∼heor∗y with K =1 the effective gravitational”con-
tor possesses group of isometries G3. stant” may vary : G6 cos2(√Kϕ)+Ksin2(√Kϕ).
When the Gauss curvature is positive (K > 0) the eff ∼
group of isometries is SU(2) (see below). The scalar-
tensor theories corresponding to this case are character-
ized by
∗Here we consider thecase ω>−3/2.
(ϕ)=acos(√Kϕ+b) †Wehaveput thesecond constant in A(ϕ) equalto one.
A
3
Below we consider in details the symmetries of the where
dilaton-matter sector for the scalar-tensor theories de-
eiγ/2 0
scribedaboveandpresentsolutions generatingformulas. (γ)= (16)
The general case when dilaton-matter sector does not D 0 e−iγ/2
(cid:18) (cid:19)
possess nontrivial symmetries is also considered.
and
cos(β/2) sin(β/2)
A. Barker’s theory (β)= SO(2). (17)
O sin(β/2) cos(β/2) ∈
(cid:18)− (cid:19)
Barker’s theory [44] is described by the functions It should be noted that the transformations (14) as-
F(Φ)=ΦandZ(Φ)=(4 3Φ)/Φ(2Φ 2)corresponding sociated with the subgroup consisting of the matrices
− −
to 2(ϕ)=cos2(ϕ). The metric (9) then is correspond to a constant shift of the scalar field σ:
A D
σ σ+const. That is why, without loss of generality,
dl2 =dϕ2+cos2(ϕ)dσ2 . (11) we→may restrict ourselves to the subgroup SO(2).
Taking as a seed solution z = eiσ0 and performing a
0
This metric can be considered as the standard metric SO(2) transformationwith the matrix (β) we obtain a
on the unit 2-sphere. It is more convenient to present new solution z which in terms of ϕ andOσ reads
the metric in the well knowncomplex form. To do so we
introduce the complex field ϕ=arcsin[sin(β)cos(σ )], (18)
0
z =cot(ϕ/2+π/4)eiσ . (12)
sin(σ )
We obtain then σ =arcsin 0 . (19)
1 sin2(β)cos2(σ )
dl2 =4(1d+zdzz¯z¯)2 . (13) q − 0
Once having the solution in the Einstein frame it is
easytorecoverthecorrespondingJordanframeone. The
The metric is invariant under the transformations
results we summarize in the following proposition:
az+b Proposition: Let (g ,σ ) be a solution to the EM-
z z = (14) µν 0
→ ′ ¯bz+a¯ CSF equations with a cosmological term Λ. Then
−
(g˜ ,Φ,σ) form a solution to the Barker-MCSF equa-
µν
where tions with a cosmological potential U(Φ) = 2ΛΦ2 where
σ is given by (19) and
a b
U = SU(2).
(cid:18)−¯b a¯(cid:19)∈ Φ−1(σ0)=1 sin2(β)cos2(σ0), (20)
−
Thereisalsoanindependentdiscretesymmetryz z¯
→
which corresponds to σ σ. g˜ =Φ 1(σ )g . (21)
→− µν − 0 µν
We note that the SU(2) transformations act nonlin-
early on the scalar fields but leave the Einstein frame Withoutgoingintodetailweformulatetheproposition
metric invariant: gµ′ν = gµν. The Jordan frame metric in terms of a perfect fluid:
g˜µν, therefore, transforms under (14) as Proposition: Let (gµν,ρ0,u(µ0)) be a solution to the
Einstein equations with an irrotational stiff perfect fluid
and a cosmological term Λ. Then (g˜ ,Φ,ρ˜,u˜ ) is a so-
µν µ
g˜ Φ =g˜ Φ.
µ′ν ′ µν lution to the Barker’s equations with an irrotational stiff
perfect fluid and cosmological potential U(Φ) = 2ΛΦ2 if
The symmetries of the dilaton-matter sector can be
Φ and g˜ are given by (20) and (21), and
used to generate new solutions from known ones. In the µν
case of Barker’s theory, any solution of Einstein equa- ρ˜=cos2(β)Φ3(σ )ρ ,
0 0
tions with a minimally coupled scalar field (EMCSF)
and cosmological term will be a solution of the Einstein u˜µ =Φ−1/2(σ0)u(µ0), (22)
frame scalar-tensor-MCSF equations with cosmological 8πG ρ = gµν∂ σ ∂ σ .
0 µ 0 ν 0
term and with vanishing ϕ or σ. Therefore, we can con- ∗ −
structBarkercounterpartstoanysolutionoftheEMCSF
equationsusingthe nonlinearactionofthe groupSU(2).
An arbitrary element U SU(2) can be presented in B. Brans-Dicke theory
∈
the form
Brans-Dicke theory is characterized by the functions
U = (δ) (β) (γ) (15) F(Φ) = Φ and Z(Φ) = ω/Φ which correspond to
D O D
4
2(ϕ) = e2αϕ where α2 = K = 1/(3+ 2ω). In the Proposition: Let (g ,ρ ,u(0)) be a solution to the
A | | µν 0 µ
Brans-Dicke case the metric (9) takes the form Einstein equations with an irrotational stiff perfect fluid
and a cosmological term Λ. Then (g˜ ,Φ,ρ˜,u˜ ) is a so-
µν µ
dl2 =dϕ2+e2αϕdσ2. (23) lution to the Brans-Dicke equations with an irrotational
stiff perfect fluid and a cosmological potential U(Φ) =
The metric (23), up to the parameter α, also appears 2ΛΦ2 if Φ and g˜ are given by (29) and (30), and
µν
in the so called dilaton-axion gravity [45]. It should be
noted,however,thatweconsider(23)inadifferentphys- ρ˜=4C2D2Φ3(σ )ρ ,
0 0
ical context. The symmetries of dilaton-axion sector are
u˜ =Φ 1/2(σ )u0, (31)
well known, but for completeness we will describe them µ − 0 µ
8πG ρ = gµν∂ σ ∂ σ .
again in our context. 0 µ 0 ν 0
∗ −
Introducing the complex field
Thetransformations(26)associatedwiththesubgroup
consisting of matrices of the form
ξ =ασ+ie αϕ, (24)
−
we can write the metric (23) in the Klein form A B
=
4 dξdξ¯ T (cid:18)0 A−1(cid:19)
dl2 = . (25)
−α2(ξ ξ¯)2 correspond to either a constant shift or rescaling the
− scalar-fields. That is why, without loss of generality, we
The dilaton-matter sector is SL(2,R) invariant: may restrict ourselves to the matrices
Aξ+B cos(β/2) sin(β/2)
ξ →ξ′ = Cξ+D (26) O(β)=(cid:18)−sin(β/2) cos(β/2)(cid:19)∈SO(2). (32)
where
A B C. A(ϕ)∼cosh( |K |ϕ) theory
L= SL(2,R).
C D ∈
(cid:18) (cid:19) p
Here we consider the scalar-tensor theories with
There is also a discrete symmetry ξ ξ¯which cor- (ϕ) = Ωcosh( K ϕ) , Ω > 0. Introducing the com-
responds to σ σ. → − Aplex field | |
TheJordan→fram−emetricg˜ transformsunder(26)as p
µν
g˜ Φ =g˜ Φ. i
µ′ν ′ µν ζ =exp(Ω K σ) tanh( K ϕ)+
Let us consider a solution (g ,σ ) to the EMCSF | | | | cosh( K ϕ)
µν 0 (cid:18) | | (cid:19)
equations. This solution is also a solution to the Ein- p p
(33)
stein frame scalar-tensor equations with ξ = ie ασ0.
0 −
The SL(2,R) transformation with the matrix L gives a the metric can be written in the Klein form
new solution z =ασ+ie αϕ where
− 4 dζdζ¯
dl2 = . (34)
eαϕ =C2e−ασ0 +D2eασ0, (27) − K (ζ ζ¯)2
| | −
ThecaseunderconsiderationisanalogoustotheBran-
1 ACe ασ0 +BDeασ0 Dicke caseandwe presentdirectly the final resultsomit-
−
σ = . (28)
α C2e ασ0 +D2eασ0 ting the intermediate details:
−
Proposition: Let (g ,σ ) be a solution to the EM-
µν 0
In this way we proved the following proposition: CSF equations with a cosmological term Λ. Then
Proposition: Let (gµν,σ0) be a solution to the EM- (g˜µν,Φ,σ)form asolutiontotheequationsofthe (ϕ)=
A
CSF equations with a cosmological term Λ. Then Ωcosh( K ϕ) theorywith a MCSF anda cosmological
(g˜µν,Φ,σ) form a solution to the Brans-Dicke equations potential U| (Φ|)=2ΛΦ2 if σ, Φ and g˜µν are given by
with a MCSF and a cosmological potential U(Φ)=2ΛΦ2 p
where σ is given by (28) and 1+ ACe√Kσ0 +BDe √Kσ0 2
| | − | |
e2Ω√K σ = , (35)
Φ−1/2(σ0)=C2e−ασ0 +D2eασ0, (29) | | (cid:16)C2e√Kσ0 +D2e √Kσ0 2(cid:17)
| | − | |
(cid:16) (cid:17)
g˜ =Φ 1(σ )g . (30)
µν − 0 µν 2
Φ 1(σ )=Ω2 1+ ACe√Kσ0 +BDe √Kσ0 ,
− 0 | | − | |
Applied to the case of a stiff perfect fluid the proposi- (cid:20) (cid:16) (cid:17) (cid:21)
tion becomes: (36)
5
g˜µν =Φ−1(σ0)gµν. (37) E. Other theories
Itisnotdifficulttorewritethispropositionforthecase
When the metric (9) has nontrivial isometries we can
of a stiff perfect fluid. The fluid energy density is given
use them to generatenew solutionsfrom knownones. In
by
the generalcase, however,the metric (9) has only trivial
symmetry σ σ +constant. That is why the solu-
ρ˜=Ω2(AD+BC)2Φ3(σ0)ρ0. (38) tion generatin−g→procedure considered above fails. Never-
theless, when the dilaton-matter sector does not possess
nontrivial symmetries we can still generate scalar-tensor
solutions starting with solutions to the EMCSF equa-
D. Theory with a flat dilaton-matter sector
tions. For this purpose we consider the geodesics of the
metric (9). It is not difficult to see that if (ϕ(τ),σ(τ)) is
As we have seen, the theory with a flatdilaton-matter
an affinely parameterized geodesic and (g ,σ ) is a so-
sector is described by (ϕ) = 1+aϕ. Let us introduce µν 0
A lutiontotheEMCSFequationswithacosmologicalterm
the complex field
Λ, then (g ,ϕ(τ) ,σ(τ) ) is a solution to the
µν |τ=σ0 |τ=σ0
ς =(1+aϕ)eiσ. (39) scalar-tensor-MCSF equations with a cosmological term
Λ.
In this way we obtain the following
The dilaton-matter sector metric then takes the stan-
Proposition: Let (ϕ(τ),σ(τ)) is an affinely param-
dard flat form
eterized geodesic of the metric (9) and (g ,σ ) is a
µν 0
dl2 =dςdς¯. (40) solution to the EMCSF equations with a cosmological
term Λ. Then (g˜ ,Φ,σ) form a solution to the scalar-
µν
The metric is invariant under the group Iso(R2) i.e. tensor equations with a MCSF and a cosmological poten-
under the transformations tial U(Φ)=2ΛF2(Φ) where
ς →ς′ =eiϑς+η (41) σ =σ(τ)|τ=σ0, (47)
where ϑ R and η =A+iB C.
∈ ∈ F 1(Φ(σ ))= 2(ϕ(τ)) , (48)
Let us take as a seed solution ς0 = 1 + aσ0. It − 0 A |τ=σ0
is not difficult to see that two of the parameters of
the group Iso(R2) can be absorbed as constant shifts g˜µν =F−1(Φ(σ0))gµν. (49)
σ σ + const, and only one essential parameter re-
→
mains. Performing a translation with η = iB we obtain The geodesic equations for the metric (9) can be for-
a new solution ς with mally solved and we obtain
(1+aϕ)= (1+aσ )2+B2, (42) dϕ
0 =C τ ,
1
1 C2/ 2(ϕ) ±
p Z − A
dτ
1 B p σ =C +C (50)
σ = arcsin (43) 2 2(ϕ(τ))
a " (1+aσ0)2+B2# Z A
where C, C and C are constants. The constants C
1 2 1
p
So we have: and C are unimportant and we shall omit them.
2
Proposition: Let (gµν,σ0) be a solution to the EM- As a concrete example we consider the theory with
CSF equations with a cosmological term Λ. Then F(Φ) = Φ and Z(Φ) = 1Φ2 3Φ+3 corresponding to
2 Φ(−Φ 1)
(g˜µν,Φ,σ)formasolutiontotheequationsoftheA(ϕ)= 2(ϕ) = 1/(1+ϕ2). Solving the−geodesic equations for
1U+(Φa)ϕ=th2eΛoΦry2wifitσhiasMgivCeSnFbyan(d43a)caonsmdological potential Athe metric dl2 =dϕ2+ 1d+σϕ22 we obtain
Φ−1(σ0)=(1+aσ0)2+B2, (44) ϕ= 1 1 C2sin(Cσ0) , (51)
C −
p
g˜ =Φ 1(σ )g . (45)
µν − 0 µν 1+C2 1 C2
σ = σ − sin(2Cσ ) (52)
In the case of a stiff perfect fluid we obtain 2C 0− 4C2 0
where 0<C2 <1.
ρ˜=B2Φ3(σ )ρ . (46)
0 0 Therefore,if(g ,σ )isasolutiontotheEMCSFequa-
µν 0
tions with a cosmological term Λ, then (g˜ ,Φ,σ) is a
µν
solution to the equations of 2(ϕ) = 1/(1+ϕ2)-theory
A
6
withaMCSFandacosmologicalpotentialU(Φ)=2ΛΦ2 3. A2(ϕ)=1/(1+ϕ2)-theory
where σ is given by (52) and
Solution generating formulas (53) and (54) give
1 C2
Φ(σ )=1+ − sin2(Cσ ),
0 C2 0 1 C2 C
g˜ =Φ 1(σ )g . (53) Φ(t)=1+ − sin2 ln(t) ,
µν − 0 µν C2 √3
(cid:18) (cid:19)
In the case of a stiff perfect fluid this solution corre- C2
8πG ρ˜= Φ3(t). (58)
sponds to the energy density ∗ 3t2
ρ˜=C2Φ3(σ0)ρ0. (54) The solution (57) (presented in coordinates different
from those used here) was derived by Gurevichet al [48]
and by Mimoso and Wands [14] using a completely dif-
ferentmethod. Although,thesolutions(56)and(58)can
III. EXAMPLES OF EXACT SOLUTIONS
be also derived by the methods developed in Refs. [13],
[14] they have not been presented so far in an explicit
Asanillustrationofthegeneraltechniquesweconsider
form. It is worth noting that the methods used here to
someparticularexamplesofexactsolutions. Here weset
derivethe homogeneouscosmologicalsolutions aremuch
Λ=0.
more elegant and powerful than the methods developed
in the previous works on the subject.
A. Homogeneous cosmological solutions
B. Inhomogeneous cosmological solutions
Our first examples are homogeneous isotropic cosmo-
logical solutions with an irrotational stiff perfect fluid.
Our next examples are much more interesting. We
Consider the metric
present new inhomogeneous stiff perfect fluid cosmolog-
ds2 = dt2+t2/3 dx2+dy2+dz2 . (55) ical solutions with everywhere regular curvature invari-
0 − ants. Thegravitationalscalarandtheenergydensityare
This is a flat Friedma(cid:0)n-Robertson-Wal(cid:1)ker metric for everywhere regular, too.
stiff perfect fluid. The fluid energy density is 8πG ρ0 = We consider the following solution to the Einstein
1/3t2 andthe4-velocityisuµ = ∂µt. Thefluidve∗locity equationswithanirrotationalstiffperfectfluid[46],[47]:
−
potential is σ = (1/√3)ln(t).
0
−
wilSlinbceecooumrpsleeetdelsyolduettieornmiisnfiedxeodntlhyebgyenthereaetxedplsicoiltutfioornms ds20 =cosh2(1−b)(2mt)cosh4b(2b−1)(mr)(−dt2+dr2)
of Φ(σ ). That is why, for brevity, we shall present only +cosh2(1−b)(2mt)sinh2(mr)cosh2(1−2b)(mr)dφ2
0
the explicit form of Φ(σ ) and the energy density. +cosh2b(2mt)cosh4b(mr)dz2, (59)
0
1. Barker’s theory σ (t)= b2 1arctan[sinh(2mt)].
0
−
p
The energy density and the four-velocity are given by
Using the solution generating procedure for Barker’s
theory we obtain
8πG ρ =
0
Φ−1(t)=1−sin2(β)cos2(√13ln(t)), 4m2(b2−1)cosh−2(2−b)(2mt)cosh−4b(2b−1)∗(mr), (60)
cos2(β)
8πG∗ρ˜= 3t2 Φ3(t). (56) u(µ0) =cosh1−b(2mt)cosh2b(2b−1)(mr). (61)
Here, m and b (b2 >1) are free parameters.
The solution is everywhere regular and satisfies the
2. Brans-Dicke theory
global hyperbolicity condition. The curvature invariants
are regular for whole spacetime. In addition, the space-
In the case of Brans-Dicke theory the transformations
time describing by this solution admits two commuting
(31) yield the following solution: spacelike Killing vectors ∂ and ∂ which are mutually
∂z ∂φ
Φ 1/2(t)=C2tα/√3+D2t α/√3, and hypersurface orthogonal.
− −
4C2D2
ρ˜= Φ3(t). (57)
3t2
7
1. Barker’s theory The solution (64) is nonsingular. The gravitational
scalar and the energy density and the curvature invari-
Usingasaseedsolution(59)andapplyingthesolution ants I˜1, I˜2, I˜3 are regular everywhere.
generatingformulas(22)weobtainthefollowingBarker’s Clearlythesolutiongeneratingtechniquesdevelopedin
theory three parametric solution: the present paper allow us to construct inhomogeneous
cosmologicalsolutionswitheverywhereregularcurvature
Φ 1(t)=1 sin2(β)cos2 b2 1arctan(sinh(2mt)) , invariants in many other scalar-tensor theories different
−
− − from those considered above.
(cid:16)p (cid:17)
ρ˜=cos2(β)Φ3(t)ρ . (62)
0 C. Static spherically symmetric solutions
When cos(β) = 0 the solution becomes a vacuum
Finally we present examples of new static, spherically
scalar-tensor solution. That is why we shall focus our
symmetric solutions.
attention on the more interesting case cos(β) = 0. In
6 AsaseedsolutionwetaketheJanis-Newman-Winicour
this case the gravitational scalar and energy density are
solution [49] to the EMCSF equations:
everywhereregularinspace-time. Moreover,wehavecal-
culated the curvature invariants
ds20 =−f02λdt2+f0−2λdr2+f02(1−λ)r2 dθ2+sin2θdφ2 ,
I˜ =C˜ C˜µναβ, I˜ =R˜ R˜µν, I˜ =R˜2 σ0 = 1−λ2ln(f0) (cid:0) ((cid:1)64)
1 µναβ 2 µν 3
wherpe f2 =1 2M and 0 λ 1.
0 − r ≤ ≤
withrespecttothemetricg˜ =Φ 1(σ )g andfound Whenwe haveasymptoticallyflatseedbackgroundsit
µν − 0 µν
that they are everywhere regular. In this sense the solu- is natural to consider the solution generating transfor-
tion (62) is nonsingular in the case cos(β)=0. mations preserving the asymptotic flatness of the field
6
configuration. It is not difficult to see that the subgroup
consisting of matrices of the form
2. Brans-Dicke theory
π π
U = ( ) (β) 1( )
−
D 2 O D 2
In the case of Brans-Dicke theory, the solution gener-
(i.e., isomorphic to SO(2) SU(2)) preserves the
ating formulas (31) give ⊂
asymptoticsinthe caseofBarker’stheory. Inthe caseof
Brans-Dicketheory, the subgrouppreserving the asymp-
Φ−1/2(t)=cos2(β/2)exp α b2−1arctan(sinh(2mt)) totics is SO(2) SL(2,R) consisting of the matrices
⊂
+sin2(β/2)exp (cid:16)α pb2 1arctan(sinh(2mt)) (cid:17), O(β).
− −
(cid:16) p (cid:17)
1. Barker’s theory
ρ˜=sin2(β)Φ3(t)ρ . (63)
0
The solution generating transformations which pre-
Inthesolution(63)wehaverestrictedourselvestothe
serve the asymptotics yield the following solution:
subgroup SO(2) SL(2,R).
⊂
Thegravitationalscalarandtheenergydensityareev-
Φ 1(f )=1 sin2(β)sin2 1 λ2ln(f ) , (65)
erywhereregularinspace-time. Thecurvatureinvariants − 0 − − 0
I˜, I˜ and I˜ are also everywhere regular. In this sense (cid:16)p (cid:17)
1 2 3 cos(β)sin √1 λ2ln(f )
the space-time described by the solution (63) is nonsin- σ =arcsin − 0 .
gular. 1 sin2(β)s(cid:0)in2 √1 λ2ln(cid:1)(f )
0
− −
q
(cid:0) (cid:1)
3. A2(ϕ)=1/(1+ϕ2)-theory
2. Brans-Dicke theory
Solution generating formulas (53) and (54) applied to
the seed solution (59) yield the following solution: The scalar-tensor image of (64) in the case of Brans-
Dicke theory is
1 C2
Φ(t)=1+ −C2 sin2 C b2−1arctan(sinh(2mt)) , Φ−1/2(f0)=cos2(β/2)f0α√1−λ2 +sin2(β/2)f0−α√1−λ2, (66)
h p i σ = 1 sin(β) f0α√1−λ2 −f0−α√1−λ2 .
ρ˜=C2Φ3(t)ρ0. 2α cos2(β/2)f0α√1−λ2 +sin2(β/2)f0−α√1−λ2
8
3. A2(ϕ)=1/(1+ϕ2)-theory the Maxwell action (in four dimensions) is conformally
invariant we can immediately write the scalar-tensor-
The solution generating transformations for the case MCSF-Maxwell action in the Einstein frame
under consideration yield the solution:
1
S = d4x(R 2gµν∂ ϕ∂ ϕ
µ ν
Φ=1+ 1−C2C2 sin2(C 1−λ2ln(f0)) , (67) 2A2(1ϕ6)πgGµν∗∂Zµσ∂νσ F−µνFµν Λ .
− − −
1+C2 p
σ = 1 λ2ln(f ) As is seen the dilaton-scalar-field sector p(cid:1)ossesses the
0
2C −
same symmetries (when they exist) as in the case of ab-
1 C2p
− sin(2C 1 λ2ln(f )) . senceofMaxwellfieldandtheycanbeemployedtogener-
− 4C2 − 0 ate new solutions with electromagnetic field from known
p ones. In the general case, when the metric associated
We recall that the Jordan frame metric in the above
with the dilaton-MCSF sector may not possess nontriv-
three examples is given by g˜ = Φ 1(f )g where the
µν − 0 µν ial symmetries we can construct exact scalar-tensor so-
explicit form of Φ(f ) depends on the particular scalar-
0 lutions in the presence of electromagnetic field using the
tensor theory.
geodesics of the dilaton-scalar-fieldsector metric.
In this section we have constructed different kinds of
exact scalar-tensor solutions in the presence of a stiff
perfect fluid or a minimally coupled scalar field. Using
ACKNOWLEDGMENTS
thesolutiongeneratingmethodsdevelopedinthepresent
work we can, of course, generate many more and much
morecomplicatedscalar-tensorsolutionstakingasaseed I would like to thank V. Rizov for his interest in this
work and stimulating discussions.
solutions the known in the literature solutions of EM-
CSF equations (for example see [30]- [35]). The physi- This work was partially supported by Sofia University
Grant No 459/2001.
calpropertiesofeverygeneratedclassscalar-tensorsolu-
tions, however, require a separate investigation.
IV. CONCLUSION
In this paper we have presented a general method for
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10
