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Astrophys Space Sci (2007) 310: 177–180
DOI 10.1007/s10509-007-9490-z
ORIGINAL ARTICLE
A higher-dimensional string cosmological model
in Brans–Dicke theory of gravitation
D.R.K. Reddy ·R.L. Naidu ·S. Atchuta Rao ·K.N. Devi
Received: 29 December 2006 / Accepted: 13 March 2007 / Published online: 6 July 2007
© Springer Science+Business Media B.V. 2007
Abstract Field equations in the presence of cosmic string
source are obtained in a scalar tensor theory of gravitation
proposed by Brans and Dicke (Phys. Rev. 124, 925 (1961))
with the aid of a five-dimensional Kaluza–Klein metric. An
exact string cosmological model is presented which rep-
resents a five-dimensional Reddy string (Astrophys. Space
Sci. 286, 2003b) in Brans–Dicke theory. Some physical
properties of the model are also discussed
Keywords Higher-dimensional string ·Cosmological
model ·Brans–Dicke theory
1 Introduction
The study of higher-dimensional space–time is important
because of the underlying idea that the cosmos at its early
stage of evolution of the universe might have had a higher-
dimensional era. This fact has attracted many researchers
to the field of higher dimensions (Witten 1984; Appelquist
et al. 1987; Chodos and Detweller 1980). Solutions of field
equations in higher-dimensional space–time are believed to
D.R.K. Reddy ()
Mathematics Department, Maharaj Vijayaram Gajapathi Raj
College of Engineering, Vizianagaram, India
e-mail: reddy_einstein@yahoo.co.in
R.L. Naidu
Department of Mathematics, GMR Institute of Technology,
GMR Nagar, Rajam, 532127, India
S. Atchuta Rao ·K.N. Devi
Department of Statistics, Andhra University, Visakhapatnam,
India
be of physical relevance possibly at the early times be-
fore the universe has undergone compactification transi-
tions. Further, Marciano (1984) has suggested that the ex-
perimental observation of fundamental constants with vary-
ing time could produce the evidence of extra dimensions.
Recently, there has been much interest in alternative the-
ories of gravitations, especially in the Brans–Dicke (1961)
theory of gravity in view of the fact that the latest inflation-
ary models.
Mathiazhagan and Johri (1984), extended inflation (La
et al. 1990; Steinhardt and Accetta 1990), hyper extended
inflation and extended chaotic inflation (Linde 1990)are
based on Brans–Dicke theory and general scalar tensor theo-
ries (Nordtvedt 1970). Brans and Dicke (1961) introduced a
scalar–tensor theory of gravitation involving a scalar field
in addition to the familiar general relativistic metric tensor
field gij . In this theory, the scalar field has the dimension of
inverse of the gravitational constant and its role is confined
to its effects on gravitational field equations. Brans–Dicke
(1961) field equations are
Gij =−8πφ−1Tij −ωφ−2φ,i φ,j −1
2gij φ,kφ,k
−φ−1(φi;j−gij φ), (1)
and
φ≡φ,k
;k=8πφ−1T(3+2ω)−1(2)
where Gij =Rij −1
2gij Ris the Einstein tensor, ωis the
dimensionless coupling constant, Tij is the stress energy of
the matter and comma and semicolon denote partial and co-
variant differentiation respectively.
Brans–Dicke cosmology, in four dimensions, has been
extensively studied by several authors. The work of Singh
178 Astrophys Space Sci (2007) 310: 177–180
and Rai (1983) gives a detailed discussion of Brans–Dicke
cosmological models in four-dimensional space–time.
In recent years there has been a lot of interest in cos-
mological models of alternative theories of gravitation with
string dust cloud source in four-dimensional space–time.
Cosmic strings are one-dimensional topological defects as-
sociated with spontaneous symmetry breaking whose plau-
sible production site is cosmological phase transitions in
the early universe (Kibble 1976; Vilenkin 1985). Lete-
lier (1983), Vilenkin (1981), Chakraborty and Chakraborty
(1992), Bali and Singh (2005) and Krori et al. (1990,1994)
are some of the authors who have investigated string cos-
mological models in general relativity while Mahanta and
Mukherjee (2001), Bhattacharjee and Baruah (2001), Sen
(2000), Reddy (2003a,2003b,2005a,2005b), Reddy and
Rao (2006a,2006b) and Reddy et al. (2006)aresomeof
the workers who have studied string cosmological models
in alternative theories of gravitation in four dimensions. Ra-
haman et al. (2003) have studied string cosmology in five
dimensional space–time based on Lyra geometry. However,
as far as our knowledge goes, much work has not been done
on string cosmological models in other alternative theories
of gravitation in higher-dimensional space–time.
Cosmic strings are important in the early stages of evo-
lutions of the universe before the particle creation and can
explain galaxy formation. Hence it is interesting to study
string cosmology in higher-dimensional space–time. In this
paper we present a string cosmology in higher-dimensional
space–time.
In this paper we present a string cosmological model in
five-dimensional space–time in Brans–Dicke (1961) theory
of gravitation. The resulting model can be considered as an
analog of Reddy string (2003a,2003b and Reddy and Rao
2006a,2006b) in higher dimensions.
2 Metric and field equations
We consider the five-dimensional line element in the form
ds2=−dt2+R2(t )(d x2+dy2+dz2)+A2(t)d ψ 2(3)
where the fifth coordinate is taken to be space-like. The
energy-momentum tensor for cosmic strings is
Tij =ρuiuj−λxixj(4)
where ρ=ρp+λis the rest energy density of cloud of
strings with particles attached to them, λis the tension den-
sity of the string and ρpis the rest energy density of the
particles, uithe cloud five-velocity and
xi=(0,0,0,0,A
−1). (5)
The direction of string will satisfy
υiυi=−1=−xixiand υixj=0(6)
in the comoving coordinate system, we have from (5)
T0
0=−ρ, T 1
1=T2
2=T3
3=0,T
4
4=−λ,
T=−(ρ +λ), T i
j=0,i=j.
(7)
Here the quantities ρ,λand φdepend on tonly.
Now with the help of (4–7), the field equations (1) and
(2) for the metric (3) reduce to
3R•2
R2+3R•A•
RA =8πφ−1ρ+ω
2
φ•2
φ2+φ••
φ+φ
φ,(8)
2R••
R+A••
A+2R•A•
RA +R•2
R2=−ω
2
φ•2
φ2+R•φ•
Rφ +φ
φ,(9)
3R••
R+3R•2
R2=8πφ−1λ−ω
2
φ•2
φ2+A•φ•
Aφ +φ
φ,(10)
φ=φ.•• +φ•3R•
R+A•
A=8π(ρ +λ)
φ(3+2ω).(11)
Also the equations of motion Tij
;j=0 are consequences of
the field equations (1) and (2) which take the simplified form
for the metric (3)as
ρ•+3ρR•
R+(ρ −λ) A•
A=0 (12)
where an over head dot denotes differentiation with respect
to t.
3 Cosmic string model
Here we have four independent field equations (8–11) con-
necting five unknown quantities R,A,λ,ρand φ. Hence
to get a determinate solution one has to assume physical
or mathematical conditions. In the literature (Letelier 1983;
Reddy 2003a,2003b; Reddy and Rao 2006a,2006b). We
have equations of state for string models
ρ=λ(geometric string or Nambu string),
ρ=(1+ω)λ (p-string),
ρ+λ=0(Reddy string).
(13)
Since the field equations are highly non-linear, we also as-
sume an analytic relation between the metric coefficients
(scale factor)
A=μRn(where μand nare constants)(14)
Astrophys Space Sci (2007) 310: 177–180 179
to get determinate solutions (Chakraborty and Chakraborty
1992).
Here we obtain a string model in five dimensions cor-
responding to ρ+λ(trace of energy momentum tensor of
string source, T=0, i.e. the sum of the rest energy density
and tension density for a cloud of string vanishes). However,
the string models corresponding to ρ=λand ρ=(1+ω)λ
could not be obtained due to the highly non-linear character
of the field equations.
Now, using (13) and (14) the field equations (8–11) yield
an exact solution given by
R=l(c3t+c4)1/l,(15)
A=l(c3t+c4)n/l ,(16)
φ=l(c3t+c4)m/l ,(17)
8πλ =−8πρ =l(c3t+c4)m/l (c3t+c4)−2
×6c2
3
l2−3c2
3
l+ωm2
2
c2
3
l2−mnc2
3
l2(18)
where
l=c2+(3+n)c3
c3,m=c2
c3(19)
and ci,i=1,2,3,4, are the constants of integration.
Thus, five-dimensional string cosmological model corre-
sponding to the above solution can be written as
ds2=−dt2+l(c3t+c4)2/ldx2+dy2+dz2
+l(c3t+c4)2n/l dψ2(20)
where the scalar field rest energy density and the tension
density of cosmic string in the model are given by (17)
and (18).
4 Some physical properties
The model given by (20) represents an exact string cosmo-
logical model in five dimensions in the framework of Brans–
Dicke theory of gravitation, when the sum of the tension
density λand the rest energy density ρof the cosmic string
vanishes. The model has no initial singularity.
In order to gain a further insight into the behaviour of the
model, physical and kinematical variables and to have rela-
tively simple picture of their explicit expressions we choose
the constants of integration as
c1=c2=c3=1,c
4=0
whichinturngivefrom(19)
l=n+4,m=1.
With this choice the explicit expressions are:
•spatial volume
V=(n +4)tn+3
n+4;(21)
•expansion scalar
θ=1
3ui
;i=n+4
3(n +4)t−2n+7
n+4;(22)
•shear scalar
σ2=1
2σij σij =1
54(n +4)2(n +4)t −4n+14
n+4.(23)
The deceleration parameter (Feinstein et al. 1995)
q=−(2n+7)(n +4)t −3n+1
n+4−1,(24)
8πρ =−8πλ =ω−8n−24
2(n +4)t−2n+7
n+4,(25)
8πρp=(ω −8n−24)(n +4)t−2n+7
n+4.(26)
The scalar field φin the model takes the form
φ=(n +4)t1
n+4.(27)
The energy conditions imply that ρ>0 and ρp>0 leav-
ing the sign of the string tension density unrestricted. The
rest energy density ρ, string tension density λand particle
density ρptend to zero as time increases indefinitely and
they possess singularities at t=0. The model (20) describes
an expanding string cosmological model. For this model the
spatial volume Vtends to infinity and the expansion scalar θ
and shear scalar tend to zero as t→∞. The negative value
of the deceleration parameter qshows that the model in-
flates. However, the scalar field in the model is free from
initial singularity.
5 Conclusions
In this paper we have presented a higher-dimensional string
cosmological model in the framework of Brans–Dicke
(1961) theory of gravitation. To get a determinate solution
we have assumed that the trace of energy momentum ten-
sor of string cloud vanishes. The model obtained represents
a five-dimensional Reddy string (Reddy 2003b) in Brans–
Dicke theory.
180 Astrophys Space Sci (2007) 310: 177–180
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