Robertson-Walker Type Lyttleton-Bondi Universe in Five Dimensional General Theory of Relativity
Department of Mathematics
Mahatma Jyotiba Phule Educational Campus
Amravati Road
Nagpur-440 010 (INDIA)
We study the Robertson-Walker type model in the Lyttleton-Bondi universe in five dimensional general theory of relativity. Some exact and physical properties of solution are discussed.
gr-qc0111084.tex
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\title {\large Robertson-Walker Type Lyttleton-Bondi Universe in Five
Dimensional General Theory of Relativity }
\author { G S Khadekar \thanks{Tel.91-0712-239461,
email:gkhadekar@yahoo.com} and  Avinash R Nagpure \thanks{Department
of Mathematics, Jawarharlal Nehru Arts, Commerce \& Science College,
Wadi (Defence), Nagpur-440 023 (INDIA)}  \\
Department of Mathematics, Nagpur University \\ 
Mahatma Jyotiba Phule Educational Campus, Amravati Road \\ 
Nagpur-440 010 (INDIA)}
\maketitle
\begin{abstract}
We study the Robertson-Walker type model in the 
Lyttleton-Bondi universe in five dimensional general 
theory of relativity. Some exact and  physical properties of 
solution are discussed. 
\end{abstract}
\section{ Introduction}
Lyttleton and Bondi [1] have developed a cosmological model assuming
that there is a continuous creation matter due to a net imbalance of
charge. To incorporate the idea of creation, the Maxwell field
equations are modified as
\begin{equation}
F_{ij} = A_{i,j} -A_{j,i}
\end{equation}
\beq
F^{ij}_{;j} = J^{i} - \lambda A^i
\eeq
\beq
J^{i}_{;i} = q_{\;;}
\eeq
where $ \lambda $ is a constant , $ A_{i} $ and $ J_{i} $ denote five
potential and current density five vector respectively, $ F_{ij} $
denote the anti-symmetric electromagnetic field tensor and $q$ the
rate of creation of a charge per unit proper volume. A semicolon 
denotes covariant differentiation.\\ The energy momentum tensor of the
field is of the form 
\beq
T^{j}_{i} =  \left( F_{i \: k} F^{k \: j} + \frac{1}{4} 
F_{k \:l} F^{k\:l}  \right) + \lambda \left ( A_{i \;j} 
- \frac{1}{2} \; \delta^{j}_{i} A_{k} A^{k} \right)
\eeq
where $ F_{ij}= 0 $ (with zero electromagnetic field) relation
\beq
J_{i} = \lambda \: A_{i;}
\eeq
and the energy momentum tensor of the field becomes
\begin{equation}
T^{j}_{i} = \lambda \left( A_{i} \; A^{j} -\frac{ 1}{2} \; \delta^{j}_{i}
A_{k} A^{k} \right)
\end{equation}
Assuming that the charge created will not affect the dynamical
characteristic of the metric and the mechanical affect of such
creation on the energy momentum tensor is nil, we can use the
Einstein's field equations 
\beq
R^{j}_{i} -  \frac{1}{2} \; \delta^{j}_{i} R = -\kappa T^{j}_{i}
\eeq
where $ \kappa (= 8 \pi G) $ is a constant. The velocity of light $ c $
is taken to be unity and $ T^{j}_{i} $ is  given by equation (6).\\
Lyttleton  and Bondi [1], Burman [2], Nduka [3],  Rao and Panda [4],
Reddy and Rao [5] are some  of the authors who have investigated
several aspects of the Lyttleton -Bondi field.
\par In this paper we consider the Kaluza-Klein type
Robertson-Walker(Chaterjee[6]) model in the Lyttleton-Bondi
universe. Some physical properties of the model are discussed. This
result is an extension of a similar one obtained by Reddy [7].
\subsection{Solutions of Field equations}  
\beq
ds^2 = dt^2 - R^{2}(t) \left[ ( 1+ \frac{b}{r} - {l}{r^2})^{-1} dr^2
+ r^2 ( d\theta_{1}^2 + sin^2 \theta_{1} d\theta_{2}^2 ) + (1 +
\frac{b}{r} - lr^2) dy^2\right]
\eeq
where $ l= +1, 0, -1$ for a closed, flat or open space,
respectively. In this situation $A_{i} $ must have the form given by 
\beq
A_{i} = (a, 0, 0, 0, \phi)
\eeq 
But since $ F_{ij} = 0 $, we have $ \frac{\partial a}{\partial t} =
\frac{\partial \phi}{\partial r}. $ \\
From equation from (8) and (9)
\beq
A ^{i} = \left[-( \frac{1+ \frac{b}{r} - l r^2)} {R^2} ) a,\; 0,\; 0,\; 0,\; 
\phi \right] 
\eeq
so that
\beq
A_{i} A^{i} = \left[-( \frac{1+ \frac{b}{r} - l r^2)} {R^2} ) a^2
+ \phi^2 \right] 
\eeq
By use of equations (9) and (11), the field equation (7) for the
metric (8) can be written as 
\beq
\frac{3}{R^2} ( R \ddot R + \dot R^2 + l ) = - \frac{\kappa
\lambda}{2}\left[( \frac{1+ \frac{b}{r} - l r^2)} {R^2} ) a^2 + \phi^2 \right]
\eeq 
\beq
\frac{3}{R^2} ( R \ddot R + \dot R^2 + l ) =  \frac{\kappa
\lambda}{2}\left[( \frac{1+ \frac{b}{r} - l r^2)} {R^2} ) a^2 - \phi^2 \right]
\eeq
\beq
\frac{6}{R^2} ( \dot R^2 + l ) =  \frac{\kappa
\lambda}{2}\left[( \frac{1+ \frac{b}{r} - l r^2)} {R^2} ) a^2 + \phi^2 \right]
\eeq 
\beq
\lambda \; a \; \phi \left( \frac{1+ \frac{b}{r} - l r^2)} {R^2}
\right)= 0
\eeq
where an overhead  dot denotes difference with respect to the time
t. Subtracting equation (12) from equation (13), we get $ a = 0 $
which in view of equation (10) implies $ \phi = \phi(t) $.
Now equation (12) to (15) reduces to 
\beq
 R \ddot R + \dot R^2 + l  = - ( \frac{\kappa
\lambda}{6}) R^2 \phi^2
\eeq 
\beq
\dot R^2 + l  = \frac{\kappa
\lambda}{12} R^2 \phi^2 
\eeq
The exact solution of the set of field equations (16) for $ l= 0$ can
be written as
\beq 
R(t) = (A t + B )^{\frac{1}{4}}, \:\: \phi = \pm \sqrt{\frac{3
A^2}{4 \kappa}}(A t + B ) ^{-1} 
\eeq
where A and B are constants of integration. The corresponding metric
of the solutions can now be written as
\beq
ds^2 = dt^2 - (A t + B )^{\frac{1}{2}} \left[ ( 1+ \frac{b}{r})^{-1} dr^2
+ r^2 ( d\theta_{1}^2 + sin^2 \theta_{1} d\theta_{2}^2 ) + (1 +
\frac{b}{r}) dy^2\right]
\eeq
\par This model is, in general, non-static, homogeneous and
isotropic. The following are some of the properties of the space time.
\begin{enumerate}
\item For $ B > 0, A > 0 $ the flat 4-space expands indefinitely from
initial singular state.
\item For $ B > 0, A < 0 $ the initially finite flat universe contracts
to a singular condition in a finite time.
\item It is interesting to note that in case (1) $ \phi $ as given by
equation (18) decreases with time where as in  case (2) it  increases
indefinitely with time. 
\end{enumerate}
For $ l= \pm 1 $ no general solution to equations (1) could be obtained.
\bibliographystyle{plain}
\begin{thebibliography}{99}
\bibitem{Lyttleton} Lyttleton R A and Bondi H (1959) {\em
Proc.Roy.Soc.A London {\bf 252}, 313 }
\bibitem{Burman} Burman R J  (1971) {\em Proc.Roy.Soc. New South Wales {\bf 104}, 4 } 
\bibitem{Nduka} Nduka A (1981) {\em Astro.Phys., Proc. {\bf B12}, 833 }
\bibitem{Rao} Rao J R and Panda H J  (1975) {\em
J.Phys. A. Math. Gen. {\bf 8},1413}
\bibitem{Reddy} Reddy D R K and  Rao  V U M   (1982) {\em
Ind.J.Pure Appl. Phys. {\bf 20}, 325}
\bibitem{Chaterjee} Chaterjee S  (1990) {\em Asrton. Astrophys. {\bf 230},1}
\bibitem{Reddy} Reddy D R K  (1985) {\em
 Astro Phys. {\bf 122}, 263}
\end{thebibliography}
\end{document}
 












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\begin{document}
\newcommand{\wst}{~{}^{{}^{*}}\llap{$\it w$}}
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\newcommand{\nudst}{~{}^{{}^{**}}\llap{$\nu$~}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\gfrc}[2]{\mbox{$ {\textstyle{\frac{#1}{#2} }\displaystyle}$}} 
\title {\large \bf Multidimensional Homogeneous Cosmological Models in
Wesson Theory of Gravitation }
\author{\Small{\bf G S Khadekar} \thanks{Tel.91-0712-23946,
 email:gkhadekar@yahoo.com} and {\bf Shilpa Samdurkar} \\
\small Department of Mathematics, Nagpur University \\
\small Mahatma Jyotiba Phule Educational Campus, Amravati Road  \\
\small Nagpur-440010 (INDIA) }
\maketitle
\begin{abstract}
Higher dimensional solutions are obtained for a homogeneous, spatially
isotropic cosmological model in Wesson theory of gravitation. Some
cosmological parameter are also calculated for this  model. 
\end{abstract}
\section{ Introduction}
Recently Wesson [1, 2] proposed a five dimensional theory of
 gravity where the rest masses varying with time. In this new theory,
 the space-time is no longer described by a four dimensional manifold
 but by a five dimensional space-time-mass (STM) Riemannian manifold
 where the mass plays the role of the fifth coordinate given by $ x^4
 =\frac{Gm}{c^2}(m= mass)$. The Einstein theory is recovered when the velocity
 $\frac{G}{c^2}\frac{dm}{dt} = 0,$ in other world, the mass is
 constant. In this regards the four dimensional theory of Einstein can
 be thought as embedded in a five dimensional STM theory.
\par This paper concerned with some possible exact solutions for a
 homogeneous, spatially isotropic n-dimensional $ (n > 5)$ cosmological
 model  in a matter free space. Although some exact vacuum solutions
 for five dimensional cosmological model have been worked out by Chi [3] and 
Chaterjee [4]. We have taken the matter density to be
 zero because otherwise  the higher dimensional field equations with
 both time and mass as a variable are too complicated to yield
 explicit solutions. Moreover for the particular case this empty space
 models provide instructive and transparent examples of various
 geometric possibilities. The paper ends with some comments and conclusions.
\par This type of problem has been discussed  in  STM theory of gravity by 
Chaterjee [4] for five dimensional case. However in our solutions, 
it is pointed out that at this stage, the implications of 
higher dimension solutions and is nterpretation are priliminary 
in nature because of some conceptual roblems associated with the new 
theory and because of the absence of
similar higher dimensional solutions in literature so for.
\subsection{Field Equations}
 The line element for a n-dimensional  homogeneous and
spatially isotropic  cosmological model is taken as \\
\begin{equation}
ds^2 = e^\nu dt^2-e^{\omega}\sum_{i=1}^{(n-2)}dx_{i}^2 + e^{\mu} dm^2
\end{equation}
where $ \mu,{\omega} $ and $\nu $ are the functions of time
and mass. Here the coordinate $x^0 = t, \: x^{1,2, \cdots, (n-2) }$  (space 
coordinate) and $x^{(n-1)} = m.$ For simplicity we have set the magnitudes of
 both c and G to unity. By applying this metric to the Einstein field equation
 $ G_{ij} = R_{ij} -\gfrc{1}{2} g_{ij} \, R = 0 $ with the assumption
$e^{\nu}= 1$  we get
\begin{equation}
G_{00} = -(n-2)(n-3) \; \frac{\dot \omega^2}{8}- (n-2) \: \frac{\dot \omega\dot 
\mu}{4} - (n-2) \, e^{-\mu}\left( \frac{\omegadst}{2}-\frac{\omegast \must}{4} + (n-1) \, \frac{\omegast^2 }{8}  \right)= 0
\end{equation}
 \begin{eqnarray}
G_{11} = e^{\omega} \left(\frac{\ddot \mu}{2}  + \frac{\dot \mu^2}{4} 
\right) +  (n-3)e^{\omega}  \left(\frac{\ddot \omega }{2} +\frac{\dot{ 
\omega} \dot {\mu}}{4} +(n-2)\frac{{\ddot \omega^2} }{8} \right)
 \nonumber \\ +  (n-3)  e^{\omega- \mu} \left( \frac{\omegadst }{2} -
 \frac{ \omegast \must}{4} +  (n-2)  \frac{ \omegast^2}{8} \right)= 0  
\end{eqnarray}
 $$ G_{11} = G_{22} = G_{33} = \cdots = G_{(n-2)(n-2)} $$
\beq
G_{0(n-1)} = (n-2) \left( \frac{\dot{\omega}^*}{2 }+ 
\frac{\omegast \: \dot{ \omega}}{4}-\frac{\dot \mu \omegast}{4}\right) = 0 
\eeq
\beq
G_{(n-1)(n-1)} = -(n-2)(n-3) \frac{\omegast^2}{8} - (n-2) \;  
e^{\mu}\left(\frac{ \ddot \omega}{2} +(n-1)\frac{\dot{\omega^2}}{8}\right)=0 
\eeq
 where  a dot and star denote, respectively partial derivative with respect to time and mass.
\subsection{Solutions}
By solving equation (4) we get
\beq
 e^{\mu}= \alpha_{1}(m)\omegast^2  e^{\omega}
\eeq
where $\alpha_{1}(m)$ is an arbitrary function of mass. Using equation (6) in 
(5), we get   
\beq
\omegast^2\left(\frac{(n-3)}{8}+\alpha_{1}(m)e^{\omega}(\frac{\ddot \omega}{2} 
+ (n-1) \; \frac{\dot \omega^2}{8})\right) = 0
\eeq
since $\omegast^2$ is not equal to zero in higher dimensional metric
it follows that
\beq
\omegadst +\frac{(n-1)}{4} \:{\omegast^2} = 
-\frac{(n-3)}{4\alpha_{1}(m)}e^{-\omega} 
\eeq
which  yield the first integral 
\beq
\dot{\omega^2} = - \frac{1}{\alpha_{1}}e^{-\omega} + C_{1}(m) e^{ - 
\frac{(n-1)}{2}\omega} 
\eeq
Equation (9) can also be written as  
\beq
{\dot X^2} = -\frac{1}{\alpha_{1}} + C_{1} X^{-2k}
\eeq
where $ X = e^{\frac{\omega}{2}} $ and $ k= \frac{(n-1)}{2} $
\beq
or\; \int{\frac{dX}{\sqrt{C_{1}a X^{2(1-k)}-1}}} =\int{\frac{dt}{2 \sqrt{a}}}
\eeq
so that
\beq
 \frac{X \sqrt{1- a\; C_{1} X^{3-n}}\; F_{1}[ \frac{1}{3-n},\;\frac{1}{2}, 
\; 1+\frac{1}{3-n},\; a\; C_{1}X^{3-n}]}{\sqrt{a\; C_{1}X^{3-n} -1}} = 
\frac{1}{2\sqrt{a}} t + C_{2}
\eeq
where $\; F_{1}$ is a hyper-geometric function whose series expansion is\\ 
$F_{1}(a, b; c; z) = \sum_{k=0}^{\infty} \frac{(a)_{k}
(b)_{k}}{(c)_{k}z^{k}} k! $ and $C_{2}$ be the arbitrary
constant of integration. \\
For  $ C_{1} =0 $ in equation (9), we get

$$ e^{\omega} = \frac{ -t^2 + 2bt -b^2}{4\alpha_{1}} $$ 
\beq
 or \; \; e^{\omega} =-\alpha t^2 + \beta t + \gamma  
\eeq
where $ \alpha = \frac{1}{4\alpha_{1}}$,  $ \beta =
\frac{1}{2\alpha_{1}}$, $ \gamma = -\frac{b^2}{4\alpha_{1}} $  are
all functions of mass.\\ Equation (13) is similar to the equation obtained by Chaterjee [4]
for five dimensional case.
\subsection{Conclusion}
\par In this paper we have considered the the n-dimensional spatially
homogeneous and isotropic cosmological model in STM theory of
gravity. The main defect of the present model appears to be the arbitrary
nature of mass functions. However, as the role of mass in the STM
theory is not yet fully understood, neither in its geometrical
concepts nor in its physical realm. We think that this new exact
higher dimensional solutions together with cosmological
considerations should bring some additional information, and as such, 
they need to be further investigated. It is our hope that the higher 
dimensional solutions  presented here can be
 used as the starting point to investigate the behaviour of the rest
 of the particles in more realistic universe models.
\bibliographystyle{plain}
\begin{thebibliography}{99}
\bibitem{Wesson} Wesson P S (1983) {\em Astro.Asrophys. {\bf 119}, 145 } 
\bibitem{Wesson} Wesson P S (1984) {\em Gen.Rel.Grav. {\bf 16}, 193 }
\bibitem{Chi} Chi L K  (1990) {\em Gen.Rel.Grav. {\bf 22}, 1347 }
\bibitem{Chaterjee} Chaterjee S (1986) {\em Gen.Rel.Grav.{\bf 1},1073}
\end{thebibliography}
\end{document}









\documentstyle[12pt,a4]{article}
\begin{document}
\newcommand{\wst}{~{}^{{}^{*}}\llap{$\it w$}}
\newcommand{\wdst}{~~~{}^{{}^{**}}\llap{$\it w$}}
\newcommand{\omegast}{~{}^{{}^{*}}\llap{$\omega$}}
\newcommand{\omegadst}{~~~{}^{{}^{**}}\llap{$\omega$}}
\newcommand{\must}{~{}^{{}^{*}}\llap{$\mu$}}
\newcommand{\mudst}{~{}^{{}^{**}}\llap{$\mu$~}}
\newcommand{\nust}{~{}^{{}^{*}}\llap{$\nu$}}
\newcommand{\nudst}{~{}^{{}^{**}}\llap{$\nu$~}}
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\gfrc}[2]{\mbox{$ {\textstyle{\frac{#1}{#2} }\displaystyle}$}} 
\title {\large \bf Multidimensional Homogeneous Cosmological Models in
Wesson Theory of Gravitation }
\author{\Small{\bf G S Khadekar} \thanks{Tel.91-0712-23946,
 email:gkhadekar@yahoo.com} and {\bf Shilpa Samdurkar} \\
\small Department of Mathematics, Nagpur University \\
\small Mahatma Jyotiba Phule Educational Campus, Amravati Road  \\
\small Nagpur-440010 (INDIA) }
\maketitle
\begin{abstract}
Higher dimensional solutions are obtained for a homogeneous, spatially
isotropic cosmological model in Wesson theory of gravitation. Some
cosmological parameter are also calculated for this  model. 
\end{abstract}
\section{ Introduction}
Recently Wesson [1, 2] proposed a five dimensional theory of
 gravity where the rest masses varying with time. In this new theory,
 the space-time is no longer described by a four dimensional manifold
 but by a five dimensional space-time-mass (STM) Riemannian manifold
 where the mass plays the role of the fifth coordinate given by $ x^4
 =\frac{Gm}{c^2}(m= mass)$. The Einstein theory is recovered when the velocity
 $\frac{G}{c^2}\frac{dm}{dt} = 0,$ in other world, the mass is
 constant. In this regards the four dimensional theory of Einstein can
 be thought as embedded in a five dimensional STM theory.
\par This paper concerned with some possible exact solutions for a
 homogeneous, spatially isotropic n-dimensional $ (n > 5)$ cosmological
 model  in a matter free space. Although some exact vacuum solutions
 for five dimensional cosmological model have been worked out by Chi [3] and 
Chaterjee [4]. We have taken the matter density to be
 zero because otherwise  the higher dimensional field equations with
 both time and mass as a variable are too complicated to yield
 explicit solutions. Moreover for the particular case this empty space
 models provide instructive and transparent examples of various
 geometric possibilities. The paper ends with some comments and conclusions.
\par This type of problem has been discussed  in  STM theory of gravity by 
Chaterjee [4] for five dimensional case. However in our solutions, 
it is pointed out that at this stage, the implications of 
higher dimension solutions and is nterpretation are priliminary 
in nature because of some conceptual roblems associated with the new 
theory and because of the absence of
similar higher dimensional solutions in literature so for.
\subsection{Field Equations}
 The line element for a n-dimensional  homogeneous and
spatially isotropic  cosmological model is taken as \\
\begin{equation}
ds^2 = e^\nu dt^2-e^{\omega}\sum_{i=1}^{(n-2)}dx_{i}^2 + e^{\mu} dm^2
\end{equation}
where $ \mu,{\omega} $ and $\nu $ are the functions of time
and mass. Here the coordinate $x^0 = t, \: x^{1,2, \cdots, (n-2) }$  (space 
coordinate) and $x^{(n-1)} = m.$ For simplicity we have set the magnitudes of
 both c and G to unity. By applying this metric to the Einstein field equation
 $ G_{ij} = R_{ij} -\gfrc{1}{2} g_{ij} \, R = 0 $ with the assumption
$e^{\nu}= 1$  we get
\begin{equation}
G_{00} = -(n-2)(n-3) \; \frac{\dot \omega^2}{8}- (n-2) \: \frac{\dot \omega\dot 
\mu}{4} - (n-2) \, e^{-\mu}\left( \frac{\omegadst}{2}-\frac{\omegast \must}{4} + (n-1) \, \frac{\omegast^2 }{8}  \right)= 0
\end{equation}
 \begin{eqnarray}
G_{11} = e^{\omega} \left(\frac{\ddot \mu}{2}  + \frac{\dot \mu^2}{4} 
\right) +  (n-3)e^{\omega}  \left(\frac{\ddot \omega }{2} +\frac{\dot{ 
\omega} \dot {\mu}}{4} +(n-2)\frac{{\ddot \omega^2} }{8} \right)
 \nonumber \\ +  (n-3)  e^{\omega- \mu} \left( \frac{\omegadst }{2} -
 \frac{ \omegast \must}{4} +  (n-2)  \frac{ \omegast^2}{8} \right)= 0  
\end{eqnarray}
 $$ G_{11} = G_{22} = G_{33} = \cdots = G_{(n-2)(n-2)} $$
\beq
G_{0(n-1)} = (n-2) \left( \frac{\dot{\omega}^*}{2 }+ 
\frac{\omegast \: \dot{ \omega}}{4}-\frac{\dot \mu \omegast}{4}\right) = 0 
\eeq
\beq
G_{(n-1)(n-1)} = -(n-2)(n-3) \frac{\omegast^2}{8} - (n-2) \;  
e^{\mu}\left(\frac{ \ddot \omega}{2} +(n-1)\frac{\dot{\omega^2}}{8}\right)=0 
\eeq
 where  a dot and star denote, respectively partial derivative with respect to time and mass.
\subsection{Solutions}
By solving equation (4) we get
\beq
 e^{\mu}= \alpha_{1}(m)\omegast^2  e^{\omega}
\eeq
where $\alpha_{1}(m)$ is an arbitrary function of mass. Using equation (6) in 
(5), we get   
\beq
\omegast^2\left(\frac{(n-3)}{8}+\alpha_{1}(m)e^{\omega}(\frac{\ddot \omega}{2} 
+ (n-1) \; \frac{\dot \omega^2}{8})\right) = 0
\eeq
since $\omegast^2$ is not equal to zero in higher dimensional metric
it follows that
\beq
\omegadst +\frac{(n-1)}{4} \:{\omegast^2} = 
-\frac{(n-3)}{4\alpha_{1}(m)}e^{-\omega} 
\eeq
which  yield the first integral 
\beq
\dot{\omega^2} = - \frac{1}{\alpha_{1}}e^{-\omega} + C_{1}(m) e^{ - 
\frac{(n-1)}{2}\omega} 
\eeq
Equation (9) can also be written as  
\beq
{\dot X^2} = -\frac{1}{\alpha_{1}} + C_{1} X^{-2k}
\eeq
where $ X = e^{\frac{\omega}{2}} $ and $ k= \frac{(n-1)}{2} $
\beq
or\; \int{\frac{dX}{\sqrt{C_{1}a X^{2(1-k)}-1}}} =\int{\frac{dt}{2 \sqrt{a}}}
\eeq
so that
\beq
 \frac{X \sqrt{1- a\; C_{1} X^{3-n}}\; F_{1}[ \frac{1}{3-n},\;\frac{1}{2}, 
\; 1+\frac{1}{3-n},\; a\; C_{1}X^{3-n}]}{\sqrt{a\; C_{1}X^{3-n} -1}} = 
\frac{1}{2\sqrt{a}} t + C_{2}
\eeq
where $\; F_{1}$ is a hyper-geometric function whose series expansion is\\ 
$F_{1}(a, b; c; z) = \sum_{k=0}^{\infty} \frac{(a)_{k}
(b)_{k}}{(c)_{k}z^{k}} k! $ and $C_{2}$ be the arbitrary
constant of integration. \\
For  $ C_{1} =0 $ in equation (9), we get

$$ e^{\omega} = \frac{ -t^2 + 2bt -b^2}{4\alpha_{1}} $$ 
\beq
 or \; \; e^{\omega} =-\alpha t^2 + \beta t + \gamma  
\eeq
where $ \alpha = \frac{1}{4\alpha_{1}}$,  $ \beta =
\frac{1}{2\alpha_{1}}$, $ \gamma = -\frac{b^2}{4\alpha_{1}} $  are
all functions of mass.\\ Equation (13) is similar to the equation obtained by Chaterjee [4]
for five dimensional case.
\subsection{Conclusion}
\par In this paper we have considered the the n-dimensional spatially
homogeneous and isotropic cosmological model in STM theory of
gravity. The main defect of the present model appears to be the arbitrary
nature of mass functions. However, as the role of mass in the STM
theory is not yet fully understood, neither in its geometrical
concepts nor in its physical realm. We think that this new exact
higher dimensional solutions together with cosmological
considerations should bring some additional information, and as such, 
they need to be further investigated. It is our hope that the higher 
dimensional solutions  presented here can be
 used as the starting point to investigate the behaviour of the rest
 of the particles in more realistic universe models.
\bibliographystyle{plain}
\begin{thebibliography}{99}
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