2
[1.1] GENERAL THEORY OF RELATIVITY
The success of Newtonian gravitation based on the inverse square law
and the Newtonian mechanics (three laws of motion) is well known.
Newtonian mechanics has glaring success and is in perfect agreement
with the experiments where low speed is concerned. However, it
collapses badly at higher speed comparable to that of light. Also the
Newton’s equations of motion are covariant under Galilean
transformations but they are not obeyed by Maxwell’s
electromagnetic equations. These two major contradictions have
doubted the universal validity of the Newtonian mechanics which
lead to the foundation of special theory of relativity by Albert
Einstein in 1905.
Special relativity unifies the concept of space and time into a single
four dimensional structure called space-time. This concept of space-
time that arises from relativity is based on two simple postulates:
(i) The speed of light in free space is constant and
(ii) The laws of physics are the same in all inertial (non-
accelerating) frames.
In special relativity the space-time is flat and hence, this theory does
not deal with gravitation. To overcome these limitations, Einstein
generalized the special theory of relativity and proposed a new theory
3
in 1916, known as general theory of relativity or Einstein’s theory of
gravitation.
The general theory of relativity is a more accurate and comprehensive
description of gravitation than the prevailing Newtonian gravitation
theory. This theory deals with non-inertial (accelerating) frame unlike
special theory of relativity which deals only with inertial frame. In
general relativity, the force of gravity is due to the curvature of
space-time which propagates as a wave. The curvature of space-time
is due to the massive object on it, such as Sun, which warps space
around its gravitational centre. In such a space, the motion of the
particles can be described in terms of gravity rather than in terms of
external forces. In the development of general theory of relativity,
Einstein was mainly guided by three general principles, viz. principle
of covariance, principle of equivalence and Mach’s principle.
The Principle of Covariance
The principle of covariance states that the laws of physics must be
independent of space-time coordinates i.e. the laws of nature must
retain their original form in all coordinates system. According to this
principle we must express all the physical laws of nature by means of
equations in the covariant form, which are independent of coordinate
systems. This can be done by expressing the laws of nature in the
4
form of tensor equations, because the tensor equation has exactly the
same form in all coordinate system.
The Principle of Equivalence
The principle of equivalence states that at any point of space-time we
can find a locally inertial system in which the laws of special theory
of relativity are valid. This principle is the actual hypothesis by which
gravitational considerations are introduced into the development of
general relativity. The experimental verification by Eotvos in 1890 at
Princeton University that the inertial mass and the gravitational mass
of the same body is equal, served as a tool to Einstein to formulate the
principle of equivalence. This principle says that no physical
experiment can distinguish whether the acceleration of a free particle
is due to gravitational field or it is due to an acceleration of a frame of
reference. Hence, this leads us to an intimate relation between metric
and gravitation. The principle of equivalence is classified into:
(a) The strong principle of equivalence and
(b) The weak principle of equivalence.
The strong principle of equivalence states that the observable laws of
nature do not depend upon the absolute values of the gravitational
potentials while the weak principle of equivalence implies equality of
inertial and gravitational mass of a closed system. In his later work
5
Einstein did not make a sharp difference between these two principles
but used mostly the strong equivalence principle.
Mach Principle
Mach’s principle states that all inertial forces are due to the
distribution of matter in the universe. This intriguing principle
inspired Einstein while he was developing the general theory of
relativity. Mach principle is based on the Machian ideas that inertia as
well as gravitation depends upon mutual action between bodies. The
importance of Mach’s principle is that it can be used to determine the
geometry of the space-time and thereby the inertial properties of test
particles from the information of density and mass energy distribution
in its neighbourhood.
In brief, according to the Mach principle:
(i) The inertia of the body must increase when ponderable
masses are piled up in its neighbourhood.
(ii) A body must experience an accelerated force when
neighbouring masses are accelerated and, in fact, the force
must be in the same direction as that of acceleration.
(iii) A rotating hallow body must generate inside of itself a
“cariolis field”, which deflects moving bodies in the sense
of the rotation, and a radial centrifugal field as well.
6
The role of Machian effects and its various interpretations in the
general theory of relativity has been discussed by many authors. But
it was Einstein who first recognized the necessity of the principle and
he has shown that above three effects are present in general relativity.
The foundation of general relativity is based on the Riemannian
metric
jiij dxdxgds 2 , ( 4,3,2,1, ji ).
Here the fundamental metric tensor ijg plays the role of gravitational
potential and gravitational field. Curvature of space-time is related to
the matter and energy through Einstein’s field equations
,8
2
12c
TGπgRRG
ijijijij
where ijG is the Einstein tensor, ijR is the Ricci tensor, R is the Ricci
scalar, ijg is the metric tensor, ijT is an energy momentum tensor,
G is the constant of gravitation and c is the speed of light.
[1.2] ALTERNATIVE THEORIES OF GRAVITATION
Einstein’s general theory of relativity is one of the most beautiful
structure of theoretical physics which describes the successful theory
of gravitation in terms of geometry. It has also served as a basis for
models of the universe. However, since Einstein first published his
theory of gravitation, there have been many criticisms on general
7
relativity because of the lack of certain ‘desirable’ features in the
theory. According to Einstein the Mach’s principle is not
substantiated by general relativity. Also the singularity problem and
some unsatisfactory features exist in general relativity. Therefore, to
overcome these difficulties several theories of gravitation have been
proposed as alternatives to Einstein’s theory of general relativity. The
most important among them are scalar-tensor theories of gravitation
formulated by Jordan (1955), Brans and Dicke (1961), Nordtvedt
(1970), Hoyle and Narlikar (1971), Ross (1972), Dunn (1974),
Schmidt et al. (1981), Saez Ballester (1985) and Motta (1997). Sen
(1957) constructed a unified field theory based on Lyra’s (1951)
modification of Riemannian geometry. Professor Rosen (1973)
proposed a bimetric theory of gravitation to get rid of the singularities
appearing in the Einstein’s theory of general relativity. Barber (1982)
proposed two ‘self-creation’ theories based on two sets of general
relativistic field equations involving matter and a scalar field.
The theories mentioned above have been developed as a consequence
of the fact that Einstein’s theory of general relativity requires some
modification in view of the certain points. The logic of the
development in each case is different and requires full analysis of the
situation which can be done, in part, by a critical mathematical survey
and seeing thereby the generality of the field equations and in part, by
8
scrutinizing the implications in view of the physics incorporated by
them. Both these ideas are interwoven and one without the other is
meaningless. With these objectives we propose to investigate, in this
thesis, some cosmological solutions of the field equations in certain
alternative theories of gravitation. In this chapter, a systematic survey
of the alternative theories of gravitation, which form the subject of
our investigation, is conducted.
(i) A Unified Field Theory Based On Lyra Geometry
Since the introduction of Einstein’s theory of gravitation, attempts
have been made to unify the field theories; such a theory would be
required for a generalization of the usual Riemannian space-time.
Weyl (1918) made one of the best attempts in this direction. He
proposed a more general theory in which electromagnetism is also
described geometrically. However this theory, based on non-
integrability of length transfer, had some unsatisfactory features and
did not gain general acceptance. Later Lyra (1951) suggested a
modification of Riemannian geometry, which may also be considered
as a modification of Weyl’s geometry, by introducing a gauge
function into the structureless manifold which removes the non-
integrability condition of the length of a vector under parallel
transport and a cosmological constant is naturally introduced from the
geometry. In the subsequent investigations Sen (1957), Sen and Dunn
9
(1971) formulated a new scalar-tensor theory of gravitation and
constructed an analog of the Einstein’s field equations based on
Lyra’s geometry.
According to Halford (1970), the present theory predicts the same
effects within observational limits, as far as the classical solar system
tests are concerned. Soleng (1987) has pointed out that the constant
displacement field in Lyra’s geometry will either include a creation
field and be equal to Hoyle’s creation field in cosmology (Hoyle,
1948; Hoyle and Narlikar, 1963, 1964) or contain a special vacuum
field which together with gauge vector term may be considered as a
cosmological term.
The field equations (in geometrized units for which c = 1, G = 1), in
normal gauge for Lyra’s manifold, obtained by Sen (1957) as
ijk
kijjiijij TπφφgφφRgR 84
3
2
3
2
1 ,
where φ is the displacement field, ijR is the Ricci tensor, R is the
Ricci scalar, ijT is the energy momentum tensor and ijg is the metric
tensor.
Interacting scalar fields for different space-times in Lyra geometry
have been studied by Several authors viz. Bhamra (1974), Karade and
Borikar (1978), Kalyanshetti and Waghmode (1982), Reddy and
Innaiah (1986), Reddy and Venkateswarlu (1987), Singh and Singh
10
(1991a, 1992), Singh and Desikan (1997), Pradhan and Pandey
(2003), Pradhan and Vishwakarma (2004), Mohanty et al. (2006),
Casana et al. (2005, 2006, 2007), Bali and Chandnani (2008), Kumar
and Singh (2008), Rao et al. (2008) and singh et al. (2009). Motivated
by these researchers, in the chapter II, we have studied spatially
homogeneous and isotropic FRW cosmological models with bulk
viscosity and zero- mass scalar fields in Lyra geometry.
(ii) Barber’s Self-Creation Theories
Several modifications to Einstein’s general theory of relativity have
been proposed and extensively studied so far by many authors to
unify gravitation and many other effects in the universe. The role of
Mach’s principle in physics is discussed in relation to the equivalence
principle. Brans-Dicke (1961) pointed out that as a consequence of a
Mach’s principle the value of gravitational constant should be
determined by the matter in the universe and they have taken this
concept as the reason for generalizing the Einstein’s theory of general
relativity to the scalar-tensor theory of gravitation. Here, the tensor
field is identified with the space-time of Riemannian geometry and
scalar field is alien to geometry. This theory does not allow the scalar
field to interact with fundamental principles and photons. However,
Barber (1982) has modified Brans-Dicke scalar-tensor theory to
develop a continuous creation of matter in the large scale structure of
11
the universe. As a result, Barber (1982) proposed two ‘self-creation’
theories based on two sets of general relativistic field equations
involving matter and a scalar field. These are generalization in some
sense of the Brans-Dicke (1961) field equations.
Barber’s First Self-Creation Theory
Barber (1982) attempted to produce a continuous creation scalar-
tensor theory by adapting the Brans-Dicke theory. Brans-Dicke theory
does not allow the scalar field to interact with fundamental particles
and photons. The modified theory creates the universe out of self-
contained gravitational, scalar and matter fields. However, the
solution of the one-body problems reveals unsatisfactory
characteristics of the first theory and this theory cannot be derived
from an action principle. Brans (1987) has pointed out that the field
equations in Barber’s first self-creation theory is not only in
disagreement with experiment but are actually inconsistence, in
general, since the equivalence principle is violated.
The field equations in Barber’s first self-creation theory are
ijjiijijij gφλ
φφλ
TπφRgR3
2
3
28
2
1;
1 □φ
and □ Tλπφ 4 ,
where λ is coupling constant to be determined from experiments.
12
Barber’s Second Self-Creation Theory
The second theory retains the attractive features of the first theory and
overcomes previous drawbacks. Like the first theory, this theory
cannot be derived from an action principle. This modified theory
creates the universe out of self-contained gravitational and matter
fields. In this theory, the gravitational coupling of the Einstein field
equations is allowed to be a variable scalar on the space-time
manifold. Barber’s second theory is a modification of general
relativity to include continuous creation and is within observational
limits. Thus, it modifies general relativity to become a variable
G-theory. In this theory the scalar field does not directly gravitate but
simply divides the matter tensor with the scalar acting as a reciprocal
gravitational constant. The scalar field is postulated to couple with the
trace of the energy momentum tensor.
The field equations in Barber’s (1982) second self-creation theory of
gravitation are
ijijij TπφRgR 182
1
and the scalar field equations is defined by
□ Tλπ
φ3
8 ,
13
where □ kkφφ ; is the invariant d’Alembertian and T is the trace of
the energy momentum tensor describing all non-gravitational and
non-scalar field matter and energy. Barber’s scalar field φ is a
function of time t due to the nature of space-time which plays the role
analogous to the reciprocal of Newtonian gravitational constant i.e.
.1Gφ A coupling constant λ has to be determined from the
experiment. The measurements of the deflection of light restrict the
value of coupling to 110λ . In the limit 0λ , the theory
approaches the Einstein’s theory in every respect. Several
cosmologists viz. Singh (1984), Shri Ram and Singh (1998), Pradhan
and Vishwakarma (2002), Panigrahi and Sahu (2003), Vishwakarma
and Narlikar (2005), Sahu and Mohanty (2006), Singh and Kumar
(2007), Singh et al. (2008), Venkateswarlu et al. (2008), Reddy and
Naidu (2009) and Pradhan et al. (2009) have studied various aspects
of different cosmological models in Barber’s second self-creation
theory. In view of the consistency of Barber’s second self-creation
theory of gravitation, we intend to investigate some of the aspects of
this theory in chapter III.
(iii) Rosen’s Bimetric Theory of Gravitation
Professor Rosen (1940, 1973) proposed to modify the formalism of
the general relativity theory by introducing into it, besides the metric
14
tensor ijg , a second metric tensor ijf corresponding to flat space. This
modification did not affect the physical predictions of the theory, but
it did improve the formalism: certain quantities which previously had
complicated transformation properties acquired a simple tensor
character. In particular, it became possible to describe gravitational
energy and momentum density by means of a tensor. This modified
form of general relativity is known to be called as bimetric theory of
relativity.
The interpretations of the two metric tensors in the bimetric theory are
not unique. One can regard the ijf as simple as an auxiliary
mathematical quantity having no direct physical or geometrical
significance, while the ijg is considered to be the fundamental metric
tensor determining the properties of space-time and hence affecting
the behavior of physical system. Alternatively, one can regard the ijf
as describing the properties of space-time, which is now considered to
be flat, while the ijg is interpreted as a gravitational potential tensor
which determines the interaction between matter and gravitation.
Rosen clearly stated his motivation in constructing his new bimetric
theory of relativity. If the existence of black hole in nature is
confirmed, this will represents a brilliant success of general relativity.
However, since there is no convincing evidence at present that a black
15
hole represents a breakdown of the familiar concepts of space-time
and hence, is something unphysical. If one has at one’s disposal the
two metric tensors, it is natural to raise the question as to whether one
can set up theories of gravitation which satisfy the covariance and
equivalence principles but which differs from the general relativity
theory.
Rosen (1940) has proposed at each point of space-time a Euclidean
metric tensor ijf in addition to the Riemannian metric tensor ijg , so
that the corresponding line elements in a coordinate system ix are
jiij dxdxgds 2
and jiij dxdxfσd 2 ,
where ds is the interval between two neighbouring events as
measured by a clock and a measuring rod. The interval σd is an
abstract geometrical quantity not directly measurable. One can regard
it as describing the geometry that exists when matter is absent.
Employing the variation principle, Rosen (1973, 1974) has obtained
the field equations of bimetric relativity as
j
ij
ij
i kTπNδN 82
1 ,
where
bahihjabj
i ggfN // )(2
1 , ij
ij NgN , f
gk
16
and j
iT is the energy momentum tensor.
For empty space-time, the field equations become
0ijN .
This theory has attracted the attention of several researchers who have
studied the various aspects of BR. To note a few are Liebscher
(1975), Yilmaz (1975, 1979), Falik and Opher (1979), Karade (1981),
Reddy and Venkateswarlu (1989), Adhav and Karade (1994),
Mohanty et al. (2002), Adhav et al. (2003), Reddy et al. (2008), and
Sahoo (2008, 2009), etc. Inspired by their work, we have taken up the
study of bimetric theory of relativity as regard to Bianchi type-I
space-time for massive string and perfect fluid distribution with
electromagnetic field and five dimensional spherically symmetric
space-time filled with scalar meson field, domain walls and cosmic
string which form the content of chapters IV and V.
[1.3] SYMMETRIES
Symmetry is described by the group of motions in such a way that
two motion groups have the similar structure. Symmetric property is
that the field is the same at every point of space. The field equations
of general theory of relativity are non-linear differential equations in
ten unknowns ( ijg ) and it is very difficult to obtain their exact
solutions. The involvement of symmetry may be plane, spherical and
17
cylindrical does reduce the number of gravitational potentials ijg and
thus helps one in simplifying the field equations to some extent. In
the case of plane symmetry the number of unknowns ijg reduces to
five only. From the work of Taub (1951) it is gathered that the
space-times with plane symmetry are quite similar to those with
spherical symmetry.
Axially symmetric gravitational fields within the frame work of
general relativity were introduced by Levi-Civita (1918). The studies
of stationary axially symmetric fields were carried out to determine
relativistic effects on the motion of a slowly rotating body.
Einstein and Rosen (1937) introduced a cylindrically symmetric
metric given by
2222222222 )( dzeφderdrdteds βββα ,
where α and β are functions of r and t only.
The above metric is widely known as Einstein-Rosen metric. Karade
and Dhoble (1979) took up the study of axially symmetric field in
bimetric relativity with Einstein-Rosen metric.
Roy and Raj Bali (1978) have obtained the solution of Einstein’s field
equations representing a non-static spherically symmetric perfect
fluid distribution which is conformally flat. Roy and Narayan (1979,
1981) have obtained some inhomogeneous solutions for plane
18
symmetric as well as cylindrically-symmetric cosmological models
for perfect fluid distribution. Karade et al. (2001) have investigated
some inhomogeneous non-static plane symmetric perfect fluid
solutions in the bimetric theory of gravitation.
[1.4] COSMOLOGY AND COSMOLOGICAL MODELS
Cosmology is a science developed in the beginning of twentieth
century rapidly. The aim of cosmology is to determine the large scale
structure of the physical universe. At first sight the universe consists
of stars, star clusters, galaxies, Nebulae, pulsars, quasars as well as
such things as cosmic rays and background radiation. Cosmology is
one of the greatest intellectual achievements of all time beginning
from its origin. Cosmology, as a common man understands, is that
branch of astronomy, which deals with the large scale structure of the
universe. The present universe is both spatially homogeneous and
isotropic. Therefore it can be well described by Friedmann-
Robertson-Walker (FRW) model. The basic problem in cosmology is
to find the cosmological models of universe and to compare the
resulting models with the present day universe using astronomical
data.
Einstein’s general theory of relativity is a satisfactory theory of
gravitation correctly predicting the motion of test particles and
photons in curved space-time, but in order to apply to the universe
19
one has to introduce simplifying assumptions and approximations.
The most powerful assumption in cosmological theory is that of
homogeneity and isotropy often referred to as the ‘cosmological
principle’. Physically, this implies that there is no preferred position,
preferred direction or preferred epoch in the universe. Thus by using
the cosmological principle, we assume that the universe is filled with
a simple macroscopic perfect fluid. It is interesting to note that there
is no necessary relationship between homogeneity and isotropy. A
manifold can be homogeneous but nowhere isotropic or it can be
isotropic around a point without being homogeneous. On the other
hand, if a space is isotropic everywhere then it is homogeneous. Since
there is ample observational evidence for isotropy, and the
Copernican principle would have us believe that, we are not the
centre of the universe and therefore observers elsewhere should also
observe isotropy, we will henceforth assume both homogeneity and
isotropy. Therefore we begin construction of cosmological models
with the idea that the universe is homogeneous and isotropic.
The cosmological principle allows us to describe the most general
homogeneous and isotropic space-time given by the Friedmann-
Robertson-Walker (FRW) metric:
22222
2
2222 sin
1)( φdθrθdr
Kr
drtRdtds .
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Here t is time like coordinate, the function R(t) is known as the scale
factor, K is a constant which by a suitable choice of r can be chosen
to have the values +1, 0 or 1 according as a universe is closed, flat
or open respectively. The coordinates ),,,( tφθr form a co-moving
coordinate system in the sense that the cosmic fluid is at rest with
respect to the coordinate system.
Friedmann (1922) was the first to investigate the evolution of the
function R(t) using Einstein’s field equations for all three curvatures.
It has been both spatially homogeneous and isotropic and therefore
can be well described by a Friedmann-Robertson-Walker (FRW)
model (Partridge and Wilkinson 1967; Ehlers et al. 1968). However,
there is evidence for a small amount of anisotropy (Boughn et al.
1981) and a small magnetic field over cosmic distance scales
(Sofue et al. 1979). This suggests a very large departure from FRW
models at early stages of evolution of the universe. Thus, it is useful
to study cosmological models which may be highly anisotropic. For
the sake of simplicity it is usual to restrict oneself to models that are
spatially homogeneous. The spatially homogeneous and anisotropic
models which are known as Bianchi models present a medium way
between FRW models and completely inhomogeneous anisotropic
universes and thus play an important role in current modern
cosmology.
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[1.5] BIANCHI SPACE-TIMES
Space-times admitting a three parameter group of automorphisms are
important in discussing cosmological models. The case where the
group is simply transitive over the three-dimensional, constant-time
subspace is particularly useful. Bianchi (1898) has shown that there
are only nine distinct sets of structure constants for groups of this type
so that the algebra may be easily used to classify homogeneous space-
times. Most of the work on cosmological solutions is in the area of
homogeneous and isotropic FRW models, because of their tractability
and their possible relevance to the real universe. However, in recent
years, much attention is being paid to the investigation of spatially
homogeneous anisotropic Bianchi cosmological models to understand
the universe at its early stage of evolution. The simplest of them are
the well known nine types of Bianchi models which are necessarily
spatially homogeneous.
Bianchi type cosmological models are important in the sense that
these are homogeneous and anisotropic from which the process of
isotropization of the universe is studied through the passage of time.
Moreover, from the mathematical or theoretical point of view
anisotropic universe have a greater generality than isotropic FRW
models. FRW universes represent only a very special class of viable
cosmological models, through the simplest and most suitable
22
interpretations of ‘fuzzy’ cosmological observational data. The
simplicity of the field equations and relative ease of solution made
Bianchi space-times useful in constructing models of spatially
homogeneous and anisotropic cosmologies. A complete list of all
exact solutions of Einstein’s equations for the Bianchi type’s I-IX
with perfect fluid is given by Krammer et al. (1980).
[1.6] HIGHER DIMENSIONAL SPACE-TIME
The exact physical situation at very early stages of the formation of
our universe provoked great interest among researchers. Several
attempts have been made to unify gravity with other fundamental
forces in nature. Kaluza and Klein (1921, 1926) unified electromag-
netism with gravity by applying Einstein’s general theory of relativity
to a five dimensional space-time manifold. This idea was
enthusiastically considered in theoretical physics and further
generalized by considering higher dimensions in the hope of
achieving unification of all interactions, including weak and strong
forces (Witten, 1984). In recent years, there has been considerable
interest in higher dimensional space-times, in which extra dimensions
are contracted to a very small size, apparently beyond our ability for
measurement. The cosmological dimensional reduction process was
proposed by Chodos and Detweiler (1980) and it is useful for more
than four dimensions. Further, Marciano (1984) has pointed out that
23
the experimental detection of time variation of fundamental constants
should be strong evidence for the existence of extra dimensions.
The latest development of super-string theory and super gravitational
theory also created interest among scientists to consider higher
dimensional space-time, for study of the early universe (Weinberg et
al., 1986). Several authors viz. Sahdev (1984), Chatterjee and Bhui
(1990, 1993) and Tan and Shen (1998) have studied physics of the
universe in higher-dimensional space-time. Overdduin and Wesson
(1997) have presented an excellent review of higher-dimensional
unified theories in which the cosmological and astrophysical
implications of extra-dimension have been discussed. All models
discussed so far are based on Einstein’s ideas of geometrization of
gravitational field and have minimal extensions of those models in the
general relativity.
[1.7] COSMOLOGICAL CONSTANT
In 1917, Einstein introduced the cosmological constant into his field
equations in order to obtain a static cosmological model since his
equations without the cosmological constant admitted only non static
solutions. Recently, there has been a lot of interest in the
cosmological term within the context of quantum field theories,
quantum gravity, super gravity theories and the inflationary universe
scenario. In general relativistic quantum field theory, the
24
cosmological constant is explained as the vacuum energy density
(Zel’dovich 1967, 1968; Fulling 1974). Negative pressure is a
property of vacuum energy, but the exact nature of dark energy
remains one of the great mysteries of the ‘Big-Bang’. The basic role
of the cosmological constant is related to the observational evidence
of high red-shift Type Ia supernovae (Permutter, et al. and Riess, et
al. 1998) for a small decreasing values of cosmological constant
( Λ presence 25610 cm ) at the present epoch. Bergmann (1968) has
studied the cosmological constant in terms of the Higgs scalar field.
Linde (1974) proposed that the term Λ is a function of temperature
and is related to the process of broken symmetries.
In modern cosmological theories the cosmological constant Λ remains
a focal point of interest. A wide range of observations now suggest
that the universe possesses a non-zero cosmological constant (Krauss
and Turner, 1995). The cosmological models without the
cosmological constant are unable to explain satisfactorily problems
like structure formation and the age of the universe (Singh et al.
1998). Recent interest in the cosmological constant term Λ has
received considerable attention among researchers for various
concepts. Some of the recent discussion on the cosmological constant
“problem” and on cosmology with a time-varying cosmological
constant by Ratra and Peebles (1988), Dolgov et al. (1990), Dolgov
25
(1997), Sahni and Starobinsky (2000) pointed out that in the absence
of any interaction with matter or radiation, the cosmological constant
remains a “constant”. However, in the presence of interactions with
matter or radiation, a solution of Einstein equations and the assumed
equation of covariant conservation of stress-energy with a time-
varying cosmological constant Λ can be found. For these solutions,
conservation of energy requires a decrease in the energy density of
the vacuum component to be compensated by a corresponding
increase in the energy density of matter or radiation.
[1.8] INFLATIONARY UNIVERSE
The exact physical situation at the very early stages of the formation
of our universe is still challenging the problem today. The primary
goal of cosmological model is to describe the time evolution of
different phases of the universe, mostly the accelerated expansion
phase of the early universe. The universe is expanding from a hot and
dense initial state so called the ‘Big-Bang’ in which the light elements
were synthesized. After ‘Big-Bang’, the universe underwent a rapid
expansion phase characterized by an exponential increase of the
volume scale factor with time. During 1960-1970s, it was claimed
that the model of the universe is decelerating. But according to Knop
et al. (2003); Riess et al. (2004) of type Ia supernova data,
observations of type Ia supernova (SNe Ia) suggest that the expansion
26
of the universe at later stage is in an accelerating phase. Recent
observations of high red-shift supernovae indicate that the universe is
accelerating at the present epoch. The basic idea of accelerated
expansion phase of the universe is known as inflationary phase.
Inflation means a period in the early universe where some field
effectively mimics a large cosmological constant and so causes a
period of rapid expansion long enough to multiply the initial length
scale many times.
In the inflationary cosmological models usually a scalar field is used
to describe the rapid expansion phase. The scalar field may be related
to cosmological constant used in the FRW model. If an inflationary
period occurs in the very early universe, the matter and radiation
densities drop very close to zero while the inflation field dominates,
but is restored during ‘reheating’ at the end of inflation when the
scalar field energy converted to radiation. We believe that there was a
period of inflation which leads to many observable properties of the
universe. In particular, inflationary model plays an important role in
solving a number of outstanding problems in cosmology like the
homogeneity, the isotropy and flatness of the observed universe. Guth
(1981), Linde (1982) and La and Steinhardt (1989) are some of the
authors who have investigated several aspects of inflationary universe
in general relativity.
27
[1.9] ZERO-MASS SCALAR FIELD
The study of interacting fields, one of the fields being a zero-mass
scalar field, is basically an attempt to look into the yet unsolved
problem of the unification of the gravitational and quantum theories.
In the last few decades there has been considerable interest focused
on the theory of gravitation representing zero-mass scalar fields
coupled with gravitational field in the last few decades. In recent
years, the zero-mass scalar field has acquired particular importance
because of a suggestion by Weinberg and Wilczek (1978) that there
should exist a pseudo scalar boson, the so called axion of negligible
mass. The work of Pecci and Quinn (1979) has lent further support to
this idea.
Bergmann and Leipnik (1957), Bramhachary (1960), Das (1962),
Gautreau (1969), Rao et al. (1972), Reddy and Innaiah (1986),
Reddy (1987) are some of the authors who have investigated various
aspects the theory of gravitation for different space-times in the
presence of zero-mass scalar fields. In particular Singh and Deo
(1986) and Verma (1987) have discussed FRW cosmological models
in the presence of zero-mass scalar fields in general relativity.
[1.10] BULK VISCOSITY
In most treatments of cosmology, cosmic fluid is considered as
perfect fluid. However, bulk viscosity is expected to play an
28
important role at certain stages of an expanding universe. At the early
stages of the evolution of the universe, when radiation is in the form
of photons as well as neutrino decoupled, the matter behaved like a
viscous fluid. Since viscosity counteracts the gravitational collapse, a
different picture of the initial stage of the universe may appear due to
dissipative process caused by viscosity. It has been widely discussed
in the literature that during the evolution of the universe, bulk
viscosity could arise in many circumstances and could lead to an
effective mechanism of galaxy formation. Bulk viscosity is associated
with the grand unified theory (GUT) may lead to inflationary
cosmology which is used to overcome lacunae of several important
problems in the standard ‘Big-Bang’ cosmology.
The study of viscous mechanism in cosmology attracted the attention
of many workers due to its significant role in the description of high
entropy of the present universe. Misner (1967, 1968) has studied the
effect of viscosity on the evolution in the cosmological models and
suggested that the strong dissipation due to the neutrino viscosity may
considerably reduce the anisotropy of the black-body radiation.
Murphy (1973) constructed isotropic homogeneous spatially-flat
cosmological model with a fluid containing bulk viscosity alone
because the shear viscosity cannot exist due to assumption of
29
isotropy. He observed that the ‘Big-Bang’ singularity of finite past
may be avoided by introduction of bulk viscosity.
Padamanabhan and Chitre (1987) have shown that the presence of
bulk viscosity leads to inflationary like solutions in general relativity.
Another characteristic of bulk viscosity is that it acts like a negative
energy field in an expanding universe (Johri and Sudharsan 1989).
Mohanty and Pradhan (1990) investigated the problem of interactions
of a gravitational field with bulk viscous fluid in FRW space-time.
Mohanty and Pradhan (1991) extended the work of Murphy (1973) by
considering the special law of variation for Hubble’s parameter
presented by Berman (1983) and solved Einstein’s field equations
when the universe is filled with viscous fluid. Pradhan and Pandey
(2003) have investigated an LRS Bianchi type-I models with bulk
viscosity in the cosmological theory based on Lyra’s geometry. The
effect of bulk viscosity on the early evolution of the universe for a
spatially homogeneous and isotropic Robertson-Walker model is
discussed by Singh (2008). The effect of bulk viscosity on the
cosmological evolution has been investigated by a number of authors
in the frame work of general theory of relativity and alternative
theories of gravitation.
30
[1.11] ELECTROMAGNETIC FIELD
Magnetic field plays a vital role in the description of the energy
distribution in the universe as it contains highly ionized matter. Large
scale magnetic fields give rise to anisotropies in the universe. It is
believed that the presence of electromagnetic field could alter the rate
of creation of particles in the anisotropic models. A cosmological
model which contains a global magnetic field is necessarily
anisotropic since the magnetic field vectors specify a preferred spatial
direction. Also, electromagnetic field directly affects the expansion
rate of the universe. Zel’dovich and Novikov (1971) have pointed out
that Galaxies and internebular spaces exhibit the presence of strong
magnetic fields. Harrison (1973) has suggested that magnetic field
could have a cosmological origin. The presence of primordial
magnetic field of cosmological origin in the early stages of the
evolution of the universe has been discussed by eminent author’s viz.
Misner et al. (1973), Melvin (1975), Asseo and Sol (1987) and Kim
et al. (1991) in his cosmological solution for dust and electromagnetic
field suggested that during the evolution of the universe, the matter
was in a highly ionized state and smoothly coupled with
electromagnetic field and consequently form a neutral matter as a
result of universe expansion. Hence in string dust universe the
presence of magnetic field is not unrealistic. The occurrence of
31
magnetic fields on galactic scale is well-established fact today, and
their importance for a variety of astrophysical phenomena is generally
acknowledged as pointed out by Zeldovich et al. (1993). As a natural
consequence, we should include magnetic fields in the energy
momentum tensor of early universe.
The energy momentum tensor for electromagnetic field is given by
Lichnerowicz (1967) in the form
j
ij
ij
ij
i hhguuhμE2
12
with μ is the magnetic permeability and ih the magnetic flux vector
defined by
ll
jklijkli hhhuF
μ
gh
2,
2,
where ijF is the electromagnetic field tensor, ijkl is the Levi-Civita
tensor density. Thorne (1967), Jacobs (1969), Collins (1972), Roy
and Prakash (1978), Bali (1986), Shri Ram and Singh (1995), Bali
and Ali (1996), Wang (2006), Pradhan et al. (2006, 2007) and Bali
and Pareek (2009) are some of the authors who have investigated
magnetized cosmological models for perfect fluid distribution in
Einstein’s general theory of relativity.
32
[1.12] COSMIC STRINGS
The astronomical consideration reveals that the present day universe
is both spatially homogeneous and isotropic. Therefore it can be well
described by Friedmann-Robertson-Walker (FRW) model. Advance
research work done by scientists and researchers lead to various
branches of cosmology such as string theory, super symmetry and
super string etc. Cosmologists are of the view that early universe is of
a different type and at some stage changed over to the present day
FRW universe.
In the last few years the study of cosmic strings has attracted
considerable interest as they are believed to play an important role
during early stages of the universe. String theory originally invented
in the Late 1960’s is an attempt to find a theory to describe the
strange force. The idea was that particles like the photon and the
neutron could be regarded as waves on a string. The presence of
strings in the early universe is a by product of Grant Unified Theories
(GUT). This does not contradict present day observations of the
universe. Cosmic strings have stress energy and coupled in a simple
way to the gravitational field. Most analysis is concerned with the
gravitational effects which arise from the presence of strings. The
general relativistic treatment of cosmic strings has been originally
given by Letelier (1979, 1983) and Stachel (1980).
33
In spontaneously broken Gauge theories and the spontaneous broken
symmetry in elementary particle physics have given rise to an
intensive study of cosmic strings. It appears that after the ‘Big-Bang’
the universe may have experienced a number of phase transitions.
These phase transitions can produce vacuum domain structures such
as domain walls, cosmic strings and monopoles. Out of these
cosmological structures, cosmic strings have excited perhaps the most
interest. They may act as gravitational lenses (Vilenkin, 1981) and
may give rise to density perturbations leading to the formation of
galaxies. Later, Letelier and Verdaguer (1988) studied a new model
of cloud formed by massive strings in the context of general
relativity. They have considered the Bianchi type-I model as they are
supported to be reasonable representation of the early universe. They
observed that during the evolution of the universe the strings
disappear and the particles become important and finally end up with
galaxies. Krori et al. (1990, 1994) studied the problem of cosmic
strings taking Bianchi types I, II, III, V, VI, VIII and IX space-times
and observed that the universe was dominated by massive strings.
The energy momentum tensor for a cloud of massive strings is given
by
jijiij xxλuuρT
34
with 1 ii
ii xxuu and 0i
i xu .
Here ρ is the rest energy density of the cloud of strings with particles
attached to them (p-strings) and λ is the string tension density. The
vector iu describes the four-velocity of a cloud of strings and ix is a
unit space-like vector in the direction of the string. If we denote the
particle energy density by pρ then λρρ p .
[1.13] DOMAIN WALLS
In recent years, symmetry is proving to be a powerful unifying tool in
particle physics and cosmology because through symmetry and
symmetry breaking, particles which appear to be different in mass,
charge etc. can be understood as different states of a single unified
field theory in which all particles and fundamentals forces of nature to
unify gravity are related through a broken symmetry. It is still a
challenging problem to know the exact physical situation at early
stages of evolution of the universe. Certain grand unified field theory
predicts topological defects, such as cosmic string, domain walls and
monopoles, which might have been formed in the early phase
transition of the universe. These defects are stable field configuration
which arises in field theories with spontaneously broken discrete
symmetries. Spontaneous symmetry breaking is an old idea, described
within the particle physics in terms of the Higgs-Kibble field
35
mechanism (Kibble, 1976). The symmetry is spontaneously broken
because the ground state is not invariant under the full symmetry of
the Lagrangian identities. Thus the expected value of Higgs field in
vacuum is non-zero. In quantum field theories, broken symmetries are
restored at sufficiently high temperature.
The well-known topological defects are domain walls which occur
when a discrete symmetry is broken at a phase transition, and the
defect density is related to the domain size. According to Hill et al.
(1989) the formation of galaxies are due to domain walls produced
during phase transition after the time of recombination of matter and
radiation. The phase transition is induced by Higgs sector of the
standard model, the defects are domain walls across which the field
flips from one minimum to the other. The defect density is related to
the domain size and the dynamics of the domain walls is governed by
the surface tension σ . It is clear that a full analysis of the role of
domain walls in the universe imposes the study of their interaction
with particles in the primordial plasma.
[1.14] STRANGE QUARK MATTER
One of the interesting consequences of phase transition in the early
universe is the formation of strange quark matter. Itoh (1970),
Bodmer (1971) and Witten (1984) proposed two ways of formation of
quark matter, namely, the quark hadron phase transition in the early
36
universe and conversion of neutron stars into strange at ultrahigh
densities. In the theories of strong interaction, quark bag modes
suppose that breaking of physical vacuum takes place inside hadrons.
As a result, vacuum energy densities inside and outside a hadron
become essentially different and the vacuum pressure on the bag wall
equilibrates the pressure of quarks, thus stabilizing the system. If the
hypothesis of the quark matter is true, then some neutron stars could
actually be strange stars, built entirely of strange matter. In this
respect, Alcock et al. (1986), Haensel et al. (1986), Yilmaz (2005,
2006), Yavuz et al. (2005), Yilmaz and Yavuz (2006), Adhav et al.
(2008) and Khadekar et al. (2009) are some of the authors who have
confined their work to the quark matters which attached to the
topological defects in general relativity.
Typically, strange quark matter is modeled with an equation of state
based on the phenomenological bag model of quark matter, in which
quark confinement is described by an energy term proportional to
volume. In the framework of this model the quark matter is composed
of massless u, d quarks, massive s quarks and electrons. In the
simplified version of the bag model, assuming that the quarks are
massless and non interacting, we then have quark pressure qq ρp3
1 ,
where qρ is the quark energy density. The total energy density is
37
cqm Bpρ
But total pressure is
cqm Bpp .
The equation of state for strange quark matter is
).4(3
1cmm Bρp
The equation of state for normal matter is given by
mm ργp )1( ,
where 21 γ is a constant.
[1.15] PROBLEMS INVESTIGATED
In this section, we mention, in brief, the problems investigated and
the results obtained in this thesis. Details of problems investigated are
given in the subsequent chapters.
Chapter II deals with spatially homogeneous and isotropic
Friedmann-Robertson-Walker cosmological models (FRW-models) in
unified field theory based on Lyra geometry in the presence of
zero- mass scalar field and bulk viscous fluid. Cosmological solutions
of the field equations are obtained with the help of special law of
variation for Hubble’s parameter and also using power law relation.
Some interesting physical consequences pertaining to the equation of
38
state ργp )1( are discussed. It has been observed that the
investigated models are free from singularities.
In Chapter III, false vacuum, radiation and stiff fluid FRW
cosmological models in Barber’s (1982) second self-creation theory
of gravitation in the presence bulk viscous fluid are investigated with
the help of special law of variation for Hubble’s parameter proposed
by Berman (1983). Models of this type are important in the self-
creation cosmology for the description of very early stage of the
universe expansion.
Chapter IV is devoted to the study of magnetized cosmological
model in Rosen’s bimetric theory of gravitation. In this chapter we
have investigated Bianchi type-I massive string barotropic perfect
fluid cosmological model filled with electromagnetic field. Some
physical and geometrical properties of the cosmological model are
briefly discussed.
In Chapter V, we have shown that the higher dimensional spherically
symmetric cosmological model in the presence of scalar meson fields
exists in Rosen’s bimetric theory of relativity. But the cosmological
models represented by domain walls and cosmic strings do not exists
in Rosen’s bimetric theory of relativity. Hence only the vacuum
models are obtained.
39
Chapter VI presents a discussion of Einstein-Rosen cylindrical
symmetric cosmological model in Einstein’s general theory of
relativity. The cosmological model is obtained for domain walls with
cosmological constant and heat flow when strange quark matter and
normal matter has been attached to the domain walls. The physical
and kinematical features of the investigated models are studied and
discussed.
Chapter VII is concerned with a magnetized cosmological model in
general theory of relativity. A spatially homogeneous and anisotropic
magnetized cosmological model is investigated for perfect fluid
distribution in Einstein’s general theory of relativity with varying
cosmological constant. The investigated model represents an
expanding, shearing and non-rotating universe. The physical and
geometrical features of the model have been discussed.
Chapter VIII is devoted to the study of Kantowski-Sachs
Inflationary universe in general relativity in the presence of mass less
scalar field with a constant flat potential. It is observed that the
investigated cosmological model is non singular, expanding and does
not approach anisotropy at late times. The physical properties of the
investigated model are discussed.