[1.1] GENERAL THEORY OF RELATIVITY. and the Newtonian mechanics (three laws of motion) is well known.

Transcription

1 INTRODUCTION

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 Special relativity unifies the concept of space and time into a single four dimensional structure called space-time. This concept of spacetime that arises from relativity is based on two simple postulates: (i) (ii) The speed of light in free space is constant and The laws of physics are the same in all inertial (nonaccelerating) 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 3

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 4

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. 5

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 ds g ij i dx dx j, ( i, j 1,, 3, 4 ). Here the fundamental metric tensor g ij 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 G ij R ij 1 R g ij 8πGT c ij, where G ij is the Einstein tensor, R ij is the Ricci tensor, R is the Ricci scalar, g ij is the metric tensor, T ij is an energy momentum tensor, G is the constant of gravitation and c is the speed of light. [1.] 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 6

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 (197), 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 (198) 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 7

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 nonintegrability 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 nonintegrability 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 8

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 k Rij gij R φiφ j gij φkφ 8πTij, 4 where φ is the displacement field, R ij is the Ricci tensor, R is the Ricci scalar, T ij is the energy momentum tensor and tensor. g ij is the metric 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 (198), Reddy and Innaiah (1986), Reddy and Venkateswarlu (1987), Singh and Singh 9

10 (1991a, 199), Singh and Desikan (1997), Pradhan and Pandey (003), Pradhan and Vishwakarma (004), Mohanty et al. (006), Casana et al. (005, 006, 007), Bali and Chandnani (008), Kumar and Singh (008), Rao et al. (008) and singh et al. (009). 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 (198) has modified Brans-Dicke scalar-tensor theory to develop a continuous creation of matter in the large scale structure of 10

11 the universe. As a result, Barber (198) 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 (198) attempted to produce a continuous creation scalartensor 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 selfcontained 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 R ij 1 g ij R 8πφ 1 T ij 3 φ λφ i; j g 3λφ ij φ and φ 4π λt, where λ is coupling constant to be determined from experiments. 11

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 (198) second self-creation theory of gravitation are R ij 1 1 gij R 8πφ T ij and the scalar field equations is defined by 8π φ λt, 3 1

13 where φ φ; k k 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. 1 φ G. A coupling constant λ has to be determined from the experiment. The measurements of the deflection of light restrict the value of coupling to λ In the limit λ 0, the theory approaches the Einstein s theory in every respect. Several cosmologists viz. Singh (1984), Shri Ram and Singh (1998), Pradhan and Vishwakarma (00), Panigrahi and Sahu (003), Vishwakarma and Narlikar (005), Sahu and Mohanty (006), Singh and Kumar (007), Singh et al. (008), Venkateswarlu et al. (008), Reddy and Naidu (009) and Pradhan et al. (009) 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 13

14 tensor g ij, a second metric tensor f ij 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 f ij as simple as an auxiliary mathematical quantity having no direct physical or geometrical significance, while the g ij 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 f ij as describing the properties of space-time, which is now considered to be flat, while the g ij 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 14

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 fij in addition to the Riemannian metric tensor g ij, so that the corresponding line elements in a coordinate system i x are ds g ij i dx dx j and dσ f ij i dx dx j, where ds is the interval between two neighbouring events as measured by a clock and a measuring rod. The interval d σ 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 1 j j Ni δi N 8π kti, where 1, N g ij Nij, j ab hj N i f ( g ghi / a )/ b k g f 15

16 and j T i is the energy momentum tensor. For empty space-time, the field equations become N 0. ij 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. (00), Adhav et al. (003), Reddy et al. (008), and Sahoo (008, 009), 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 ( g ij ) and it is very difficult to obtain their exact solutions. The involvement of symmetry may be plane, spherical and 16

17 cylindrical does reduce the number of gravitational potentials g ij and thus helps one in simplifying the field equations to some extent. In the case of plane symmetry the number of unknowns g ij 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 ds e αβ β β ( dt dr ) r e dφ e dz, 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 17

18 symmetric as well as cylindrically-symmetric cosmological models for perfect fluid distribution. Karade et al. (001) 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 18

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: dr ds dt R ( t) r dθ r sin θ dφ. 1 Kr 19

20 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 ( r, θ, φ, t) form a co-moving coordinate system in the sense that the cosmic fluid is at rest with respect to the coordinate system. Friedmann (19) 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. 0

21 [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 spacetimes. 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 1

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 (191, 196) unified electromagnetism 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 3

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 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 4

25 (1997), Sahni and Starobinsky (000) 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 timevarying 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 s, it was claimed that the model of the universe is decelerating. But according to Knop et al. (003); Riess et al. (004) of type Ia supernova data, observations of type Ia supernova (SNe Ia) suggest that the expansion 5

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 (198) and La and Steinhardt (1989) are some of the authors who have investigated several aspects of inflationary universe in general relativity. 6

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 (196), Gautreau (1969), Rao et al. (197), 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 7

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 8

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 (003) 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 (008). 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. 9

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 30

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 E j i μ h uiu j 1 g j i h h i j with μ is the magnetic permeability and h i the magnetic flux vector defined by h i g μ ijkl F kl u j, h l l h h, where F ij is the electromagnetic field tensor, ijkl is the Levi-Civita tensor density. Thorne (1967), Jacobs (1969), Collins (197), Roy and Prakash (1978), Bali (1986), Shri Ram and Singh (1995), Bali and Ali (1996), Wang (006), Pradhan et al. (006, 007) and Bali and Pareek (009) are some of the authors who have investigated magnetized cosmological models for perfect fluid distribution in Einstein s general theory of relativity. 31

32 [1.1] 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). 3

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 T ij ρu u i j λ x x i j 33

34 i i with u u x x 1 and u 0. i i 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 i u describes the four-velocity of a cloud of strings and i x i i x is a unit space-like vector in the direction of the string. If we denote the particle energy density by ρ p then ρ ρ λ. 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 34

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 35

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 (005, 006), Yavuz et al. (005), Yilmaz and Yavuz (006), Adhav et al. (008) and Khadekar et al. (009) 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 1 p q ρq, 3 where ρ q is the quark energy density. The total energy density is 36

37 ρ m p q B c But total pressure is p m p B. q c The equation of state for strange quark matter is p m 1 ( 3 ρ m 4B c ). The equation of state for normal matter is given by p ( γ 1), m ρ m where 1 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 37

38 state p ( γ 1) ρ 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 (198) 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 selfcreation 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. 38

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. 39

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