INHOMOGENEOUS COSMOLOGICAL PERFECT FLUID MODELS IN MODIFIED THEORY OF GENERAL RELATIVITY WITH TIME DEPENDENT-TERM R. N. Patra 1 1 P.G.
Department of Mathematics, Berhampur University, Odisha, India
1. INTRODUCTION Barber
(1982) proposed his second
self-creation theory of gravity, which modifies general relativity, in an effort to outperform Einstein's theory. The ideal
fluid in this simplified theory of general relativity just splits the matter
tensor as a reciprocal of the gravitational constant G, rather than gravitating
directly. The local impacts shown in observational studies are predicted by
this idea. Furthermore, by analysing the behaviour of photons and degenerate matter
entities, this theory may be supported or denied. An accurate measurement of
the deflection of light and radio waves passing near the sun, along with the
discovery of anomalous precessions in pulsar orbits over central masses, would
validate or invalidate such a notion. The theory predicts the perihelia of
planets with the same precision as general relativity and in that regard; it is in agreement with observation to within 1%. In the limit In every way, this revised theory is similar to Einstein's theory. Many authors have examined the modified theory of general relativity from various perspectives. The Friedman-Barber field equations have been solved by Pimentel (1985) using the power law dependency of the scalar field on the scale factor as an assumption. In generalising Pimentel's work Pimentel (1985), Venkateswarlu & Reddy (1990), Soleng (1987) has obtained solutions for the vacuum-dominated, dust-filled universe of the flat FRW spase-time. Bianchi cosmological solutions of VIo type are found in the works of Reddy & Venkateswarlu (1989), both in vacuum and with perfect fluid pressure equivalent to the energy density. When the source of the gravitational field is a perfect fluid, Venkateswarlu & Reddy (1990) have also built spatially homogenous and anisotropic Bianchi type-1 cosmological macro models. Space homogeneous and anisotropic Bianchi type-II and III cosmological models have been obtained by Shanthi & Rao (1991) in both vacuum and stiff fluid conditions. Carvalho (1996) obtained a homogenous and isotropic model of the primitive universe in which the gamma parameter of the "gamma law" state equation continually varies with cosmological time. He also presented a unified description of the primitive universe between the inflationary period and the epoch dominated by radiation. Shri Ram & Singh (1998) have obtained a spatially homogeneous and isotropic R-W model of the universe in the presence of perfect fluid by using the ‘gamma law” equation of state. Mohanty et al. (2000) have obtained vacuum and Zeldovich fluid models for plane symmetric anisotropic homogeneous space-time. Mohanty et al. (2002), Mohanty et al. (2003) have obtained an anisotropic homogeneous Bianchi Type-1 cosmological micro model in Barber’s second theory of gravitation wherein the scalar field describes the elementary particles and their interactions Srivastav & Sinha (1998). Also, they have obtained a micro and macro cosmological model in the presence of a massless scalar field interacted with perfect fluid. Panigrahi & Sahu (2003), Panigrahi & Sahu (2002), Panigrahi & Sahu (2003), Panigrahi & Sahu (2004) have obtained plane symmetric inhomogeneous macro models in Barber’s second theory of gravitation. Sahu and Bianchi Type-1 vacuum models have been obtained by Sahu & Panigrahi (2003). Sahu & Panigrahi (2006), Sahu et al. (2010) have investigated Masonic perfect fluid models in modified theory of general relativity. The vitality energy tensor of matter, which is produced by a idealize liquid, is ordinarily the subject of examination for relativistic models. But to get more reasonable models, one must consider the consistency component in cosmology has pulled in the consideration of numerous analysts because it can account for tall entropy of the display universe Weinberg (1971), Weinberg (1972). The tall entropy by baryon and the momentous degree of isotropy of microwave infinite foundation radiation recommends that dissipative impacts in cosmology ought to be considered. Furthermore, it's over here. Thick impacts are anticipated to happen due to a few forms. These are the decoupling of neutrinos amid the radiation time and the decay of matter and radiation amid the recombination time Kolb & Turner (1990), gravitational string generation Turok (1988) and Barrow (1988) and molecule creation impact within the terrific unification time. Murphy (1973) illustrated that the presentation of bulk thickness can anticipate the peculiarity of the enormous bang. therefore, one would need to consider the nearness of fabric dissemination other than the idealize liquid to get practical cosmological models (see Gron (1990) for an audit of cosmological models with bulk consistency). To our information none of the creators has examined the
altered hypothesis of common relativity for plane symmetric inhomogeneous space
time in nearness of idealize liquid with time subordinate term 2. Field Equations Here we consider the space time portrayed by inhomogeneous
metric of the frame
Where A, B are functions of ‘x’ and’ t’. The field equations in Barbers second self-creation theory with time dependent cosmological constant are
Where The vitality force tensor
Together with In commoving co-ordinate system the surviving components of the field equations (2)-(5) for the space time (1) are
Here after wards the prime (') and the subscript “4” denotes partial differentiation w.r.to x and t respectively. In order to solve the field equations for obtaining solutions in explicit forms, we may consider different equation of state. As the metric potentials are functions of x and t, it is difficult to solve the field equations (6)-(10) for non-static case. Hence, we consider the following particular cases. Case-1: In this case the field equations (6)-(10) reduces to
Here we have the system of four equations in five unknowns. In order to make the system consistent, we take the help of gamma law equation of state 2.1. VACUUM
MODEL For this case equations (11)-(14) reduce to
Using equation (16) in equation (15), we obtain
Integrating equation (19), we obtain
Where Now using equation (20) in equation (17) we get
For simplification if we consider
With the help of equation (20), equation (22) reduces to
Integrating we get Again integrating
Where If we consider
Integrating equation (2.4), we obtain
Again integrating equation (25), we get
Where If we consider
Where Further, if we are consider
Hence the Vacuum cosmological model in second self
creation theory of Barber can determined for any arbitrary metric potential 2.2. Radiating
Model: In this case, equating equation (11) and (12), we find
On integration, equation (29) yields
where
Further using equation (30) and
For the simplification, if we consider
This yield on intrgration
Where If we consider
Which on integration yields
Where Using (34) and (36) in equation (31), we get
Or
Also, if
Where Further, if we consider
Hence the radiating cosmological model in second self
creation theory of Barber can be determined for any arbitrary metric potential Using (39) and (40) in equation (31), we can get another
two values of 2.3. Zeldevich Model: In this case the model doesn’t exist. 3. Conclusion In this paper a plane symmetric inhomogeneous cosmological model has been constructed by taking perfect fluid along with time-depended cosmological constant term. Also, I have studied the consistency of this theory to the case of a perfect fluid in three different cases. The Vacuum and radiating cosmological model exists in second self creation theory of Barber and can be determined for any arbitrary metric potential, but in case of Zel’dovich model it doesn’t exist.
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