Quest Journals Journal of Research in Applied Mathematics Volume 7 ~ Issue 11 (2021) pp: 33-39 ISSN(Online) : 2394-0743 ISSN (Print): 2394-0735 www.questjournals.org *Corresponding Author: Prashant R. Dhongle 33 | Page Research Paper The de-Sitter Model with Dark Energy Prashant R. Dhongle Seth Kesarimal Porwal College, Kamptee, Nagpur, India ABSTRACT: The de-Sitter model with dark energy has been found in stationary space-time and it is realized that the model is completely filled with dark matter and it represents dark energy star, for the value of M greater than zero. In particular, for zero value of M, the model goes over to de-Sitter empty space-time and then it describes open flat universe of special relativity. The model is the generalization of empty de-Sitter universe. Further geometrical and physical aspects of the model are studied. KEYWORDS: General Relativity, Einstein Field Equations, Dark Energy, Dark Matter. Received 08 November, 2021; Revised: 22 November, 2021; Accepted 24 November, 2021 © The author(s) 2021. Published with open access at www.questjournals.org I. INTRODUCTION General relativity has developed into an essential tool in Cosmology and modern Astrophysics and it is the basis of current cosmological models of the expanding universe. The spherical symmetry has its own importance in general theory of relativity due to its comparative simplicity. Many important space-times like Schwarzschild solutions (exterior and interior), the Robertson – Walker model etc. are spherically symmetric in nature. Most of the spherically symmetric space-times are also stationary space-times. Stationary space-times or time – independent gravitational fields are very significant in general relativity and in it the metric tensor components ij g , are all independent of the time coordinate, i.e., there exists a coordinate system in which we can express the metric tensor independent of the time coordinate. Borkar et al. [1] have studied the spherically symmetric space-time with charged perfect fluid distribution in stationary space-time. Borkar et al. [2] and Pandya et al. [3] have studied the spherically symmetric space-time in bimetric theory of gravitation. In the universe, 96 % of energy content is in exotic form, out of which 70 % of energy is repulsive in nature, called dark energy and 23 % of energy is attractive in nature, called dark matter [4] and [5]. Although dark matter is the most popular theory to explain the various astronomical observations of galaxies and galaxy clusters, there has been no direct observational evidence of dark matter. It has been noted that the names “dark matter” and “dark energy” serve mainly as expressions of human ignorance. Dark matter is matter that is inferred to exist from gravitational effects on visible matter and background radiation, but is undetectable by emitted or scattered electromagnetic radiation [6] whereas dark energy is a hypothetical form of energy that permeates all of space and tends to increase the rate of expansion of universe [7]. Generally, dark energy has a strong negative pressure (i.e., effects, acting repulsively) in order to explain the observed acceleration in the expansion rate of the universe. The models of dark energy have created a lot of interest in the research area and many researchers ([8] to [16]) have developed the dark energy cosmological models of the universe. Dark energy is one of the popular ways to explain recent observations and experiments that the universe appears to be expanding at an accelerating rate. A star is self luminous, massive celestial body that generates the nuclear energy in its core. Dark energy star is the hypothetical compact star, which is believed to be the alternative explanation for observations of astronomical black hole candidate. The theory states that, the in falling matter from the event horizon is converted into vacuum energy or dark energy. The dark energy star theory proposed by Chapline [17] is the generalization of picture of gravastar (gravitational vacuum star) and dark energy star has interior solution with matter and is governed by equation of state 3 1 ,       p which is matched with exterior vacuum Schwarzschild solution. Many researchers are studying the nature and properties of the different cosmological models of the universe in stationary space-times. Beig et al. [18] have thrown light on the far – field behavior of stationary space-times. Garcia et al. [19] have discussed the conformally flat stationary axisymmetric space-times. Bartolo et al. [20] have studied the existence and multiplicity results for orthogonal trajectories on stationary space- times under intrinsic assumptions, with some examples and applications. Many other researchers ([21] to [28])