In the present article, we have constructed static anisotropic compact star models of Einstein field equations for the spherical symmetric metric of embedding class one. By assuming the particular form of metric function ν, We have solved the Einstein field equations for anisotropic matter distribution. The anisotropic models are representing the realistic compact objects such as SAX J 1808.4-3658 (SS1), Her X -1, Vela X-12, PSR J1614-2230 and Cen X -3. We have reported our results in details for compact star Her X-1 on the ground of physical properties such as pressure, density, velocity of sound, energy conditions, TOV equation and red-shift etc. Along with these, we have also discussed about stability of the compact star models. Finally we made the comparison between our anisotropic stars with the realistic objects on the key aspects as central density, central pressure, compactness and surface red-shift.

In the present paper we have constructed a new relativistic anisotropic compact star model having a spherically symmetric metric of embedding class one. Here we have assumed an arbitrary form of metric function e λ and solved the Einstein's relativistic field equations with the help of Karmarkar condition for an anisotropic matter distribution. The physical properties of our model such as pressure, density, mass function, surface red-shift, gravitational red-shift are investigated and the stability of the stellar configuration is discussed in details. Our model is free from central singularities and satisfies all energy conditions. The model we present here satisfy the static stability criterion i.e. dM/dρc > 0 for 0 ≤ ρc ≤ 4.04 × 10 17 g/cm 3 (stable region) and for ρc ≥ 4.04 × 10 17 g/cm 3 , the region is unstable i.e., dM/dρc ≤ 0.

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In this article, we study the possibility of sustaining static and spherically symmetric traversable wormhole geometries admitting conformal motion in Einstein gravity, which presents a more systematic approach to search a relation between matter and geometry. In wormhole physics, the presence of exotic matter is a fundamental ingredient and we show that this exotic source can be dark energy type which support the existence of wormhole spacetimes. In this work we model a wormhole supported by dark energy which admits conformal motion. We also discuss the possibility of the detection of wormholes in the outer regions of galactic halos by means of gravitational lensing. Studies of the total gravitational energy for the exotic matter inside a static wormhole configuration are also performed.