Static fluid spheres admitting Karmarkar condition

Abstract

We explore a new relativistic anisotropic solution of the Einstein field equations for compact stars based on embedding class one condition. For this purpose, we use the embedding class one methodology by employing the Karmarkar condition. Employing this methodology, we obtain a particular differential equation that connects both the gravitational potentials and . We solve this particular differential equation choosing a simple form of generalized gravitational potential to describe a complete structure of the space-time within the stellar configuration. After determining this space-time geometry for the stellar models, we discuss thermodynamical observables including radial and tangential pressures, matter density, red-shift, velocity of sound, etc., in the stellar models. We also perform a complete graphical analysis, which shows that our models satisfy all the physical and mathematical requirements of ultra-high dense collapsed structures. Further, we discuss the moment of inertia and M-R curve for rotating and non-rotating stars.

Embedding with Vaidya geometry

“…Hence the embedding condition for the generalized Vaidya metric leads to a restriction on the mass function m(ν, r ) given by (29). Note that the condition (29) does not allow for the mass function to depend on the variable ν only with m(ν).…”

“…Hence the conventional Vaidya metric (1) is not embeddable in 5-dimensional flat spacetime but particular forms of the generalized Vaidya metric are embeddable. It is interesting to observe that the dependence in the retarded time coordinate ν is arbitrary in equation (29). The dependence in the radial coordinate r is fully known and the mass function has the form…”

“…Embedding leads to an explicit functional form for the mass function m(ν, r ). The condition (29) was obtained from the Gauss and Codazzi equations which are necessary and sufficient conditions for the metric to be Class I embeddable. Thus we can state the following theorem:…”

“…[13] for a complete solution with both redshift and shape functions. Several other studies of stellar models with the embedding Class I condition are contained in the works [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]. A general algorithm to describe embeddable anisotropic compact stellar models was discovered by Murad [30].…”

An EGD model in the background of embedding class I space–time

“…This condition provides an extensive tool for solving Einstein's field equations and investigate new relativistic astrophysical compact stellar structures. Recently, several solutions were obtained in the context of charged and anisotropic matter distributions with well defined compact structures for class I space-time [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26] (and references contained therein).…”