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

Abstract This work is devoted to the study of relativistic anisotropic compact objects. To obtain this class of solutions of the Einstein field equations, we have developed a general scheme to generate the metric of the space–time describing the interior of the compact structure. This approach is based on the class I space–time and the extended gravitational decoupling by means of an extended geometric deformation (EGD). The class I condition provides a differential equation relating both metric potential $$\nu $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ν</mml:mi> </mml:math> and $$\lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>λ</mml:mi> </mml:math> , whilst the EGD translates the metric potentials to $$ \nu =\xi +\beta \,h(r)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ν</mml:mi> <mml:mo>=</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> <mml:mspace/> <mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and $$ \lambda =-\ln [\mu +\beta \,f(r)]$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mo>ln</mml:mo> <mml:mo>[</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> <mml:mspace/> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mo>)</mml:mo> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> , where h ( r ) and f ( r ) are the deformation functions and $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> a dimensionless constant. In this case the pair $$\{\xi ,\mu \}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> represents the seed solution satisfying the class I condition without any deformation. Once the deformed metric potentials are inserted into the class I, the main task is to obtain h ( r ) or f ( r ). So, in this case a particular ansatz for h ( r ) is considered in conjunction with $$\beta =0.5$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>β</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0.5</mml:mn> </mml:mrow> </mml:math> to get f ( r ). In order to check feasibility of our model, we have performed a thoroughly physical, mathematical and graphical analysis.

Перевод пока недоступен