MGD solution under Class I generator

Abstract

We present an anisotropic solution of Einstein field equations through the minimally deformed Class I generator. To do this, first we apply the MGD approach under the transformation $Y\longrightarrow H(r)+\alpha \,{\varPsi }(r)$ in the original field equations, which leads to two sets of equations. Later on, we employ this said transformation into differential equation corresponding to the Class I condition. In this way, we achieve a first-order nonlinear differential equation in ${\varPsi }(r)$ (called a generating equation) that depends on the gravitational potentials X(r) and Y(r). So, we assume two different well-behaved ansatzes for X(r) and obtained Y(r) using the Class I condition. There after, we solved this generating equation analytically by using these gravitational potentials and obtained the deformation function ${\varPsi }(r)$. The function ${\varPsi }(r)$ vanishes at $r=0$ and is decreasing monotonically outward. The constant parameters involved in solutions are obtained by using well-known Schwarzschild exterior solution. Finally, we have discussed several mathematical and physical analysis of the solutions via graphical representation. The results have been also parented in tabular forms. The physical analysis confirms that our anisotropic solutions describe the realistic stellar models and may be useful in modeling of the astrophysical compact objects.

Appendix

$$\begin{aligned} \phi _{11}(r)= & {} 8 \cos ^3(A r^2) [-8 (4 A r^2 \sin (A r^2)\\&+\cos (A r^2)) (-128 A^2 C F^2 r^4+2 A b F r^4+56 C \cos (2 A r^2)\\&+28 C \cos (4 A r^2)+8 C \cos (6 A r^2)+C \cos (8 A r^2)+35 C)\\&-\cos (A r^2) (8 A F r^2+4 \cos (2 A r^2)\\&+\cos (4 A r^2)+3) (64 A C F r^2+32 C \cos (2 A r^2)+8 C \cos (4 A r^2)-b r^2+24 C)],\\ \phi _{12}(r)= & {} (8 A F r^2+4 \cos (2 A r^2)+\cos (4 A r^2)+3)^2 [64 A C F r^2+32 C \cos (2 A r^2)\\&+8 C \cos (4 A r^2)-b r^2+24 C]^2,\\ \phi _{13}(r)= & {} (8 A F r^2+4 \cos (2 A r^2)+\cos (4 A r^2)+3)\,[64 A C F r^2+32 C \cos (2 A r^2)\\&+8 C \cos (4 A r^2)-b r^2+24 C],\\ \phi _{14}(r)= & {} 16 \cos (A r^2) [(4 A r^2 \sin (A r^2)+\cos (A r^2)) (4 A r^2+\sin (2 A r^2)\\&+\cos (2 A r^2)+1) (2 A b F r^4-128 A^2 C F^2 r^4\\&+56 C \cos (2 A r^2)+28 C \cos (4 A r^2)+8 C \cos (6 A r^2)\\&+C \cos (8 A r^2)+35 C)+2 A r^2 (8 A F r^2+4 \cos (2 A r^2)\\&+\cos (4 A r^2)+3) (2 A r^2 \sin (A r^2)\\&+\cos (A r^2)) (64 A C F r^2+32 C \cos (2 A r^2)\\&+8 C \cos (4 A r^2)-b r^2+24 C)],\\ \phi _{15}(r)= & {} [\tan (A r^2)+1]\, [8 A F r^2+4 \cos (2 A r^2)\\&+\cos (4 A r^2)+3]^2 [64 A C F r^2+32 C \cos (2 A r^2)+8 C \cos (4 A r^2)\\&-b r^2+24 C]^2.\\ \phi _{21}= & {} 8 \cosh ^3(A r^2) \{8 (4 A r^2 \sinh (A r^2)\\&-\cosh (A r^2)) [2 A b F r^4-128 A^2 C F^2 r^4+56 C \cosh (2 A r^2)\\&+28 C\cosh (4 A r^2)+8 C \cosh (6 A r^2)+C \cosh (8 A r^2)+35 C]\\&-\cosh (A r^2) [8 A F r^2+4 \cosh (2 A r^2)\\&+\cosh (4 A r^2)+3] [64 A C F r^2+32 C \cosh (2 A r^2)\\&+8 C \cosh (4 A r^2)-b r^2+24 C]\},\\ \phi _{22}= & {} (8 A F r^2+4 \cosh (2 A r^2)+\cosh (4 A r^2)+3)^2 [64 A C F r^2\\&+32 C \cosh (2 A r^2)+8 C \cosh (4 A r^2)\\&-b r^2+24 C]^2 \\ \phi _{23}= & {} (8 A F r^2+4 \cosh (2 A r^2)+\cosh (4 A r^2)+3) (64 A C F r^2+32 C \cosh (2 A r^2)\\&+8 C \cosh (4 A r^2)-b r^2+24 C)\\ \phi _{24}= & {} 4 \{-8 r \cosh (A r^2) [\sinh (A r^2)\\&+\cosh (A r^2)] [4 A r^2 \sinh (A r^2)\\&-\cosh (A r^2)] [(2 A r^2+1) \cosh (A r^2)\\&-2 A r^2 \sinh (A r^2)] (-128 A^2 C F^2 r^4+2 A b F r^4+56 C \cosh (2 A r^2)\\&+28 C \cosh (4 A r^2)+8 C \cosh (6 A r^2)\\&+C \cosh (8 A r^2)+35 C)-8 A r^3 \cosh ^2(A r^2) (2 A r^2 \tanh (A r^2)-1) (8 A F r^2\\&+4 \cosh (2 A r^2)+\cosh (4 A r^2)\\&+3) (64 A C F r^2+32 C \cosh (2 A r^2)+8 C \cosh (4 A r^2)-b r^2+24 C)\},\\ \phi _{25}= & {} r (\tanh (A r^2)+1) [8 A F r^2+4 \cosh (2 A r^2)+\cosh (4 A r^2)+3]^2 [64 A C F r^2\\&+32 C \cosh (2 A r^2)+8 C \cosh (4 A r^2)\\&-b r^2+24 C]^2. \end{aligned}$$