Spherically symmetric Buchdahl-type model via extended gravitational decoupling

Abstract

In this article, we have obtained a gravitationally decoupled Class-I anisotropic solution for compact stars using the Buchdahl-type space–time geometry. The anisotropic solution is obtained by solving Einstein’s field equations via a complete geometric deformation (CGD) approach. This CGD approach transforms both gravitational potentials by introducing two unknown functions that govern the equations of motion for extra sources. The solutions for these deformation functions are derived using mimic constraint to density and equation of state (EoS) between extra source components rather than imposing a particular ansatz for them. To ensure that the solution describes a physically realisable stellar structure, we have tested the physical viability of the solution based on its regularity and stability conditions. We observed that the decoupling parameter suppresses the pressure, energy density and mass of the stellar objects. Also, the radii for several known astrophysical objects have been predicted for different values of the decoupling constant. The obtained results show that gravitational decoupling yields more compact objects than pure Einstein’s GR.

Appendix

$$\begin{aligned}{} & {} \theta _{11}(r)=-a^3 r^4 \\{} & {} \quad \times \left( B \sqrt{a r^2+1}\left( \beta (\alpha +1) \left( b r^2+3\right) \right. \right. \\{} & {} \quad \left. \left. -3 \left( b \alpha r^2+\alpha \right) \right) +A \left( \beta (\alpha +1) \left( br^2+3\right) \right. \right. \\{} & {} \quad \left. \left. -\alpha \left( b r^2+1\right) \right) \right) +a^2 r^2 \\{} & {} \quad \times \Big [B \sqrt{a r^2+1} \left( \beta \left( br^2+3\right) \left( \alpha \left( 2 b r^2-1\right) \right. \right. \\{} & {} \quad \left. \left. +br^2-2\right) +\left( b r^2+1\right) \left( b \gamma \alpha r^4+b \gamma r^4\right. \right. \\{} & {} \quad \left. \left. -6 b \alpha r^2+\gamma \alpha r^2+\gamma r^2+\alpha \right) \right) \\{} & {} \quad +A \left( \beta \left( br^2+3\right) \left( \alpha \left( 2 b r^2-1\right) +br^2-2\right) \right. \\{} & {} \quad \left. +\left( b r^2+1\right) \left( b \gamma \alpha r^4+b \gamma r^4-2 b \alpha r^2+\gamma \alpha r^2\right. \right. \\{} & {} \quad \left. \left. +\gamma r^2+\alpha \right) \right) \Big ],\\{} & {} \theta _{12}(r)= a \Big [B \sqrt{a r^2+1}\big (-\beta \big (b r^2+3\big )\\{} & {} \quad \times \big (b^2 \alpha r^4-2 b (\alpha +1)r^2+1\big )\\{} & {} \quad -\big (r^2 \big (b r^2+1\big ) \big (\gamma \big (b r^2+1\big )\big (b \alpha r^2-\alpha -2\big )\\{} & {} \quad +b \alpha \big (2-3 b r^2\big )\big )\big )\big ) \\{} & {} \quad +A \big \{-\beta \big (b r^2+3\big )\big (b^2 \alpha r^4-2 b (\alpha +1) r^2+1\big )\\{} & {} \quad -\big [r^2 \big (br^2+1\big ) \big (\gamma \big (b r^2+1\big ) \\{} & {} \quad +\big (b \alpha r^2-\alpha -2\big )+b \alpha \big (2-br^2\big )\big )\big ]\big \}\Big ]\\{} & {} \quad -\big (B \sqrt{a r^2+1}+A\big )\Big [b^3\alpha r^4 \big (\beta +\gamma r^2-1\big ) \\{} & {} \quad +b^2 r^2 \big (\beta (3\alpha -1)+2 \gamma \alpha r^2-\gamma r^2-\alpha \big ) \\{} & {} \quad +b \big (\gamma (\alpha -2)r^2-3\beta \big )-\gamma \Big ], \end{aligned}$$

$$\begin{aligned} A_1(R){} & {} =-b^3 \alpha ^2 R^4 \sqrt{a R^2+1} \left( \beta +\gamma R^2-1\right) \\{} & {} \quad +b \sqrt{a R^2+1} \left( 3 \beta \alpha -\gamma (\alpha -2) \alpha R^2+1\right) +\gamma \alpha \sqrt{a R^2+1},\\ A_2(R){} & {} =-\sqrt{a^2 \left( a R^2+1\right) \left( b R^2+1\right) ^2 \left( a^2 \left( \alpha ^2-1\right) R^4-2 a \left( b \alpha ^2 R^4+R^2\right) +b^2 \alpha ^2 R^4-1\right) ^2} \\{} & {} \quad -a^3 (\alpha +1) R^4 \sqrt{a R^2+1} \left( \alpha \left( b \beta R^2-2 b R^2+3 \beta -2\right) +2 b R^2+2\right) +b^2 R^2 \sqrt{a R^2+1}\\{} & {} \quad \times \left( \beta \left( \alpha -3 \alpha ^2\right) +\alpha ^2 \left( 1-2 \gamma R^2\right) +\gamma \alpha R^2+1\right) ,\\ A_3(R)= & {} a \sqrt{a R^2+1} \big [-\beta \alpha \left( b R^2+3\right) \left( b^2 \alpha R^4-2 b (\alpha +1) R^2+1\right) -\left( b R^2+1\right) \\{} & {} \quad \times \big (b^2 \alpha ^2 R^4 \left( \gamma R^2-2\right) -2 b R^2 (\gamma \alpha R^2 -\alpha ^2+1)-\gamma \alpha (\alpha +2) R^2+2\big )\big ], \end{aligned}$$

$$\begin{aligned}{} & {} A_4(R)=a^2 R^2 \sqrt{a R^2+1} \\{} & {} \quad \times \big [\beta \alpha \left( b R^2+3\right) \left( \alpha \left( 2 b R^2-1\right) +b R^2-2\right) \\{} & {} \quad +\left( b R^2+1\right) \big (\alpha ^2 \left( b \gamma R^4-4 b R^2+\gamma R^2+1\right) \\{} & {} \quad +\gamma \alpha R^2 \left( b R^2+1\right) +b R^2-4\big )\big ],\\{} & {} A_5(R)=a^2 (\alpha +1) R^2 \\{} & {} \quad \left( \alpha \left( \beta \left( b R^2+3\right) -b R^2-1\right) +b R^2+1\right) \\{} & {} \quad +b^3 \alpha ^2 R^4 \left( \beta +\gamma R^2-1\right) +b^2 R^2 \big (\beta (3 \alpha -1) \alpha \\{} & {} \quad +\alpha ^2 \left( 2 \gamma R^2-1\right) -\gamma \alpha R^2-1\big )\\{} & {} \quad +b \left( -3 \beta \alpha +\gamma (\alpha -2) \alpha R^2-1\right) -\gamma \alpha ,\\{} & {} A_6(R)=-a \left( \beta \alpha \left( b R^2+3\right) \left( b (2 \alpha +1) R^2-1\right) \right. \\{} & {} \quad \left. +\left( b R^2+1\right) \left( b R^2 \left( \alpha ^2 \left( \gamma R^2-2\right) +\gamma \alpha R^2+1\right) \right. \right. \\{} & {} \quad \left. \left. +\gamma \alpha (\alpha +1) R^2-1\right) \right) ,\\{} & {} B_1(R)=\ln \left( a R^2+1\right) \Big [\frac{b \gamma }{a^2}\\{} & {} \quad +\frac{\frac{2 b}{a}-2}{\left( \frac{A}{B} \right) ^2}-\frac{b (\beta -1)+\gamma }{a}+3 \beta -1\Big ]\\{} & {} \quad +\frac{B^2}{a^2 A^2}\Bigg \{\frac{A}{B} \left( \frac{4 a (b-a)}{\sqrt{a R^2+1}}-\frac{A}{B} b \gamma \left( a R^2+1\right) \right) \\{} & {} \quad -4 a \left[ \left( \frac{A}{B} \right) ^2-1\right] (a-b) \log \left( \sqrt{a R^2+1}+\frac{A}{B} \right) \Bigg \}\\{} & {} \quad -2 \beta \ln \left( a b R^2+a\right) . \end{aligned}$$