Compact objects by extended gravitational decoupling in f(G, T) gravity

Abstract

In this paper, we investigate the anisotropic interior spherically symmetric solutions by utilizing the extended gravitational decoupling method in the background of f(G, T) gravity, where G and T signify the Gauss–Bonnet term and trace of the stress-energy tensor, respectively. The anisotropy in the interior geometry arises with the inclusion of an additional source in the isotropic configuration. In this technique, the temporal and radial potentials are decoupled which split the field equations into two independent sets. Both sets individually represent the isotropic and anisotropic configurations, respectively. The solution corresponding to the first set is determined by using the Krori–Barua metric potentials, whereas the second set contains unknowns which are solved with the help of some constraints. The ultimate anisotropic results are evaluated by combining the solutions of both distributions. The influence of decoupling parameter is examined on the matter variables as well as anisotropic factor. We illustrate the viable and stable features of the constructed solutions by using energy constraints and three stability criteria, respectively. Finally, we conclude that the obtained solutions are viable as well stable for the whole domain of the decoupling parameter.

Appendices

Appendix A

The extra curvature terms in f(G, T) are given as

$$\begin{aligned}{} & {} T^{0\textsf {(Cor)}}_{0}=\frac{1}{8\pi }\bigg [-\frac{1}{2} G^2+\bigg (\frac{4 e^{-2 \vartheta } \varphi ''}{r^2}\nonumber \\{} & {} \quad -\frac{4 e^{-\vartheta } \varphi ''}{r^2}-\frac{2 e^{-\vartheta } \varphi '^2}{r^2}+\frac{2 e^{-\vartheta } \varphi ' \vartheta '}{r^2}\nonumber \\{} & {} \quad +\frac{2 e^{-2 \vartheta } \varphi '^2}{r^2}-\frac{6 e^{-2 \vartheta } \varphi ' \vartheta '}{r^2}\bigg )G+\bigg (\frac{12 e^{-2 \vartheta } \vartheta '}{r^2}-\frac{4 e^{-\vartheta }\vartheta '}{r^2}\bigg )G'\nonumber \\{} & {} \quad \bigg (-\frac{8 e^{-2 \vartheta }}{r^2}+\frac{8 e^{-\vartheta } }{r^2}\bigg )G''\bigg ], \end{aligned}$$

(A1)

$$\begin{aligned}{} & {} T^{1\textsf {(Cor)}}_{1}=\frac{1}{8 \pi } \bigg [\frac{1}{2} G^2+ \bigg (-\frac{4 e^{-2 \vartheta } \varphi ''}{r^2}+\frac{4 e^{-\vartheta } \varphi ''}{r^2}\nonumber \\{} & {} \quad +\frac{6 e^{-2 \vartheta } \varphi ' \vartheta '}{r^2}-\frac{2 e^{-\vartheta } \varphi ' \vartheta '}{r^2}\nonumber \\{} & {} \quad -\frac{2 e^{-2 \vartheta } \varphi '^2}{r^2}+\frac{2 e^{-\vartheta } \varphi '^2}{r^2}\bigg )G+ \bigg (\frac{12 e^{-2\vartheta } \varphi '}{r^2}-\frac{4 e^{-\vartheta } \varphi '}{r^2}\bigg )G'\bigg ], \end{aligned}$$

(A2)

$$\begin{aligned}{} & {} \quad T^{2\textsf {(Cor)}}_{2}=\frac{1}{8 \pi }\bigg [\frac{1}{2} G^2+ \bigg (-\frac{4 e^{-2\vartheta } \varphi ''}{r^2}\nonumber \\{} & {} \quad +\frac{4 e^{-\vartheta } \varphi ''}{r^2}+\frac{2 e^{-\vartheta } \varphi '^2}{r^2}-\frac{2 e^{-2 \vartheta } \varphi '^2}{r^2}\nonumber \\{} & {} \quad -\frac{2 e^{-\vartheta } \varphi ' \vartheta '}{r^2}+\frac{6 e^{-2 \vartheta } \varphi ' \vartheta '}{r^2}\bigg )G+ \bigg (-\frac{6 e^{-2 \vartheta } \varphi ' \vartheta '}{r}+\frac{4 e^{-2 \vartheta } \varphi ''}{r}\nonumber \\{} & {} \quad +\frac{2 e^{-2 \vartheta } \varphi '^2}{r}\bigg )G'+ \frac{4 e^{-2 \vartheta } \varphi '}{r}G''\bigg ]. \end{aligned}$$

(A3)

The Gauss–Bonnet term as well as its higher derivatives turn out to be

$$\begin{aligned}{} & {} G=\frac{1}{r^2}\bigg [2 e^{-2 \vartheta } \left( \left( e^{\vartheta }-3\right) \vartheta '\varphi ' -\left( 2 \vartheta ''+\varphi '^2\right) \left( e^{\vartheta }-1\right) \right) \bigg ], \end{aligned}$$

(A4)

$$\begin{aligned}{} {} G'&=\frac{-1}{r^3}\bigg [2e^{-2 \vartheta } \big ( -\left( \vartheta ''\left( e^{\vartheta }-3\right) \right. \nonumber \\{} & {} \quad \left. -2\varphi '' \left( e^{\vartheta }-1\right) \right) r \varphi '+r \varphi '\left( e^{\vartheta }-6\right) \vartheta '^2\big )\nonumber \\{} & {} \quad +\vartheta ' \left( r \left( -\left( e^{\vartheta }-2\right) \right) \varphi '^2+2\varphi 'r \left( e^{\vartheta }-3\right) \right. \nonumber \\{} & {} \quad \left. -\left( 3 e^{\vartheta }-7\right) r \varphi ''\right) -2\varphi '^2 \big (e^{\vartheta } -1\big ) \end{aligned}$$

(A5)

$$\begin{aligned}{} & {} -2 \left( 2 \varphi ''-r \varphi ^{(3)}\right) \left( e^{\vartheta }-1\right) \bigg )\bigg ],\nonumber \\{} {} G''&=\frac{1}{r^4}\bigg [ 2e^{-2 \vartheta } \bigg (6-6 e^{\vartheta }+\varphi '^2 \big (r^2 \big (e^{\vartheta }-2\big ) \vartheta ''\big )\nonumber \\{} & {} \quad -2 \bigg (\varphi '' \big (6 \big (e^{\vartheta }-1\big )-r^2 \big (2 e^{\vartheta }\nonumber \\{} & {} -\quad 5\big ) \vartheta ''\big )+\varphi ''^2r^2 \big (e^{\vartheta }-1\big ) +r \big (-4 \varphi ^{(3)}\nonumber \\{} & {} \quad +r \varphi ^{(4)}\big ) \big (e^{\vartheta }-1\big )\bigg )+r^2 \big (e^{\vartheta }-12\big )\nonumber \\{} & {} \quad \times \varphi ' \vartheta '^3+\vartheta ' \bigg (\varphi ' \big (-3 r^2 \big (e^{\vartheta }-6\big ) \vartheta ''\nonumber \\{} & {} \quad +4 r^2 \big (e^{\vartheta }-2\big ) \varphi ''+6 \big (e^{\vartheta }-3\big )\big )-4\nonumber \\{} & {} \quad \times r\varphi '^2 \big (e^{\vartheta }-2\big ) +r \big ( \big (5 e^{\vartheta }-11\big )r \varphi ^{(3)}\nonumber \\{} & {} \quad -4 \varphi ''\big (3 e^{\vartheta }-7\big )\big )\bigg )-r \vartheta '^2 \bigg (4 r \big (e^{\vartheta }\nonumber \\{} & {} \quad -5\big ) \varphi ''-4 \big (e^{\vartheta }-6\big ) \varphi '\bigg )+r \big (e^{\vartheta }-4\big ) \varphi '^2\nonumber \\{} & {} \quad + \varphi ' r \bigg (r \big (\big (e^{\vartheta }-3\big ) \vartheta ^{(3)}-2 \varphi ^{(3)}\nonumber \\{} & {} \quad \times \big (e^{\vartheta }-1\big )\big )-4 \big (e^{\vartheta }-3\big ) \vartheta ''+8 \big (e^{\vartheta }-1\big ) \varphi ''\bigg )\bigg )\bigg ]. \end{aligned}$$

(A6)

Appendix B

The radial component of adiabatic index corresponding to solutions I and II are

$$\begin{aligned}\check{\Gamma _{r}}&=\{8 \pi (\psi +4 \pi ) r^2 \big (2 (L+X)-T^{0\textsf {(Cor)}}_{0} e^{r^2 X}\nonumber \\&\quad -T^{1\textsf {(Cor)}}_{1} e^{r^2 X}\big ) \big (\alpha \psi ^2+12 \pi \alpha \psi \nonumber \\&\quad +32 \pi ^2 \alpha +4 \pi \psi +2 \alpha \psi ^2 L r^4 X\nonumber \\&\quad +24 \pi \alpha \psi L r^4 X+64 \pi ^2 \alpha L r^4 X+12 \pi \psi L r^4 X\nonumber \\&\quad +64 \pi ^2 L r^4 X+4 \pi \psi r^4 X^2+\alpha \psi ^2 r^2 X\nonumber \\&\quad -\alpha \psi ^2 e^{r^2 X}+12 \pi \alpha \psi r^2 X-12 \pi \alpha \psi e^{r^2 X}\nonumber \\&\quad +32 \pi ^2 \alpha r^2 X-32 \pi ^2 \alpha e^{r^2 X}+4 \pi \psi r^2 X\nonumber \\&\quad -4 \pi \psi e^{r^2 X}+32 \pi ^2 r^2 X-32 \pi ^2 e^{r^2 X}\nonumber \\ {}&+\quad \pi \psi r^3 e^{r^2 X} {T^{0\textsf {(Cor)}}_{0}}^{'}+\pi (3 \psi +16 \pi ) r^3 e^{r^2 X} {T^{1\textsf {(Cor)}}_{1}}^{'}\nonumber \\&\quad +32 \pi ^2\big )\}\{\big (\alpha \psi ^2+12 \pi \alpha \psi \nonumber \\&\quad +32 \pi ^2 \alpha +4 \pi \psi +2 \alpha \psi ^2 L r^2+24 \pi \alpha \psi L r^2\nonumber \\&\quad +64 \pi ^2 \alpha L r^2+12 \pi \psi L r^2+64 \pi ^2 L r^2\nonumber \\&\quad -2 \pi \psi r^2 T^{0\textsf {(Cor)}}_{0} e^{r^2 X}-2 \pi (3 \psi +16 \pi ) r^2 T^{1\textsf {(Cor)}}_{1} e^{r^2 X}\nonumber \\&\quad -\alpha \psi ^2 e^{r^2 X}-12 \pi \alpha \psi e^{r^2 X}\nonumber \\&\quad -32 \pi ^2 \alpha e^{r^2 X}+4 \pi \psi r^2 X-4 \pi \psi e^{r^2 X}\nonumber \\&\quad -32 \pi ^2 e^{r^2 X}+32 \pi ^2\big ) \big (-\alpha \psi ^2-12 \pi \alpha \psi \nonumber \\&\quad -32 \pi ^2 \alpha -4 \pi \psi -2 \alpha \psi ^2 L r^4 X-24 \pi \alpha \psi L r^4 X\nonumber \\&\quad -64 \pi ^2 \alpha L r^4 X+4 \pi \psi L r^4 X\nonumber \\&\quad +12 \pi \psi r^4 X^2+64 \pi ^2 r^4 X^2-\alpha \psi ^2 r^2 X\nonumber \\&\quad +\alpha \psi ^2 e^{r^2 X}-12 \pi \alpha \psi r^2 X+12 \pi \alpha \psi e^{r^2X}\nonumber \\&\quad -32 \pi ^2 \alpha r^2 X+32 \pi ^2 \alpha e^{r^2 X}-4 \pi \psi r^2X\nonumber \\&\quad +4 \pi \psi e^{r^2 X}-32 \pi ^2 r^2 X+32 \pi ^2 e^{r^2 X}\nonumber \\&\quad +\pi (3 \psi +16 \pi ) r^3 e^{r^2 X} {T^{0\textsf {(Cor)}}_{0}}^{'}\nonumber \\&\quad +\pi \psi r^3 e^{r^2 X} {T^{1\textsf {(Cor)}}_{1}}^{'}-32 \pi ^2\big )\}^{-1}, \end{aligned}$$

(B1)

$$\begin{aligned} \check{\Gamma _{r}}&=\{8 \pi (\psi +4 \pi ) r^2 \big (2 (L+X)-T^{0\textsf {(Cor)}}_{0} e^{r^2 X}\nonumber \\&\quad -T^{1\textsf {(Cor)}}_{1} e^{r^2 X}\big ) \big (-\alpha \psi ^2-12 \pi \alpha \psi \nonumber \\&\quad -32 \pi ^2 \alpha +4 \pi \psi +12 \pi \psi L r^4 X+64 \pi ^2 L r^4 X\nonumber \\&\quad +2 \alpha \psi ^2 r^4 X^2+24 \pi \alpha \psi r^4 X^2\nonumber \\&\quad +64 \pi ^2 \alpha r^4 X^2+4 \pi \psi r^4 X^2-\alpha \psi ^2 r^2 X\nonumber \\&\quad +\alpha \psi ^2 e^{r^2 X}+32 \pi ^2 \alpha e^{r^2 X}+4 \pi \psi r^2 X\nonumber \\&\quad -12 \pi \alpha \psi r^2 X+12 \pi \alpha \psi e^{r^2 X}-32 \pi ^2 \alpha r^2 X\nonumber \\&\quad -4 \pi \psi e^{r^2 X}+32 \pi ^2 r^2 X-32 \pi ^2 e^{r^2 X}\nonumber \\&\quad +\pi \psi r^3 e^{r^2 X} {T^{0\textsf {(Cor)}}_{0}}^{'}+\pi (3 \psi +16 \pi ) r^3 e^{r^2 X} {T^{1\textsf {(Cor)}}_{1}}^{'}\nonumber \\&\quad +32 \pi ^2\big )\}\{\big (-\alpha \psi ^2-12 \pi \alpha \psi \nonumber \\&\quad -32 \pi ^2 \alpha +4 \pi \psi +12 \pi \psi L r^2+64 \pi ^2 L r^2\nonumber \\&\quad -2 \pi \psi r^2 T^{0\textsf {(Cor)}}_{0} e^{r^2 X}-2 \pi (3 \psi +16 \pi ) r^2\nonumber \\&\quad \times T^{1\textsf {(Cor)}}_{1} e^{r^2 X}+2 \alpha \psi ^2 r^2 X+\alpha \psi ^2 e^{r^2 X}\nonumber \\&\quad +24 \pi \alpha \psi r^2 X+12 \pi \alpha \psi e^{r^2 X}+64 \pi ^2 \alpha r^2 X\nonumber \\&\quad +32 \pi ^2 \alpha e^{r^2 X}+4 \pi \psi r^2 X-4 \pi \psi e^{r^2 X}\nonumber \\&\quad -32 \pi ^2 e^{r^2 X}+32 \pi ^2\big ) \big (\alpha \psi ^2+12 \pi \alpha \psi +32 \pi ^2 \alpha \nonumber \\&\quad -4 \pi \psi +4 \pi \psi L r^4 X-2 \alpha \psi ^2 r^4 X^2\nonumber \\&\quad -24 \pi \alpha \psi r^4 X^2-64 \pi ^2 \alpha r^4 X^2+12 \pi \psi r^4 X^2\nonumber \\&\quad +64 \pi ^2 r^4 X^2+\alpha \psi ^2 r^2 X-\alpha \psi ^2 e^{r^2 X}\nonumber \\&\quad +12 \pi \alpha \psi r^2 X-12 \pi \alpha \psi e^{r^2 X}+32 \pi ^2 \alpha r^2 X\nonumber \\&\quad -32 \pi ^2 \alpha e^{r^2 X}-4 \pi \psi r^2 X+4 \pi \psi e^{r^2 X}\nonumber \\&\quad -32 \pi ^2 r^2 X+32 \pi ^2 e^{r^2 X}+\pi (3 \psi +16 \pi ) r^3 \nonumber \\&\quad \times e^{r^2 X} {T^{0\textsf {(Cor)}}_{0}}^{'}\nonumber \\&\quad +\pi \psi r^3 e^{r^2 X} {T^{1\textsf {(Cor)}}_{1}}^{'}-32 \pi ^2\big )\}^{-1}. \end{aligned}$$

(B2)

The expressions of radial velocity in the case of first and second solution are given as

$$\begin{aligned} \nu ^{2}_{r}&=\{\alpha \psi ^2+12 \pi \alpha \psi +32 \pi ^2 \alpha +4 \pi \psi +2 \alpha \psi ^2 L r^4 X\nonumber \\&\quad +24 \pi \alpha \psi L r^4 X+64 \pi ^2 \alpha L r^4 X\nonumber \\&\quad +12 \pi \psi L r^4 X+64 \pi ^2 L r^4 X+4 \pi \psi r^4 X^2+\alpha \psi ^2 r^2 X\nonumber \\&\quad -\alpha \psi ^2 e^{r^2 X}+12 \pi \alpha \psi r^2 X\nonumber \\&\quad -12 \pi \alpha \psi e^{r^2 X}+32 \pi ^2 \alpha r^2 X-32 \pi ^2 \alpha e^{r^2 X}\nonumber \\&\quad +4 \pi \psi r^2 X-4 \pi \psi e^{r^2 X}+32 \pi ^2 r^2 X\nonumber \\&\quad -32 \pi ^2 e^{r^2 X}+\pi \psi r^3 e^{r^2 X} {T^{0\textsf {(Cor)}}_{0}}^{'}\nonumber \\&\quad +\pi (3 \psi +16 \pi ) r^3 e^{r^2 X} {T^{1\textsf {(Cor)}}_{1}}^{'}+32 \pi ^2\}\{-\alpha \psi ^2\nonumber \\&\quad -12 \pi \alpha \psi -32 \pi ^2 \alpha -4 \pi \psi -2 \alpha \psi ^2 L r^4 X\nonumber \\&\quad -24 \pi \alpha \psi L r^4 X-64 \pi ^2 \alpha L r^4 X\nonumber \\&\quad +4 \pi \psi L r^4 X+12 \pi \psi r^4 X^2+64 \pi ^2 r^4 X^2\nonumber \\&\quad -\alpha \psi ^2 r^2 X+\alpha \psi ^2 e^{r^2 X}-12 \pi \alpha \psi r^2 X\nonumber \\&\quad +12 \pi \alpha \psi e^{r^2 X}-32 \pi ^2 \alpha r^2 X+32 \pi ^2 \alpha e^{r^2 X}\nonumber \\&\quad -4 \pi \psi r^2 X+4 \pi \psi e^{r^2 X}-32 \pi ^2 r^2 X\nonumber \\&\quad +32 \pi ^2 e^{r^2 X}+\pi (3 \psi +16 \pi ) r^3 e^{r^2 X} {T^{0\textsf {(Cor)}}_{0}}^{'}\nonumber \\&\quad +\pi \psi r^3 e^{r^2 X} {T^{1\textsf {(Cor)}}_{1}}^{'}-32 \pi ^2\}^{-1}, \end{aligned}$$

(B3)

$$\begin{aligned}\nu ^{2}_{r}&=\{-\alpha \psi ^2-12 \pi \alpha \psi -32 \pi ^2 \alpha +4 \pi \psi \nonumber \\&\quad +12 \pi \psi L r^4 X+64 \pi ^2 L r^4 X+2 \alpha \psi ^2 r^4 X^2\nonumber \\&\quad +24 \pi \alpha \psi r^4 X^2+64 \pi ^2 \alpha r^4 X^2+4 \pi \psi r^4 X^2\nonumber \\&\quad -\alpha \psi ^2 r^2 X+\alpha \psi ^2 e^{r^2 X}-12 \pi \alpha \psi r^2 X\nonumber \\&\quad +12 \pi \alpha \psi e^{r^2 X}-32 \pi ^2 \alpha r^2 X+32 \pi ^2 \alpha e^{r^2 X}\nonumber \\&\quad +4 \pi \psi r^2 X-4 \pi \psi e^{r^2 X}+32 \pi ^2 r^2 X\nonumber \\&\quad -32 \pi ^2 e^{r^2 X}+\pi \psi r^3 e^{r^2 X} {T^{0\textsf {(Cor)}}_{0}}^{'}\nonumber \\&\quad +\pi (3 \psi +16 \pi ) r^3 e^{r^2 X} {T^{1\textsf {(Cor)}}_{1}}^{'}+32 \pi ^2\}\{\alpha \psi ^2\nonumber \\&\quad +12 \pi \alpha \psi -4 \pi \psi +4 \pi \psi L r^4 X-2 \alpha \psi ^2 r^4 X^2\nonumber \\&\quad -24 \pi \alpha \psi r^4 X^2-64 \pi ^2 \alpha r^4 X^2\nonumber \\&\quad +12 \pi \psi r^4 X^2+32 \pi ^2 \alpha +64 \pi ^2 r^4 X^2\nonumber \\&\quad +\alpha \psi ^2 r^2 X-\alpha \psi ^2 e^{r^2 X}+12 \pi \alpha \psi r^2 X\nonumber \\&-12 \pi \alpha \psi e^{r^2 X}+32 \pi ^2 \alpha r^2 X-32 \pi ^2 \alpha e^{r^2 X}\nonumber \\&\quad -4 \pi \psi r^2 X+4 \pi \psi e^{r^2 X}-32 \pi ^2 r^2 X\nonumber \\&\quad +32 \pi ^2 e^{r^2 X}+\pi (3 \psi +16 \pi ) r^3 e^{r^2 X} {T^{0\textsf {(Cor)}}_{0}}^{'}\nonumber \\&\quad +\pi \psi r^3 e^{r^2 X} {T^{1\textsf {(Cor)}}_{1}}^{'}-32 \pi ^2\}^{-1}. \end{aligned}$$

(B4)