Anisotropic stars in $f({\textit{G}},{\textit{T}})$ gravity under class I space-time

Abstract

In this paper, we studied the possible existence of anisotropic spherically symmetric solutions in the arena of modified $f(\textit{G}, \textit{T})$-gravity theory. To supply exact solutions of the field equations, we consider that the gravitational Lagrangian can be expressed as the generic function of the quadratic Gauss–Bonnet invariant $\textit{G}$ and the trace of the stress–energy tensor $\textit{T}$, i.e., $f(\textit{G},\textit{T}) = \textit{G}^2 + \chi \textit{T}$, where $\chi$ is a coupling parameter. We ansatz the gravitational potential: $g_{rr} \equiv e^{\lambda (r)}$ from the relationship quasi-local mass function, $e^{-\lambda }=1-\frac{2m(r)}{r}$, and we obtained the gravitational potential: $g_{tt} \equiv e^{\nu (r)}$ via the embedding class one procedure. In this regard, we investigated that the new solution is well analyzed and well comported through various physical and mathematical tests, which confirmed the physical viability and the stability of the system. The present investigation uncovers that the $f(\textit{G},\textit{T})$-gravity via embedding class one approach is a well acceptable to describe compact systems, and we successfully compared the effects of all the necessary physical requirements with the standard results of $f(\textit{G})$-gravity, which can be retrieved at $\chi = 0$.

Appendix A

The explicit expressions for G, $G^{\prime }$ and $G^{\prime \prime }$ are given as,

$$\begin{aligned} G(r)= & {} \frac{8 b B (a-b)^{3/2} \sqrt{b r^2+1}}{\left( a r^2+1\right) ^3 \left( B \sqrt{a-b} \sqrt{b r^2+1}+A b\right) }\\ G\,'(r)= & {} \frac{24 b B r (a-b)}{\left( a r^2+1\right) ^4 \sqrt{b r^2+1} \left( B \sqrt{a-b} \sqrt{b r^2+1}+A b\right) ^3} \Big [a^2 B \Big \{6 B \sqrt{a-b} \left( b r^2+1\right) ^2+A b \\&\quad (11 b r^2+12) \sqrt{b r^2+1}\Big \}+a b \Big \{A^2 b \sqrt{a-b} \left( 5 b r^2+6\right) -6 \sqrt{a-b} \left( b B r^2+B\right) ^2-\\&\quad A b B\sqrt{b r^2+1} \left( 11 b r^2+13\right) \Big \}+A b^3 \left( B \sqrt{b r^2+1}-A \sqrt{a-b}\right) \Big ],\\ G\,''(r)= & {} -\frac{24 b B (a-b)}{\left( a r^2+1\right) ^5 \left( b r^2+1\right) ^{3/2} \left( B \sqrt{a-b} \sqrt{b r^2+1}+A b\right) ^4} \Big [42 a^4 B^3 r^2 \left( b r^2+1\right) ^{7/2}+ \\&\quad a^3 B \Big \{A b B r^2 \sqrt{a-b} \left( 112 b^3 r^6+353 b^2 r^4+367 b r^2+126\right) +2 A^2 b^2 r^2 (50 b^2 r^4 \\&+\,115 b r^2+63) \sqrt{b r^2+1}-6 B^2 \left( 14 b r^2+1\right) \left( b r^2+1\right) ^{7/2}\Big \}+a^2 b \Big \{A^3 b^2 r^2 \sqrt{a-b}\\&\quad \left( 30 b^2 r^4+73 b r^2+42\right) -A B^2 \sqrt{a-b} \left( 112 b^4 r^8+387 b^3 r^6+447 b^2 r^4+190 b r^2+18\right) \\&-\,2 A^2 b B \sqrt{b r^2+1} \left( 50 b^3 r^6+138 b^2 r^4+91 b r^2+9\right) +6 B^3 \left( 7 b r^2+2\right) \left( b r^2+1\right) ^{7/2}\Big \}\\&+\,A b^3 \left( A^2 b \sqrt{a-b}+B^2 \sqrt{a-b} \left( 2 b^2 r^4+b r^2-1\right) +2 A b B \left( b r^2-1\right) \sqrt{b r^2+1}\right) +a b^2 \\&\quad \Big \{-2 A^3 b \sqrt{a-b} \left( 9 b^2 r^4+11 b r^2+3\right) +A B^2 \sqrt{a-b} \left( 34 b^3 r^6+78 b^2 r^4+63 b r^2+19\right) \\&+2 A^2 b B \left( 23 b^2 r^4+27 b r^2+10\right) \sqrt{b r^2+1}-6 B^3 \left( b r^2+1\right) ^{7/2}\Big \} \Big ] \end{aligned}$$