Physical analysis of spherical stellar structures in $f(\textrm{Q},\textrm{T})$ theory

Abstract

This paper explores the viability and stability of compact stellar objects characterized by anisotropic matter in the framework of $f(\textrm{Q},\textrm{T})$ theory, where $\textrm{Q}$ denotes non-metricity and $\textrm{T}$ represents the trace of the energy-momentum tensor. We consider a specific model of this theory to obtain explicit expressions for the field equations governing the behavior of matter and geometry in this context. Furthermore, the Karmarkar condition is employed to assess the configuration of static spherically symmetric structures. The values of unknown constants in the metric potentials are determined through matching conditions of the interior and exterior spacetimes. Various physical quantities such as fluid parameters, energy constraints, equation of state parameters, mass, compactness and redshift are graphically analyzed to evaluate the viability of the considered compact stars. The Tolman–Oppenheimer–Volkoff equation is used to examine the equilibrium state of the stellar models. Moreover, the stability of the proposed compact stars is investigated through sound speed and adiabatic index methods. This study concludes that the proposed compact stars analyzed in this theoretical framework are viable and stable, as all the required conditions are satisfied.

Appendices

Appendix A: Non-metricity scalar

According to Eqs. (18) and (19), we have

$$\begin{aligned} \textrm{Q}\equiv & {} -\textrm{g}^{\mu \nu } (\textrm{L}^{\lambda }_{\xi \mu }\textrm{L}^{\xi }_{\nu \lambda } - \textrm{L}^{\lambda }_{\xi \lambda }\textrm{L}^{\xi }_{\mu \nu }), \\ \textrm{L}^{\lambda }_{\xi \mu }= & {} -\frac{1}{2} \textrm{g}^{\lambda \varsigma }(\textrm{Q}_{\mu \xi \varsigma } +\textrm{Q}_{\xi \varsigma \mu }-\textrm{Q}_{\varsigma \mu \xi }), \\ \textrm{L}^{\xi }_{\nu \lambda }= & {} -\frac{1}{2} \textrm{g}^{\xi \varsigma }(\textrm{Q}_{\lambda \nu \varsigma } +\textrm{Q}_{\nu \varsigma \lambda }-\textrm{Q}_{\varsigma \lambda \nu }), \\ \textrm{L}^{\lambda }_{\xi \mu }= & {} -\frac{1}{2} \textrm{g}^{\lambda \varsigma }(\textrm{Q}_{\lambda \xi \varsigma } +\textrm{Q}_{\xi \varsigma \lambda }-\textrm{Q}_{\varsigma \lambda \xi }), \\= & {} -\frac{1}{2}(\bar{\textrm{Q}}_{\xi } +\textrm{Q}_{\xi }-\bar{\textrm{Q}}_{\xi })=-\frac{1}{2} \textrm{Q}_{\xi }, \\ \textrm{L}^{\xi }_{\mu \nu }= & {} -\frac{1}{2}\textrm{g}^{\xi \varsigma }(\textrm{Q}_{\nu \mu \varsigma } +\textrm{Q}_{\mu \varsigma \nu }-\textrm{Q}_{\varsigma \nu \mu }). \end{aligned}$$

Thus, we have

$$\begin{aligned} -\textrm{g}^{\mu \nu }\textrm{L}^{\lambda }_{\xi \mu } \textrm{L}^{\xi }_{\nu \lambda }= & {} -\frac{1}{4}\textrm{g}^{\mu \nu }\textrm{g}^{\lambda \varsigma } \textrm{g}^{\xi \varsigma } (\textrm{Q}_{\mu \xi \varsigma }+\textrm{Q}_{\xi \varsigma \mu } -\textrm{Q}_{\varsigma \mu \xi }) \\{} & {} \times (\textrm{Q}_{\lambda \nu \varsigma }+\textrm{Q}_{\nu \varsigma \lambda } -\textrm{Q}_{\varsigma \lambda \nu }), \\= & {} -\frac{1}{4}(\textrm{Q}^{\nu \varsigma \lambda }+\textrm{Q}^{\varsigma \lambda \nu } -\textrm{Q}^{\lambda \nu \varsigma }) \\{} & {} \times (\textrm{Q}_{\lambda \nu \varsigma }+\textrm{Q}_{\nu \varsigma \lambda } -\textrm{Q}_{\varsigma \lambda \nu }), \\= & {} -\frac{1}{4}(2\textrm{Q}^{\nu \varsigma \lambda }\textrm{Q}_{\varsigma \lambda \nu } - \textrm{Q}^{\nu \varsigma \lambda }\textrm{Q}_{\nu \varsigma \lambda }), \\ \textrm{g}^{\mu \nu }\textrm{L}^{\lambda }_{\xi \lambda }\textrm{L}^{\xi }_{\mu \nu }= & {} \frac{1}{4}\textrm{g}^{\mu \nu }\textrm{g}^{\xi \varsigma }\textrm{Q}_{\varsigma } (\textrm{Q}_{\nu \mu \varsigma }+\textrm{Q}_{\mu \varsigma \nu } -\textrm{Q}_{\varsigma \nu \mu }), \\= & {} \frac{1}{4}\textrm{Q}^{\varsigma }(2\bar{\textrm{Q}_{\varsigma }}- \textrm{Q}_{\varsigma }),\ \\ \textrm{Q}= & {} -\frac{1}{4}(\textrm{Q}^{\lambda \nu \varsigma }\textrm{Q}_{\lambda \nu \varsigma } +2\textrm{Q}^{\lambda \nu \varsigma \lambda }\textrm{Q}_{\varsigma \lambda \nu } \\{} & {} -2\textrm{Q}^{\varsigma }\bar{\textrm{Q}_{\varsigma }}+\textrm{Q}^{\varsigma }\textrm{Q}_{\varsigma }). \end{aligned}$$

According to Eq. (24), we obtain

$$\begin{aligned} \textrm{P}^{\lambda \mu \nu }= & {} \frac{1}{4}[-\textrm{Q}^{\lambda \mu \nu } +\textrm{Q}^{\mu \lambda \nu }+\textrm{Q}^{\nu \lambda \mu } +\textrm{Q}^{\lambda }\textrm{g}^{\mu \nu } \\{} & {} -\bar{\textrm{Q}^{\lambda }}\textrm{g}^{\mu \nu }-\frac{1}{2} (\textrm{g}^{\lambda \mu } \textrm{Q}^{\nu }+\textrm{g}^{\lambda \nu }\textrm{Q}^{\mu })], \\ -\textrm{Q}_{\lambda \mu \nu }\textrm{P}^{\lambda \mu \nu }= & {} -\frac{1}{4}[-\textrm{Q}_{\lambda \mu \nu }\textrm{Q}^{\lambda \mu \nu } \\{} & {} +\textrm{Q}_{\lambda \mu \nu }\textrm{Q}^{\mu \lambda \nu } +\textrm{Q}^{\nu \lambda \mu } \textrm{Q}_{\lambda \mu \nu }+\textrm{Q}_{\lambda \mu \nu } \textrm{Q}^{\lambda }\textrm{g}^{\mu \nu } \\{} & {} -\textrm{Q}_{\lambda \mu \nu }\bar{\textrm{Q}^{\lambda }} \textrm{g}^{\mu \nu } -\frac{1}{2}\textrm{Q}_{\lambda \mu \nu }(\textrm{g}^{\lambda \mu } \textrm{Q}^{\nu } +\textrm{g}^{\lambda \nu }\textrm{Q}^{\mu })], \\= & {} -\frac{1}{4}(-\textrm{Q}_{\lambda \mu \nu }\textrm{Q}^{\lambda \mu \nu } +2\textrm{Q}_{\lambda \mu \nu }\textrm{Q}^{\mu \lambda \nu }+\textrm{Q}^{\lambda } \textrm{Q}_{\lambda }-2\tilde{\textrm{Q}^{\lambda }}\textrm{Q}_{\lambda }), \\= & {} \textrm{Q}. \end{aligned}$$

Appendix B: Variation of non-metricity scalar

All the non-metricity tensors are given as

$$\begin{aligned} \textrm{Q}_{\lambda \mu \nu }= & {} \nabla _{\lambda }\textrm{g}_{\mu \nu }, \\ \textrm{Q}^{\lambda }~_{\mu \nu }= & {} \textrm{g}^{\lambda \xi } \textrm{Q}_{\xi \mu \nu } =\textrm{g}^{\lambda \xi }\nabla _{\xi }\textrm{g}_{\mu \nu } =\nabla ^{\lambda }\textrm{g}_{\mu \nu }, \\ \textrm{Q}_{\lambda ~~\nu }^{~~\mu }= & {} \textrm{g}^{\mu \varsigma }\textrm{Q}_{\lambda \varsigma \nu } =\textrm{g}^{\mu \varsigma }\nabla _{\lambda }\textrm{g}_{\varsigma \nu } =-\textrm{g}_{\mu \varsigma }\nabla _{\lambda }\textrm{g}^{\mu \varsigma }, \\ \textrm{Q}_{\lambda \mu }^{~~\nu }= & {} \textrm{g}^{\nu \varsigma }\textrm{Q}_{\lambda \mu \varsigma } =\textrm{g}^{\nu \varsigma }\nabla _{\lambda }\textrm{g}_{\mu \varsigma } =-\textrm{g}_{\mu \varsigma }\nabla _{\lambda }\textrm{g}^{\nu \varsigma }, \\ \textrm{Q}^{\lambda \mu }_{~~\nu }= & {} \textrm{g}^{\mu \varsigma }\textrm{g}^{\lambda \xi }\nabla _{\xi }\textrm{g} _{\varsigma \nu } =\textrm{g}^{\mu \varsigma }\nabla ^{\lambda }\textrm{g}_{\nu \varsigma } =-\textrm{g}_{\varsigma \nu }\nabla ^{\lambda }\textrm{g}^{\mu \varsigma }, \\ \textrm{Q}^ {\lambda ~~\nu } _{~\mu }= & {} \textrm{g}^{\nu \varsigma }\textrm{g}^{\lambda \xi }\nabla _{\xi }\textrm{g} _{\mu \varsigma } =\textrm{g}^{\nu \varsigma }\nabla ^{\lambda }\textrm{g}_{\mu \varsigma } =-\textrm{g}_{\mu \varsigma }\nabla ^{\lambda }\textrm{g}^{\nu \varsigma }, \\ \textrm{Q}_{\lambda }^{~~\mu \nu }= & {} \textrm{g}^{\mu \varsigma }\textrm{g}^{\nu \xi }\nabla _{\lambda }\textrm{g} _{\varsigma \xi } =-\textrm{g}^{\mu \varsigma }\textrm{g}_{\varsigma \xi }\nabla _{\lambda } \textrm{g}^{\nu \varsigma } =-\nabla _{\lambda }\textrm{g}^{\mu \nu }. \end{aligned}$$

By using Eq. (25), we have

$$\begin{aligned} \delta \textrm{Q}= & {} -\frac{1}{4} \delta (-\textrm{Q}^{\lambda \nu \varsigma } \textrm{Q}_{\lambda \nu \varsigma }+2\textrm{Q}^{\lambda \nu \varsigma } \textrm{Q}_{\varsigma \lambda \nu }-2\textrm{Q}^{\varsigma } \bar{\textrm{Q}_{\varsigma }}+\textrm{Q}^{\varsigma }\textrm{Q}_{\varsigma }), \\= & {} -\frac{1}{4}(-\delta \textrm{Q}^{\lambda \nu \varsigma } \textrm{Q}_{\lambda \nu \varsigma } - \textrm{Q}^{\lambda \nu \varsigma }\delta \textrm{Q}_{\lambda \nu \varsigma } + 2\delta \textrm{Q}_{\lambda \nu \varsigma }\textrm{Q}^{\varsigma \lambda \nu } \\{} & {} + 2 \textrm{Q}^{\lambda \nu \varsigma }\delta \textrm{Q}_{\varsigma \lambda \nu }-2\delta \textrm{Q}^{\varsigma }\bar{\textrm{Q}_{\varsigma }}+\delta \textrm{Q}^{\varsigma }\textrm{Q}_{\varsigma }-2 \textrm{Q}^{\varsigma }\delta \bar{\textrm{Q}_{\varsigma }} + \textrm{Q}^{\varsigma }\delta \textrm{Q}_{\varsigma }), \\= & {} -\frac{1}{4}[\textrm{Q}_{\lambda \nu \varsigma }\nabla ^{\lambda }\delta \textrm{g}^{\nu \varsigma }-\textrm{Q}^{\lambda \nu \varsigma } \nabla _{\lambda }\delta \textrm{g}_{\nu \varsigma }-2\textrm{Q}_{\varsigma \lambda \nu } \nabla ^{\lambda }\delta \textrm{g}^{\nu \varsigma } \\{} & {} +2\textrm{Q}^{\lambda \nu \varsigma }\nabla _{\varsigma }\delta \textrm{g}_{\lambda \nu }+ 2\bar{\textrm{Q}_{\varsigma }}\textrm{g}^{\mu \nu }\nabla ^{\varsigma }\delta \textrm{g}_{\mu \nu }+2\textrm{Q}^{\varsigma }\nabla ^{\xi }\delta \textrm{g}_{\varsigma \xi } \\{} & {} +2\bar{\textrm{Q}_{\varsigma }} \textrm{g}_{\mu \nu }\nabla ^{\varsigma }\delta \textrm{g}^{\mu \nu }-\textrm{Q}_{\varsigma }\nabla ^{\xi }\textrm{g} ^{\mu \nu }\delta \textrm{g}_{\mu \nu }-\textrm{Q}_{\varsigma }\textrm{g}_{\mu \nu }\nabla ^{\varsigma }\delta \textrm{g}^{\mu \nu } \\{} & {} -\textrm{Q}_{\varsigma } \textrm{g}^{\mu \nu } \nabla _{\varsigma } \delta \textrm{g}_{\mu \nu } -\textrm{Q}^{\varsigma }\textrm{g}_{\mu \nu }\nabla _{\varsigma }\delta \textrm{g}_{\mu \nu }]. \end{aligned}$$

We use the following relations to simplify the above equation

$$\begin{aligned} \delta \textrm{g}_{\mu \nu }= & {} -\textrm{g}_{\mu \lambda } \delta \textrm{g}^{\lambda \xi }\textrm{g}_{\xi \nu }-\textrm{Q}^ {\lambda \nu \varsigma } \nabla _{\lambda }\delta \textrm{g}_{\nu \varsigma }, \\ \delta \textrm{g}_{\nu \varsigma }= & {} -\textrm{Q}^{\lambda \nu \varsigma } \nabla _{\lambda }(-\textrm{g}_{\nu \mu }\delta \textrm{g}^{\mu \xi }\textrm{g}_{\xi \varsigma }), \\= & {} 2\textrm{Q}_{~~\varsigma }^{\lambda \nu }\textrm{Q}_{\lambda \nu \mu }\delta \textrm{g}^{\mu \varsigma } + \textrm{Q}_{\lambda \xi \varsigma }\nabla ^{\lambda }\textrm{g}^{\mu \varsigma } \\= & {} 2\textrm{Q}_{~~\nu }^{\lambda \xi }\textrm{Q}_{\lambda \xi \nu }\delta \textrm{g}^{\mu \nu }+\textrm{Q}_{\lambda \nu \varsigma }\nabla ^{\lambda } \textrm{g}^{\nu \varsigma }, \\ 2\textrm{Q}^{\lambda \nu \varsigma }\nabla _{\varsigma }\delta \textrm{g}_{\lambda \nu }= & {} -4\textrm{Q}_{\mu }^{~\xi \varsigma }\textrm{Q}_{\varsigma \xi \nu } \delta \textrm{g}^{\mu \nu }-2 \textrm{Q}_{\nu \varsigma \lambda }\nabla ^{\lambda }\delta \textrm{g}^{\nu \varsigma }, \\ -2\textrm{Q}^{\varsigma } \nabla ^{\xi } \delta \textrm{g}_{\varsigma \xi }= & {} 2\textrm{Q}^{\lambda } \textrm{Q}_{\nu \lambda \mu } \delta \textrm{g}^{\mu \nu }+ 2\textrm{Q}_{\mu }\bar{\textrm{Q}_{\nu }} \delta \textrm{g}^{\mu \nu } \\{} & {} +2\textrm{Q}_{\nu }\textrm{g}_{\lambda \varsigma }\nabla ^{\lambda } \textrm{g}^{\nu \varsigma }. \end{aligned}$$

Thus, we have

$$\begin{aligned} \delta \textrm{Q}=2\textrm{P}_{\lambda \nu \varsigma }\nabla ^{\lambda }\delta \textrm{g}^{\nu \varsigma }-(\textrm{P}_{\mu \lambda \xi }\textrm{Q}_{\nu } ^{\lambda \xi }-2 \textrm{P}_{\lambda \xi \nu }\textrm{Q}^{\lambda \xi } _{\nu })\delta \textrm{g}^{\mu \nu }, \end{aligned}$$

where

$$\begin{aligned} 2\textrm{P}_{\lambda \nu \varsigma }= & {} -\frac{1}{4}[2\textrm{Q} _{\lambda \nu \varsigma }-2\textrm{Q}_{\varsigma \lambda \nu } -2\textrm{Q}_{\nu \varsigma \lambda } \\{} & {} +2(\bar{\textrm{Q}}_{\lambda }-\textrm{Q}_{\lambda }) \textrm{g}_{\nu \varsigma }+2\textrm{Q}_{\nu }\textrm{g}_{\lambda \xi }], \\ 4(\textrm{P}_{\mu \lambda \xi }\textrm{Q}_{\nu }^{~~\lambda \xi }-2 \textrm{P}_{\lambda \xi \nu }\textrm{Q}^{\lambda \xi } _{~~\nu })= & {} 2\textrm{Q}^{\lambda \xi } _{~~\nu }\textrm{Q}_{\lambda \xi \mu }-4\textrm{Q}_{\mu } ^{~~\lambda \xi }\textrm{Q}_{\xi \lambda \nu }+2\textrm{Q}_{\lambda \mu \nu } \bar{\textrm{Q}}^{\lambda } \\{} & {} -\textrm{Q}^{\lambda }\textrm{Q}_{\lambda \mu \nu } +2\textrm{Q}^{\lambda }\textrm{Q}_{\nu \lambda \mu }+2\textrm{Q} _{\mu }\bar{\textrm{Q}_{\nu }}. \end{aligned}$$

Appendix C

The radial and tangential components of sound speed are given by

$$\begin{aligned} u_{sr}= & {} \bigg [a^{7}d^{2}r^{2}(3-2\eta )+4a^{6}cbdr^{2} (1+br^{2})(2\eta -3)+8cb^{3}(1+br^{2})^{10}d(-6 \\{} & {} -13\eta +br^{2}(17\eta -6))+8a^{2}cb^{2}d(1+br^{2})^{6} \big (-9-32\eta +br^{2}(-18+41\eta \\{} & {} +br^{2}(23\eta -9))\big )-2ab^{2}(1+br^{2})^{8}(d^{2}(1+br^{2}) (-23\eta -6+br^{2}(17\eta -6)) \\{} & {} +8c^{2}b(-3-14\eta +br^{2}(26\eta -3)))-4a^{4}cbd(1+br^{2}) ^{3}(3+14\eta +br^{2}(24 \\{} & {} -46\eta +br^{2}(9+40\eta )))+(1+br^{2})^{4}2a^{3}b(d^{2}(1+br^{2}) (12+36\eta +br^{2}(12 \\{} & {} -59\eta +25br^{2}\eta ))+2c^{2}b(14\eta +3+br^{2}(18-44\eta +br^{2} (15+38\eta )))) \\{} & {} +a^{5}(1+br^{2})^{2}\big (4c^{2}(3-2\eta )b^{2}r^{2}+d^{2}(3+14\eta +br^{2}(30-48\eta +br^{2} \\{} & {} \times (46\eta -9)))\big )\bigg ]\bigg [-40cb^{3}d(br^{2}-5)(1+br^{2}) ^{10}\eta +a^{7}d^{2}r^{2}(2\eta -3) \\{} & {} -4a^{6}cbr^{2}(1+br^{2})d(-3+2\eta )+10ab^{2}(1+br^{2})^{8} \big (d^{2}(br^{2}-7)(1+br^{2})\eta \\{} & {} -8c^{2}b(br^2-1)(2\eta -3)\big )+40a^{2}cb^{2}d(1+br^{2})^{6}(6-2 \eta -br^{2}\eta +b^{2}r^{4} \\{} & {} \times (11\eta -6))-4a^{4}cbd(1+br^{2})^{3}(5(-3+2\eta )+br^{2} (6-14\eta +br^{2} \\{} & {} \times (44\eta -51)))-2a^{3}b(1+br^{2})^{4}\big (-2c^{2}b(5+(-2 +17br^{2}))br^{2}(2\eta -3) \\{} & {} +5d^{2}(1+br^{2})(6+br^{2}(-7\eta +br^{2}(17\eta -6)))\big )+a^{5} (br^{2}+1+b(-7\eta \\{} & {} +br^{2}(17\eta -6)))+a^{5}(1+br^{2})^{2}(4(2\eta -3)c^{2}b^{2} r^{2}+d^{2}(5(2\eta -3)+br^{2} \\{} & {} \times (6-24\eta +br^{2}(74\eta -51))))\bigg ]^{-1}, \\ u_{st}= & {} \bigg [2(2a^{7}d^{2}r^{2}\eta -8a^{6}cbdr^{2}(1+br^{2}) \eta -4cb^{3}d(1+br^{2})^{10}((7\eta -6)r^{2}b \\{} & {} +12+\eta )+ab^{2}(1+br^{2})^{8}\big [-8c^{2}b(-6-8\eta +br^{2}(9+2\eta )) +d^{2}(1+br^{2}) \\{} & {} \times (18-\eta +br^{2}(7\eta -6))+2a^{2}cb^{2}d(1+br^{2})^{6}(-33-34\eta +br^{2} \\{} & {} \times (24-38\eta +br^{2}(16\eta -3)))-2a^{4}cbd(1+br^{2})^{3} (6+8\eta +br^{2} (-9 \\{} & {} +20\eta +br^{2}(15+28\eta )))+a^{5}(1+br^{2})^{2}(8c^{2}b^{2} r^{2}\eta +d^{2}(3+4\eta +br^{2} \\{} & {} \times (12\eta -3+br^{2}(3+8\eta ))))+a^{3}b(1+br^{2})^{4} (4c^{2}b(3+4\eta +br^{2}(-6 \\{} & {} +8\eta +br^{2}(9+16\eta )))-d^{2}(1+br^{2})(-3(7+6\eta )+br^{2} (-19\eta +27 \\{} & {} +br^{2}(23\eta -24))))\big ]\bigg ]\bigg [-40cb^{3}d(br^{2}-5) (1+br^{2})^{10}\eta +a^{7}(2\eta -3) \\{} & {} \times d^{2}r^{2}-4a^{6}cbdr^{2}(1+br^{2})(2\eta -3)+10ab^{2}(1+br^{2}) ^{8}(d^{2}(br^{2}-7) \\{} & {} \times (1+br^{2})\eta -8c^{2}b(br^{2}-1)(2\eta -3))+40a^{2}cdb^{2}d(1+br^{2})^{6}(-2\eta \\{} & {} +6+b^{2}r^{4}(11\eta -6)-br^{2}\eta )-4a^{4}cbd((1+br^{2})^{3}(5(-3+2\eta )+br^{2} \\{} & {} \times (6+br^{2}(44\eta -51)-14\eta ))-2a^{3}b(1+br^{2})^{4}(-2c^{2}b((-2+17br^{2}) \\{} & {} \times br^{2}+5)(-3+2\eta )+5d^{2}(1+br^{2})(6+br^{2}(-7\eta +br^{2}(-6+17\eta )))) \\{} & {} +a^{5}(1+br^{2})^{2}(4(2\eta -3)c^{2}b^{2}r^{2}+d^{2}(5(2\eta -3)+br^{2}(6-24\eta +br^{2} \\{} & {} \times (74\eta -51)))))\bigg ]^{-1}. \end{aligned}$$

Appendix D

The radial and tangential components of adiabatic index are given by

$$\begin{aligned} \Gamma _{r}= & {} \bigg [-6(1+br^{2})^{3}(2bd(1+br^{2})^{4} -2acb(1+br^{2})(3br^{2}-1)+a^{2}d \\{} & {} \times (5br^{2}-1))(2\eta -1)(a^{7}d^{2}r^{2}(3-2\eta ) +4a^{6}cbdr^{2}(2\eta -3) \\{} & {} \times (1+br^{2})+8cb^{3}d(1+br^{2})^{10}(-6-13\eta +br^{2}(17\eta -6))+8a^{2} \\{} & {} \times cb^{2}d(1+br^{2})^{6}(-9-32\eta +br^{2}(-18+41 \eta +br^{2}(23\eta -9))) \\{} & {} -2ab^{2}(1+br^{2})^{8}(d^{2}(1+br^{2})(-6-23\eta +br^{2} (17\eta -6))+8c^{2}b \\{} & {} \times (-3-14\eta +br^{2}(26\eta -3)))-4a^{4}cbd(1+br^{2})^{3} (3+14\eta +br^{2} \\{} & {} \times (24-46\eta +br^{2}(9+40\eta )))+2a^{3}b(1+br^{2})^{4} (d^{2}(1+br^{2})(36\eta \\{} & {} +12+br^{2}(12-59\eta +25br^{2}\eta ))+2c^{2}b(3+14\eta +br^{2} (18-44\eta \\{} & {} +br^{2}(15+38\eta ))))+a^{5}(1+br^{2})^{2}(4c^{2}b^{2}r^{2} (3-2\eta )+d^{2}(3+14\eta \\{} & {} +br^{2}(30-48\eta +br^{2}(46\eta -9)))))\bigg ]\bigg [(a^{4}dr^{2} (2\eta -3)-2a^{3}cbr^{2} \\{} & {} (1+br^{2})(2\eta -3)+3a^{2}d(1+br^{2})^{3}(-1-2\eta +5br^{2} (2\eta -1)) \\{} & {} +2bd(1+br^{2})^{6}(-3(2+\eta )+br^{2}(17\eta -6))-2acb(1+br^{2}) ^{4}(-3 \\{} & {} -6\eta +br^{2}(26\eta -3)))(a^{7}d^{2}r^{2}(3-2\eta )+40cb^{3} d(1+br^{2})^{10}\eta \\{} & {} \times (br^{2}-5)+4a^{6}cbdr^{2}(1+br^{2})(2\eta -3)+10ab^{2} (1+br^{2})^{8} \\{} & {} \times (d^{2}(7+br^{2}(6-br^{2}))\eta +8c^{2}b(br^{2}-1) (2\eta -3))-40a^{2}cb^{2}d \\{} & {} \times (1+br^{2})^{6}(6-2\eta -br^{2}\eta +b^{2}r^{4}(11\eta -6)) +4a^{4}cbd(1+br^{2})^{3} \\{} & {} \times (5(2\eta -3)+br^{2}(6-14\eta +br^{2}(44\eta -51)))+2a^{3} b(1+br^{2})^{4} \\{} & {} \times (-2c^{2}b(5+br^{2}(17br^{2}-2))(2\eta -3)+5d^{2}(1+br^{2}) (6+br^{2} \\{} & {} \times (-7\eta +br^{2}(17\eta -6))))-a^{5}(1+br^{2})^{2}(4c^{2} b^{2}r^{2}(2\eta -3) \\{} & {} +d^{2}(5(2\eta -3)+br^{2}(6-24\eta +br^{2}(74\eta -51)))))\bigg ]^{-1}, \\ \Gamma _{t}= & {} \bigg [6(a^{2}d-2acb(1+br^{2})-2bd(1+br^{2})^{3}) (a^{2}r^{2}-2(br^{2}-1) \\{} & {} \times (1+br^{2})^{3})(2\eta -1)(2a^{7}d^{2}r^{2}\eta -8a^{6}cbdr^{2} (1+br^{2})\eta -4cd \\{} & {} \times b^{3}(1+br^{2})^{10}(12+\eta +br^{2}(7\eta -6))+ab^{2} (1+br^{2})^{8}(-8c^{2}b \\{} & {} \times (-6-8\eta +br^{2}(9+2\eta ))+d^{2}(1+br^{2})(18-\eta +br^{2} (7\eta -6))) \\{} & {} \times 2a^{2}cb^{2}d(1+br^{2})^{6}(-33-34\eta +br^{2}(24-38\eta +br^{2}(16\eta -3))) \\{} & {} -2a^{4}cbd(1+br^{2})^{3}(6+8\eta +br^{2}(-9+20\eta +br^{2}(15 +28\eta )))+a^{5} \\{} & {} \times (1+br^{2})^{2}(8c^{2}b^{2}r^{2}\eta +d^{2}(3+4\eta +br^{2} (-3+12\eta +br^{2}(3+8\eta )))) \\{} & {} +a^{3}b(1+br^{2})^{4}(4c^{2}b(3+4\eta +br^{2}(-6+8\eta +br^{2}(9 +16\eta )))-d^{2} \\{} & {} \times (1+br^{2})(-3(7+6\eta )+br^{2}(27-19\eta +br^{2}(23\eta -24) ))))\bigg ] \\{} & {} \times \bigg [(4a^{4}dr^{2}\eta -8a^{3}cbr^{2}(1+br^{2})\eta +3a^{2} d(1+br^{2})^{3}(1+2\eta +br^{2} \\{} & {} \times (2\eta -1))+2acb(1+br^{2})^{4}(-3-6\eta +br^{2}(9+2\eta )) +2(1+br^{2})^{6} \\{} & {} \times bd(3(2+\eta )+br^{2}(7\eta -6)))(-40cb^{3}d(br^{2}-5)(1+br^{2}) ^{10}\eta \\{} & {} +a^{7}d^{2}r^{2}(2\eta -3)-4a^{6}cbdr^{2}(1+br^{2})(2\eta -3)+10ab^{2} (1+br^{2})^{8} \\{} & {} \times (d^{2}(br^{2}-7)(1+br^{2})\eta -8c^{2}b(br^{2}-1)(2\eta -3) +40a^{2}cb^{2}d \\{} & {} \times (1+br^{2})^{6}(6-2\eta -br^{2}\eta +b^{2}r^{4}(11\eta -6)) -4a^{4}cdb(1+br^{2})^{3} \\{} & {} \times (5(2\eta -3)+br^{2}(6-14\eta +br^{2}(44\eta -51)))-2a^{3} b(1+br^{2})^{4}(-2 \\{} & {} \times c^{2}b(5+br^{2}(17br^{2}-2))(2\eta -3)+5d^{2}(1+br^{2}) (6+br^{2}(-7\eta \\{} & {} +br^{2}(17\eta -6))))+a^{5}(1+br^{2})^{2}(4c^{2}b^{2}r^{2} (2\eta -3)+d^{2}(5(2\eta -3) \\{} & {} +br^{2}(6-24\eta +br^{2}(74\eta -51))))))\big ]^{-1}. \end{aligned}$$