Complexity analysis of charged dynamical dissipative cylindrical structure in modified gravity

Abstract

This article focuses on the formulation of some scalar factors which are uniquely expressed in terms of matter variables for dynamical charged dissipative cylindrical geometry in a standard gravity model $\mathcal {R}+\Phi \mathcal {Q}$ ($\Phi$ is the coupling parameter, $\mathcal {Q}=\mathcal {R}_{\varphi \vartheta }\mathcal {T}^{\varphi \vartheta }$) and calculates four scalars by orthogonally decomposing the Riemann tensor. We find that only $\mathcal {Y}_{TF}$ involves inhomogeneous energy density, heat flux, charge and pressure anisotropy coupled with modified corrections, and thus call it as complexity factor for the considered distribution. Two evolutionary modes are discussed to study the dynamics of cylinder. We then take the homologous condition with $\mathcal {Y}_{TF}=0$ to calculate unknown metric potentials in the absence as well as presence of heat dissipation. The stability criterion of the later condition is also checked throughout the evolution by applying some constraints. We conclude that the effects of charge and modified theory yield more complex system.

Appendices

Appendix A

The $f(\mathcal {R},\mathcal {T},\mathcal {Q})$ corrections in the field equations (12)–(15) are

$$\begin{aligned} \mathcal {T}^{(\mathcal {C})}_{00}&=\frac{\Phi }{8\pi \big (1-\frac{\Phi s^2}{2C^4}\big )}\bigg \{\mu \bigg (\frac{3\ddot{B}}{2B^2}-\frac{3\dot{A}\dot{B}}{2AB} +\frac{3\ddot{C}}{C}-\frac{\dot{A}\dot{C}}{AC}-\frac{\dot{C}^2}{C^2}-\frac{3AA''}{2B^2}+\frac{2A^{\prime 2}}{B^2}\nonumber \\&\quad -\,\frac{1}{2}A^2\mathcal {R}-\frac{AA'B'}{2B^3}-\frac{2\dot{B}\dot{C}}{BC}-\frac{3AA'C'}{B^2C}\bigg )-\dot{\mu }\bigg (\frac{3\dot{A}}{A} +\frac{\dot{C}}{C}+\frac{\dot{B}}{2B}\bigg )+\frac{\mu ''A^2}{2B^2}\nonumber \\&\quad -\,\mu '\bigg (\frac{A^2B'}{2B^3}-\frac{A^2C'}{B^2}\bigg )+P_{r}\bigg (-\frac{\ddot{B}}{2B}+\frac{AA''}{2B^2}-\frac{AA'B'}{2B^3}-\frac{A^2C''}{B^2C} -\frac{A^2C^{\prime 2}}{B^2C^2}\nonumber \\&\quad +\,\frac{2A^2B'C'}{B^3C}-\frac{\dot{B}\dot{C}}{BC}-\frac{2A^2B'\dot{C}}{B^3C}+\frac{2A^{\prime 2}}{B^2}\bigg )+\frac{\dot{P}_{r}\dot{B}}{2B} +P'_{r}\bigg (\frac{A^2B'}{2B^3}-\frac{2A^2C'}{B^2C}\bigg )\nonumber \\&\quad -\,\frac{P''_{r}A}{2B^2}-P_{\bot }\bigg (\frac{\ddot{C}}{C}+\frac{\dot{C}^2}{C}-\frac{\dot{A}\dot{C}}{AC} -\frac{AA'C'}{B^2C}-\frac{A^2C''}{B^2C}+\frac{A^2B'C'}{B^3C}-\frac{A^2C^{\prime 2}}{B^2C^2}\nonumber \\&\quad +\,\frac{\dot{B}\dot{C}}{BC}\bigg )-\frac{3\dot{P}_{\bot }\dot{C}}{C}+\frac{P'_{\bot }A^2C'}{B^2C}-\varsigma \bigg (\frac{9\dot{A}A'}{2AB} -\frac{2A\dot{C}'}{BC}+\frac{3A'\dot{C}}{BC}+\frac{5A\dot{B}C'}{B^2C} \\ &\quad+\,\frac{3A'\dot{B}}{B^2}+\frac{\dot{A}C'}{BC}+\frac{2A\dot{C}C'}{BC^2}\bigg )+\frac{2\dot{\varsigma }A'}{B}-\frac{2\varsigma 'A\dot{C}}{BC} +\frac{A^2\mathcal {Q}}{2}\bigg \},\nonumber\end{aligned}$$

(A1)

$$\begin{aligned} \mathcal {T}^{(\mathcal {C})}_{01}&=\frac{\Phi }{8\pi \big (1-\frac{\Phi s^2}{2C^4}\big )}\bigg \{\mu \bigg (-\frac{\dot{A}A'}{A^2}-\frac{3\dot{A}'}{2A} +\frac{2\dot{C}'}{C}-\frac{2\dot{B}C'}{BC}-\frac{2A'\dot{C}}{AC}\bigg )-\frac{3\dot{\mu }A'}{2A}\nonumber \\&\quad +\,\mu '\bigg (\frac{\dot{C}}{C}-\frac{\dot{B}}{2B}\bigg )+\frac{\dot{\mu }'}{2}+P_{r}\bigg (\frac{A^{\prime 2}}{2B^2}-\frac{A'\dot{B}}{2AB} -\frac{2\dot{C}'}{C}+\frac{2\dot{B}C'}{BC}+\frac{2A'\dot{C}}{AC}\bigg )\nonumber \\&\quad +\,\dot{P}_{r}\bigg (\frac{A'}{2A}-\frac{C'}{C}\bigg )+\frac{P'_{r}\dot{B}}{2B}-\frac{\dot{P}'_{r}}{2}+\frac{\dot{P}_{\bot }C'}{C} +\frac{P'_{\bot }\dot{C}}{C}+\varsigma \bigg (\frac{1}{2}AB\mathcal {R}-\frac{2B\ddot{C}}{AC}\nonumber \\&\quad -\,\frac{\ddot{B}}{A}+\frac{A''}{B}+\frac{2B\dot{A}\dot{C}}{A^2C}+\frac{2AC''}{BC}-\frac{2AB'C'}{B^2C}+\frac{\dot{A}\dot{B}}{A^2} -\frac{A'B'}{B^2}\bigg )\bigg \}, \end{aligned}$$

(A2)

$$\begin{aligned} \mathcal {T}^{(\mathcal {C})}_{11}&=\frac{\Phi }{8\pi \big (1-\frac{\Phi s^2}{2C^4}\big )}\bigg \{\mu \bigg (\frac{B\ddot{B}}{2A^2}-\frac{B\dot{A}\dot{B}}{2A^3} -\frac{B^2\ddot{C}}{A^2C}+\frac{B^2\dot{A}\dot{C}}{A^3C}-\frac{B^2\dot{C}^2}{A^2C^2}-\frac{A''}{2A}\nonumber \\&\quad +\,\frac{A'B'}{2AB}-\frac{A'C'}{AC}\bigg )+\dot{\mu }\bigg (\frac{B^2\dot{A}}{2A^3}-\frac{2B^2\dot{C}}{A^2C}\bigg ) +\frac{\mu 'A'}{2A}-\frac{\ddot{\mu }B^2}{2A^2}+P_{r}\bigg (\frac{1}{2}B^2\mathcal {R}\nonumber \\&\quad +\,\frac{3B\dot{A}\dot{B}}{2A^3}-\frac{3B\ddot{B}}{2A^2}-\frac{2A'C'}{AC}+\frac{3A''}{2A}-\frac{3A'B'}{2AB}-\frac{2B'C'}{BC} +\frac{3C''}{C}-\frac{3B\dot{B}\dot{C}}{A^2C}\nonumber \\&\quad -\,\frac{2B'\dot{C}}{BC}-\frac{C^{\prime 2}}{C^2}\bigg )+\dot{P}_{r}\bigg (\frac{B^2\dot{C}}{A^2C}-\frac{B^2\dot{A}}{2A^3}\bigg ) -P'_{r}\bigg (\frac{A'}{2A}+\frac{C'}{C}\bigg )+\frac{\ddot{P}_{r}B^2}{2A^2}\nonumber \\&\quad +\,P_{\bot }\bigg (\frac{A'C'}{AC}-\frac{B^2\ddot{C}}{A^2C}-\frac{B^2\dot{C}^2}{A^2C^2}+\frac{B^2\dot{A}\dot{C}}{A^3C}+\frac{C''}{C} -\frac{B'C'}{BC}+\frac{C^{\prime 2}}{C^2}-\frac{B\dot{B}\dot{C}}{A^2C}\bigg )\nonumber \\&\quad -\,\frac{\dot{P}_{\bot }B^2\dot{C}}{A^2C}-\frac{P'_{\bot }C'}{C}+\varsigma \bigg (\frac{2B\dot{C}'}{AC}-\frac{B\dot{A}C'}{A^2C} -\frac{3BA'\dot{C}}{A^2C}-\frac{2B\dot{C}C'}{AC^2}-\frac{4\dot{B}C'}{AC}\bigg )\nonumber \\&\quad +\frac{B^2\mathcal {Q}}{2}-\frac{2\dot{\varsigma }BC'}{AC}\bigg \}, \end{aligned}$$

(A3)

$$\begin{aligned} \mathcal {T}^{(\mathcal {C})}_{22}&=\frac{\Phi }{8\pi \big (1-\frac{\Phi s^2}{2C^4}\big )}\bigg \{\mu \bigg (-\frac{C^2\ddot{B}}{2A^2B}+\frac{C^2\dot{A}\dot{B}}{2A^3B} -\frac{C^2A''}{2AB^2}+\frac{C^2A'B'}{2AB^3}-\frac{C\dot{B}\dot{C}}{A^2B}\nonumber \\&\quad -\,\frac{CA'C'}{AB^2}\bigg )+\dot{\mu }\bigg (\frac{C^2\dot{A}}{2A^3}-\frac{C^2\dot{B}}{A^2B}-\frac{C\dot{C}}{A^2}\bigg ) -\frac{\mu 'C^2A'}{2AB^2}-\frac{\ddot{\mu }C^2}{2A^2}+P_{r}\bigg (\frac{C^2\dot{A}\dot{B}}{2A^3B}\nonumber \\&\quad -\,\frac{C^2\ddot{B}}{2A^2B}-\frac{CA'C'}{AB^2}-\frac{C^2A''}{2AB^2}+\frac{C^2A'B'}{2AB^3}+\frac{CB'C'}{B^3}-\frac{C\dot{B}\dot{C}}{A^2B} -\frac{2CB'\dot{C}}{B^3}\bigg )\nonumber \\&\quad -\,\frac{\dot{P}_{r}C^2\dot{B}}{2A^2B}+P'_{r}\bigg (-\frac{C^2A'}{AB^2}+\frac{C^2B'}{2B^3}-\frac{CC'}{B^2}\bigg )-\frac{P''_{r}C^2}{2B^2} +P_{\bot }\bigg (\frac{1}{2}C^2\mathcal {R}\nonumber \\&\quad +\,\frac{2CC''}{B^2}-\frac{2C\ddot{C}}{A^2}-\frac{2\dot{C}^2}{A^2}+\frac{2C^{\prime 2}}{B^2}-\frac{2CB'C'}{B^3}+\frac{2CA'C'}{AB^2} -\frac{2C\dot{B}\dot{C}}{A^2B}-2\nonumber \\&\quad +\,\frac{2C\dot{A}\dot{C}}{A^3}\bigg )+\dot{P}_{\bot }C^2\bigg (\frac{\dot{B}}{2A^2B}-\frac{\dot{A}}{2A^3}\bigg )+P'_{\bot }C^2\bigg (-\frac{A'}{2AB^2} +\frac{B'}{2B^3}\bigg )\nonumber \\&\quad +\,\frac{\ddot{P}_{\bot }C^2}{2A^2}-\frac{P''_{\bot }C^2}{2B^2}+\varsigma \bigg (-\frac{C^2\dot{A}'}{A^2B}+\frac{C^2\dot{A}A'}{A^3B} -\frac{C\dot{A}C'}{A^2B}+\frac{C^2\dot{B}B'}{AB^3}-\frac{C^2\dot{B}'}{AB^2}\nonumber \\&\quad -\,\frac{CA'\dot{C}}{A^2B}-\frac{2C\dot{B}C'}{AB^2}\bigg )-\dot{\varsigma }\bigg (\frac{C^2A'}{A^2B}+\frac{CC'}{AB}\bigg ) -\varsigma '\bigg (\frac{C^2\dot{B}}{AB^2}+\frac{C\dot{C}}{AB}\bigg )-\frac{\dot{\varsigma }'C^2}{AB} +\,\frac{C^2\mathcal {Q}}{2}\bigg \}, \end{aligned}$$

(A4)

$$\begin{aligned} \mathcal {E}^{(\mathcal {C})}_{00}&=\frac{\Phi s^2}{8\pi AB^3C^5\big (1-\frac{\Phi s^2}{2C^4}\big )}\bigg \{2A^3BC''+A^2BCA''-2AB^2\dot{B}\dot{C}-2A^3B'C'\nonumber \\&\quad -\,A^2CA'B'-AB^2C\ddot{B}+B^2C\dot{A}\dot{B}+\frac{A^3B^3C\mathcal {R}}{2}\bigg \}, \end{aligned}$$

(A5)

$$\begin{aligned} \mathcal {E}^{(\mathcal {C})}_{01}&=\frac{\Phi s^2}{8\pi ABC^5\big (1-\frac{\Phi s^2}{2C^4}\big )}\big \{AB\dot{C}'-BA'\dot{C}-A\dot{B}C'\big \}, \end{aligned}$$

(A6)

$$\begin{aligned} \mathcal {E}^{(\mathcal {C})}_{11}&=\frac{\Phi s^2}{8\pi A^3BC^5\big (1-\frac{\Phi s^2}{2C^4}\big )}\bigg \{2AB^3\ddot{C}+AB^2C\ddot{B}-A^2BCA''-2A^2BA'C' \\ &\quad -\,2B^3\dot{A}\dot{C}+A^2CA'B'-B^2C\dot{A}\dot{B}-\frac{A^3B^3C\mathcal {R}}{2}\bigg \}, \end{aligned}$$

(A7)

$$\begin{aligned} \mathcal {E}^{(\mathcal {C})}_{22}&=\frac{\Phi s^2\mathcal {R}}{16\pi C^2\big (1-\frac{\Phi s^2}{2C^4}\big )}. \end{aligned}$$

(A8)

The terms $\mathbb {Z}_1$ and $\mathbb {Z}_2$ in Eqs. (16) and (17) are

$$\begin{aligned} \mathbb {Z}_1&=\frac{2\Phi }{16\pi +\Phi \mathcal {R}}\bigg [\big (\frac{\varsigma B\mathcal {R}^{10}}{A}\big )^.+\big (\frac{\varsigma B\mathcal {R}^{11}}{A}\big )'-\big (\mu \mathcal {R}^{00}\big )^.-\big (\mu \mathcal {R}^{01}\big )'+\frac{2s^2\dot{C}\mathcal {G}^{00}}{C^5}\nonumber \\&\quad -\,\mathcal {G}^{01}\bigg (\frac{ss'}{C^4}-\frac{2s^2C'}{C^5}\bigg )+\frac{1}{2A^2}\bigg \{\mathcal {R}_{00}\bigg (\frac{\dot{\mu }}{A^2} -\frac{2\mu \dot{A}}{A^3}\bigg )+2\bigg (\frac{\dot{\varsigma }}{AB}-\frac{\varsigma \dot{A}}{A}-\frac{\varsigma \dot{B}}{B}\bigg )\nonumber \\&\quad \times \,\mathcal {R}_{01}+\mathcal {R}_{11}\bigg (\frac{\dot{P}_r}{B^2} -\frac{2P_r\dot{B}}{B^3}\bigg )+2\mathcal {R}_{22}\bigg (\frac{\dot{P}_\bot }{C^2}-\frac{2P_\bot \dot{C}}{C^3}\bigg )\bigg \} -\frac{\mu \dot{\mathcal {R}}}{2A^2}\bigg ], \end{aligned}$$

(A9)

$$\begin{aligned} \mathbb {Z}_2&=\frac{2\Phi }{16\pi +\Phi \mathcal {R}}\bigg [\big (P_r\mathcal {R}^{10}\big )^.+\big (P_r\mathcal {R}^{11}\big )' -\big (\frac{\varsigma A\mathcal {R}^{00}}{B}\big )^.-\big (\frac{\varsigma A\mathcal {R}^{11}}{B}\big )'+\frac{2s^2\dot{C}\mathcal {G}^{10}}{C^5}\nonumber \\&\quad -\,\mathcal {G}^{11}\bigg (\frac{ss'}{C^4}-\frac{2s^2C'}{C^5}\bigg )-\frac{1}{2B^2}\bigg \{\mathcal {R}_{00}\bigg (\frac{\mu '}{A^2}-\frac{2\mu A'}{A^3}\bigg )+2\bigg (\frac{\varsigma '}{AB}-\frac{\varsigma A'}{A}-\frac{\varsigma B'}{B}\bigg )\nonumber \\&\quad \times \,\mathcal {R}_{01}+\mathcal {R}_{11}\bigg (\frac{P'_r}{B^2}-\frac{2P_rB'}{B^3}\bigg )+2\mathcal {R}_{22}\bigg (\frac{P'_\bot }{C^2} -\frac{2P_\bot C'}{C^3}\bigg )\bigg \}-\frac{P_r\mathcal {R}'}{2B^2}\bigg ]. \end{aligned}$$

(A10)

Appendix B

The scalars (38)–(41) encompass modified corrections which are

$$\begin{aligned} \chi _{1}^{(\mathcal {C})}&=-\frac{8\pi \Phi }{1-\frac{\Phi s^2}{2C^4}}\bigg [\bigg \{\Box \Omega ^{\rho }_{\vartheta } +\frac{1}{2}\nabla _{\varphi }\nabla ^{\rho }\Omega ^{\varphi }_{\vartheta } +\frac{1}{2}\nabla _{\varphi }\nabla _{\vartheta }\Omega ^{\varphi \rho }\bigg \}h^{\vartheta } _{\rho }-\big (\mathcal {R}^{\rho }_{\varphi }h^{\varphi }_{\rho }\nonumber \\&\quad +\,\mathcal {R}_{\varphi \vartheta }h^{\vartheta \varphi }\big )\bigg (P-\frac{\Pi }{3}+\frac{s^2}{8\pi C^4}\bigg )-\frac{\mathcal {Q}}{2}+\frac{1}{2}\nabla _{\varphi }\nabla _{\xi }\Omega ^{\varphi \xi }-\frac{1}{2}\Box (\mu -3P)\nonumber \\&\quad -\,\nabla _{\varphi }\nabla _{\rho }\Omega ^{\varphi \rho }+2\mathcal {R}\bigg (P-\frac{\Pi }{3}+\frac{s^2}{8\pi C^4}\bigg )-2g^{\rho \vartheta }\mathcal {R}^{\varphi \xi }\frac{\partial ^2\mathbb {L}_{\mathcal {M}}}{\partial g^{\rho \vartheta }\partial g^{\varphi \xi }}\bigg ], \end{aligned}$$

(B1)

$$\begin{aligned} \chi _{2}^{(\mathcal {C})}&=\frac{8\pi \Phi }{1-\frac{\Phi s^2}{2C^4}}\bigg [\frac{1}{2}\nabla _{\varphi }\nabla _{\xi }\Omega ^{\varphi \xi }\big \{\Box (\mu -3P) -\mathcal {K}^{\varphi }\mathcal {K}^{\vartheta }\Box \Omega _{\varphi \vartheta } +3\mathcal {K}_{\varphi }\mathcal {K}^{\vartheta }\Box \Omega ^{\varphi }_{\vartheta }\big \}\nonumber \\&\quad +\,\frac{3\mathcal {Q}}{2}+4\mathcal {R}^{\vartheta }_{\varphi }\bigg \{\bigg (P+\frac{s^2}{24\pi C^4}\bigg )h^{\varphi }_{\vartheta }+2\bigg (\Pi -\frac{s^2}{4\pi C^4}\bigg ) \bigg (\mathcal {W}^{\varphi }\mathcal {W}_{\vartheta }-\frac{1}{3}h^{\varphi }_{\vartheta }\bigg )\nonumber \\&\quad +\,\varsigma \mathcal {K}^{\varphi }\mathcal {W}_{\vartheta }\bigg \}+6\mathcal {R}^{\vartheta }_{\varphi }\bigg \{\bigg (\mu -\frac{s^2}{8\pi C^4}\bigg )\mathcal {K}^{\varphi }\mathcal {K}_{\vartheta }+\varsigma \mathcal {W}^{\varphi }\mathcal {K}_{\vartheta }\bigg \} +6\mathcal {R}_{\varphi \vartheta }\nonumber \\&\quad \times \,\bigg \{\bigg (\mu -\frac{s^2}{8\pi C^4}\bigg )\mathcal {K}^{\varphi }\mathcal {K}^{\vartheta }+\varsigma \mathcal {K}^{\vartheta }\mathcal {W}^{\varphi }\bigg \} -2\nabla _{\varphi }\nabla ^{\vartheta }\Omega ^{\varphi }_{\vartheta } +3\mathcal {K}_{\vartheta }\mathcal {K}^{\xi }\nabla _{\varphi }\nabla ^{\vartheta }\Omega ^{\varphi }_{\xi }\nonumber \\&\quad +\,3\mathcal {K}_{\vartheta }\mathcal {K}^{\xi }\nabla _{\varphi }\nabla _{\xi }\Omega ^{\varphi \vartheta } -2\mathcal {K}^{\vartheta }\mathcal {K}^{\xi }\nabla _{\varphi }\nabla _{\vartheta }\Omega ^{\varphi }_{\xi } -4h^{\vartheta \beta }\mathcal {R}^{\varphi \xi }\frac{\partial ^2\mathbb {L}_{\mathcal {M}}}{\partial g^{\vartheta \beta }\partial g^{\varphi \xi }}\nonumber \\&\quad +\,\frac{1}{2}\Box (\mu -3P)+\nabla _{\varphi }\nabla _{\vartheta } \Omega ^{\varphi \vartheta }+2g^{\vartheta \xi }\mathcal {R}^{\alpha \rho }\frac{\partial ^2\mathbb {L}_{\mathcal {M}}}{\partial g^{\vartheta \xi }\partial g^{\alpha \rho }}\bigg ], \end{aligned}$$

(B2)

$$\begin{aligned} \chi _{\varphi \vartheta }^{(\mathcal {C})}&=-\frac{4\pi \Phi }{1-\frac{\Phi s^2}{2C^4}}\bigg [-\frac{1}{2}\big \{h^{\lambda }_{\varphi }h^{\xi }_{\vartheta }\Box \Omega _{\lambda \xi }-\Box \Omega _{\varphi \vartheta } -\mathcal {K}_{\varphi }\mathcal {K}_{\vartheta }\mathcal {K}_{\lambda }\mathcal {K}^{\delta }\Box \Omega ^{\lambda }_{\delta }\big \} -(h^{\lambda }_{\varphi }\mathcal {R}_{\lambda \rho }\nonumber \\&\quad -\,\mathcal {R}_{\varphi \rho })\bigg \{\bigg (P+\frac{s^2}{24\pi C^4}\bigg )h^{\rho }_{\vartheta }+\bigg (\Pi -\frac{s^2}{4\pi C^4}\bigg )\bigg (\mathcal {W}^{\rho }\mathcal {W}_{\vartheta }-\frac{1}{3}h^{\rho }_{\vartheta }\bigg ) +\varsigma \mathcal {K}^{\rho }\mathcal {W}_{\vartheta }\bigg \}\nonumber \\&\quad -\,(h^{\xi }_{\vartheta }\mathcal {R}_{\rho \xi }-\mathcal {R}_{\rho \vartheta }) \bigg \{\bigg (P+\frac{s^2}{24\pi C^4}\bigg )h^{\rho }_{\varphi } +\bigg (\Pi -\frac{s^2}{4\pi C^4}\bigg )\bigg (\mathcal {W}^{\rho }\mathcal {W}_{\varphi }-\frac{1}{3}h^{\rho }_{\varphi }\bigg )\nonumber \\&\quad +\,\varsigma \mathcal {K}^{\rho }\mathcal {W}_{\varphi }\bigg \} +\frac{1}{2}\{h^{\lambda }_{\varphi }h^{\xi }_{\vartheta }\nabla _{\rho }\nabla _{\lambda }\Omega ^{\rho }_{\xi } +h^{\lambda }_{\varphi }h^{\xi }_{\vartheta }\nabla _{\rho }\nabla _{\xi }\Omega ^{\rho }_{\lambda } -\nabla _{\rho }\nabla _{\varphi }\Omega ^{\rho }_{\vartheta }\nonumber \\&\quad -\,\nabla _{\rho } \nabla _{\vartheta }\Omega ^{\rho }_{\varphi }-\mathcal {K}_{\varphi }\mathcal {K}_{\vartheta }\mathcal {K}_{\lambda }\mathcal {K}^{\delta } \nabla _{\rho }\nabla ^{\lambda }\Omega ^{\rho }_{\delta } -\mathcal {K}_{\varphi }\mathcal {K}_{\vartheta }\mathcal {K}_{\lambda }\mathcal {K}^{\delta }\nabla _{\rho }\nabla _{\delta }\Omega ^{\rho \lambda }\} \nonumber \\&\quad +\,2\mathcal {R}^{\rho \beta }h^{\lambda }_{\varphi }\bigg \{h^{\xi }_{\vartheta }\frac{\partial ^2\mathbb {L}_{\mathcal {M}}}{\partial g^{\lambda \xi }\partial g^{\rho \beta }} -\frac{\partial ^2\mathbb {L}_{\mathcal {M}}}{\partial g^{\epsilon \vartheta }\partial g^{\rho \beta }}\bigg \}\bigg ]. \end{aligned}$$

(B3)

The complexity-free and homologous conditions in the absence of dissipation flux are

$$\begin{aligned}&\epsilon -\frac{2\pi }{C^4}\left( \frac{\Phi \mathcal {R}}{2}+1\right) \big [\chi _9^{-1}\big \{\chi _7\big (2C^4-s^2\Phi \big )-\chi _4^{-1}\big (\chi _9 \big (\chi _1\big (s^2\Phi -2C^4\big )+2C^4s^2\nonumber \\&\quad \times \, \chi _6\big )+\chi _3\big (\chi _7\big (s^2 \Phi -2C^4\big )-2C^4s^2\chi _{11}\big )\big )+2C^4s^2\chi _{11}\big \}-2s^2-\big \{\chi _9\big (\big (\chi _5\chi _9\nonumber \\&\quad +\,\chi _3\big (\chi _{10}+8\pi \big )\big )\chi _{15}-\chi _4 \big (-8\pi \chi _{14}+\chi _{10}\big (\chi _{14}-8\pi \big )+\chi _9\chi _{16}+64\pi ^2\big )\big )\big \}^{-1}\nonumber \\&\quad \times \,\big \{\big (8\pi -\chi _{10}\big )\big (\chi _{15}\big (\chi _9\big (\chi _1\big (s^2\Phi -2C^4\big )+2C^4s^2 \chi _6\big )+\chi _3\big (\chi _7\big (s^2\Phi -2C^4\big )\nonumber \\&\quad -\,2C^4s^2\chi _{11}\big )\big )+\chi _4 \big (\chi _7\big (8\pi -\chi _{14}\big )\big (s^2\Phi -2C^4\big )+\chi _9\big (\chi _{12} \big (2C^4-s^2\Phi \big )+2C^4\nonumber \\&\quad \times \, s^2\chi _{17}\big )+2C^4s^2\chi _{11}\big (\chi _{14}-8\pi \big )\big )\big )\big \}-\big \{\chi _4\chi _9\big (\big (-\chi _5\chi _9 -\chi _3\big (\chi _{10}-8\pi \big )\big )\chi _{15}\nonumber \\&\quad +\,\chi _4\big (-8\pi \chi _{14}+\chi _{10}\big (\chi _{14}-8\pi \big )+\chi _9\chi _{16}+64\pi ^2\big )\big )\big \}^{-1} \big \{\big (\chi _5\chi _9+\chi _3 \big (\chi _{10}-8 \pi \big )\big )\nonumber \\&\quad \times \,\big (\chi _{15} \big (\chi _9 \big (\chi _1 \big (s^2 \Phi -2 C^4\big )+2 C^4 s^2\chi _6\big )+\chi _3 \big (\chi _7 \big (s^2 \Phi -2 C^4\big )-2 C^4 s^2 \chi _{11}\big )\big )\nonumber \\&\quad +\,\chi _4 \big (\chi _7 \big (8 \pi -\chi _{14}\big ) \big (s^2 \Phi -2 C^4\big )+\chi _9 \big (\chi _{12} \big (2 C^4-s^2 \Phi \big )+2 C^4 s^2 \chi _{17}\big )+2 C^4 s^2\nonumber \\&\quad \times \, \chi _{11} \big (\chi _{14}-8 \pi \big )\big )\big )\big \} -\big \{\big (\big (-\chi _5 \chi _9-\chi _3 \big (\chi _{10}-8 \pi \big )\big ) \chi _{15}+\chi _4 \big (-8 \pi \chi _{14}+\chi _{10}\nonumber \\&\quad \times \, \big (\chi _{14}-8 \pi \big )+\chi _9 \chi _{16}+64 \pi ^2\big )\big ) \big (\big (\big (-\chi _5 \chi _9-\chi _3 \big (\chi _{10}-8 \pi \big )\big ) \chi _{15}-\chi _4 \big (8 \pi \chi _{14}\nonumber \\&\quad -\,\chi _{10} \big (\chi _{14}-8 \pi \big )-\chi _9 \chi _{16}-64 \pi ^2\big )\big ) \big (-\chi _4 \chi _8 \chi _{20}+\chi _3 \chi _8 \big (\chi _{21}-8 \pi \big )+\chi _9 \big (-\chi _4\nonumber \\&\quad \times \, \chi _{19}-8 \pi \chi _{21}+\chi _2 \big (\chi _{21}-8 \pi \big )+64 \pi ^2\big )\big )+\big (\big (\chi _3 \chi _8+\big (\chi _2-8 \pi \big ) \chi _9\big ) \chi _{15}-\chi _4 \nonumber \\&\quad \times \,\big (\chi _9 \chi _{13}+\chi _8 \big (\chi _{14}-8 \pi \big )\big )\big ) \big (\chi _3 \big (8 \pi -\chi _{10}\big ) \big (8 \pi -\chi _{21}\big )+\chi _5 \chi _9 \big (\chi _{21}-8 \pi \big )+\chi _4 \nonumber \\&\quad \times \,\big (\big (8 \pi -\chi _{10}\big ) \chi _{20}-\chi _9 \chi _{22}\big )\big )\big )\big \}^{-1}\big \{\big (\big (\chi _5 \chi _8+\chi _2 \big (8 \pi -\chi _{10}\big )+8 \pi \big (\chi _{10}-8 \pi \big )\big ) \chi _{15}\nonumber \\&\quad +\,\chi _4 \big (\big (\chi _{10}-8 \pi \big ) \chi _{13}-\chi _8 \chi _{16}\big )\big ) \big (\big (\chi _3 \big (8 \pi -\chi _{10}\big ) \big (8 \pi -\chi _{21}\big )+\chi _5 \chi _9 \big (\chi _{21}-8 \pi \big )\nonumber \\&\quad +\,\chi _4 \big (\big (8 \pi -\chi _{10}\big ) \chi _{20}-\chi _9 \chi _{22}\big )\big ) \big (\chi _{15} \big (\chi _9 \big (\chi _1 \big (s^2 \Phi -2 C^4\big )+2 C^4 s^2 \chi _6\big )+\chi _3 \big (\chi _7 \nonumber \\&\quad \times \,\big (s^2 \Phi -2 C^4\big )-2 C^4 s^2 \chi _{11}\big )\big )-\chi _4 \big (\chi _7 \big (8 \pi -\chi _{14}\big ) \big (2 C^4-s^2 \Phi \big )+\chi _9 \big (\chi _{12} \big (s^2 \Phi \nonumber \\&\quad -\,2 C^4\big )-2 C^4 s^2 \chi _{17}\big )+2 C^4 s^2 \chi _{11} \big (8 \pi -\chi _{14}\big )\big )\big )+\big (\big (-\chi _5 \chi _9-\chi _3 \big (\chi _{10}-8 \pi \big )\big )\nonumber \\&\quad \times \, \chi _{15}+\chi _4 \big (-8 \pi \chi _{14}+\chi _{10} \big (\chi _{14}-8 \pi \big )+\chi _9 \chi _{16}+64 \pi ^2\big )\big ) \big (\chi _1 \chi _9 \big (8 \pi -\chi _{21}\big )\nonumber \\&\quad \times \, \big (2 C^4-s^2 \Phi \big )+\chi _3 \big (8 \pi -\chi _{21}\big ) \big (\chi _7 \big (2 C^4-s^2 \Phi \big )+2 C^4 s^2 \chi _{11}\big )-16 \pi C^4 s^2 \chi _6 \chi _9\nonumber \\&\quad +2 C^4 s^2 \chi _4 \chi _{11} \chi _{20}+2 C^4 s^2 \chi _6 \chi _9 \chi _{21}-2 C^4 s^2 \chi _4 \chi _9 \chi _{23}+2 C^4 \chi _4 \chi _9 \chi _{18}+2 C^4 \chi _4 \nonumber \\&\quad \times \,\chi _7 \chi _{20}-s^2 \Phi \chi _4 \chi _9 \chi _{18}-s^2 \Phi \chi _4 \chi _7 \chi _{20}\big )\big )\big \}+\big \{\big (\big (-\chi _5 \chi _9-\chi _3 \big (\chi _{10}-8 \pi \big )\big ) \chi _{15}\nonumber \\&\quad +\,\chi _4 \big (-8 \pi \chi _{14}+\chi _{10} \big (\chi _{14}-8 \pi \big )+\chi _9 \chi _{16}+64 \pi ^2\big )\big ) \big (\big (\big (-\chi _5 \chi _9- \big (\chi _{10}-8 \pi \big )\nonumber \\&\quad \times \,\chi _3\big ) \chi _{15}+\chi _4 \big (-8 \pi \chi _{14}+\chi _{10} \big (\chi _{14}-8 \pi \big )+\chi _9 \chi _{16}+64 \pi ^2\big )\big ) \big (-\chi _4 \chi _8 \chi _{20}\nonumber \\&\quad +\,\chi _3 \chi _8 \big (\chi _{21}-8 \pi \big )+\chi _9 \big (-\chi _4 \chi _{19}-8 \pi \chi _{21}+\chi _2 \big (\chi _{21}-8 \pi \big )+64 \pi ^2\big )\big )+\big (\big (\chi _3 \nonumber \\&\quad \times \,\chi _8+\big (\chi _2-8 \pi \big ) \chi _9\big ) \chi _{15}-\chi _4 \big (\chi _9 \chi _{13}+\chi _8 \big (\chi _{14}-8 \pi \big )\big )\big ) \big (\chi _3 \big (8 \pi -\chi _{10}\big ) \big (8 \pi \nonumber \\&\quad -\,\chi _{21}\big )+\chi _5 \chi _9 \big (\chi _{21}-8 \pi \big )+\chi _4 \big (\big (8 \pi -\chi _{10}\big ) \chi _{20}-\chi _9 \chi _{22}\big )\big )\big )\big \}^{-1}\big \{\big (-\chi _3 \chi _{13} \chi _{10}\nonumber \\&\quad -\,8 \pi \chi _{14} \chi _{10}+64 \pi ^2 \chi _{10}+8 \pi \chi _3 \chi _{13}+64 \pi ^2 \chi _{14}-\chi _5 \big (\chi _9 \chi _{13}+\chi _8 \big (\chi _{14}-8 \pi \big )\big )\nonumber \\&\quad +\chi _3 \chi _8 \chi _{16}-8 \pi \chi _9 \chi _{16}-512 \pi ^3+ \big (-8 \pi \chi _{14}+\chi _{10} \big (\chi _{14}-8 \pi \big )+\chi _9 \chi _{16}+64 \pi ^2\big )\nonumber \\&\quad \times \,\chi _2\big ) \big (\big (\chi _3 \big (8 \pi -\chi _{10}\big ) \big (8 \pi -\chi _{21}\big )+\chi _5 \chi _9 \big (\chi _{21}-8 \pi \big )+\chi _4 \big (\big (8 \pi -\chi _{10}\big ) \chi _{20}-\chi _9 \nonumber \\&\quad \times \,\chi _{22}\big )\big ) \big (\chi _{15} \big (\chi _9 \big (\chi _1 \big (s^2 \Phi -2 C^4\big )+2 C^4 s^2 \chi _6\big )+ \big (\chi _7 \big (s^2 \Phi -2 C^4\big )-2 C^4 s^2 \chi _{11}\big )\nonumber \\&\quad \times \,\chi _3\big )-\chi _4 \big (\chi _7 \big (8 \pi -\chi _{14}\big ) \big (2 C^4-s^2 \Phi \big )+\chi _9 \big (\chi _{12} \big (s^2 \Phi -2 C^4\big )-2 C^4 s^2 \chi _{17}\big )\nonumber \\&\quad +\,2 C^4 s^2 \chi _{11} \big (8 \pi -\chi _{14}\big )\big )\big )+\big (\big (-\chi _5 \chi _9-\chi _3 \big (\chi _{10}-8 \pi \big )\big ) \chi _{15}+\chi _4 \big (-8 \pi \chi _{14}\nonumber \\&\quad +\,\chi _{10} \big (\chi _{14}-8 \pi \big )+\chi _9 \chi _{16}+64 \pi ^2\big )\big ) \big (\chi _1 \chi _9 \big (8 \pi -\chi _{21}\big ) \big (2 C^4-s^2 \Phi \big )+\chi _3 \big (8 \pi \nonumber \\&\quad -\,\chi _{21}\big ) \big (\chi _7 \big (2 C^4-s^2 \Phi \big )+2 C^4 s^2 \chi _{11}\big )-16 \pi C^4 s^2 \chi _6 \chi _9+2 C^4 s^2 \chi _4 \chi _{11} \chi _{20}+2 C^4\nonumber \\&\quad \times s^2 \chi _6 \chi _9 \chi _{21}-2 C^4 s^2 \chi _4 \chi _9 \chi _{23}+2 C^4 \chi _4 \chi _9 \chi _{18}+2 C^4 \chi _4 \chi _7 \chi _{20}-s^2 \Phi \chi _4 \chi _9 \chi _{18}\nonumber \\&\quad -\,s^2 \Phi \chi _4 \chi _7 \chi _{20}\big )\big )\big \}\big ]=0, \end{aligned}$$

(B4)

$$\begin{aligned}&\big (\chi _4\big (8\pi -\chi _{14}\big )+\chi _3\chi _{15}\big )\big (-\chi _7\big (2C^4-s^2\Phi \big )-2 C^4s^2\chi _{11}\big )+\chi _9\big (\chi _{15}\big (\chi _1\big (s^2\Phi \nonumber \\&\quad -\,2C^4\big )+2C^4s^2\chi _6\big )+\chi _4 \big (\chi _{12}\big (2C^4-s^2\Phi \big )+2C^4s^2\chi _{17}\big )\big )+\big \{\big (\big (-\chi _5\chi _9-\chi _3\nonumber \\&\quad \times \,\big (\chi _{10}-8\pi \big )\big )\chi _{15}+\chi _4\big (-8\pi \chi _{14}+\chi _{10}\big (\chi _{14}-8\pi \big )+\chi _9\chi _{16}+64\pi ^2\big )\big )\big (-\chi _4\nonumber \\&\quad \times \,\chi _8\chi _{20}+\chi _3\chi _8 \big (\chi _{21}-8\pi \big )+\chi _9\big (-\chi _4\chi _{19}-8\pi \chi _{21}+\chi _2 \big (\chi _{21}-8\pi \big )+64\pi ^2\big )\big )\nonumber \\&\quad \,+\big (\big (\chi _3\chi _8+\big (\chi _2-8\pi \big ) \chi _9\big )\chi _{15}-\chi _4\big (\chi _9\chi _{13}+\chi _8\big (\chi _{14}-8\pi \big )\big )\big )\big (\chi _3 \big (8\pi -\chi _{10}\big )\nonumber \\&\quad \times \,\big (8\pi -\chi _{21}\big )+\chi _5\chi _9\big (\chi _{21}-8\pi \big )+\chi _4\big (\big (8\pi -\chi _{10}\big )\chi _{20}-\chi _9\chi _{22}\big )\big )\big \}^{-1} \big \{\big (\chi _4 \big (\chi _9\nonumber \\&\quad \times \,\chi _{13}+\chi _8 \big (\chi _{14}-8 \pi \big )\big )-\big (\chi _3 \chi _8+\big (\chi _2-8 \pi \big ) \chi _9\big ) \chi _{15}\big ) \big (\big ( \big (8 \pi -\chi _{10}\big ) \big (8 \pi -\chi _{21}\big )\nonumber \\&\quad \times \,\chi _3+\chi _5 \chi _9 \big (\chi _{21}-8 \pi \big )+\chi _4 \big (\big (8 \pi -\chi _{10}\big ) \chi _{20}-\chi _9 \chi _{22}\big )\big ) \big (\chi _{15} \big (-\big (\chi _9 \big (\chi _1 \big (2 C^4\nonumber \\&\quad -\,s^2 \Phi \big )-2 C^4 s^2 \chi _6\big )+\chi _3 \big (\chi _7 \big (2 C^4-s^2 \Phi \big )+2 C^4 s^2 \chi _{11}\big )\big )\big )-\chi _4 \big (\chi _7 \big (8 \pi -\chi _{14}\big )\nonumber \\&\quad \times \,\big (2 C^4-s^2 \Phi \big )+\chi _9 \big (\chi _{12} \big (s^2 \Phi -2 C^4\big )-2 C^4 s^2 \chi _{17}\big )+2 C^4 s^2 \chi _{11} \big (8 \pi -\chi _{14}\big )\big )\big )\nonumber \\&\quad +\,\big (\big (-\chi _5 \chi _9-\chi _3 \big (\chi _{10}-8 \pi \big )\big ) \chi _{15}+\chi _4 \big (-8 \pi \chi _{14}+\chi _{10} \big (\chi _{14}-8 \pi \big )+\chi _9 \chi _{16}\nonumber \\&\quad +\,64 \pi ^2\big )\big ) \big (\chi _1 \chi _9 \big (8 \pi -\chi _{21}\big ) \big (2 C^4-s^2 \Phi \big )+\chi _3 \big (8 \pi -\chi _{21}\big ) \big (\chi _7 \big (2 C^4-s^2 \Phi \big )+2\nonumber \\&\quad \times \, C^4 s^2 \chi _{11}\big )-16 \pi C^4 s^2 \chi _6 \chi _9+2 C^4 s^2 \chi _4 \chi _{11} \chi _{20}+2 C^4 s^2 \chi _6 \chi _9 \chi _{21}-2 C^4 s^2 \chi _4 \chi _9\nonumber \\&\quad \times \, \chi _{23}+2 C^4 \chi _4 \chi _9 \chi _{18}+2 C^4 \chi _4 \chi _7 \chi _{20}-s^2 \Phi \chi _4 \chi _9 \chi _{18}-s^2 \Phi \chi _4 \chi _7 \chi _{20}\big )\big )\big \}=0. \end{aligned}$$

(B5)

The homologous condition in the presence of heat dissipation is

$$\begin{aligned} \bar{\varsigma }&=\frac{8\pi }{8\pi -\chi _{10}}\bigg [\frac{\chi _7}{8\pi }\bigg (1-\frac{s^2\Phi }{2C^4}\bigg )+\frac{s^2\chi _{11}}{8\pi } +\big \{16 \pi C^4 \big (\big (\big (-\chi _5 \chi _9-\chi _3 \big (\chi _{10}-8 \pi \big )\big )\nonumber \\&\quad \times \,\chi _{15}+\chi _4 \big (-8 \pi \chi _{14}+\chi _{10} \big (\chi _{14}-8 \pi \big )+\chi _9 \chi _{16}+64 \pi ^2\big )\big ) \big (-\chi _4 \chi _8 \chi _{20}+\chi _3 \chi _8\nonumber \\&\quad \times \, \big (\chi _{21}-8 \pi \big )+\chi _9 \big (-\chi _4 \chi _{19}-8 \pi \chi _{21}+\chi _2 \big (\chi _{21}-8 \pi \big )+64 \pi ^2\big )\big )+\big (\big (\chi _3 \chi _8\nonumber \\&\quad +\,\big (\chi _2-8 \pi \big ) \chi _9\big ) \chi _{15}-\chi _4 \big (\chi _9 \chi _{13}+\chi _8 \big (\chi _{14}-8 \pi \big )\big )\big ) \big (\chi _3 \big (8 \pi -\chi _{10}\big ) \big (8 \pi -\chi _{21}\big )\nonumber \\&\quad +\,\chi _5 \chi _9 \big (\chi _{21}-8 \pi \big )+\chi _4 \big (\big (8 \pi -\chi _{10}\big ) \chi _{20}-\chi _9 \chi _{22}\big )\big )\big )\big \}^{-1}\big \{\chi _8 \big (\big (\chi _3 \big (8 \pi -\chi _{10}\big ) \nonumber \\&\quad \times \,\big (8 \pi -\chi _{21}\big )+\chi _5 \chi _9 \big (\chi _{21}-8 \pi \big )+\chi _4 \big (\big (8 \pi -\chi _{10}\big ) \chi _{20}-\chi _9 \chi _{22}\big )\big ) \big (\chi _{15} \big (\chi _9 \big (2 s^2\nonumber \\&\quad \times \, C^4 \chi _6-\chi _1 \big (2 C^4-s^2 \Phi \big )\big )-\chi _3 \big (\chi _7 \big (2 C^4-s^2 \Phi \big )+2 C^4 s^2 \chi _{11}\big )\big )-\chi _4 \big (\chi _7 \big (8 \pi \nonumber \\&\quad -\,\chi _{14}\big ) \big (2 C^4-s^2 \Phi \big )+\chi _9 \big (\chi _{12} \big (s^2 \Phi -2 C^4\big )-2 C^4 s^2 \chi _{17}\big )+2 C^4 \big (8 \pi -\chi _{14}\big )\nonumber \\&\quad \times \, s^2 \chi _{11}\big )\big )+\big (\big (-\chi _5 \chi _9-\chi _3 \big (\chi _{10}-8 \pi \big )\big ) \chi _{15}+\chi _4 \big (-8 \pi \chi _{14}+\chi _{10} \big (\chi _{14}-8 \pi \big )\nonumber \\&\quad +\,\chi _9 \chi _{16}+64 \pi ^2\big )\big ) \big (\chi _1 \chi _9 \big (8 \pi -\chi _{21}\big ) \big (2 C^4-s^2 \Phi \big )+\chi _3 \big ( \big (2 C^4-s^2 \Phi \big )\chi _7+2 C^4\nonumber \\&\quad \times \, s^2 \chi _{11}\big )\big (8 \pi -\chi _{21}\big )-16 \pi C^4 s^2 \chi _6 \chi _9+2 C^4 s^2 \chi _4 \chi _{11} \chi _{20}+2 C^4 s^2 \chi _6 \chi _9 \chi _{21}-2 s^2\nonumber \\&\quad \times \, C^4 \chi _4 \chi _9 \chi _{23}+2 C^4 \chi _4 \chi _9 \chi _{18}+2 C^4 \chi _4 \chi _7 \chi _{20}-s^2 \Phi \chi _4 \chi _9 \chi _{18}-s^2 \Phi \chi _4 \chi _7 \chi _{20}\big )\big )\big \}\bigg ]\nonumber \\&\quad -\,\frac{\sigma C'}{4\pi BC}. \end{aligned}$$

(B6)

The values of $\chi _i$, $i=1,2,3,...,23$ used in Eqs. (B4)–(B6) are

$$\begin{aligned} \chi _1&=-\frac{1}{B^2}\bigg (\frac{2C''}{C}-\frac{2B'C'}{BC}-\frac{B^2}{C^2}+\frac{C^{\prime 2}}{C^2}\bigg ) +\frac{\dot{C}}{C}\bigg (\frac{2\dot{B}}{B}+\frac{\dot{C}}{C}\bigg ), \end{aligned}$$

(B7)

$$\begin{aligned} \chi _2&=\Phi \bigg (\frac{2\dot{B}\dot{C}}{BC}-\frac{3\ddot{B}}{2B^2}+\frac{\dot{C}^2}{C^2}-\frac{3\ddot{C}}{C} +\frac{\mathcal {R}}{2}\bigg ), \end{aligned}$$

(B8)

$$\begin{aligned} \chi _3&=\Phi \bigg (\frac{2B'\dot{C}}{B^3C}-\frac{2B'C'}{B^3C}+\frac{\dot{B}\dot{C}}{BC}+\frac{\ddot{B}}{2B}+\frac{C^{\prime 2}}{B^2 C^2}+\frac{C''}{B^2C}\bigg ), \end{aligned}$$

(B9)

$$\begin{aligned} \chi _4&=\Phi \bigg (\frac{B'C'}{B^3C}+\frac{\dot{B}\dot{C}}{BC}-\frac{C^{\prime 2}}{B^2C^2}-\frac{C''}{B^2C}+\frac{\dot{C}^2}{C} +\frac{\ddot{C}}{C}\bigg ), \end{aligned}$$

(B10)

$$\begin{aligned} \chi _5&=\Phi \bigg (\frac{5\dot{B}C'}{BC^2}+\frac{2C'\dot{C}}{BC^2}-\frac{\dot{C}'}{BC}\bigg ), \end{aligned}$$

(B11)

$$\begin{aligned} \chi _6&=\frac{1}{C^4}+\frac{\Phi }{B^3C^5}\bigg (2\dot{B}B^2\dot{C}+2B'C'+\ddot{B}B^2C-2BC''-\frac{\mathcal {R}B^3C}{2}\bigg ), \end{aligned}$$

(B12)

$$\begin{aligned} \chi _7&=-\frac{1}{B}\bigg (\frac{2\dot{B}C'}{BC}-\frac{2\dot{C}'}{C}\bigg ), \quad \chi _8=\chi _9=\Phi \bigg (\frac{2\dot{C}'}{BC}-\frac{2\dot{B}C'}{B^2C}\bigg ), \end{aligned}$$

(B13)

$$\begin{aligned} \chi _{10}&=\Phi \bigg (\frac{2C''}{B^2C}-\frac{2B'C'}{B^3C}-\frac{\ddot{B}}{B}-\frac{2\ddot{C}}{C}+\frac{\mathcal {R}}{2}\bigg ), \end{aligned}$$

(B14)

$$\begin{aligned} \chi _{11}&=\frac{\Phi }{B^2}\left( \dot{B}C'-B\dot{C}'\right) , \quad \chi _{12}=\frac{C^{\prime 2}}{B^2C^2}-\frac{2\ddot{C}}{C}-\frac{\dot{C}^2}{C^2}-\frac{1}{C^2}, \end{aligned}$$

(B15)

$$\begin{aligned} \chi _{13}&=\Phi \bigg (\frac{\ddot{B}}{2B}-\frac{\ddot{C}}{C}-\frac{\dot{C}^2}{C^2}\bigg ), \end{aligned}$$

(B16)

$$\begin{aligned} \chi _{14}&=\Phi \bigg (\frac{3C''}{B^2C}-\frac{2B'C'}{B^3C}-\frac{2B'\dot{C}}{B^3C}-\frac{3\dot{B}\dot{C}}{BC}-\frac{3 \ddot{B}}{2B}-\frac{C^{\prime 2}}{B^2C^2}+\frac{\mathcal {R}}{2}\bigg ), \end{aligned}$$

(B17)

$$\begin{aligned} \chi _{15}&=\Phi \bigg (\frac{C^{\prime 2}}{B^2C^2}-\frac{B'C'}{B^3C}-\frac{\dot{B}\dot{C}}{BC}+\frac{C''}{B^2C}-\frac{\dot{C}^2}{C^2} -\frac{\ddot{C}}{C}\bigg ), \end{aligned}$$

(B18)

$$\begin{aligned} \chi _{16}&=\Phi \bigg (\frac{2\dot{C}'}{BC}-\frac{4\dot{B}C'}{B^2C}-\frac{2C'\dot{C}}{BC^2}\bigg ), \end{aligned}$$

(B19)

$$\begin{aligned} \chi _{17}&=\frac{1}{C^4}-\frac{\Phi }{B^3C^5}\left( \ddot{B}B^2C+2B^3\ddot{C}-\frac{\mathcal {R}B^3C}{2}\right) , \end{aligned}$$

(B20)

$$\begin{aligned} \chi _{18}&=\frac{1}{B^2}\bigg (\frac{C''}{C}-\frac{B'C'}{BC}\bigg )-\frac{\dot{B}\dot{C}}{BC}-\frac{\ddot{B}}{B}-\frac{\ddot{C}}{C}, \end{aligned}$$

(B21)

$$\begin{aligned} \chi _{19}&=-\Phi \bigg (\frac{\dot{B}\dot{C}}{BC}+\frac{\ddot{B}}{2B}\bigg ), \quad \chi _{20}=\Phi \bigg (\frac{B'C'}{B^3C}-\frac{2B'\dot{C}}{B^3C}-\frac{\dot{B}\dot{C}}{BC}-\frac{\ddot{B}}{2B}\bigg ), \end{aligned}$$

(B22)

$$\begin{aligned} \chi _{21}&=\Phi \bigg (\frac{2C^{\prime 2}}{B^2C^2}-\frac{2B'C'}{B^3C}-\frac{2\dot{B}\dot{C}}{BC}+\frac{2C''}{B^2C}-\frac{2\dot{C}^2}{C^2} -\frac{2\ddot{C}}{C}-\frac{2}{C^2}+\frac{\mathcal {R}}{2}\bigg ), \end{aligned}$$

(B23)

$$\begin{aligned} \chi _{22}&=\Phi \bigg (\frac{B'\dot{B}}{B^3}-\frac{2\dot{B}C'}{B^2C}-\frac{\dot{B}'}{B^2}\bigg ), \quad \chi _{23}=\frac{1}{C^4}\bigg (1+\frac{\Phi \mathcal {R}}{2}\bigg ). \end{aligned}$$

(B24)