Complexity factor for static cylindrical objects in f(G, T) gravity

Abstract

The purpose of this paper is to investigate the definition of complexity for static anisotropic cylindrical objects in the background of f(G, T) gravity, where G and T stand for Gauss–Bonnet term and trace of the energy–momentum tensor, respectively. We develop the modified field equations, Tolman–Oppenheimer–Volkoff equation, mass distribution and structure scalars. The complexity is calculated from the splitting of the Riemann tensor in terms of a complexity factor which is associated with the physical characteristics (anisotropic pressure, inhomogeneous energy density) of the system. The zero complexity condition is derived as a constraint to estimate the behaviour of two compact objects corresponding to Gokhroo and Mehra ansatz and polytropic equation of state. We conclude that the addition of f(G, T) terms contribute to the increment in the complexity of the system.

Appendix A

The terms Z and $\zeta$ are given as

$$\begin{aligned} Z&{=}&\frac{f_{T}}{k^{2}-f_{T}}\left[ (-p_{r}B^2-pB^2)\frac{f'_{T}}{f_{T}} {+}({-}2p_{r}B^2-pB^2)'\right. \\&{-}\left. \frac{B^2}{2}(\rho +3p) \right] ,\\ \zeta= & {} -\frac{1}{8\pi }\left[ \frac{2}{3}\Pi f_{T}+\frac{f}{2} +\frac{12A'B'f_{G}}{r^2AB^5}+\frac{12A'f_{G}'}{r^2AB^4}\right. \\&-\left. \frac{4A''f_{G}}{r^2AB^4} \right] . \end{aligned}$$

The contribution of the modified terms in the structure scalars are described by

$$\begin{aligned} N^{GT}_{\nu \mu }= & {} \left[ 2R_{m\delta }R^{m\gamma }f_{G}+2R^{lm}R^{\gamma }_{l\delta m}f_{G}-RR^{\gamma }_{\delta }f_{G} \right. \\&\left. -2R_{\delta lmn}R^{lmn\gamma }f_{G}\right. \\&+\left. 2R^{\gamma }_{\delta }\Box f_{G}+R\nabla ^{\gamma }\nabla _{\delta }f_{G} -2R^{m\gamma }\nabla _{\delta }\nabla _{m}f_{G}\right. \\&\left. -2R^{m}_{\delta }\nabla ^{\gamma }\nabla _{m}f_{G}\right. \\&-\left. 2R^{\gamma }_{l\delta m}\nabla ^{m}\nabla ^{l}f_{G}\right] \epsilon _{\nu \mu \gamma }u^{\delta }, \\ D^{GT}_{\nu \mu }= & {} 2\left[ R_{m\mu }R^{m}_{\nu }f_{G}+ R^{lm}R_{l\mu m\nu }f_{G}\right. \\&-\left. \frac{1}{2}RR_{\nu \mu }f_{G}-\frac{1}{2}R_{\mu lmn}R^{lmn}_{\nu }f_{G}\right. \\&+\left. \frac{1}{2}R \nabla _{\nu }\nabla _{\mu }f_{G}+R_{\nu \mu }\Box f_{G}-R^{m}_{\nu }\nabla _{\mu }\nabla _{m}f_{G}\right. \\&-\left. R^{m}_{\mu } \nabla _{\nu }\nabla _{m}f_{G} -R_{l\mu m\nu }\nabla ^{m}\nabla ^{l}f_{G}\right] \\&+4R^{lm}h_{\nu \mu }\nabla _{m}\nabla _{l}f_{G}-2Rh_{\nu \mu }\Box f_{G}\\&+2\left[ -R_{m\delta }R^{m}_{\nu }f_{G}- R^{lm}R_{l\delta m\nu }f_{G}\right. \\&+\left. \frac{1}{2}R_{\delta lmn}R^{lmn}_{\nu }f_{G}+\frac{1}{2}RR_{\nu \mu }f_{G} -R_{\delta \nu }\Box f_{G}\right. \\&+\left. R_{l\delta m\nu }\nabla ^{m}\nabla ^{m}f_{G}-\frac{1}{2}R \nabla _{\nu }\nabla _{\delta }f_{G}\right. \\&\left. +R^{m}_{\nu }\nabla _{\delta }\nabla _{m}f_{G} +R^{m}_{\delta }\nabla _{\nu }\nabla _{m}f_{G}\right] u_{\mu }u^{\delta }\\&+2\left[ -R_{m\mu }R^{m\gamma }f_{G} +\frac{1}{2}R_{\mu lmn}R^{lmn\gamma }f_{G}\right. \\&\left. - R^{lm}R^{\gamma }_{l\mu m}f_{G}+\frac{1}{2}RR^{\gamma }_{\mu }f_{G}-R^{\gamma }_{\mu }\Box f_{G}\right. \\&\left. {+}R^{m\gamma }\nabla _{\mu }\nabla _{m}f_{G}{-}\frac{1}{2}R \nabla ^{\gamma }\nabla _{\mu }f_{G} {+}R^{m}_{\mu }\nabla ^{\gamma }\nabla _{m}f_{G}\right. \\&\left. {+}R^{\gamma }_{m\mu l}\nabla ^{m}\nabla ^{l}\right] u_{\nu }u_{\gamma }{+} 2\Bigg [R_{m\delta }R^{m\gamma }f_{G}\\&-\left. \frac{1}{2}R_{\delta lmn}R^{lmn \gamma }f_{G}+R^{lm}R^{\gamma }_{l\delta m}f_{G}-\frac{1}{2}RR^{\gamma }_{\delta }f_{G}\right. \\&+\left. R^{\gamma }_{\delta }\Box f_{G}+\frac{1}{2}R \nabla ^{\gamma }\nabla _{\delta }f_{G}-R^{\gamma }_{l\delta m}\nabla ^{m}\nabla ^{l}f_{G}\right. \\&-R^{m\gamma }\nabla _{\delta }\nabla _{m}f_{G}-R^{m}_{\delta }\nabla ^{\gamma }\nabla _{m} f_{G}\Bigg ] g_{\nu \mu }u_{\gamma }u^{\delta }\\&-\frac{1}{3}\left[ -2R^{2}f_{G} -2R^{\beta }_{lmn}R^{lmn}_{\beta }f_{G} -2R\Box f_{G}\right. \\&{+}\left. 4R^{m\alpha }R_{m\alpha }f_{G} {+}4R^{lm}R^{\alpha }_{m\alpha l}f_{G}{+}16R^{lm}\nabla _{m}\nabla _{l}f_{G}\right. \\&-\left. 4R^{m\beta }\nabla _{\beta }\nabla _{m}f_{G} -4R^{m\alpha }\nabla _{\alpha }\nabla _{m}f_{G}\right. \\&\left. -4R^{\alpha }_{l\alpha m}\nabla ^{m}\nabla ^{l}f_{G}\right] h_{\nu \mu } -\frac{1}{6}fh_{\nu \mu },\\ F^{GT}= & {} 2\left[ R_{m\mu }R^{m}_{\nu }f_{G}+ R^{lm}R_{m\mu l\nu }f_{G}-\frac{1}{2}RR_{\nu \mu }f_{G}\right. \\&-\left. \frac{1}{2}R_{\mu \mu lmn}R^{lmn}_{\nu }f_{G}+R_{\nu \mu }\Box f_{G}\right. \\&+\left. \frac{1}{2}R \nabla _{\nu }\nabla _{\mu }f_{G} -R^{m}_{\nu }\nabla _{\mu }\nabla _{m}f_{G}\right. \\&-\left. R^{m}_{\mu }\nabla _{\nu } \nabla _{m}f_{G}-R_{m\mu l\nu }\nabla ^{m}\nabla ^{l}f_{G}\right] g^{\nu \mu }\\&+12R^{lm}\nabla _{m}\nabla _{l}f_{G}-6R\Box f_{G}\\&+2\left[ -R_{m\delta }R^{m}_{\nu }f_{G}- R^{lm}R_{m\delta l\nu }f_{G}\right. \\&\left. +\frac{1}{2}RR_{\nu \delta }f_{G}\frac{1}{2}R \nabla _{\nu }\nabla _{\delta }f_{G} -R_{\delta \nu }\Box f_{G}\right. \\&+\left. R_{m\delta l\nu }\nabla ^{m}\nabla ^{l}f_{G} +R^{m}_{\nu }\nabla _{\delta }\nabla _{m}f_{G}\right. \\&\left. +R^{m}_{\delta }\nabla _{\nu }\nabla _{m}f_{G}+\frac{1}{2}R_{\delta lmn}R^{lmn}_{\nu }f_{G}\right] u_{\mu }u^{\delta }g^{\nu \mu }\\&+2\left[ -R_{m\mu }R^{m\gamma }f_{G}- R^{lm}R^{\gamma }_{m\mu l}f_{G}{+}\frac{1}{2}RR^{\gamma }_{\mu }f_{G}\right. \\&{+}\left. \frac{1}{2}R_{\mu lmn}R^{lmn\gamma }f_{G}{-}R^{\gamma }_{\mu }\Box f_{G}+R^{m\gamma }\nabla _{\mu }\nabla _{m}f_{G}\right. \\&{+}\left. R^{m}_{\mu }\nabla ^{\gamma }\nabla _{m}f_{G}{+}R^{\gamma }_{m\mu l}\nabla ^{m}\nabla ^{l}f_{G}\right. \\&-\left. \frac{1}{2}R \nabla ^{\gamma }\nabla _{\mu }f_{G}\right] u_{\nu }u_{\gamma }g^{\nu \mu }+ 2\left[ R_{m\delta }R^{m\gamma }f_{G}\right. \\&+\left. R^{lm}R^{\gamma }_{m\delta l}f_{G}-R^{m\gamma }\nabla _{\delta }\nabla _{m}f_{G}\right. \\&-\left. \frac{1}{2}R_{\delta lmn}R^{lmn \gamma }f_{G}+R^{\gamma }_{\delta }\Box f_{G}+\frac{1}{2}R \nabla ^{\gamma }\nabla _{\delta }f_{G}\right. \\&\left. -\frac{1}{2}RR^{\gamma }_{\delta }f_{G}-R^{m}_{\delta }\nabla ^{\gamma }\nabla _{m} f_{G}\right. \\&-\left. R^{\gamma }_{m\delta l}\nabla ^{m}\nabla ^{l}f_{G}\right] g_{\nu \mu }u_{\gamma }u^{\delta }g^{\nu \mu }\\&-\left[ 4R^{l\alpha }R_{l\alpha }f_{G}+4R^{lm}R^{\alpha }_{l\alpha m}f_{G}-2R^{2}f_{G}\right. \\&-\left. 2R^{\beta }_{lmn}R^{lmn}_{\beta }f_{G}{-}2R\Box f_{G}{+}16R^{lm}\nabla _{m}\nabla _{l}f_{G}\right. \\&-\left. 4R^{m\alpha }\nabla _{\alpha }\nabla _{m}f_{G}\right. \\&-\left. 4R^{m\beta }\nabla _{\beta }\nabla _{m}f_{G} -4R^{\alpha }_{l\alpha m}\nabla ^{m}\nabla ^{l}f_{G}\right] - \frac{1}{2}f,\\ \end{aligned}$$

$$\begin{aligned} Q^{GT}_{(\nu \mu )}= & {} \left[ 2R_{md}R^{m}_{c}f_{G}+2R^{lm}R_{ldmc}f_{G}\right. \\&-\left. RR_{cd}f_{G} -R_{dlmn}R^{lmn}_{c}f_{G}+2R_{cd}\Box f_{G}\right. \\&+\left. R\nabla _{c}\nabla _{d}f_{G}-2R^{m}_{c} \nabla _{d}\nabla _{m}f_{G}\right. \\&-\left. 2R^{m}_{d}\nabla _{c}\nabla _{m}f_{G} -2R_{ldmc}\nabla ^{m}\nabla ^{l}f_{G}\right] h^{c}_{\nu }h^{d}_{\mu }\\&+2\left[ R_{m\delta }R^{m\gamma }f_{G}+ R^{lm}R^{\gamma }_{m\delta l}f_{G}\right. \\&-\left. \frac{1}{2}RR^{\gamma }_{\delta }f_{G} -\frac{1}{2}R_{\delta lmn}R^{lmn \gamma }f_{G}\right. \\&+\left. R^{\gamma }_{\delta }\Box f_{G}+\frac{1}{2}R \nabla ^{\gamma }\nabla _{\delta }f_{G}\right. \\&-\left. R^{m\gamma }\nabla _{\delta }\nabla _{m}f_{G} -R^{m}_{\delta }\nabla ^{\gamma }\nabla _{m}f_{G}\right. \\&-\left. R^{\gamma }_{m\delta l}\nabla ^{m}\nabla ^{l}f_{G}\right] h_{\nu \mu }u_{\gamma }u^{\delta }\\&-2\left[ R_{m\mu }R^{m}_{\nu }f_{G}+ R^{lm}R_{m\mu l\nu }f_{G} -\frac{1}{2}RR_{\nu \mu }f_{G}\right. \\&-\left. \frac{1}{2}R_{\mu lmn}R^{lmn}_{\nu }f_{G}+R_{\nu \mu }\Box f_{G}+\frac{1}{2}R \nabla _{\nu }\nabla _{\mu }f_{G}\right. \\&-\left. R^{m}_{\nu }\nabla _{\mu }\nabla _{m}f_{G} \right. \\&-\left. R^{m}_{\mu }\nabla _{\nu }\nabla _{m}f_{G}-R_{m\mu l\nu }\nabla ^{m} \nabla ^{l}f_{G}\right] \\&-2\left[ -R_{m\delta }R^{m}_{\nu }f_{G}- R^{lm}R_{m\delta l\nu }f_{G}\right. \\&+\left. \frac{1}{2}RR_{\nu \delta }f_{G}+\frac{1}{2}R_{\delta lmn}R^{lmn}_{\nu }f_{G}\right. \\&-\left. R_{\delta \nu }\Box f_{G}+R_{m\delta l\nu }\nabla ^{m}\nabla ^{l}f_{G}\right. \\&+\left. R^{m}_{\nu }\nabla _{\delta }\nabla _{m}f_{G} +R^{m}_{\delta }\nabla _{\nu }\nabla _{m}f_{G}\right. \\&-\left. \frac{1}{2}R \nabla _{\nu }\nabla _{\delta }f_{G}\right] u_{\mu }u^{\delta }\\&-2\left[ -R_{m\mu }R^{m\gamma }f_{G}-R^{lm}R^{\gamma }_{m\mu l}f_{G}\right. \\&\left. +\frac{1}{2}RR^{\gamma }_{\mu }f_{G}\frac{1}{2}R_{\mu lmn}R^{lmn\gamma }f_{G}-R^{\gamma }_{\mu }\Box f_{G}\right. \\&+\left. R^{m\gamma }\nabla _{\mu }\nabla _{m}f_{G}+R^{m}_{\mu } \nabla ^{\gamma }\nabla _{m}f_{G}\right. \\&\left. +R^{\gamma }_{m\mu l}\nabla ^{m}\nabla ^{l}f_{G}-\frac{1}{2}R \nabla ^{\gamma }\nabla _{\mu }f_{G}\right] u_{\nu }u_{\gamma }\\&-2\left[ R_{m\delta }R^{m\gamma }f_{G}+ R^{lm}R^{\gamma }_{m\delta l}f_{G}-\frac{1}{2}RR^{\gamma }_{\delta }f_{G}\right. \\&-\left. \frac{1}{2}R_{\delta lmn}R^{lmn \gamma }f_{G} +R^{\gamma }_{\delta }\Box f_{G}\right. \\&\left. +\frac{1}{2}R \nabla ^{\gamma }\nabla _{\delta }f_{G}-R^{m\gamma }\nabla _{\delta }\nabla _{m}f_{G}\right. \\&\left. -R^{m}_{\delta }\nabla ^{\gamma }\nabla _{m}f_{G}-R^{\gamma }_{m\delta l}\nabla ^{m}\nabla ^{l}f_{G}\right] u_{\gamma }u^{\delta }g_{\nu \mu }, \end{aligned}$$

$$\begin{aligned} M^{GT}_{\nu \mu }= & {} \left[ \frac{1}{2}R_{m\epsilon }R^{m p}f_{G}+\frac{1}{2}R^{lm}R^{p}_{m\epsilon l}f_{G}\right. \\&-\left. \frac{1}{4}RR^{p}_{\epsilon }f_{G}-\frac{1}{4}R_{\epsilon lmn}R^{lmn p}f_{G}+\frac{1}{2} R^{p}_{\epsilon }\Box f_{G}\right. \\&+\left. \frac{1}{4}R\nabla ^{p}\nabla _{\epsilon }f_{G} -\frac{1}{4}R^{lp}\nabla _{\epsilon }\nabla _{m}f_{G}\right. \\&-\left. \frac{1}{2}R^{m}_{\epsilon }\nabla ^{p}\nabla _{m} f_{G} -\frac{1}{2}R^{p}_{m\epsilon l}\nabla ^{m}\nabla ^{l}f_{G}\right] \epsilon _{p \delta \mu }\epsilon ^{\epsilon \delta }_{\nu }\\&+\left[ -\frac{1}{2}R_{m\delta }R^{lp}f_{G}-\frac{1}{2}R^{lm}R^{p}_{m\delta l}f_{G}\right. \\&+\left. \frac{1}{4}RR^{p}_{\delta }f_{G}+\frac{1}{4}R_{\delta lmn}R^{lmnp}f_{G}-\frac{1}{2}R^{p}_{\delta }\Box f_{G}\right. \\&-\left. \frac{1}{4}R\nabla ^{p}\nabla _{\delta }f_{G} +\frac{1}{4}R^{lp}\nabla _{\delta }\nabla _{m}f_{G}\right. \\&+\left. \frac{1}{2}R^{m}_{\delta }\nabla ^{p}\nabla _{m}f_{G} +\frac{1}{2}R^{p}_{m\delta l}\nabla ^{m}\nabla ^{l}f_{G}\right] \epsilon _{p\epsilon \mu }\epsilon ^{\epsilon \delta }_{\nu }\\&+\left[ -\frac{1}{2}R_{m\epsilon }R^{m\gamma }f_{G}-\frac{1}{2}R^{lm} R^{\gamma }_{m\epsilon l}f_{G} \right. \\&+\left. \frac{1}{4}RR^{\gamma }_{\epsilon }f_{G}+\frac{1}{4}R_{\epsilon lmn}R^{lmn\gamma }f_{G}-\frac{1}{2}R^{\gamma }_{\epsilon }\Box f_{G}\right. \\&\left. -\frac{1}{4}R\nabla ^{\gamma }\nabla _{\epsilon }f_{G} +\frac{1}{4}R^{m\gamma }\nabla _{\epsilon }\nabla _{m}f_{G}\right. \\&+\left. \frac{1}{2}R^{m}_{\epsilon }\nabla ^{\gamma }\nabla _{m} f_{G}+\frac{1}{2}R^{\gamma }_{m\epsilon l}\nabla ^{m} \nabla ^{l}f_{G}\right] \epsilon _{\delta \gamma \mu } \epsilon ^{\epsilon \delta }_{\nu }\\&+\left[ \frac{1}{2}R_{m\delta }R^{m\gamma }f_{G} +\frac{1}{2}R^{lm}R^{\gamma }_{m\delta l}f_{G}-\frac{1}{4}RR^{\gamma }_{\delta }f_{G}\right. \\&\left. -\frac{1}{4}R_{\delta lmn}R^{lmn\gamma }f_{G}+\frac{1}{2}R^{\gamma }_{\delta }\Box f_{G}+\frac{1}{4}R\nabla ^{\gamma }\nabla _{\delta }f_{G}\right. \\&-\left. \frac{1}{4}R^{m\gamma }\nabla _{\delta }\nabla _{m}f_{G} -\frac{1}{2}R^{m}_{\delta }\nabla ^{\gamma }\nabla _{m}f_{G}\right. \\&-\left. \frac{1}{2}R^{\gamma }_{m\delta l}\nabla ^{m}\nabla ^{l}f_{G}\right] \epsilon _{\epsilon \gamma \mu }\epsilon ^{\epsilon \delta }_{\nu }-4R^{lm}\nabla _{m}\nabla _{l}h_{\nu \mu }f_{G}\\&{+}2Rh_{\nu \mu }\Box f_{G} {+}\frac{1}{3}\bigg [-\left( \rho +p\right) f_{T}{+}4R^{m\alpha }R_{m\alpha }f_{G}\\&{+}\left. 4R^{lm}R^{\alpha }_{l\alpha m}f_{G}{-}2R^{2}f_{G} {-}2R^{\beta }_{lmn}R^{lmn}_{\beta }f_{G}\right. \\&-\left. 2R\Box f_{G} +16R^{lm}\nabla _{m}\nabla _{l}f_{G} -4R^{m\alpha }\nabla _{\alpha }\nabla _{m}f_{G}\right. \\&-4R^{m\beta }\nabla _{\beta }\nabla _{m}f_{G}-4R^{\alpha }_{l\alpha m}\nabla ^{m}\nabla ^{l}f_{G}\bigg ]h_{\nu \mu }\\&+\frac{1}{6}fh_{\nu \mu },\\ O^{GT}= & {} \left[ \frac{1}{2}R_{m\epsilon }R^{m p}f_{G}+\frac{1}{2}R^{lm}R^{p}_{m\epsilon l}f_{G}\right. \\&-\left. \frac{1}{4}RR^{p}_{\epsilon }f_{G}-\frac{1}{4}R_{\epsilon lmn}R^{lmn p}f_{G}+\frac{1}{2} R^{p}_{\epsilon }\Box f_{G}\right. \\&\left. +\frac{1}{4}R\nabla ^{p}\nabla _{\epsilon }f_{G} -\frac{1}{4}R^{lp}\nabla _{\epsilon }\nabla _{m}f_{G} \right. \\&\left. -\frac{1}{2}R^{m}_{\epsilon }\nabla ^{p}\nabla _{m} f_{G}\right. \\&-\left. \frac{1}{2}R^{p}_{m\epsilon l}\nabla ^{m}\nabla ^{l}f_{G}\right] g^{\nu \mu }\epsilon _{p \delta \mu }\epsilon ^{\epsilon \delta }_{\nu }\\&+\left[ -\frac{1}{2}R_{m\delta }R^{lp}f_{G} -\frac{1}{2}R^{lm}R^{p}_{m\delta l}f_{G}\right. \\&+\left. \frac{1}{4}RR^{p}_{\delta }f_{G}+\frac{1}{4}R_{\delta lmn}R^{lmnp}f_{G}\right. \\&-\left. \frac{1}{2}R^{p}_{\delta }\Box f_{G}-\frac{1}{4}R\nabla ^{p}\nabla _{\delta }f_{G}+\frac{1}{4}R^{lp}\nabla _{\delta }\nabla _{m}f_{G}\right. \\&\left. +\frac{1}{2} R^{m}_{\delta }\nabla ^{p}\nabla _{m}f_{G} +\frac{1}{2}R^{p}_{m\delta l}\nabla ^{m}\nabla ^{l}f_{G}\right] \\&\times g^{\nu \mu }\epsilon _{p\epsilon \mu }\epsilon ^{\epsilon \delta }_{\nu }\\&+\left[ -\frac{1}{2}R_{m\epsilon }R^{m\gamma }f_{G} -\frac{1}{2}R^{lm}R^{\gamma }_{m\epsilon l}f_{G}\right. \\&+\left. \frac{1}{4}RR^{\gamma }_{\epsilon }f_{G} +\frac{1}{4}R_{\epsilon lmn}R^{lmn\gamma }f_{G}\right. \\&-\left. \frac{1}{2}R^{\gamma }_{\epsilon }\Box f_{G}-\frac{1}{4}R\nabla ^{\gamma }\nabla _{\epsilon }f_{G} +\frac{1}{4}R^{m\gamma }\nabla _{\epsilon }\nabla _{m}f_{G}\right. \\&+\left. \frac{1}{2}R^{m}_{\epsilon }\nabla ^{\gamma }\nabla _{m} f_{G} +\frac{1}{2}R^{\gamma }_{m\epsilon l}\nabla ^{m} \nabla ^{l}f_{G}\right] \\&\times g^{\nu \mu } \epsilon _{\delta \gamma \mu }\epsilon ^{\epsilon \delta }_{\nu }\\&+\left[ \frac{1}{2}R_{m\delta }R^{m\gamma }f_{G} +\frac{1}{2}R^{lm}R^{\gamma }_{m\delta l}f_{G}\right. \\&-\left. \frac{1}{4}RR^{\gamma }_{\delta }f_{G}-\frac{1}{4}R_{\delta lmn}R^{lmn\gamma }f_{G}+\frac{1}{2}R^{\gamma }_{\delta }\Box f_{G}\right. \\&\left. +\frac{1}{4}R\nabla ^{\gamma }\nabla _{\delta }f_{G} -\frac{1}{4}R^{m\gamma }\nabla _{\delta }\nabla _{m}f_{G}\right. \\&\left. -\frac{1}{2}R^{m}_{\delta }\nabla ^{\gamma }\nabla _{m}f_{G} -\frac{1}{2}R^{\gamma }_{m\delta l}\nabla ^{m}\nabla ^{l}f_{G}\right] \\&\times g^{\nu \mu } \epsilon _{\epsilon \gamma \mu }\epsilon ^{\epsilon \delta }_{\nu }\\&- 12R^{lm}\nabla _{m}\nabla _{l}h_{\nu \mu }f_{G} +6R\Box f_{G}\\&+\left[ -\left( \rho +p\right) f_{T}+4R^{l\nu }R_{l\nu }f_{G}\right. \\&+\left. 4R^{lm}R^{\nu }_{l\nu m}f_{G}-2R^{\mu }_{lmn}R^{lmn}_{\mu }f_{G}\right. \\&\left. -2R^{2}f_{G}-2R\Box f_{G}\right. \\&+\left. 16R^{lm}\nabla _{l}\nabla _{m}f_{G} -4R^{l\mu }\nabla _{\mu }\nabla _{l}f_{G}\right. \\&\left. -4R^{l\nu }\nabla _{\nu }\nabla _{l}f_{G}\right. \\&-\left. 4R^{\nu }_{l\nu m}\nabla ^{l}\nabla ^{m}f_{G}\right] +{2}f. \end{aligned}$$