Electromagnetic effects on the complexity of static cylindrical object in f(G, T) gravity

Abstract

In this paper, we investigate complexity of anisotropic cylindrical object under the influence of electromagnetic field in f(G, T) theory, where G and T indicate the Gauss–Bonnet term and trace of the stress–energy tensor, respectively. For this purpose, we calculate the modified field equations, non-conservation equation and mass distributions that assist in comprehending the structure of astrophysical objects. The Riemann tensor is divided into different structure scalars, among which one is called the complexity factor. This factor is used to measure complexity of the system due to the involvement of inhomogeneous energy density, anisotropic pressure and charge. The vanishing of the complexity factor is employed as a constraint to formulate charged static solutions for the Gokhroo–Mehra model and polytropic equation of state. We conclude that the presence of charge reduces the complexity of the anisotropic system.

Appendix A

The modified term Z is expressed as

$$\begin{aligned} \textsf {Z} & = \frac{f_{T}}{k^{2}-f_{T}}\left[ \left( -p_{r}-p\right) \frac{f'_{T}}{f_{T}}-\frac{\textsf {q}\textsf {q}^{'}}{4\pi r^4}+\left( -2p_{r}L^2-pL^2\right) '-\frac{L^2}{2}(-\varrho +3p)^{'} \right] . \end{aligned}$$

The following describes the impact of additional terms in the scalar structure

$$\begin{aligned} \textsf {N}^{GT}_{\psi \chi } & = \left[ 2R_{m\delta }R^{m\gamma }f_{G}-RR^{\gamma }_{\delta }f_{G} +2R^{lm}R^{\gamma }_{l\delta m}f_{G}-2R_{\delta lmn}R^{lmn\gamma }f_{G}\right. \\&+\,\left. 2R^{\gamma }_{\delta }\Box f_{G}-2R^{m\gamma }\nabla _{\delta }\nabla _{m}f_{G}+R\nabla ^{\gamma }\nabla _{\delta }f_{G} -2R^{m}_{\delta }\nabla ^{\gamma }\nabla _{m}f_{G}\right. \\&-\,\left. 2R^{\gamma }_{l\delta m}\nabla ^{m}\nabla ^{l}f_{G}\right] \epsilon _{\psi \chi \gamma }\mathrm {u}^{\delta }, \\ \textsf {D}^{GT}_{\psi \chi } & = 2\left[ R_{m\chi }R^{m}_{\psi }f_{G}-\frac{1}{2}RR_{\psi \chi }f_{G}+ R^{lm}R_{l\chi m\psi }f_{G}-\frac{1}{2}R_{\chi lmn}R^{lmn}_{\psi }f_{G}+R_{\psi \chi }\Box f_{G}\right. \\&+\,\left. \frac{1}{2}R \nabla _{\psi }\nabla _{\chi }f_{G} -R^{m}_{\psi }\nabla _{\chi }\nabla _{m}f_{G} -R^{m}_{\chi }\nabla _{\psi }\nabla _{m}f_{G}-R_{l\chi m\psi }\nabla ^{m}\nabla ^{l}f_{G}\right] \\&+\,4R^{lm}h_{\psi \chi }\nabla _{m}\nabla _{l}f_{G}-2Rh_{\psi \chi }\Box f_{G}+2\left[ -R_{m\delta }R^{m}_{\psi }f_{G}- R^{lm}R_{l\delta m\psi }f_{G}\right. \\&+\,\left. \frac{1}{2}R_{\delta lmn}R^{lmn}_{\psi }f_{G}+\frac{1}{2}RR_{\psi \chi }f_{G} -R_{\delta \psi }\Box f_{G}+R_{l\delta m\psi }\nabla ^{m}\nabla ^{m}f_{G}\right. \\&-\,\left. \frac{1}{2}R \nabla _{\psi }\nabla _{\delta }f_{G}+R^{m}_{\psi }\nabla _{\delta }\nabla _{m}f_{G} +R^{m}_{\delta }\nabla _{\psi }\nabla _{m}f_{G}\right] \mathrm {u}_{\chi }\mathrm {u}^{\delta } +2\left[ -R_{m\chi }R^{m\gamma }f_{G}\right. \\&+\,\left. \frac{1}{2}R_{\chi lmn}R^{lmn\gamma }f_{G}- R^{lm}R^{\gamma }_{l\chi m}f_{G}+\frac{1}{2}RR^{\gamma }_{\chi }f_{G}-R^{\gamma }_{\chi }\Box f_{G}+R^{m\gamma }\nabla _{\chi }\nabla _{m}f_{G}\right. \\&-\,\left. \frac{1}{2}R \nabla ^{\gamma }\nabla _{\chi }f_{G}+R^{m}_{\chi }\nabla ^{\gamma }\nabla _{m}f_{G}+R^{\gamma }_{m\chi l}\nabla ^{m}\nabla ^{l}\right] \mathrm {u}_{\psi }\mathrm {u}_{\gamma }+ 2\left[ R_{m\delta }R^{m\gamma }f_{G}\right. \\&-\,\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}+R^{\gamma }_{\delta }\Box f_{G}+\frac{1}{2}R \nabla ^{\gamma }\nabla _{\delta }f_{G}\right. \\&-\,\left. R^{\gamma }_{l\delta m}\nabla ^{m}\nabla ^{l}f_{G}-R^{m\gamma }\nabla _{\delta }\nabla _{m}f_{G} -R^{m}_{\delta }\nabla ^{\gamma }\nabla _{m} f_{G}\right] g_{\psi \chi }\mathrm {u}_{\gamma }\mathrm {u}^{\delta } -\frac{1}{3}\left[ -2R^{2}f_{G}\right. \\&-\,\left. 2R^{\beta }_{lmn}R^{lmn}_{\beta }f_{G} -2R\Box f_{G}+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}-4R^{\alpha }_{l\alpha m}\nabla ^{m}\nabla ^{l}f_{G}\right] h_{\psi \chi } -\frac{1}{6}fh_{\psi \chi },\\ \textsf {F}^{GT} & = 2\left[ R_{m\chi }R^{m}_{\psi }f_{G}+ R^{lm}R_{m\chi l\psi }f_{G}-\frac{1}{2}RR_{\psi \chi }f_{G}-\frac{1}{2}R_{\chi \chi lmn}R^{lmn}_{\psi }f_{G}+R_{\psi \chi }\Box f_{G}\right. \\&+\,\left. \frac{1}{2}R \nabla _{\psi }\nabla _{\chi }f_{G}-R^{m}_{\psi }\nabla _{\chi }\nabla _{m}f_{G} -R^{m}_{\chi }\nabla _{\psi }\nabla _{m}f_{G}-R_{m\chi l\psi }\nabla ^{m}\nabla ^{l}f_{G}\right] g^{\psi \chi }\\&+\,12R^{lm}\nabla _{m}\nabla _{l}f_{G}-6R\Box f_{G}+2\left[ -R_{m\delta }R^{m}_{\psi }f_{G}- R^{lm}R_{m\delta l\psi }f_{G}+\frac{1}{2}RR_{\psi \delta }f_{G}\right. \\&-\,\left. \frac{1}{2}R \nabla _{\psi }\nabla _{\delta }f_{G}-R_{\delta \psi }\Box f_{G}+R_{m\delta l\psi }\nabla ^{m}\nabla ^{l}f_{G}+R^{m}_{\psi }\nabla _{\delta }\nabla _{m}f_{G} +R^{m}_{\delta }\nabla _{\psi }\nabla _{m}f_{G}\right. \\&+\,\left. \frac{1}{2}R_{\delta lmn}R^{lmn}_{\psi }f_{G}\right] \mathrm {u}_{\chi }\mathrm {u}^{\delta }g^{\psi \chi } +2\left[ -R_{m\chi }R^{m\gamma }f_{G}- R^{lm}R^{\gamma }_{m\chi l}f_{G}+\frac{1}{2}RR^{\gamma }_{\chi }f_{G}\right. \\&+\,vvvvv\left. \frac{1}{2}R_{\chi lmn}R^{lmn\gamma }f_{G}-R^{\gamma }_{\chi }\Box f_{G}+R^{m\gamma }\nabla _{\chi }\nabla _{m}f_{G}+R^{m}_{\chi } \nabla ^{\gamma }\nabla _{m}f_{G}+R^{\gamma }_{m\chi l}\nabla ^{m}\nabla ^{l}f_{G}\right. \\&-\,\left. \frac{1}{2}R \nabla ^{\gamma }\nabla _{\chi }f_{G}\right] \mathrm {u}_{\psi }\mathrm {u}_{\gamma }g^{\psi \chi }+ 2\left[ R_{m\delta }R^{m\gamma }f_{G}+ 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}-\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_{\psi \chi }\mathrm {u}_{\gamma }\mathrm {u}^{\delta }g^{\psi \chi } -\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} -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,\\ \textsf {Q}^{GT}_{(\psi \chi )} & = \left[ 2R_{md}R^{m}_{c}f_{G}+2R^{lm}R_{ldmc}f_{G}-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}-2R^{m}_{d}\nabla _{c}\nabla _{m}f_{G} -2R_{ldmc}\nabla ^{m}\nabla ^{l}f_{G}\right] h^{c}_{\psi }h^{d}_{\chi }\\&+\,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}-\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}-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_{\psi \chi } \mathrm {u}_{\gamma }\mathrm {u}^{\delta }-2\left[ R_{m\chi }R^{m}_{\psi }f_{G}+ R^{lm}R_{m\chi l\psi }f_{G}-\frac{1}{2}RR_{\psi \chi }f_{G}\right. \\&-\,\left. \frac{1}{2}R_{\chi lmn}R^{lmn}_{\psi }f_{G}+R_{\psi \chi }\Box f_{G}+\frac{1}{2}R \nabla _{\psi }\nabla _{\chi }f_{G}-R^{m}_{\psi }\nabla _{\chi }\nabla _{m}f_{G}\right. \\&-\,\left. R^{m}_{\chi }\nabla _{\psi }\nabla _{m}f_{G}-R_{m\chi l\psi }\nabla ^{m}\nabla ^{l}f_{G}\right] -2\left[ -R_{m\delta }R^{m}_{\psi }f_{G}- R^{lm}R_{m\delta l\psi }f_{G}\right. \\&+\,\left. \frac{1}{2}RR_{\psi \delta }f_{G}+\frac{1}{2}R_{\delta lmn}R^{lmn}_{\psi }f_{G}-R_{\delta \psi }\Box f_{G}+R_{m\delta l\psi }\nabla ^{m}\nabla ^{l}f_{G}\right. \\&+\,\left. R^{m}_{\psi }\nabla _{\delta }\nabla _{m}f_{G} +R^{m}_{\delta }\nabla _{\psi }\nabla _{m}f_{G}-\frac{1}{2}R \nabla _{\psi }\nabla _{\delta }f_{G}\right] \mathrm {u}_{\chi } \mathrm {u}^{\delta }-2\left[ -R_{m\chi }R^{m\gamma }f_{G}\right. \\&-\,\left. R^{lm}R^{\gamma }_{m\chi l}f_{G}+\frac{1}{2}RR^{\gamma }_{\chi }f_{G}+\frac{1}{2}R_{\chi lmn}R^{lmn\gamma }f_{G}-R^{\gamma }_{\chi }\Box f_{G}\right. \\&+\,\left. R^{m\gamma }\nabla _{\chi }\nabla _{m}f_{G}+R^{m}_{\chi }\nabla ^{\gamma }\nabla _{m}f_{G}+R^{\gamma }_{m\chi l}\nabla ^{m}\nabla ^{l}f_{G}-\frac{1}{2}R \nabla ^{\gamma }\nabla _{\chi }f_{G}\right] \mathrm {u}_{\psi }\mathrm {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}-\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}-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] \mathrm {u}_{\gamma }\mathrm {u}^{\delta }g_{\psi \chi },\\ \textsf {M}^{GT}_{\psi \chi } & = \left[ \frac{1}{2}R_{m\epsilon }R^{m p}f_{G}+\frac{1}{2}R^{lm}R^{p}_{m\epsilon l}f_{G}-\frac{1}{4}RR^{p}_{\epsilon }f_{G}-\frac{1}{4}R_{\epsilon lmn}R^{lmn p}f_{G}\right. \\&+\,\left. \frac{1}{2} R^{p}_{\epsilon }\Box f_{G}+\frac{1}{4}R\nabla ^{p}\nabla _{\epsilon }f_{G} -\frac{1}{4}R^{lp}\nabla _{\epsilon }\nabla _{m}f_{G} -\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] \epsilon _{p \delta \chi }\epsilon ^{\epsilon \delta }_{\psi } +\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}-\frac{1}{4}R\nabla ^{p}\nabla _{\delta }f_{G}\right. \\&+\,\left. \frac{1}{4}R^{lp}\nabla _{\delta }\nabla _{m}f_{G} +\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 \chi }\epsilon ^{\epsilon \delta }_{\psi }\\&+\,\left[ -\frac{1}{2}R_{m\epsilon }R^{m\gamma }f_{G} -\frac{1}{2}R^{lm}R^{\gamma }_{m\epsilon l}f_{G} +\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} +\frac{1}{2}R^{m}_{\epsilon }\nabla ^{\gamma }\nabla _{m} f_{G}\right. \\&+\,\left. \frac{1}{2}R^{\gamma }_{m\epsilon l}\nabla ^{m} \nabla ^{l}f_{G}\right] \epsilon _{\delta \gamma \chi } \epsilon ^{\epsilon \delta }_{\psi }+\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}+\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}-\frac{1}{2}R^{\gamma }_{m\delta l}\nabla ^{m}\nabla ^{l}f_{G}\right] \epsilon _{\epsilon \gamma \chi }\epsilon ^{\epsilon \delta }_{\psi }\\&-\,4R^{lm}\nabla _{m}\nabla _{l}h_{\psi \chi }f_{G} +2Rh_{\psi \chi }\Box f_{G} +\frac{1}{3}\left[ \left( \varrho +p\right) f_{T}+4R^{m\alpha }R_{m\alpha }f_{G}\right. \\&+\,\left. 4R^{lm}R^{\alpha }_{l\alpha m}f_{G}-2R^{2}f_{G}-2R^{\beta }_{lmn}R^{lmn}_{\beta }f_{G}-2R\Box f_{G}\right. \\&+\,\left. 16R^{lm}\nabla _{m}\nabla _{l}f_{G}-4R^{m\alpha }\nabla _{\alpha }\nabla _{m}f_{G} -4R^{m\beta }\nabla _{\beta }\nabla _{m}f_{G}\right. \\&-\,\left. 4R^{\alpha }_{l\alpha m}\nabla ^{m}\nabla ^{l}f_{G}\right] h_{\psi \chi }+\frac{1}{6}fh_{\psi \chi },\\ \textsf {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}-\frac{1}{4}RR^{p}_{\epsilon }f_{G}-\frac{1}{4}R_{\epsilon lmn}R^{lmn p}f_{G}\right. \\&+\,\left. \frac{1}{2} R^{p}_{\epsilon }\Box f_{G}+\frac{1}{4}R\nabla ^{p}\nabla _{\epsilon }f_{G} -\frac{1}{4}R^{lp}\nabla _{\epsilon }\nabla _{m}f_{G}-\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^{\psi \chi }\epsilon _{p \delta \chi }\epsilon ^{\epsilon \delta }_{\psi }+\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}-\frac{1}{4}R\nabla ^{p}\nabla _{\delta }f_{G}\right. \\&+\,\left. \frac{1}{4}R^{lp}\nabla _{\delta }\nabla _{m}f_{G}+\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] g^{\psi \chi }\epsilon _{p\epsilon \chi }\epsilon ^{\epsilon \delta }_{\psi }\\&+\,\left[ -\frac{1}{2}R_{m\epsilon }R^{m\gamma }f_{G}-\frac{1}{2}R^{lm}R^{\gamma }_{m\epsilon l}f_{G} +\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} +\frac{1}{2}R^{m}_{\epsilon }\nabla ^{\gamma }\nabla _{m} f_{G}\right. \\&+\,\left. \frac{1}{2}R^{\gamma }_{m\epsilon l}\nabla ^{m} \nabla ^{l}f_{G}\right] g^{\psi \chi }\epsilon _{\delta \gamma \chi } \epsilon ^{\epsilon \delta }_{\psi }+\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}+\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}-\frac{1}{2}R^{\gamma }_{m\delta l}\nabla ^{m}\nabla ^{l}f_{G}\right] g^{\psi \chi }\epsilon _{\epsilon \gamma \chi }\epsilon ^{\epsilon \delta }_{\psi }\\&-\,12R^{lm}\nabla _{m}\nabla _{l}h_{\psi \chi }f_{G} +6R\Box f_{G} +\left[ \left( \varrho +p\right) f_{T}+4R^{l\psi }R_{l\psi }f_{G}\right. \\&+\,\left. 4R^{lm}R^{\psi }_{l\psi m}f_{G}-2R^{\chi }_{lmn}R^{lmn}_{\chi }f_{G}-2R^{2}f_{G}-2R\Box f_{G}\right. \\&+\,\left. 16R^{lm}\nabla _{l}\nabla _{m}f_{G} -4R^{l\chi }\nabla _{\chi }\nabla _{l}f_{G}-4R^{l\psi }\nabla _{\psi }\nabla _{l}f_{G}\right. \\&-\,\left. 4R^{\psi }_{l\psi m}\nabla ^{l}\nabla ^{m}f_{G}\right] +\frac{1}{2}f. \end{aligned}$$