Complexity for self-gravitating fluid distributions in f(G, T) gravity

Abstract

This paper is devoted to analyze the effects of charge on the complexity for static self-gravitating fluids in the background of f(G, T) gravity, where G and T represent the Gauss–Bonnet term and trace of energy momentum tensor, respectively. We work out the modified field equations, Darmois junction conditions, relation between the Reimann and Weyl tensor, mass function, Tolman mass in f(G, T) gravity with electromagnetic effects. After orthogonal breaking of the Riemann tensor, we evaluated all the structure scalars and identified $Y_{TF}$ as a complexity of the system. We also evaluated couple of static solutions with the zero contribution of the obtained complexity condition to analyze the structure and evolution of compact objects. Finally, we deduced that effective charge terms decrease the complexity of the system.

Appendices

Appendix A

$$\begin{aligned} \mu ^\mathrm{eff}= & {} \mu +\frac{1}{8\pi }\left[ (\mu +P)f_{T}+\frac{f}{2} -\frac{6\lambda 'f_{G}'}{r^{2}\mathrm{e}^{2\lambda }} -\frac{4f_{G}''}{r^{2}\mathrm{e}^{\lambda }}+\frac{2\lambda 'f_{G}'}{r^{2}\mathrm{e}^{\lambda }} +\frac{4f_{G}''}{r^{2}\mathrm{e}^{2\lambda }}\right. \\&-\left. \frac{\nu '^{2}f_{G}}{r^{2}\mathrm{e}^{2\lambda }}-\frac{\lambda '\nu 'f_{G}}{r^{2}\mathrm{e}^{\lambda }} +\frac{3\lambda '\nu 'f_{G}}{r^{2}\mathrm{e}^{\lambda }} +\frac{\nu '^{2}f_{G}}{r^{2}\mathrm{e}^{\lambda }}-\frac{2\nu ''f_{G}}{r^{2}\mathrm{e}^{2\lambda }} +\frac{2\nu ''f_{G}}{r^{2}\mathrm{e}^{\lambda }}\right] ,\\ P_{r}^\mathrm{eff}= & {} P_{r}+\frac{1}{8\pi }\left[ \frac{2}{3}\Pi f_{T}-\frac{f}{2}+\frac{\nu '^{2}f_{G}}{r^{2}\mathrm{e}^{2\lambda }} +\frac{2\nu ''f_{G}}{r^{2}\mathrm{e}^{2\lambda }} -\frac{2\nu ''f_{G}}{r^{2}\mathrm{e}^{\lambda }} -\frac{\nu '^{2}f_{G}}{r^{2}\mathrm{e}^{\lambda }}\right. \\&-\left. \frac{6\nu 'f_{G}'}{r^{2}\mathrm{e}^{2\lambda }}+\frac{\lambda '\nu 'f_{G}}{r^{2}\mathrm{e}^{\lambda }} -\frac{2\nu '^{2}f_{G}'}{r\mathrm{e}^{2\lambda }} +\frac{2\nu 'f_{G}'}{r^{2}\mathrm{e}^{\lambda }}+\frac{2\nu '\lambda 'f_{G}'}{r\mathrm{e}^{2\lambda }} -\frac{3\lambda '\nu 'f_{G}}{r^{2}\mathrm{e}^{2\lambda }}\right] ,\\ P^\mathrm{eff}_{\bot }= & {} P_\bot +\frac{1}{8\pi }\left[ -\left( \frac{\Pi }{3}+\frac{q^2}{4\pi r^4}\right) f_{T}-\frac{f}{2}+\frac{\nu '^{2}f_{G}}{r^{2}\mathrm{e}^{2\lambda }} +\frac{3\lambda '\nu 'f_{G}'}{r\mathrm{e}^{2\lambda }}-\frac{\nu '^{2}f_{G}}{r^{2}\mathrm{e}^{\lambda }} \right. \\&+\left. \frac{\nu '^{2}f_{G}'}{r\mathrm{e}^{2\lambda }} -\frac{2\nu 'f_{G}''}{r\mathrm{e}^{2\lambda }} -\frac{2\nu ''f_{G}'}{r\mathrm{e}^{2\lambda }}-\frac{3\lambda '\nu 'f_{G}}{r^{2}\mathrm{e}^{2\lambda }}-\frac{2\nu ''f_{G}}{r^{2}\mathrm{e}^{\lambda }} +\frac{2\nu ''f_{G}}{r^{2}\mathrm{e}^{2\lambda }}+\frac{\lambda '\nu 'f_{G}}{r^{2}\mathrm{e}^{\lambda }}\right] ,\\ \chi _{1}= & {} \frac{-1}{8\pi }\left[ \frac{2}{3}\Pi {\tilde{f}}_{T}-\frac{{\tilde{f}}}{2}+\frac{\nu '^{2}{\tilde{f}}_{G}}{r^{2}\mathrm{e}^{2\lambda }} -\frac{\nu '^{2}{\tilde{f}}_{G}}{r^{2}\mathrm{e}^{\lambda }}\right. +\left. \frac{2\nu ''{\tilde{f}}_{G}}{r^{2}\mathrm{e}^{2\lambda }} -\frac{3\lambda '\nu '{\tilde{f}}_{G}}{r^{2}\mathrm{e}^{2\lambda }} -\frac{2\nu ''{\tilde{f}}_{G}}{r^{2}\mathrm{e}^{\lambda }} +\frac{\lambda '\nu '{\tilde{f}}_{G}}{r^{2}\mathrm{e}^{\lambda }}\right] ,\\ Z= & {} \frac{f_{T}}{k^{2}-f_{T}}\left[ \mathrm{e}^{\lambda }\left( -P_{r}+P-\frac{q^2}{8\pi r^4}\right) \frac{f_{T,r}}{f_{T}}+\left( -2P_{r}\mathrm{e}^{\lambda }+P\mathrm{e}^{\lambda }-\frac{q^2}{8\pi r^4}\mathrm{e}^{\lambda }\right) _{,r}\right. \\&+\frac{\mathrm{e}^{\lambda }}{2}\left[ (\mu -3P) \right. +\left. \left. \frac{1}{8\pi }\left( (\mu +P)f_{T}-\frac{q^2}{3\pi r^4}f_{T}+2f-2R^{2}f_{G} \right. \right. \right. \\&+4R_{l\nu }R^{l\nu }f_{G}+4R^{lm}R^{\nu }_{l\nu m}f_{G}-\left. \left. \left. 2R^{\mu }_{lmn}R^{lmn}_{\mu }f_{G} -2R\Box f_{G}+16R^{lm}\nabla _{l}\nabla _{m}f_{G} \right. \right. \right. \\&-4R^{l\mu }\nabla _{l}\nabla _{\mu }f_{G} -4R^{l\nu }\nabla _{\nu }\nabla _{l}f_{G}\left. \left. \left. -4R^{\nu }_{l\nu m}\nabla ^{l}\nabla ^{m}f_{G}\right) \right] _{,r}\right] . \end{aligned}$$

Appendix B

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

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

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