Study of static charged spherical structure in f(R, T, Q) gravity

Abstract

We generalize the definition of complexity for self-gravitating object proposed by Herrera (Phys Rev D 97(4):044010, 2018), for the case of charged spherical matter distribution in f(R, T, Q) theory of gravity, where R and T represent the Ricci scalar and the trace of the energy–momentum tensor, respectively, and $Q\equiv R_{\mu \nu }T^{\mu \nu }$. We split the Riemann tensor orthogonally to determine the modified structure scalars, and one of these scalars has been found to be involved in the emergence of complexity of the system. We consider the vanishing of the complexity factor to study some mathematical models of field equations under the influence of dark source terms of f(R, T, Q) gravity. By applying the restriction, $f(R,T,Q)=R$, one can get all these results in general relativity.

Appendices

Appendix A

The effective physical and charge components appearing in Eqs. (13)–(15) are

$$\begin{aligned} \mu ^\mathrm{(eff)}= & {} \mu \left[ 1+2f_{T}+f_{Q}\left( \frac{1}{2}R-\frac{3\nu '^2}{8e^{\lambda }}-\frac{3\nu '}{2re^{\lambda }}+\frac{5\lambda '\nu '}{8e^{\lambda }} -\frac{3\nu ''}{4e^{\lambda }}\right) \right. \nonumber \\&+\left. f'_{Q}\left( \frac{1}{re^{\lambda }}-\frac{\lambda '}{4e^{\lambda }}\right) +\frac{f''_{Q}}{2e^{\lambda }}\right] +\mu '\left[ f_{Q}\left( \frac{1}{re^{\lambda }}-\frac{\lambda '}{4e^{\lambda }}\right) +\frac{f'_{Q}}{e^{\lambda }}\right] +\frac{\mu ''f_{Q}}{2e^{\lambda }}\nonumber \\&+P_{r}\left[ f_{Q}\left( \frac{\nu '^2}{8e^{\lambda }}-\frac{1}{r^2e^{\lambda }}-\frac{\lambda '\nu '}{8e^{\lambda }}+\frac{\lambda '}{2re^{\lambda }} +\frac{\nu ''}{4e^{\lambda }}\right) +f'_{Q}\left( \frac{\lambda '}{4e^{\lambda }}-\frac{2}{re^{\lambda }}\right) \right. \nonumber \\&-\left. \frac{f''_{Q}}{2e^{\lambda }}\right] +P'_{r}\left[ f_{Q}\left( \frac{\lambda '}{4e^{\lambda }}-\frac{2}{re^{\lambda }}\right) -\frac{f'_{Q}}{e^{\lambda }}\right] -\frac{P''_{r}f_{Q}}{2e^{\lambda }}\nonumber \\&+P_{\bot }\left[ f_{Q}\left( \frac{\nu '}{2re^{\lambda }}+\frac{1}{r^2e^{\lambda }}-\frac{\lambda '}{2re^{\lambda }}\right) +\frac{f'_{Q}}{re^{\lambda }}\right] +\frac{P'_{\bot }f_{Q}}{re^{\lambda }}+\frac{R}{2}\left( \frac{f}{R}-f_{R}\right) \nonumber \\&+f'_{R}\left( \frac{2}{re^{\lambda }}-\frac{\lambda '}{2e^{\lambda }}\right) +\frac{f''_{R}}{e^{\lambda }}, \end{aligned}$$

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$$\begin{aligned} P_{r}^\mathrm{(eff)}= & {} \mu \left[ -f_{T}+f_{Q}\left( \frac{\nu '^2}{8e^{\lambda }}+\frac{\nu '}{2re^{\lambda }}-\frac{\lambda '\nu '}{8e^{\lambda }}+\frac{\nu ''}{4e^{\lambda }}\right) -\frac{f'_{Q}\nu '}{4e^{\lambda }}\right] -\frac{\mu '\nu 'f_{Q}}{4}\nonumber \\&+P_{r}\left[ 1+f_{T}+f_{Q}\left( \frac{1}{2}R-\frac{3\nu '^2}{8e^{\lambda }}+\frac{\nu '}{re^{\lambda }}+\frac{1}{r^2e^{\lambda }}+\frac{3\lambda '\nu '}{8e^{\lambda }} +\frac{3\lambda '}{2re^{\lambda }}\right. \right. \nonumber \\&-\left. \frac{3\nu ''}{4e^{\lambda }}\right) +f'_{Q}\left( \frac{1}{re^{\lambda }}-\frac{\nu '}{4e^{\lambda }}+\left. \frac{\nu '}{2e^{\lambda }}\right) \right] +P'_{r}\left[ f_{Q}\left( \frac{1}{re^{\lambda }} +\frac{\nu '}{4e^{\lambda }}\right) \right] \nonumber \\&+P_{\bot }\left[ f_{Q}\left( -\frac{\nu '}{2re^{\lambda }}-\frac{1}{r^2e^{\lambda }}+\frac{\lambda '}{2re^{\lambda }}\right) +\frac{f'_{Q}}{re^{\lambda }}\right] +\frac{P'_{\bot }f_{Q}}{r} \nonumber \\&-\frac{R}{2}\left( \frac{f}{R}-f_{R}\right) -f'_{R}\left( \frac{\nu '}{2e^{\lambda }}+\frac{2}{re^{\lambda }}\right) , \end{aligned}$$

(83)

$$\begin{aligned} P_{\bot }^\mathrm{(eff)}= & {} \mu \left[ -f_{T}+f_{Q}\left( \frac{\nu '^2}{8e^{\lambda }}+\frac{\nu '}{2re^{\lambda }}-\frac{\lambda '\nu '}{8e^{\lambda }}+\frac{\nu ''}{4e^{\lambda }}\right) +\frac{f'_{Q}\nu '}{4e^{\lambda }}\right] +\frac{\mu '\nu 'f_{Q}}{4e^{\lambda }}\nonumber \\&+P_{r}\left[ f_{Q}\left( \frac{\nu '^2}{8e^{\lambda }}+\frac{\nu '}{2re^{\lambda }}-\frac{\lambda '\nu '}{8e^{\lambda }}+\frac{\nu ''}{4e^{\lambda }}\right) +f'_{Q}\left( \frac{\nu '}{2e^{\lambda }}+\frac{1}{re^{\lambda }}-\frac{\lambda '}{4e^{\lambda }}\right) \right. \nonumber \\&+\left. \frac{f''_{Q}}{2e^{\lambda }}\right] +P'_{r}\left[ f_{Q}\left( \frac{\nu '}{2e^{\lambda }}+\frac{1}{re^{\lambda }}-\frac{\lambda '}{4e^{\lambda }}\right) +\frac{f'_{Q}}{e^{\lambda }}\right] +\frac{P''_{r}f_{Q}}{2e^{\lambda }}\nonumber \\&+P_{\bot }\left[ 1+f_{T}+f_{Q}\left( \frac{1}{2}R-\frac{2}{r^2e^{\lambda }}+\frac{\lambda '}{re^{\lambda }}-\frac{\nu '}{re^{\lambda }}+\frac{2}{r^2}\right) \right. \nonumber \\&+\left. f'_{Q}\left( \frac{\nu '}{4e^{\lambda }}-\frac{\lambda '}{4e^{\lambda }} \right) +\frac{f''_{Q}}{2e^{\lambda }}\right] +P'_{\bot }\left[ f_{Q}\left( \frac{\nu '}{4e^{\lambda }}-\frac{\lambda '}{4e^{\lambda }}+\frac{2}{re^{\lambda }}\right) +\frac{f'_{Q}}{e^{\lambda }}\right] \nonumber \\&+\frac{P''_{\bot }f_{Q}}{2e^{\lambda }}-\frac{R}{2}\left( \frac{f}{R}-f_{R}\right) +f'_{R}\left( \frac{\lambda '}{2e^{\lambda }}-\frac{1}{re^{\lambda }}-\frac{\nu '}{2e^{\lambda }}\right) -\frac{f''_{R}}{e^{\lambda }}, \end{aligned}$$

(84)

$$\begin{aligned} Q_{0}^\mathrm{(eff)}= & {} (f_{T}+\frac{1}{2}Rf_{Q}+1)\frac{q^2}{8\pi r^4}-\frac{q^2f_{T}}{2r^4}+f_{Q}\left( \frac{qq''}{4\pi r^4e^\lambda } -\frac{\lambda 'qq'}{8\pi r^4e^\lambda }-\frac{qq'}{\pi r^5e^\lambda }\right. \nonumber \\&+\left. \frac{q'^2}{4\pi r^4e^\lambda }+\frac{3q^2}{4\pi r^6e^\lambda }+\frac{\lambda 'q^2}{8\pi r^5e^\lambda }-\frac{\nu 'q^2}{8\pi r^5e^\lambda } +\frac{\lambda '\nu 'q^2}{16\pi r^4e^\lambda }-\frac{\nu '^2q^2}{16\pi r^4e^\lambda }\right. \nonumber \\&-\left. \frac{\nu ''q^2}{8\pi r^4e^\lambda }\right) +f'_{Q}\left( \frac{qq'}{2\pi r^4e^\lambda }-\frac{\lambda 'q^2}{16\pi r^4e^\lambda }-\frac{q^2}{2\pi r^5e^\lambda }-\frac{\nu 'q^2}{16\pi r^4e^\lambda }\right) \nonumber \\&+\frac{f''_{Q}q^2}{8\pi r^4e^\lambda }, \end{aligned}$$

(85)

$$\begin{aligned} Q_{1}^\mathrm{(eff)}= & {} -(f_{T}+\frac{1}{2}Rf_{Q}+1)\frac{q^2}{8\pi r^4}+\frac{q^2f_{T}}{2r^4}+f_{Q}\left( \frac{\nu 'q^2}{\pi r^5e^\lambda } -\frac{\nu 'qq'}{8\pi r^4e^\lambda }\right. \nonumber \\&-\left. \frac{\lambda 'q^2}{8\pi r^5e^\lambda }-\frac{q^2}{8\pi r^6e^\lambda }-\frac{\lambda 'qq'}{8\pi r^4e^\lambda }-\frac{\lambda '\nu 'q^2}{16\pi r^4e^\lambda }+\frac{\nu '^2q^2}{16\pi r^4e^\lambda } +\frac{\nu ''q^2}{8\pi r^4e^\lambda }\right) \nonumber \\&-\frac{f'_{Q}\nu 'q^2}{16\pi r^4e^\lambda },\end{aligned}$$

(86)

$$\begin{aligned} Q_{2}^\mathrm{(eff)}= & {} (f_{T}+\frac{1}{2}Rf_{Q}+1)\frac{q^2}{8\pi r^4}+\frac{q^2f_{T}}{2r^4}+f_{Q}\left( -\frac{qq'}{4\pi r^5e^\lambda } +\frac{\lambda 'q^2}{8\pi r^5e^\lambda }\right. \nonumber \\&-\frac{\nu 'q^2}{8\pi r^5e^\lambda }+\frac{q^2}{4\pi r^6}+\left. \frac{q^2}{4\pi r^6e^\lambda }\right) -\frac{f'_{Q}q^2}{8\pi r^5e^\lambda }. \end{aligned}$$

(87)

The quantity Z arising in Eq. (17) is

$$\begin{aligned} Z =&\frac{2}{\left( 2+Rf_{Q}+2f_{T}\right) }\left[ f'_{Q}e^{-\lambda }\left( \frac{P_{r}^\mathrm{(eff)}+Q_{1}^\mathrm{(eff)}}{H}\right) \left( \frac{\nu '}{r}-\frac{e^\lambda }{r^2}+\frac{1}{r^2}\right) \right. \nonumber \\&+f_{Q}e^{-\lambda }\left( \frac{P_{r}^\mathrm{(eff)}+Q_{1}^\mathrm{(eff)}}{H}\right) \left( \frac{\nu ''}{r}-\frac{\lambda '}{r^2}-\frac{\nu '\lambda '}{r}-\frac{\nu '}{r^2}+\frac{2e^\lambda }{r^3}-\frac{2}{r^3}\right) \nonumber \\&+\frac{f_{Q}e^{-\lambda }}{2}\left( \frac{P_{r}^\mathrm{(eff)}+Q_{1}^\mathrm{(eff)}}{H}\right) '\left( \frac{\nu '\lambda '}{4} -\frac{\nu '^2}{4}-\frac{\nu ''}{4}+\frac{\lambda '}{r}-\frac{f_{T}}{2}\right) \nonumber \\&-f_{T}\left( \mu '-\frac{qq'}{r^4}+\frac{2q^2}{r^5}\right) -f'_{T}\left( \mu -\frac{q^2}{2r^4}\right) -\left( \frac{\mu ^\mathrm{(eff)}+Q_{0}^\mathrm{(eff)}}{H}\right) '\nonumber \\&\times \left\{ \frac{f_{Q}e^{-\lambda }}{8}\left( -\nu '\lambda '+\nu '^2+2\nu ''+\frac{4\nu '}{r}\right) +\frac{f_{T}}{2}\right\} -\left( \frac{P_{\bot }^\mathrm{(eff)}+Q_{2}^\mathrm{(eff)}}{H}\right) '\nonumber \\&\times \left\{ \frac{f_{Q}e^{-\lambda }}{r}\left( \frac{\lambda '}{2}-\frac{\nu '}{2}\frac{e^\lambda }{r}\right. \right. -\left. \left. \frac{1}{r}\right) -f_{T}\right\} +\left( \frac{1}{r^2}-\frac{e^{-\lambda }}{r^2}-\frac{\nu 'e^{-\lambda }}{r}\right) \nonumber \\&\times \left\{ f_{Q}\left( \mu '-\frac{q^2}{2r^4}\right) +f'_{Q}\left( \mu -\frac{qq'}{r^4} +\frac{2q^2}{r^5} \right) \right\} \nonumber \\&+\left. \left( \frac{P_{r}^\mathrm{(eff)}+Q_{1}^\mathrm{(eff)}}{H}\right) f'_{T}\right] . \end{aligned}$$

(88)

The terms $D_{0}$ and $D_{1}$ in Eqs. (25) and (26) are given as

$$\begin{aligned} D_{0}&=\mu \left[ -{\tilde{f}}_{T}+{\tilde{f}}_{Q}\left( \frac{\nu '^2}{8e^{\lambda }}+\frac{\nu '}{2re^{\lambda }} -\frac{\lambda '\nu '}{8e^{\lambda }}+\frac{\nu ''}{4e^{\lambda }}\right) \right] -\frac{\mu '\nu '{\tilde{f}}_{Q}}{4}+P_{r}\left[ {\tilde{f}}_{T}\right. \nonumber \\&\quad +\left. {\tilde{f}}_{Q}\left( \frac{1}{2}R- \frac{3\nu '^2}{8e^{\lambda }}+\frac{\nu '}{re^{\lambda }}+\frac{1}{r^2e^{\lambda }}+\frac{3\lambda '\nu '}{8e^{\lambda }} +\frac{3\lambda '}{2re^{\lambda }}-\frac{3\nu ''}{4e^{\lambda }}\right) \right] \nonumber \\&\quad +P'_{r}\left[ {\tilde{f}}_{Q}\left( \frac{1}{re^{\lambda }} +\frac{\nu '}{4e^{\lambda }}\right) \right] + P_{\bot }\left[ {\tilde{f}}_{Q}\left( -\frac{\nu '}{2re^{\lambda }}-\frac{1}{r^2e^{\lambda }} +\frac{\lambda '}{2re^{\lambda }}\right) \right] \nonumber \\&\quad +\frac{P'_{\bot }{\tilde{f}}_{Q}}{r}-\frac{R}{2}\left( \frac{f}{R}-{\tilde{f}}_{R}\right) , \end{aligned}$$

(89)

$$\begin{aligned} D_{1}=&-({\tilde{f}}_{T}+\frac{1}{2}R{\tilde{f}}_{Q})\frac{q^2}{8\pi r^4}+\frac{q^2{\tilde{f}}_{T}}{2r^4}+{\tilde{f}}_{Q}\left( \frac{\nu 'q^2}{\pi r^5e^\lambda } -\frac{\nu 'qq'}{8\pi r^4e^\lambda }-\frac{\lambda 'q^2}{8\pi r^5e^\lambda }\right. \nonumber \\&\quad -\left. \frac{q^2}{8\pi r^6e^\lambda }-\frac{\lambda 'qq'}{8\pi r^4e^\lambda }-\frac{\lambda '\nu 'q^2}{16\pi r^4e^\lambda }+\frac{\nu '^2q^2}{16\pi r^4e^\lambda }+\frac{\nu ''q^2}{8\pi r^4e^\lambda }\right) . \end{aligned}$$

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Appendix B

The modified correction terms appeared in structure scalars (54), (56) and (57) are

$$\begin{aligned} \chi _{1}^{(D)}= & {} \frac{4\pi }{H}\left[ \left\{ h^{\epsilon }_{\rho }\Box (f_{Q}X^{\rho }_{\epsilon })-2h^{\epsilon }_{\rho }\nabla ^{\rho }\nabla _{\epsilon }f_{R} -h^{\epsilon }_{\rho }\nabla _{\mu }\nabla ^{\rho }(f_{Q}X^{\mu }_{\epsilon }) \right. \right. \nonumber \\&-\left. h^{\epsilon }_{\rho }\nabla _{\mu }\nabla _{\epsilon }(f_{Q}X^{\mu \rho })\right\} -2f_{Q}\left( R^{\rho }_{\mu }h^{\mu }_{\rho }+R_{\mu \epsilon }h^{\epsilon \mu }\right) \left\{ \left( P+\frac{q^2}{24\pi r^4}\right) \right. \nonumber \\&-\left. \left. \frac{1}{3}\left( \Pi -\frac{q^2}{4\pi r^4}\right) \right\} \right] +\frac{8\pi }{H}\left[ \left\{ \frac{R}{2}\left( \frac{f}{R}-f_{R}\right) +\left( \mu -\frac{q^2}{2r^4}\right) f_{T}\right. \right. \nonumber \\&-\left. \frac{1}{2}\nabla _{\mu }\nabla _{\nu }(f_{Q}X^{\mu \nu })\right\} -\frac{1}{2}\Box \{f_{Q}(\mu -3P)\}+2Rf_{Q}\left\{ \left( P+\frac{q^2}{24\pi r^4}\right) \right. \nonumber \\&-\left. \frac{1}{3}\left( \Pi -\frac{q^2}{4\pi r^4}\right) \right\} +\nabla _{\mu }\nabla _{\rho }(f_{Q}X^{\mu \rho })+2g^{\rho \epsilon }(f_{Q}R^{\mu \nu }+f_{T}g^{\mu \nu })\nonumber \\&\times \left. \frac{\partial ^2L_{m}}{\partial g^{\rho \epsilon }\partial g^{\mu \nu }}\right] , \end{aligned}$$

(91)

$$\begin{aligned} \chi _{2}^{(D)}= & {} -\frac{8\pi }{H}\left\{ \frac{R}{2}\left( \frac{f}{R}-f_{R}\right) +\left( \mu -\frac{q^2}{2r^4}\right) f_{T}-\frac{1}{2}\nabla _{\mu }\nabla _{\nu }(f_{Q}X^{\mu \nu })\right\} \nonumber \\&+\frac{4\pi }{H}\left[ -\frac{1}{2}\left\{ \Box (f_{Q}X)-u^{\alpha }u^{\delta }\Box (f_{Q}X_{\alpha \delta })-u^{\beta }u_{\gamma }\Box (f_{Q}X^{\gamma }_{\beta }) \right. \right. \nonumber \\&+\left. 4u_{\gamma }u^{\delta }\Box (f_{Q}X^{\gamma }_{\delta })\right\} +\left( \Box f_{R}-u^{\alpha }u^{\delta }\nabla _{\alpha }\nabla _{\delta }f_{R}-u^{\beta }u_{\gamma }\nabla ^{\gamma }\nabla _{\beta }f_{R} \right. \nonumber \\&+\left. 4u_{\gamma }u^{\delta }\nabla ^{\gamma }\nabla _{\delta }f_{R}\right) +2f_{Q}R^{\beta }_{\mu }\left\{ \left( P+\frac{q^2}{24\pi r^4}\right) h^{\mu }_{\beta }-\left( \Pi -\frac{q^2}{4\pi r^4}\right) \right. \nonumber \\&\times \left. \left( s^{\mu }s_{\beta } +\frac{1}{3}h^{\mu }_{\beta }\right) \right\} -3f_{Q}\left( R^{\gamma }_{\mu }u^{\mu }u_{\gamma }+R_{\mu \delta } u^{\mu }u^{\delta }\right) \left( \mu +\frac{q^2}{8\pi r^4}\right) \nonumber \\&+\frac{1}{2}\{\nabla _{\mu }\nabla _{\alpha }(f_{Q}X^{\mu \alpha })+\nabla _{\mu }\nabla _{\beta }(f_{Q}X^{\mu \beta }) +4u_{\gamma }u^{\delta }\nabla _{\mu }\nabla ^{\gamma }(f_{Q}X^{\mu }_{\delta })\nonumber \\&+4u_{\gamma }u^{\delta }\nabla _{\mu }\nabla _{\delta }(f_{Q}X^{\mu \gamma }) -u^{\alpha }u^{\delta }\nabla _{\mu }\nabla _{\alpha }(f_{Q}X^{\mu }_{\delta }) -u_{\gamma }u^{\beta }\nabla _{\mu }\nabla _{\beta }(f_{Q}X^{\gamma \mu })\nonumber \\&-u_{\gamma }u^{\beta }\nabla _{\mu }\nabla ^{\gamma }(f_{Q}X^{\mu }_{\beta }) -u^{\alpha }u^{\delta }\nabla _{\mu }\nabla _{\delta }(f_{Q}X^{\mu }_{\alpha })\}+2h^{\epsilon \beta }(f_{Q}R^{\mu \nu }+f_{T}g^{\mu \nu })\nonumber \\&\times \left. \frac{\partial ^2L_{m}}{\partial g^{\epsilon \beta }\partial g^{\mu \nu }}\right] +\frac{8\pi }{H}\left[ \frac{1}{2}\Box \{f_{Q}(\mu -3P)\}+2f_{Q}R_{\mu \epsilon }X^{\mu \epsilon } -\nabla _{\mu }\nabla _{\epsilon }(f_{Q}X^{\mu \epsilon })\right. \nonumber \\&-\left. 2g^{\epsilon \xi }(f_{Q}R^{\mu \nu }+f_{T}g^{\mu \nu })\frac{\partial ^2L_{m}}{\partial g^{\epsilon \xi }\partial g^{\mu \nu }}\right] , \end{aligned}$$

(92)

$$\begin{aligned} \chi _{\alpha \beta }^{(D)}= & {} -\frac{2\pi }{H}\left[ h^{\lambda }_{\alpha }h^{\pi }_{\beta }\Box (f_{Q}X_{\lambda \pi })-\Box (f_{Q}X_{\alpha \beta }) -u_{\alpha }u_{\beta }u_{\gamma }u^{\delta }\Box (f_{Q}X^{\gamma }_{\delta })\right] \nonumber \\&+\frac{4\pi }{H}\left[ (h^{\lambda }_{\alpha }h^{\pi }_{\beta }\nabla _{\pi }\nabla _{\lambda }f_{R}-\nabla _{\alpha }\nabla _{\beta }f_{R}-u_{\alpha }u_{\beta }u_{\gamma }u^{\delta }\nabla ^{\gamma }\nabla _{\delta }f_{R}) \right. \nonumber \\&+f_{Q}(R_{\lambda \mu }h^{\lambda }_{\alpha }-R_{\alpha \mu })\left\{ \left( P+\frac{q^2}{24\pi r^4}\right) h^{\mu }_{\beta }-\left( \Pi -\frac{q^2}{4\pi r^4}\right) \right. \nonumber \\&\times \left. \left( s^{\mu }s_{\beta }+\frac{1}{3}h^{\mu }_{\beta }\right) \right\} +f_{Q}(R_{\mu \pi }h^{\pi }_{\beta }-R_{\mu \beta })\left\{ \left( P+\frac{q^2}{24\pi r^4}\right) h^{\mu }_{\alpha }\right. \nonumber \\&-\left. \left( \Pi -\frac{q^2}{4\pi r^4}\right) \left( s^{\mu }s_{\alpha }+\frac{1}{3}h^{\mu }_{\alpha }\right) \right\} +\frac{1}{2}\{h^{\lambda }_{\alpha }h^{\pi }_{\beta }\nabla _{\mu }\nabla _{\lambda } (f_{Q}X^{\mu }_{\pi })\nonumber \\&+h^{\lambda }_{\alpha }h^{\pi }_{\beta }\nabla _{\mu }\nabla _{\pi }(f_{Q}X^{\mu }_{\lambda }) -\nabla _{\mu }\nabla _{\alpha }(f_{Q}X^{\mu }_{\beta }) -\nabla _{\mu }\nabla _{\beta }(f_{Q}X^{\mu }_{\alpha })\nonumber \\&-u_{\alpha }u_{\beta }u_{\gamma }u^{\delta }\nabla _{\mu }\nabla ^{\gamma }(f_{Q}X^{\mu }_{\delta }) -u_{\alpha }u_{\beta }u_{\gamma }u^{\delta }\nabla _{\mu }\nabla _{\delta }(f_{Q}X^{\mu \gamma })\}\nonumber \\&+\left. 2(f_{Q}R^{\mu \nu }+f_{T}R^{\mu \nu })h^{\epsilon }_{\alpha }\left\{ h^{\pi }_{\beta }\frac{\partial ^2L_{m}}{\partial g^{\epsilon \pi }\partial g^{\mu \nu }} -\frac{\partial ^2L_{m}}{\partial g^{\epsilon \beta }\partial g^{\mu \nu }}\right\} \right] . \end{aligned}$$

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