Appendix 1
According to the product law of differentiation, the covariant differentiation is very similar to ordinary differentiations. But there is an important property of ordinary differentiation, i.e., if we perform two differentiations in succession, their order does not matter but it does not hold for covariant differentiation in general.
$$\begin{aligned} A_{\mu ,\nu }=A_{\mu :\nu }-\varGamma ^{\alpha } _{\mu \nu }A_{\alpha }. \end{aligned}$$
(65)
This is symmetric w.r.t. \(\mu \) and \(\nu \). So in this case, the order of covariant differentiations does not matter. Now, we take a vector \(A_{\nu }\) and apply two covariant differentiations on it.
$$\begin{aligned} A_{\nu ;\rho ;\sigma }&= A_{\nu ;\rho ,\sigma }- \varGamma ^{\alpha } _{\nu \sigma }A_{\alpha ;\rho } - \varGamma ^{\alpha } _{\rho \sigma }A_{\nu ;\alpha },\\&= \left( A_{\nu ,\rho }-{\varGamma ^{\alpha }}_{\nu \rho } A_{\alpha }\right) ,\sigma -{\varGamma ^{\alpha }} _{\nu \sigma } \left( A_{\alpha ,\rho } - {\varGamma ^{\beta }} _{\alpha \rho }A_{\beta }\right) \\&-{\varGamma ^{\alpha }}_{\rho \sigma }\left( A_{\nu ,\alpha } - {\varGamma ^{\beta }}_{\nu \alpha } A_{\beta }\right) ,\\&= A_{\nu ,\rho ,\sigma }-\varGamma ^{\alpha }_{\nu \rho }A_{\alpha ,\sigma }- \varGamma ^{\alpha }_{\nu \sigma }A_{\alpha ,\rho } -\varGamma ^{\alpha }_{\rho \sigma }A_{\nu ,\alpha }\\&- A_{\beta }\left( \varGamma ^{\beta }_{\nu \rho ,\sigma }-\varGamma ^{\alpha } _{\nu \rho }\varGamma ^{\beta }_{\alpha \rho }-\varGamma ^{\alpha } _{\rho \sigma }\varGamma ^{\beta }_{\nu \alpha }\right) . \end{aligned}$$
Interchanging \({\rho }\) and \({\sigma }\) here and subtract from the previous expression. The result is
$$\begin{aligned} A_{\nu ;\rho ;\sigma }-A_{\nu ;\sigma ;\rho }=A_{\beta } {R^{\beta }}_{\nu \rho \sigma }, \end{aligned}$$
(66)
where
$$\begin{aligned} {R^{\beta }}_{\nu \rho \sigma }={\varGamma ^{\beta }}_{\nu \sigma ,\rho } -{\varGamma ^{\beta }}_{\nu \rho ,\sigma }+{\varGamma ^{\alpha }}_{\nu \sigma } {\varGamma ^{\beta }}_{\alpha \rho }-{\varGamma ^{\alpha }}_{\nu \rho } {\varGamma ^{\beta }}_{\alpha \sigma }. \end{aligned}$$
(67)
The left-hand side of Eq. (67) is a tensor. It follows that the right-hand side must be a tensor. Thus, by quotient theorem \({{R}}^{\beta }_{\nu \rho \sigma }\) is tensor. It is called curvature tensor. It is antisymmetric in its lower index, i.e.,
$$\begin{aligned} {R^{\beta }}_{\nu \rho \sigma }=-{R^{\beta }}_{\nu \sigma \rho }. \end{aligned}$$
(68)
also from Eq. (68), we have
$$\begin{aligned} {R^{\beta }}_{\nu \rho \sigma }+{R^{\beta }}_{\rho \sigma \nu }+{R^{\beta }}_{\sigma \nu \rho }=0. \end{aligned}$$
(69)
Let us lower the suffix \({{\beta }}\) and put it as the first suffix. We get,
$$\begin{aligned} R_{\mu \nu \rho \sigma }=g_{\mu \beta }{R^{\beta }}_{\nu \rho \sigma }=g_{\mu \beta } {\varGamma ^{\beta }}_{\nu \sigma ,\rho }+{\varGamma ^{\alpha }}_{\nu \sigma } \varGamma _{\mu \alpha \rho }-\langle \rho \sigma \rangle . \end{aligned}$$
where the symbol \(\langle \rho \sigma \rangle \) is used to denote the proceeding terms with \(\rho \) and \(\sigma \) interchanged.
$$\begin{aligned} R_{\mu \nu \rho \sigma }&= \varGamma _{\mu \nu \sigma ,\rho } - g_{\mu \beta ,\rho }{\varGamma ^{\beta }}_{\nu \sigma }+ \varGamma _{\mu \beta \rho }{\varGamma ^{\beta }}_{\nu \sigma }- \langle \rho \sigma \rangle ,\\&= \varGamma _{\mu \nu \sigma ,\rho }-\varGamma _{\beta \mu \rho } {\varGamma ^{\beta }}_{\nu \sigma }-\langle \rho \sigma \rangle . \end{aligned}$$
Hence,
$$\begin{aligned} R_{\mu \nu \rho \sigma }=\frac{1}{2}(g_{\mu \sigma ,\nu \rho } -g_{\nu \sigma ,\mu \rho }-g_{\mu \rho ,\nu \sigma }+g_{\nu \rho ,\mu \sigma }) +\varGamma _{\beta \mu \sigma } \varGamma ^{\beta } _{\nu \rho }-\varGamma _{\mu \beta \rho }{\varGamma ^{\beta }}_{\nu \sigma }. \end{aligned}$$
(70)
So further symmetries now show up, namely
$$\begin{aligned} R_{\mu \nu \rho \sigma }=-R_{\nu \mu \rho \sigma }, \end{aligned}$$
(71)
and
$$\begin{aligned} R_{\mu \nu \rho \sigma }=R_{\rho \sigma \mu \nu }=R_{\sigma \rho \mu \nu }. \end{aligned}$$
(72)
The result of all these symmetries is that of the 256 components of \(R_{\mu \nu \rho \sigma }\), and only 20 are independent.
Let us contract two of the suffixes in \(R_{\mu \nu \rho \sigma }\). If we take two suffixes with respect to which it is antisymmetrical, we get zero, of course. If we take any other two, we get the same result, because of these symmetries,
Let us take the first and last suffixes
$$\begin{aligned} {R^{\mu }}_{\nu \rho \mu } = R_{\nu \rho }. \end{aligned}$$
(73)
It is called Ricci tensor. Contracting Eq. (67) by \(g^{\mu \sigma }\), we get
$$\begin{aligned} R_{\nu \rho }=R_{\rho \nu }. \end{aligned}$$
(74)
It can be contracted again and formed
$$\begin{aligned} g^{\nu \rho }R_{\nu \rho }={R^{\nu }}_{\nu }=R. \end{aligned}$$
(75)
This is a scalar and is called scaler curvature or total curvature. It is defined in such a way that it is positive for the surface of a sphere in three dimensions.
Appendix 2
The corresponding non-vanishing components of Riemann tensor for the Gödel-type metrics are given as:
$$\begin{aligned} R_{0002}&= R_{2020} = \frac{H(r)H^{\prime ^{2}}(r)}{4B^{2}(r)},\\ R_{1010}&= \frac{H^{\prime ^{2}}(r)}{4B^{2}(r)},\\ R_{1012}&= \frac{-2B(r)B^{\prime }(r)H^{\prime }(r) + H(r)H^{\prime ^{2}}(r) + 2B^{2}(r)H^{\prime \prime }(r)}{4B^{2}(r)},\\ R_{0101}&= \frac{B(r)H^{\prime ^{2}}(r) + H(r)(-2B^{\prime }(r)H^{\prime }(r)+2B(r)H^{\prime \prime }(r))}{4B^{3}(r)},\\ R_{0121}&= \frac{1}{2B^{3}(r)}\left[ H^{2}(r)B^{\prime }(r)H^{\prime }(r) \right. \\&\left.\quad + B^{2}(r)(B^{\prime }(r)H^{\prime }(r) + 2H(r)B^{\prime \prime }(r))\right. \\&\left. \quad - B^{3}(r)H^{\prime \prime }(r) - B(r)H(r)\left( 2H^{\prime ^{2}}(r) + H(r)H^{\prime \prime }(r)\right) \right] ,\\ R_{2110}&= \frac{B^{\prime }(r)H^{\prime }(r) - B(r)H^{\prime \prime }(r)}{2B^{3}(r)},\\ R_{2121}&= \frac{B(r)\left( 3H^{\prime ^{2}}(r) - 4B(r)B^{\prime \prime }(r)\right) + H(r)(- 2B^{\prime }(r)H^{\prime }(r) + 2B(r)H^{\prime \prime }(r))}{4B^{3}(r)},\\ R_{0220}&= \frac{\left( B^{2}(r) - H^{2}(r)\right) H^{\prime ^{2}}(r)}{4B^{2}(r)},\\ R_{1210}&= \frac{1}{4}\left( \frac{H^{\prime }(r)(- 2B(r)B^{\prime }(r) + H(r)H^{\prime }(r))}{B^{2}(r)} \right. \\&\left. \quad + 2H^{\prime \prime }(r)\right) ,\\ R_{1221}&= \frac{H(r)B^{\prime }(r)H^{\prime }(r)}{B(r)} \\&\quad- \frac{3}{4}H^{\prime ^{2}}(r) - \frac{H^{2}(r)H^{\prime ^{2}}(r)}{4B^{2}(r)} + B(r)B^{\prime \prime }(r) - H(r)H^{\prime \prime }(r),\\ R_{2020}&= -\frac{H^{\prime ^{2}}(r)}{4B^{2}(r)},\\ R_{2202}&= -\frac{H(r)H^{\prime ^{2}}(r)}{4B^{2}(r)}. \end{aligned}$$
The surviving components of C-tensor are given as:
$$\begin{aligned} C^{00}&= \frac{1}{2B^{5}(r)}\left( H^{2}(r)\partial _3\varTheta H^{\prime }(r)\left( 2B^{\prime ^{2}}(r) + H^{\prime ^{2}}(r)\right) \right. \nonumber \\&\left.\quad - B(r)H(r)\left( 6\partial _3\varTheta B^{\prime }(r)H^{\prime ^{2}}(r)\right. \right. \nonumber \\&\left. \left. \quad+ H(r)(2\partial _3\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left.\quad +B^{\prime }(r)(\partial _3\partial _1\varTheta H^{\prime }(r)+ 2\partial _3\varTheta H^{\prime \prime }(r)))\right) \right. \nonumber \\&\left. \quad - B^{3}(r)(3\partial _3\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ B^{\prime }(r)(\partial _3\partial _1\varTheta H^{\prime }(r)+ \partial _3\varTheta H^{\prime \prime }(r))\right. \nonumber \\&\left. \quad+ 2H(r)(\partial _3\partial _1\varTheta B^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ \partial _3\varTheta B^{3}(r)))+ B^{4}(r)(\partial _3\partial _1\varTheta H^{\prime \prime }(r) + \partial _3\varTheta H^{3}(r))\right. \nonumber \\&\left. \quad + B^{2}(r)\left( \partial _3\varTheta B^{\prime ^{2}}(r)H^{\prime }(r)+ 2\partial _3\varTheta H^{\prime ^{3}}(r) \right. \right. \nonumber \\&\left. \left.\quad + 2H(r)\partial _3\varTheta B^{\prime }(r)B^{\prime \prime }(r)\nonumber \right. \right. \\&\left. \left.\quad + 2H(r)H^{\prime }(r)(\partial _3\partial _1\varTheta H^{\prime }(r)+3\partial _3\varTheta H^{\prime \prime }(r))\right. \right. \nonumber \\&\left. \left. \quad+H^{2}(r)(\partial _3\partial _1\varTheta H^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad+ \partial _3\varTheta H^{3}(r))\right) \right) ,\nonumber \\ C^{01}&= \frac{1}{4B^{4}(r)}(H(r)(H(r)\partial _3\partial _0\varTheta - \partial _3\partial _2\varTheta )B^{\prime }(r)H^{\prime }(r)\nonumber \\&\quad+ B^{2}(r)(\partial _3\partial _0\varTheta B^{\prime }(r)H^{\prime }(r)\nonumber \\&\quad+2(H(r)\partial _3\partial _0\varTheta - \partial _3\partial _2\varTheta )B^{\prime \prime }(r))\nonumber \\&\quad - B^{3}(r)\partial _3\partial _0\varTheta H^{\prime \prime }(r)-B(r)(H(r)\partial _3\partial _0\varTheta \nonumber \\&\quad - \partial _3\partial _2\varTheta )(2H^{\prime }(r)\nonumber \\&\quad + H(r)H^{\prime \prime }(r))),\nonumber \\ C^{02}&= \frac{1}{2B^{5}(r)}\left( -H(r)\partial _3\varTheta H^{\prime }(r)\left( 2B^{\prime ^{2}}(r) + H^{\prime ^{2}}(r)\right) \right. \nonumber \\&\left. \quad + B(r)\left( 3\partial _3\varTheta B^{\prime }(r)H^{\prime ^{2}}(r)\right. \right. \nonumber \\&\left. \left. \quad + H(r)(2\partial _3\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left.\quad +B^{\prime }(r)(\partial _3\partial _1\varTheta H^{\prime }(r) + 2\partial _3\varTheta H^{\prime \prime }(r)))\right) + B^{3}(r)(\partial _3\partial _1\varTheta B^{\prime \prime }(r)\right. \nonumber \\&\left.\quad +\partial _3\varTheta B^{(3)}(r))- B^{2}(r)\left( \partial _3\partial _1\varTheta H^{\prime ^{2}}(r)+ \partial _3\varTheta B^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad + H(r)\partial _3\partial _1\varTheta H^{\prime \prime }(r)+ 3\partial _3\varTheta H^{\prime }(r)H^{\prime \prime }(r)+ H(r)\partial _3\varTheta H^{3}(r)\right) \right) ,\nonumber \\ C^{03}&= \frac{1}{4B^{5}(r)}\left( 3H(r)(- H(r)\partial _0\varTheta +\partial _2\varTheta )B^{\prime ^{2}}(r)H^{\prime }(r)\right. \nonumber \\&\left. \quad+ B(r)\left( - 2\partial _2\varTheta B^{\prime }(r)H^{\prime ^{2}}(r)+ H^{2}(r)(\partial _0\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left.\quad + B^{\prime }(r)(\partial _1\partial _0\varTheta H^{\prime }(r)+ 3\partial _0\varTheta H^{\prime \prime }(r)))- H(r)(\partial _2\varTheta H^{\prime }(r)B^{\prime \prime }(r) \right. \right. \nonumber \\&\left. \left. \quad+ B^{\prime }(r)(\partial _2\partial _1\varTheta H^{\prime }(r)- 6\partial _0\varTheta H^{\prime ^{2}}(r)+ 3\partial _2\varTheta H^{\prime \prime }(r)))\right) \right. \nonumber \\&\left. \quad+ B^{3}(r)(3\partial _0\varTheta H^{\prime }(r)B^{\prime \prime }(r)+ B^{\prime }(r)(\partial _1\partial _0\varTheta H^{\prime }(r) +\partial _0\varTheta H^{\prime \prime }(r))\right. \nonumber \\&\left.\quad + 2H(r)(\partial _1\partial _0\varTheta B^{\prime \prime }(r)+ \partial _0\varTheta B^{3}(r))) - B^{4}(r)(\partial _1\partial _0\varTheta H^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ \partial _0\varTheta H^{(3)}(r))- B^{2}(r)\left( \partial _0\varTheta B^{\prime ^{2}}(r)H^{\prime }(r) + 2\partial _0\varTheta H^{\prime ^{3}}(r)\right. \right. \nonumber \\&\left. \left.\quad + 2H(r)\partial _0\varTheta B^{\prime }(r)B^{\prime \prime }(r)- 2\partial _2\varTheta H^{\prime }(r)H^{\prime \prime }(r)+ H^{2}(r)(\partial _1\partial _0\varTheta H^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad+ \partial _0\varTheta H^{(3)}(r))+ H(r)\left( 2\partial _1\partial _0\varTheta H^{\prime ^{2}}(r)-\partial _2\partial _1\varTheta H^{\prime \prime }(r)\right. \right. \right. \nonumber \\&\left. \left. \left. \quad+ 6\partial _0\varTheta H^{\prime }(r)H^{\prime \prime }(r)- \partial _2\varTheta H^{(3)}(r)\right) \right) \right) ,\nonumber \\ C^{11}&= \frac{1}{2B^{3}(r)}\left( \partial _{3}\varTheta \left( - B^{\prime ^2}(r)H^{\prime }(r)+ H^{\prime ^{3}}(r) - B(r)H^{\prime }(r)B^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad+ B(r)B^{\prime }(r) H^{\prime \prime }(r)\right) \right) ,\nonumber \\ C^{12}&= \frac{1}{4B^{4}(r)}(H(r)\partial _3 \partial _0 \varTheta - \partial _3 \partial _2 \varTheta )( - B^{\prime }(r)H^{\prime }(r) + B(r)H^{\prime \prime }(r)),\nonumber \\ C^{13}&= \frac{1}{4B^{4}(r)}\left( -(H^{2}(r)\partial _0 \partial _0\varTheta -2H(r)\partial _2\partial _0\varTheta +\partial _2\partial _2 \varTheta )B^{\prime }(r) H^{\prime }(r)\right. \nonumber \\&\left. \quad- B^{2}(r)(\partial _0\partial _0\varTheta B^{\prime }(r)H^{\prime }(r)+2(H(r)\partial _0\partial _0\varTheta - \partial _2\partial _0\varTheta )B^{\prime \prime }(r))\right. \nonumber \\&\left. \quad+ B^{3}(r)\partial _0\partial _0\varTheta H^{\prime \prime }(r)+ B(r)\left( -2\partial _2\partial _0\varTheta H^{\prime ^{2}}(r) + H^{2}(r)\partial _0\partial _0\varTheta H^{\prime \prime }(r)\right. \right. \nonumber \\&\left. \left. \quad+\partial _2\partial _2\varTheta H^{\prime \prime }(r)+ 2H(r)\left( \partial _0\partial _0\varTheta H^{\prime ^{2}}(r) - \partial _2\partial _0\varTheta H^{\prime \prime }(r)\right) \right) \right) ,\nonumber \\ C^{22}&= \frac{1}{2B^{5}(r)}\left( {\partial _3}\varTheta H^{\prime }(r)\left( 2B^{\prime ^{2}}(r) + H^{\prime ^{2}}(r)\right) -B(r)(2{\partial _3}\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ B^{\prime }(r)(\partial _3 \partial _1 \varTheta H^{\prime }(r)+ 2\partial _3 \varTheta H^{\prime \prime }(r))) + B^{2}(r)(\partial _3 \partial _1 \varTheta H^{\prime \prime }(r)\right. \nonumber \\&\left.\quad +\partial _3 \varTheta H^{(3)}(r))\right) ,\nonumber \\ C^{23}&= \frac{1}{4B^{5}(r)}\left( 3(H(r)\partial _0 \varTheta -\partial _2 \varTheta ) B^{\prime ^{2}}(r)H^{\prime }(r) \right. \nonumber \\&\left. \quad+ B(r)({\partial _2}\varTheta H^{\prime }(r)B^{\prime \prime }(r)\right. \nonumber \\&\left.\quad + B^{\prime }(r)\left( \partial _2\partial _1\varTheta H^{\prime }(r) - 4 \partial _0\varTheta H^{\prime ^{2}}(r) + 3 \partial _2\varTheta H^{\prime \prime }(r)\right) \right. \nonumber \\&\left. \quad - H(r)(\partial _0 \varTheta H^{\prime }(r)B^{\prime \prime }(r)+ B^{\prime }(r)(\partial _1\partial _0\varTheta H^{\prime }(r) + 3 \partial _0 \varTheta H^{\prime \prime }(r))))\right. \nonumber \\&\left. \quad -2B^{3}(r)(\partial _1 \partial _0 \varTheta B^{\prime \prime }(r)+ \partial _0 \varTheta B^{(3)}(r))+ B^{2}(r)(2 \partial _1 \partial _0 \varTheta H^{\prime ^{2}}(r)\right. \nonumber \\&\left. \quad+2 \partial _0 \varTheta B^{\prime }(r)B^{\prime \prime }(r)+ H(r)\partial _1 \partial _0 \varTheta H^{\prime \prime }(r)- \partial _2\partial _1\varTheta H^{\prime \prime }(r)\right. \nonumber \\&\left. \quad+ 4\partial _0\varTheta H^{\prime }(r)H^{\prime \prime }(r)+ H(r)\partial _0\varTheta H^{(3)}(r)-\partial _2\varTheta H^{(3)}(r))\right) . \end{aligned}$$
(76)
The nonzero components of C-tensor for the spherically symmetric metrics are given by
$$\begin{aligned}&C_{02}=C_{20}=-C_{03}=-C_{30}\nonumber \\&\quad = -\frac{l}{2A^5 B^4} \csc \theta \left( -3\partial _{3}\varTheta B A^{\prime 2}B^{\prime } + A(-\partial _{3} \varTheta A^{\prime }B^{\prime 2}\right. \nonumber \\&\left. \qquad + B(\partial _{3} \varTheta B^{\prime }A^{\prime \prime }+A^{\prime }( \partial _{3}\partial _{1}\varTheta B^{\prime }+ 3\partial _{3} \varTheta B^{\prime \prime })))+ A^2(\partial _{3}\partial _{1}\varTheta B^{\prime 2}\right. \nonumber \\&\left. \qquad + \partial _{3} \varTheta B^{\prime }A^{\prime \prime }+B(- \partial _{3}\partial _{1}\varTheta B^{\prime \prime } - \partial _{3} \varTheta (B^{\prime \prime \prime }+ A^{\prime }(\dot{A}\dot{B} - B\ddot{A}))))\right. \nonumber \\&\left. \qquad + A^{3}(\partial _{3} \varTheta B^{\prime }\dot{A}\dot{B}+B(\dot{A}(\partial _{3}\partial _{1}\varTheta \dot{B} + \partial _{3} \varTheta \dot{B^{\prime }}) + \partial _{3} \varTheta (\dot{B}\dot{A}^{\prime } - 2B^{\prime }\ddot{A}))\right. \nonumber \\&\left. \qquad - B^{2}(\partial _{3}\partial _{1}\varTheta \ddot{A} + \partial _{3} \varTheta \ddot{A^{\prime }}))+ A^{4}(- \partial _{3}\partial _{1}\varTheta - \partial _{3}\partial _{1}\varTheta \dot{B}^{2} - 2\partial _{3} \varTheta \dot{B}\dot{B}^{\prime }\right. \nonumber \\&\left. \qquad +\partial _{3} \varTheta B^{\prime }\ddot{B}+ B(\partial _{3}\partial _{1}\varTheta \ddot{B} + \partial _{3} \varTheta \ddot{B}^{\prime }))\right) , \end{aligned}$$
(77)
$$\begin{aligned}&C_{12}=C_{21}=-C_{13}=-C_{31}\nonumber \\&\quad =\frac{l}{2A^5B^4} \csc \theta \Bigg (- 3\partial _{2} \varTheta BA^{\prime 2}B^{\prime } + A(-\partial _{2} \varTheta A^{\prime }B^{\prime 2}\nonumber \\&\qquad +B(\partial _{2} \varTheta B^{\prime }A^{\prime \prime }+A^{\prime }( \partial _{2}\partial _{0}\varTheta B^{\prime }+3\partial _{2} \varTheta B^{\prime \prime })))+ A^2(\partial _{2}\partial _{0}\varTheta B^{\prime 2} + \partial _{2} \varTheta B^{\prime }A^{\prime \prime }\nonumber \\&\qquad +B(- \partial _{2}\partial _{0}\varTheta B^{\prime \prime } - \partial _{2} \varTheta (B^{\prime \prime \prime }+ A^{\prime }(\dot{A}\dot{B} - B\ddot{A}))))+ A^{3}(\partial _{2} \varTheta B^{\prime }\dot{A}\dot{B}\nonumber \\&\qquad +B(\dot{A}(\partial _{2}\partial _{0}\varTheta \dot{B}+\partial _{2} \varTheta \dot{B^{\prime }}) + \partial _{2} \varTheta (\dot{B}\dot{A}^{\prime }-2B^{\prime }\ddot{A}))- B^{2}(\partial _{2}\partial _{0}\varTheta \ddot{A} + \partial _{2} \varTheta \ddot{A^{\prime }}))\nonumber \\&\qquad +A^{4}\Bigg (- \partial _{2}\partial _{0}\varTheta -\partial _{2}\partial _{0}\varTheta \dot{B}^{2} - 2\partial _{2} \varTheta \dot{B}\dot{B}^{\prime }+ \partial _{2} \varTheta B^{\prime }\ddot{B}\nonumber \\&\qquad + B(\partial _{2}\partial _{0}\varTheta \ddot{B} + \partial _{2} \varTheta \ddot{B}^{\prime })\Bigg )\Bigg ). \end{aligned}$$
(78)