metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{A} & 0 & 0 & 0 \\ 0 & B & 0 & 0 \\ 0 & 0 & {r}^{2} & 0 \\ 0 & 0 & 0 & {{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1} - {\cos\left( \theta\right)}}\right)}}\end{matrix} \right]}}$
metric inverse:
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{\frac{1}{A}} & 0 & 0 & 0 \\ 0 & \frac{1}{B} & 0 & 0 \\ 0 & 0 & \frac{1}{{r}^{2}} & 0 \\ 0 & 0 & 0 & \frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1} - {\cos\left( \theta\right)}}\right)}}}\end{matrix} \right]}}$
metric derivative:
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{ A_{,{{t}}}} & -{ A_{,{{r}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ B_{,{{t}}} & B_{,{{r}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & {{2}} {{r}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & {{2}} {{r}} {{{\sin\left( \theta\right)}^{2}}} & {{2}} {{{r}^{2}}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} & 0\end{matrix} \right]}\end{matrix} \right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} -{{\frac{1}{2}} { A_{,{{t}}}}} & -{{\frac{1}{2}} { A_{,{{r}}}}} & 0 & 0 \\ -{{\frac{1}{2}} { A_{,{{r}}}}} & -{{\frac{1}{2}} { B_{,{{t}}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} {\frac{1}{2}} { A_{,{{r}}}} & {\frac{1}{2}} { B_{,{{t}}}} & 0 & 0 \\ {\frac{1}{2}} { B_{,{{t}}}} & {\frac{1}{2}} { B_{,{{r}}}} & 0 & 0 \\ 0 & 0 & -{r} & 0 \\ 0 & 0 & 0 & -{{{r}} {{{\sin\left( \theta\right)}^{2}}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & r & 0 \\ 0 & r & 0 & 0 \\ 0 & 0 & 0 & -{{{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{matrix} \right]} \\ \overset{b\downarrow c\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {{r}} {{{\sin\left( \theta\right)}^{2}}} \\ 0 & 0 & 0 & {{{r}^{2}}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} \\ 0 & {{r}} {{{\sin\left( \theta\right)}^{2}}} & {{{r}^{2}}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} & 0\end{matrix} \right]}\end{matrix} \right]}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} \frac{ A_{,{{t}}}}{{{2}} {{A}}} & \frac{ A_{,{{r}}}}{{{2}} {{A}}} & 0 & 0 \\ \frac{ A_{,{{r}}}}{{{2}} {{A}}} & \frac{ B_{,{{t}}}}{{{2}} {{A}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} \frac{ A_{,{{r}}}}{{{2}} {{B}}} & \frac{ B_{,{{t}}}}{{{2}} {{B}}} & 0 & 0 \\ \frac{ B_{,{{t}}}}{{{2}} {{B}}} & \frac{ B_{,{{r}}}}{{{2}} {{B}}} & 0 & 0 \\ 0 & 0 & -{{\frac{1}{B}} {r}} & 0 \\ 0 & 0 & 0 & -{{\frac{1}{B}} {{{r}} {{{\sin\left( \theta\right)}^{2}}}}}\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & \frac{1}{r} & 0 & 0 \\ 0 & 0 & 0 & -{{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{r} \\ 0 & 0 & 0 & \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)} \\ 0 & \frac{1}{r} & \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)} & 0\end{matrix} \right]\end{matrix} \right]}}$
connection coefficients derivative:
${{{{{ \Gamma} ^a} _b} _c} _{,d}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{{A}} {{ A_{,{{t}}{{t}}}}}} - {{ A_{,{{t}}}}^{2}}}{{{2}} {{{A}^{2}}}} & \frac{{{{A}} {{ A_{,{{r}}{{t}}}}}} - {{{ A_{,{{r}}}}} {{ A_{,{{t}}}}}}}{{{2}} {{{A}^{2}}}} & 0 & 0 \\ \frac{{{{A}} {{ A_{,{{r}}{{t}}}}}} - {{{ A_{,{{r}}}}} {{ A_{,{{t}}}}}}}{{{2}} {{{A}^{2}}}} & \frac{{{{A}} {{ A_{,{{r}}{{r}}}}}} - {{ A_{,{{r}}}}^{2}}}{{{2}} {{{A}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{{A}} {{ A_{,{{r}}{{t}}}}}} - {{{ A_{,{{r}}}}} {{ A_{,{{t}}}}}}}{{{2}} {{{A}^{2}}}} & \frac{{{{A}} {{ A_{,{{r}}{{r}}}}}} - {{ A_{,{{r}}}}^{2}}}{{{2}} {{{A}^{2}}}} & 0 & 0 \\ \frac{{{{A}} {{ B_{,{{t}}{{t}}}}}} - {{{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}{{{2}} {{{A}^{2}}}} & \frac{{{{A}} {{ B_{,{{r}}{{t}}}}}} - {{{ A_{,{{r}}}}} {{ B_{,{{t}}}}}}}{{{2}} {{{A}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{{B}} {{ A_{,{{r}}{{t}}}}}} - {{{ A_{,{{r}}}}} {{ B_{,{{t}}}}}}}{{{2}} {{{B}^{2}}}} & \frac{{{{B}} {{ A_{,{{r}}{{r}}}}}} - {{{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{{{2}} {{{B}^{2}}}} & 0 & 0 \\ \frac{{{{B}} {{ B_{,{{t}}{{t}}}}}} - {{ B_{,{{t}}}}^{2}}}{{{2}} {{{B}^{2}}}} & \frac{{{{B}} {{ B_{,{{r}}{{t}}}}}} - {{{ B_{,{{r}}}}} {{ B_{,{{t}}}}}}}{{{2}} {{{B}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{{B}} {{ B_{,{{t}}{{t}}}}}} - {{ B_{,{{t}}}}^{2}}}{{{2}} {{{B}^{2}}}} & \frac{{{{B}} {{ B_{,{{r}}{{t}}}}}} - {{{ B_{,{{r}}}}} {{ B_{,{{t}}}}}}}{{{2}} {{{B}^{2}}}} & 0 & 0 \\ \frac{{{{B}} {{ B_{,{{r}}{{t}}}}}} - {{{ B_{,{{r}}}}} {{ B_{,{{t}}}}}}}{{{2}} {{{B}^{2}}}} & \frac{{{{B}} {{ B_{,{{r}}{{r}}}}}} - {{ B_{,{{r}}}}^{2}}}{{{2}} {{{B}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{{r}} {{ B_{,{{t}}}}}}{{B}^{2}} & \frac{{-{B}} + {{{r}} {{ B_{,{{r}}}}}}}{{B}^{2}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ B_{,{{t}}}}}}{{B}^{2}} & \frac{{-{B}} + {{{r}} {{ B_{,{{r}}}}}} + {{{{B}} {{{\cos\left( \theta\right)}^{2}}}} - {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}}{{B}^{2}} & -{{\frac{1}{B}} {{{2}} {{r}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & -{\frac{1}{{r}^{2}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & -{\frac{1}{{r}^{2}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & {1} - {{{2}} {{{\cos\left( \theta\right)}^{2}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & -{\frac{1}{{r}^{2}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -{\frac{1}{{\sin\left( \theta\right)}^{2}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & -{\frac{1}{{r}^{2}}} & 0 & 0 \\ 0 & 0 & -{\frac{1}{{\sin\left( \theta\right)}^{2}}} & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
connection coefficients squared:
${{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{{A}} {{{ A_{,{{r}}}}^{2}}}} + {{{B}} {{{ A_{,{{t}}}}^{2}}}}}{{{4}} {{B}} {{{A}^{2}}}} & \frac{{{\left({{{{A}} {{ B_{,{{t}}}}}} + {{{B}} {{ A_{,{{t}}}}}}}\right)}} {{ A_{,{{r}}}}}}{{{4}} {{B}} {{{A}^{2}}}} & 0 & 0 \\ \frac{{{\left({{{{A}} {{ B_{,{{t}}}}}} + {{{B}} {{ A_{,{{t}}}}}}}\right)}} {{ A_{,{{r}}}}}}{{{4}} {{B}} {{{A}^{2}}}} & \frac{{{{A}} {{{ B_{,{{t}}}}^{2}}}} + {{{B}} {{{ A_{,{{r}}}}^{2}}}}}{{{4}} {{B}} {{{A}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{\left({{{{A}} {{ B_{,{{t}}}}}} + {{{B}} {{ A_{,{{t}}}}}}}\right)}} {{ A_{,{{r}}}}}}{{{4}} {{B}} {{{A}^{2}}}} & \frac{{{{B}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{{{4}} {{B}} {{{A}^{2}}}} & 0 & 0 \\ \frac{{{{A}} {{{ B_{,{{t}}}}^{2}}}} + {{{B}} {{{ A_{,{{r}}}}^{2}}}}}{{{4}} {{B}} {{{A}^{2}}}} & \frac{{{\left({{{{A}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}}}}}}\right)}} {{ B_{,{{t}}}}}}{{{4}} {{B}} {{{A}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{\frac{{{r}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}} & 0 \\ 0 & 0 & -{\frac{{{r}} {{ B_{,{{t}}}}}}{{{2}} {{A}} {{B}}}} & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}} \\ 0 & 0 & 0 & -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ B_{,{{t}}}}}}{{{2}} {{A}} {{B}}}} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{\left({{{{A}} {{ B_{,{{t}}}}}} + {{{B}} {{ A_{,{{t}}}}}}}\right)}} {{ A_{,{{r}}}}}}{{{4}} {{A}} {{{B}^{2}}}} & \frac{{{{B}} {{{ A_{,{{r}}}}^{2}}}} + {{{A}} {{{ B_{,{{t}}}}^{2}}}}}{{{4}} {{A}} {{{B}^{2}}}} & 0 & 0 \\ \frac{{{{B}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} + {{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{{{4}} {{A}} {{{B}^{2}}}} & \frac{{{\left({{{{A}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}}}}}}\right)}} {{ B_{,{{t}}}}}}{{{4}} {{A}} {{{B}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} \frac{{{{B}} {{{ A_{,{{r}}}}^{2}}}} + {{{A}} {{{ B_{,{{t}}}}^{2}}}}}{{{4}} {{A}} {{{B}^{2}}}} & \frac{{{\left({{{{A}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}}}}}}\right)}} {{ B_{,{{t}}}}}}{{{4}} {{A}} {{{B}^{2}}}} & 0 & 0 \\ \frac{{{\left({{{{A}} {{ B_{,{{r}}}}}} + {{{B}} {{ A_{,{{r}}}}}}}\right)}} {{ B_{,{{t}}}}}}{{{4}} {{A}} {{{B}^{2}}}} & \frac{{{{B}} {{{ B_{,{{t}}}}^{2}}}} + {{{A}} {{{ B_{,{{r}}}}^{2}}}}}{{{4}} {{A}} {{{B}^{2}}}} & 0 & 0 \\ 0 & 0 & -{\frac{1}{B}} & 0 \\ 0 & 0 & 0 & -{{\frac{1}{B}} {{\sin\left( \theta\right)}^{2}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{\frac{{{r}} {{ B_{,{{t}}}}}}{{{2}} {{{B}^{2}}}}} & 0 \\ 0 & 0 & -{\frac{{{r}} {{ B_{,{{r}}}}}}{{{2}} {{{B}^{2}}}}} & 0 \\ 0 & -{\frac{1}{B}} & 0 & 0 \\ 0 & 0 & 0 & -{{\frac{1}{B}} {{{r}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & -{\frac{{{r}} {{ B_{,{{t}}}}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{{B}^{2}}}}} \\ 0 & 0 & 0 & -{\frac{{{r}} {{ B_{,{{r}}}}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{{B}^{2}}}}} \\ 0 & 0 & 0 & {\frac{1}{B}} {{{r}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}} \\ 0 & -{{\frac{1}{B}} {{\sin\left( \theta\right)}^{2}}} & -{{\frac{1}{B}} {{{r}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}} & \frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{{r}^{2}} & 0 \\ \frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}} & \frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}} & 0 & 0 \\ 0 & 0 & 0 & -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{{r}^{2}} & 0 & 0 \\ 0 & 0 & -{\frac{1}{B}} & 0 \\ 0 & 0 & 0 & -{{\cos\left( \theta\right)}^{2}}\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}} \\ 0 & 0 & 0 & -{{\frac{1}{B}} {{\sin\left( \theta\right)}^{2}}} \\ 0 & -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}} & -{{\cos\left( \theta\right)}^{2}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}} & \frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{1}{{r}^{2}} \\ 0 & 0 & 0 & \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} \\ \frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}} & \frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}} & \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} \\ 0 & 0 & 0 & \frac{{\cos\left( \theta\right)}^{2}}{{\sin\left( \theta\right)}^{2}} \\ 0 & \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} & -{\frac{1}{B}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & \frac{1}{{r}^{2}} & \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} & 0 \\ 0 & \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}} & \frac{{\cos\left( \theta\right)}^{2}}{{\sin\left( \theta\right)}^{2}} & 0 \\ 0 & 0 & 0 & {\frac{1}{B}}{\left({{{-{1}} - {{{B}} {{{\cos\left( \theta\right)}^{2}}}}} + {{\cos\left( \theta\right)}^{2}}}\right)}\end{matrix} \right]}\end{matrix} \right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{-{{{A}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{B}} {{{ A_{,{{r}}}}^{2}}}} - {{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{ B_{,{{t}}{{t}}}}}} - {{{B}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{{{4}} {{B}} {{{A}^{2}}}} & 0 & 0 \\ \frac{{{{{{B}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} - {{{A}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}} - {{{B}} {{{ A_{,{{r}}}}^{2}}}}} + {{{A}} {{{ B_{,{{t}}}}^{2}}}}}{{{4}} {{B}} {{{A}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{\frac{{{r}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}} & 0 \\ 0 & 0 & -{\frac{{{r}} {{ B_{,{{t}}}}}}{{{2}} {{A}} {{B}}}} & 0 \\ \frac{{{r}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}} & \frac{{{r}} {{ B_{,{{t}}}}}}{{{2}} {{A}} {{B}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}}} \\ 0 & 0 & 0 & -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ B_{,{{t}}}}}}{{{2}} {{A}} {{B}}}} \\ 0 & 0 & 0 & 0 \\ \frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}{{{2}} {{A}} {{B}}} & \frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ B_{,{{t}}}}}}{{{2}} {{A}} {{B}}} & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{-{{{B}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{ B_{,{{t}}{{t}}}}}} - {{{A}} {{{ B_{,{{t}}}}^{2}}}}} + {{{B}} {{{ A_{,{{r}}}}^{2}}}}}{{{4}} {{A}} {{{B}^{2}}}} & 0 & 0 \\ \frac{{-{{{B}} {{{ A_{,{{r}}}}^{2}}}}} + {{{{A}} {{{ B_{,{{t}}}}^{2}}}} - {{{2}} {{A}} {{B}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}} - {{{A}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{B}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}}{{{4}} {{A}} {{{B}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & \frac{{{r}} {{ B_{,{{t}}}}}}{{{2}} {{{B}^{2}}}} & 0 \\ 0 & 0 & \frac{{{r}} {{ B_{,{{r}}}}}}{{{2}} {{{B}^{2}}}} & 0 \\ -{\frac{{{r}} {{ B_{,{{t}}}}}}{{{2}} {{{B}^{2}}}}} & -{\frac{{{r}} {{ B_{,{{r}}}}}}{{{2}} {{{B}^{2}}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & \frac{{{r}} {{ B_{,{{t}}}}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{{B}^{2}}}} \\ 0 & 0 & 0 & \frac{{{r}} {{ B_{,{{r}}}}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{{B}^{2}}}} \\ 0 & 0 & 0 & 0 \\ -{\frac{{{r}} {{ B_{,{{t}}}}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{{B}^{2}}}}} & -{\frac{{{r}} {{ B_{,{{r}}}}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{{B}^{2}}}}} & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{\frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}}} & 0 \\ 0 & 0 & -{\frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}}} & 0 \\ \frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}} & \frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & -{\frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}}} & 0 \\ 0 & 0 & -{\frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}}} & 0 \\ \frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}} & \frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{B}}{\left({{-{1}} + {B} + {{{\cos\left( \theta\right)}^{2}} - {{{B}} {{{\cos\left( \theta\right)}^{2}}}}}}\right)} \\ 0 & 0 & {\frac{1}{B}}{\left({{{{1} - {B}} - {{\cos\left( \theta\right)}^{2}}} + {{{B}} {{{\cos\left( \theta\right)}^{2}}}}}\right)} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & -{\frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}}} \\ 0 & 0 & 0 & -{\frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}}} \\ 0 & 0 & 0 & 0 \\ \frac{ A_{,{{r}}}}{{{2}} {{B}} {{r}}} & \frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & -{\frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}}} \\ 0 & 0 & 0 & -{\frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}}} \\ 0 & 0 & 0 & 0 \\ \frac{ B_{,{{t}}}}{{{2}} {{B}} {{r}}} & \frac{ B_{,{{r}}}}{{{2}} {{B}} {{r}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & {\frac{1}{B}}{\left({{1} - {B}}\right)} \\ 0 & 0 & {\frac{1}{B}}{\left({{-{1}} + {B}}\right)} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & \frac{{-{{{A}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{B}} {{{ A_{,{{r}}}}^{2}}}} - {{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{ B_{,{{t}}{{t}}}}}} - {{{B}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}} & 0 & 0 \\ \frac{{{{{{B}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} - {{{A}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}} - {{{B}} {{{ A_{,{{r}}}}^{2}}}}} + {{{A}} {{{ B_{,{{t}}}}^{2}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & -{\frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}} & 0 \\ 0 & 0 & -{\frac{ B_{,{{t}}}}{{{2}} {{A}} {{B}} {{r}}}} & 0 \\ \frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}} & \frac{ B_{,{{t}}}}{{{2}} {{A}} {{B}} {{r}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & -{\frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}} \\ 0 & 0 & 0 & -{\frac{ B_{,{{t}}}}{{{2}} {{A}} {{B}} {{r}}}} \\ 0 & 0 & 0 & 0 \\ \frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}} & \frac{ B_{,{{t}}}}{{{2}} {{A}} {{B}} {{r}}} & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & \frac{{{{{B}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} - {{{A}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{ B_{,{{t}}{{t}}}}}}} + {{{{A}} {{{ B_{,{{t}}}}^{2}}}} - {{{B}} {{{ A_{,{{r}}}}^{2}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}} & 0 & 0 \\ \frac{{{{{B}} {{{ A_{,{{r}}}}^{2}}}} - {{{A}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{2}} {{A}} {{B}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{B}} {{ A_{,{{r}}{{r}}}}}}} + {{{{A}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{B}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & \frac{ B_{,{{t}}}}{{{2}} {{r}} {{{B}^{2}}}} & 0 \\ 0 & 0 & \frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}} & 0 \\ -{\frac{ B_{,{{t}}}}{{{2}} {{r}} {{{B}^{2}}}}} & -{\frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & \frac{ B_{,{{t}}}}{{{2}} {{r}} {{{B}^{2}}}} \\ 0 & 0 & 0 & \frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}} \\ 0 & 0 & 0 & 0 \\ -{\frac{ B_{,{{t}}}}{{{2}} {{r}} {{{B}^{2}}}}} & -{\frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}} & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & \frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}} & 0 \\ 0 & 0 & \frac{ B_{,{{t}}}}{{{2}} {{A}} {{B}} {{r}}} & 0 \\ -{\frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}} & -{\frac{ B_{,{{t}}}}{{{2}} {{A}} {{B}} {{r}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & -{\frac{ B_{,{{t}}}}{{{2}} {{r}} {{{B}^{2}}}}} & 0 \\ 0 & 0 & -{\frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}} & 0 \\ \frac{ B_{,{{t}}}}{{{2}} {{r}} {{{B}^{2}}}} & \frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{-{1}} + {B}}{{{B}} {{{r}^{2}}}} \\ 0 & 0 & \frac{{1} - {B}}{{{B}} {{{r}^{2}}}} & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 & \frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}} \\ 0 & 0 & 0 & \frac{ B_{,{{t}}}}{{{2}} {{A}} {{B}} {{r}}} \\ 0 & 0 & 0 & 0 \\ -{\frac{ A_{,{{r}}}}{{{2}} {{A}} {{B}} {{r}}}} & -{\frac{ B_{,{{t}}}}{{{2}} {{A}} {{B}} {{r}}}} & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & -{\frac{ B_{,{{t}}}}{{{2}} {{r}} {{{B}^{2}}}}} \\ 0 & 0 & 0 & -{\frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}}} \\ 0 & 0 & 0 & 0 \\ \frac{ B_{,{{t}}}}{{{2}} {{r}} {{{B}^{2}}}} & \frac{ B_{,{{r}}}}{{{2}} {{r}} {{{B}^{2}}}} & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{1} - {B}}{{{B}} {{{r}^{2}}}} \\ 0 & 0 & \frac{{-{1}} + {B}}{{{B}} {{{r}^{2}}}} & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]\end{matrix} \right]}}$
Ricci curvature, $\sharp\flat$:
${{{ R} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{{B}} {{r}} {{{ A_{,{{r}}}}^{2}}}} - {{{A}} {{r}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{2}} {{A}} {{B}} {{r}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}{{r}}}}}}} + {{{{{A}} {{r}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{B}} {{r}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} - {{{4}} {{A}} {{B}} {{ A_{,{{r}}}}}}}}{{{4}} {{r}} {{{A}^{2}}} {{{B}^{2}}}} & -{\frac{ B_{,{{t}}}}{{{A}} {{B}} {{r}}}} & 0 & 0 \\ \frac{ B_{,{{t}}}}{{{r}} {{{B}^{2}}}} & \frac{{-{{{A}} {{r}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{B}} {{r}} {{{ A_{,{{r}}}}^{2}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{r}} {{ B_{,{{t}}{{t}}}}}} - {{{B}} {{r}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} + {{{A}} {{r}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{4}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{4}} {{r}} {{{A}^{2}}} {{{B}^{2}}}} & 0 & 0 \\ 0 & 0 & \frac{{-{{{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{{A}} {{r}} {{ B_{,{{r}}}}}} - {{{2}} {{A}} {{B}}}} + {{{2}} {{A}} {{{B}^{2}}}}}{{{2}} {{A}} {{{B}^{2}}} {{{r}^{2}}}} & 0 \\ 0 & 0 & 0 & \frac{{-{{{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{{A}} {{r}} {{ B_{,{{r}}}}}} - {{{2}} {{A}} {{B}}}} + {{{2}} {{A}} {{{B}^{2}}}}}{{{2}} {{A}} {{{B}^{2}}} {{{r}^{2}}}}\end{matrix} \right]}}$
Gaussian curvature:
${R} = {\frac{{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} - {{{4}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} + {{{4}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{2}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}}$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{4}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{{4}} {{B}} {{{A}^{2}}}} - {{{4}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{8}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & -{\frac{ B_{,{{t}}}}{{{A}} {{B}} {{r}}}} & 0 & 0 \\ \frac{ B_{,{{t}}}}{{{r}} {{{B}^{2}}}} & \frac{{-{{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{{4}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{4}} {{B}} {{{A}^{2}}}}}{{{8}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 \\ 0 & 0 & \frac{{-{{{4}} {{B}} {{{A}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}{{{8}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 \\ 0 & 0 & 0 & \frac{{-{{{4}} {{B}} {{{A}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}{{{8}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\end{matrix} \right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{-{{B}^{2}}} + {{B} - {{{r}} {{ B_{,{{r}}}}}}}}{{{{B}^{2}}} {{{r}^{2}}}} & -{\frac{ B_{,{{t}}}}{{{A}} {{B}} {{r}}}} & 0 & 0 \\ \frac{ B_{,{{t}}}}{{{r}} {{{B}^{2}}}} & \frac{{-{{{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{B}} {{{A}^{2}}}}}{{{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 \\ 0 & 0 & \frac{{{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 \\ 0 & 0 & 0 & \frac{{{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\end{matrix} \right]}}$
Schouten, $\sharp\flat$:
${{{ P} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{{-{{{2}} {{{A}^{2}}} {{{B}^{2}}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} - {{{4}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{{2}} {{B}} {{{A}^{2}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & -{\frac{ B_{,{{t}}}}{{{2}} {{A}} {{B}} {{r}}}} & 0 & 0 \\ \frac{ B_{,{{t}}}}{{{2}} {{r}} {{{B}^{2}}}} & \frac{{{{-{{{2}} {{{A}^{2}}} {{{B}^{2}}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{2}} {{B}} {{{A}^{2}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} + {{{4}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 \\ 0 & 0 & \frac{{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{8}} {{B}} {{{A}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{2}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 \\ 0 & 0 & 0 & \frac{{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{8}} {{B}} {{{A}^{2}}}}} + {{{8}} {{{A}^{2}}} {{{B}^{2}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}\end{matrix} \right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 \\ \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & \frac{{{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & \frac{{{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 \\ \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{{{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 \\ 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} \\ 0 & 0 & 0 & 0 \\ 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & \frac{{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 \\ 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} \\ 0 & 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0\end{matrix} \right]} \\ \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & \frac{{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} \\ 0 & 0 & 0 & 0 \\ 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0 & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} \\ 0 & 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} & 0\end{matrix} \right]} & \overset{c\downarrow d\rightarrow}{\left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}\end{matrix} \right]}}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[ \begin{matrix} \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{12}} {{A}} {{B}} {{{r}^{2}}}} & 0 & 0 \\ \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{12}} {{A}} {{B}} {{{r}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & \frac{{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{A}} {{{B}^{2}}}} & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{A}} {{{B}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & \frac{{{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}} + {{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{4}} {{B}} {{{A}^{2}}}}} + {{{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{A}} {{{B}^{2}}}} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{4}} {{B}} {{{A}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} - {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} + {{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{A}} {{{B}^{2}}}} & 0 & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{12}} {{A}} {{B}} {{{r}^{2}}}} & 0 & 0 \\ \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{12}} {{A}} {{B}} {{{r}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{{{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}}{{{24}} {{B}} {{{A}^{2}}}} & 0 \\ 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{B}} {{{A}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{{{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{4}} {{B}} {{{A}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} - {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}}{{{24}} {{B}} {{{A}^{2}}}} \\ 0 & 0 & 0 & 0 \\ 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{4}} {{B}} {{{A}^{2}}}}} + {{{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{B}} {{{A}^{2}}}} & 0 & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & \frac{{{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{A}} {{{B}^{2}}}} & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{A}} {{{B}^{2}}}} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} + {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} + {{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}}{{{24}} {{B}} {{{A}^{2}}}} & 0 \\ 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} + {{{{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{4}} {{B}} {{{A}^{2}}}} + {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{B}} {{{A}^{2}}}} & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{A}} {{{r}^{4}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{A}} {{{r}^{4}}} {{{ B_{,{{t}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{2}} {{A}} {{B}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{3}}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{4}}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{A}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}}} - {{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}}}} + {{{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{B}} {{{r}^{4}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} - {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{ B_{,{{r}}}}}}} + {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}}} \\ 0 & 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{A}} {{{r}^{4}}} {{{ B_{,{{t}}}}^{2}}}} - {{{A}} {{{r}^{4}}} {{{ B_{,{{t}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{{2}} {{A}} {{B}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{3}}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{4}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{A}} {{{r}^{4}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{B}} {{{r}^{4}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} - {{{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{{2}} {{{A}^{2}}} {{{r}^{3}}} {{ B_{,{{r}}}}}} - {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}}} & 0\end{matrix} \right] \\ \left[ \begin{matrix} 0 & 0 & 0 & \frac{{-{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} - {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{4}} {{B}} {{{A}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} - {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}}} + {{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{A}} {{{B}^{2}}}} \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{A}} {{{r}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} - {{{4}} {{B}} {{{A}^{2}}}}} + {{{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} + {{{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{{{24}} {{A}} {{{B}^{2}}}} & 0 & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{-{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}}} + {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}} + {{{{{4}} {{{A}^{2}}} {{{B}^{2}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{t}}}}^{2}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}} - {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} - {{{4}} {{B}} {{{A}^{2}}}}} + {{{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{B}} {{{r}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} + {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}}{{{24}} {{B}} {{{A}^{2}}}} \\ 0 & 0 & 0 & 0 \\ 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{A}} {{{r}^{2}}} {{{ B_{,{{t}}}}^{2}}}} - {{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{{2}} {{A}} {{B}} {{{r}^{2}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} - {{{2}} {{A}} {{B}} {{r}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{A}} {{{r}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}}} + {{{{A}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{2}}} {{{ A_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{4}} {{B}} {{{A}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{B}} {{{r}^{2}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} - {{{B}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} + {{{{2}} {{r}} {{{A}^{2}}} {{ B_{,{{r}}}}}} - {{{2}} {{r}} {{{A}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}}{{{24}} {{B}} {{{A}^{2}}}} & 0 & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{{-{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}}} + {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{A}} {{{r}^{4}}} {{{ B_{,{{t}}}}^{2}}}} - {{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{{2}} {{A}} {{B}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{3}}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{4}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}} - {{{A}} {{{r}^{4}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}} - {{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}}}} + {{{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}} + {{{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}}} - {{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{B}} {{{r}^{4}}} {{ B_{,{{t}}}}} {{ A_{,{{t}}}}}} - {{{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} + {{{{2}} {{{A}^{2}}} {{{r}^{3}}} {{ B_{,{{r}}}}}} - {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}}} \\ 0 & 0 & \frac{{{{{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}}} - {{{4}} {{{A}^{2}}} {{{B}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{A}} {{{r}^{4}}} {{{ B_{,{{t}}}}^{2}}}}} + {{{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ B_{,{{t}}}}^{2}}}} - {{{2}} {{A}} {{B}} {{{r}^{4}}} {{ A_{,{{r}}{{r}}}}}}} + {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}{{r}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{3}}} {{ A_{,{{r}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}}}} + {{{{2}} {{A}} {{B}} {{{r}^{4}}} {{ B_{,{{t}}{{t}}}}}} - {{{2}} {{A}} {{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{t}}{{t}}}}}}} + {{{{A}} {{{r}^{4}}} {{ B_{,{{r}}}}} {{ A_{,{{r}}}}}} - {{{A}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{r}}}}} {{ B_{,{{r}}}}}}} + {{{{{B}} {{{r}^{4}}} {{{ A_{,{{r}}}}^{2}}}} - {{{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ A_{,{{r}}}}^{2}}}}} - {{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}}}} + {{{{4}} {{B}} {{{A}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{B}} {{{r}^{4}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}}} + {{{{B}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ A_{,{{t}}}}} {{ B_{,{{t}}}}}} - {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{ B_{,{{r}}}}}}} + {{{2}} {{{A}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ B_{,{{r}}}}}}}{{{12}} {{{A}^{2}}} {{{B}^{2}}}} & 0\end{matrix} \right] & \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]\end{matrix} \right]}}$