metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{\exp\left({{{2}} {{\Phi}}}\right)}& 0& 0& 0\\ 0& \exp\left({{{2}} {{\Lambda}}}\right)& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& {{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}\end{array}\right]}}$
metric inverse:
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{\frac{1}{\exp\left({{{2}} {{\Phi}}}\right)}}& 0& 0& 0\\ 0& \frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& 0& 0& \frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}\end{array}\right]}}$
metric derivative:
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{{{2}} {{\exp\left({{{2}} {{\Phi}}}\right)}} {{ \Phi_{,{{r}}}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& {{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{ \Lambda_{,{{r}}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{2}} {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{{{\exp\left({{{2}} {{\Phi}}}\right)}} {{ \Phi_{,{{r}}}}}}& 0& 0\\ -{{{\exp\left({{{2}} {{\Phi}}}\right)}} {{ \Phi_{,{{r}}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} {{\exp\left({{{2}} {{\Phi}}}\right)}} {{ \Phi_{,{{r}}}}}& 0& 0& 0\\ 0& {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{ \Lambda_{,{{r}}}}}& 0& 0\\ 0& 0& -{r}& 0\\ 0& 0& 0& -{{{r}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& r& 0\\ 0& r& 0& 0\\ 0& 0& 0& -{{{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {{r}} {{{\sin\left( \theta\right)}^{2}}}\\ 0& 0& 0& {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}\\ 0& {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} 0& \Phi_{,{{r}}}& 0& 0\\ \Phi_{,{{r}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} {{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{ \Phi_{,{{r}}}}}& 0& 0& 0\\ 0& \Lambda_{,{{r}}}& 0& 0\\ 0& 0& -{\frac{r}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0\\ 0& 0& 0& -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& \frac{1}{r}& 0& 0\\ 0& 0& 0& -{{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{r}\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}\\ 0& \frac{1}{r}& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}& 0\end{array}\right]\end{matrix}\right]}}$
connection coefficients derivative:
${{{{{ \Gamma} ^a} _b} _c} _{,d}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \Phi_{,{{r}}{{r}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \Phi_{,{{r}}{{r}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{\left({{ \Phi_{,{{r}}{{r}}}} + {{{2}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{2}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \Lambda_{,{{r}}{{r}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& \frac{{-{1}} + {{{2}} {{r}} {{ \Lambda_{,{{r}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& \frac{{-{1}} + {{\cos\left( \theta\right)}^{2}} + {{{2}} {{r}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& -{\frac{{{2}} {{r}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& {1}{-{{{2}} {{{\cos\left( \theta\right)}^{2}}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& -{\frac{1}{{r}^{2}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& -{\frac{1}{{\sin\left( \theta\right)}^{2}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 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{array}\right]}\end{array}\right]}}$
connection coefficients squared:
${{{{{{ \Gamma} ^a} _e} _c}} {{{{{ \Gamma} ^e} _b} _d}}} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{{ \Phi_{,{{r}}}}^{2}}}& 0& 0& 0\\ 0& { \Phi_{,{{r}}}}^{2}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}& 0& 0\\ { \Phi_{,{{r}}}}^{2}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{{{r}} {{ \Phi_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{{ \Phi_{,{{r}}}}^{2}}}& 0& 0\\ {{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} {{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{{ \Phi_{,{{r}}}}^{2}}}& 0& 0& 0\\ 0& { \Lambda_{,{{r}}}}^{2}& 0& 0\\ 0& 0& -{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0\\ 0& 0& 0& -{\frac{{\sin\left( \theta\right)}^{2}}{\exp\left({{{2}} {{\Lambda}}}\right)}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{{{r}} {{ \Lambda_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0\\ 0& -{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0& 0\\ 0& 0& 0& -{\frac{{{r}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}\\ 0& 0& 0& \frac{{{r}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}{\exp\left({{{2}} {{\Lambda}}}\right)}\\ 0& -{\frac{{\sin\left( \theta\right)}^{2}}{\exp\left({{{2}} {{\Lambda}}}\right)}}& -{\frac{{{r}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{ \Phi_{,{{r}}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& {\frac{1}{r}} { \Lambda_{,{{r}}}}& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& \frac{1}{{r}^{2}}& 0& 0\\ 0& 0& -{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0\\ 0& 0& 0& -{{\cos\left( \theta\right)}^{2}}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\\ 0& 0& 0& -{\frac{{\sin\left( \theta\right)}^{2}}{\exp\left({{{2}} {{\Lambda}}}\right)}}\\ 0& -{{\frac{1}{r}} {{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}& -{{\cos\left( \theta\right)}^{2}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{ \Phi_{,{{r}}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{{r}^{2}}\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}\\ 0& {\frac{1}{r}} { \Lambda_{,{{r}}}}& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 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}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 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}} + {{\cos\left( \theta\right)}^{2}}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R} ^a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {-{ \Phi_{,{{r}}{{r}}}}}{-{{ \Phi_{,{{r}}}}^{2}}} + {{{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}& 0& 0\\ { \Phi_{,{{r}}{{r}}}} + {{ \Phi_{,{{r}}}}^{2}}{-{{{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{\frac{{{r}} {{ \Phi_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0\\ 0& 0& 0& 0\\ \frac{{{r}} {{ \Phi_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& {{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{\left({{-{ \Phi_{,{{r}}{{r}}}}}{-{{ \Phi_{,{{r}}}}^{2}}} + {{{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}\right)}}& 0& 0\\ {{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{\left({{ \Phi_{,{{r}}{{r}}}} + {{ \Phi_{,{{r}}}}^{2}}{-{{{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{{r}} {{ \Lambda_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0\\ 0& -{\frac{{{r}} {{ \Lambda_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}\\ 0& 0& 0& 0\\ 0& -{\frac{{{r}} {{{\sin\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& -{{\frac{1}{r}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{ \Phi_{,{{r}}}}}}}& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{ \Phi_{,{{r}}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} { \Lambda_{,{{r}}}}}& 0\\ 0& {\frac{1}{r}} { \Lambda_{,{{r}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}} + {{\cos\left( \theta\right)}^{2}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}\\ 0& 0& \frac{{1}{-{{\cos\left( \theta\right)}^{2}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{{\frac{1}{r}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{ \Phi_{,{{r}}}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{r}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{ \Phi_{,{{r}}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} { \Lambda_{,{{r}}}}}\\ 0& 0& 0& 0\\ 0& {\frac{1}{r}} { \Lambda_{,{{r}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}}}{\exp\left({{{2}} {{\Lambda}}}\right)}\\ 0& 0& \frac{{-{1}} + {\exp\left({{{2}} {{\Lambda}}}\right)}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& \frac{{-{ \Phi_{,{{r}}{{r}}}}}{-{{ \Phi_{,{{r}}}}^{2}}} + {{{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0& 0\\ \frac{{ \Phi_{,{{r}}{{r}}}} + {{ \Phi_{,{{r}}}}^{2}}{-{{{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& -{\frac{ \Phi_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}& 0\\ 0& 0& 0& 0\\ \frac{ \Phi_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& -{\frac{ \Phi_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{ \Phi_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& \frac{{ \Phi_{,{{r}}{{r}}}} + {{ \Phi_{,{{r}}}}^{2}}{-{{{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0& 0\\ \frac{{-{ \Phi_{,{{r}}{{r}}}}}{-{{ \Phi_{,{{r}}}}^{2}}} + {{{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{ \Lambda_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0\\ 0& -{\frac{ \Lambda_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{ \Lambda_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}\\ 0& 0& 0& 0\\ 0& -{\frac{ \Lambda_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& \frac{ \Phi_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0\\ 0& 0& 0& 0\\ -{\frac{ \Phi_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{\frac{ \Lambda_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}& 0\\ 0& \frac{ \Lambda_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}} + {\exp\left({{{2}} {{\Lambda}}}\right)}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\\ 0& 0& \frac{{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& \frac{ \Phi_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{ \Phi_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{ \Lambda_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}\\ 0& 0& 0& 0\\ 0& \frac{ \Lambda_{,{{r}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\\ 0& 0& \frac{{-{1}} + {\exp\left({{{2}} {{\Lambda}}}\right)}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$
Ricci curvature, $\sharp\flat$:
${{{ R} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{-{{{2}} {{ \Phi_{,{{r}}}}}}}{-{{{r}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{r}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{r}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0& 0& 0\\ 0& \frac{{{{2}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{r}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{r}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{{{r}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0& 0\\ 0& 0& \frac{{-{1}} + {\exp\left({{{2}} {{\Lambda}}}\right)} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{1}} + {\exp\left({{{2}} {{\Lambda}}}\right)} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\end{array}\right]}}$
Gaussian curvature:
${R} = {\frac{{{2}} {{\left({{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{r}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}\right)}}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}}$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{1}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{r}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0& 0\\ 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{r}} {{ \Lambda_{,{{r}}}}}} + {{{2}} {{r}} {{ \Phi_{,{{r}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0\\ 0& 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\end{array}\right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}}{-{{{2}} {{r}} {{ \Lambda_{,{{r}}}}}}}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0& 0\\ 0& \frac{{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{2}} {{r}} {{ \Phi_{,{{r}}}}}}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0\\ 0& 0& \frac{{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\\ 0& 0& 0& \frac{{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\end{array}\right]}}$
Schouten, $\sharp\flat$:
${{{ P} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}}{-{{{2}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{r}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{4}} {{r}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0& 0\\ 0& \frac{{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}}{-{{{2}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{2}} {{r}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{4}} {{r}} {{ \Lambda_{,{{r}}}}}} + {{{2}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0\\ 0& 0& \frac{{-{2}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\\ 0& 0& 0& \frac{{-{2}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\end{array}\right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${{{{{ C} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0\\ \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0\\ \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\\ 0& \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\\ 0& \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\\ 0& 0& \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}\\ 0& 0& \frac{{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Weyl, $\flat\flat\flat\flat$:
${{{{{ C} _a} _b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& \frac{{{\exp\left({{{2}} {{\Phi}}}\right)}} {{\left({{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}}}{{{3}} {{{r}^{2}}}}& 0& 0\\ \frac{{{\exp\left({{{2}} {{\Phi}}}\right)}} {{\left({{-{1}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}\right)}}}{{{3}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& {\frac{1}{6}} {{{\exp\left({{{2}} {{\Phi}}}\right)}} {{\left({{1}{-{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}}{-{{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{r}} {{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{ \Phi_{,{{r}}}}}}{-{{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}\right)}}}& 0\\ 0& 0& 0& 0\\ {\frac{1}{6}} {{{\exp\left({{{2}} {{\Phi}}}\right)}} {{\left({{-{1}} + {\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{r}} {{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{ \Phi_{,{{r}}}}}}} + {{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& {\frac{1}{6}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{\left({{-{1}} + {{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{6}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{\left({{1}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}\right)}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& \frac{{{\exp\left({{{2}} {{\Phi}}}\right)}} {{\left({{-{1}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}\right)}}}{{{3}} {{{r}^{2}}}}& 0& 0\\ \frac{{{\exp\left({{{2}} {{\Phi}}}\right)}} {{\left({{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}}}{{{3}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {\frac{1}{6}}{\left({{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}& 0\\ 0& {\frac{1}{6}}{\left({{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}\right)}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{6}}{\left({{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{\cos\left( \theta\right)}^{2}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}\right)}\\ 0& 0& 0& 0\\ 0& {\frac{1}{6}}{\left({{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{\cos\left( \theta\right)}^{2}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}\right)}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& {\frac{1}{6}} {{{\exp\left({{{2}} {{\Phi}}}\right)}} {{\left({{-{1}} + {\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{r}} {{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{ \Phi_{,{{r}}}}}}} + {{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}}}& 0\\ 0& 0& 0& 0\\ {\frac{1}{6}} {{{\exp\left({{{2}} {{\Phi}}}\right)}} {{\left({{1}{-{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}}{-{{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{r}} {{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{ \Phi_{,{{r}}}}}}{-{{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}\right)}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {\frac{1}{6}}{\left({{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}\right)}& 0\\ 0& {\frac{1}{6}}{\left({{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{{r}^{2}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{{r}^{3}}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{-{{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{4}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}\\ 0& 0& \frac{{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{{r}^{3}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}} + {{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{4}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& {\frac{1}{6}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{\left({{1}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}\right)}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ {\frac{1}{6}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{\left({{-{1}} + {{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\sin\left( \theta\right)}^{2}}}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{6}}{\left({{-{1}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{\cos\left( \theta\right)}^{2}} + {\exp\left({{{2}} {{\Lambda}}}\right)}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}\right)}\\ 0& 0& 0& 0\\ 0& {\frac{1}{6}}{\left({{1} + {{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{\cos\left( \theta\right)}^{2}}}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}\right)}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{{r}^{2}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{{r}^{3}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}} + {{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{4}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}\\ 0& 0& \frac{{-{{r}^{2}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{{r}^{3}}} {{ \Phi_{,{{r}}}}}}{-{{{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}}}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{-{{{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{4}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{3}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\end{array}\right]}}$
Plebanski, $\sharp\sharp\flat\flat$:
${{{{{ P} ^a} ^b} _c} _d} = {\overset{a\downarrow b\rightarrow[{c\downarrow d\rightarrow}]}{\left[\begin{array}{cccc} \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\\ 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& 0& 0\\ 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{12}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{{-{1}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}} + {{\cos\left( \theta\right)}^{2}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{\exp\left({{{4}} {{\Lambda}}}\right)}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{3}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Phi_{,{{r}}}}} {{ \Lambda_{,{{r}}}}}}}}{{{6}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& \frac{{1}{-{{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}}}}{-{{\cos\left( \theta\right)}^{2}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{4}}}} + {\exp\left({{{4}} {{\Lambda}}}\right)}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}}}}{-{{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{{r}^{2}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}}} + {{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{{r}^{4}}} {{{ \Phi_{,{{r}}{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{4}}}}}{-{{{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}} + {{{2}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}} + {{{3}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{{r}^{4}}} {{{ \Lambda_{,{{r}}}}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{2}} {{{r}^{4}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{{ \Phi_{,{{r}}}}^{3}}} {{ \Lambda_{,{{r}}}}}}{-{{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{2}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}} + {{{2}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}} {{ \Phi_{,{{r}}{{r}}}}} {{ \Lambda_{,{{r}}}}}}}{{{6}} {{\exp\left({{{4}} {{\Lambda}}}\right)}} {{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Einstein, $\flat\flat$:
${{{ G} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{{{\exp\left({{{2}} {{\Phi}}}\right)}} {{\left({{1}{-{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} + {{{2}} {{r}} {{\frac{1}{\exp\left({{{2}} {{\Lambda}}}\right)}}} {{ \Lambda_{,{{r}}}}}}}\right)}}}{{r}^{2}}& 0& 0& 0\\ 0& \frac{{1}{-{\exp\left({{{2}} {{\Lambda}}}\right)}} + {{{2}} {{r}} {{ \Phi_{,{{r}}}}}}}{{r}^{2}}& 0& 0\\ 0& 0& \frac{{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0\\ 0& 0& 0& \frac{{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{r}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{ \Phi_{,{{r}}}}}} + {{{{r}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}{-{{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{{r}^{2}}} {{{ \Phi_{,{{r}}}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}}{-{{{{r}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}} + {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}\end{array}\right]}}$
Ricci, $\flat\flat$:
${{{ R} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{r}} {{{\exp\left({{{2}} {{\left({{-{\Lambda}} + {\Phi}}\right)}}}\right)}} {{\left({{{{2}} {{ \Phi_{,{{r}}}}}} + {{{r}} {{ \Phi_{,{{r}}{{r}}}}}} + {{{r}} {{{ \Phi_{,{{r}}}}^{2}}}}{-{{{r}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}}\right)}}}& 0& 0& 0\\ 0& {\frac{1}{r}}{\left({{{{2}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}{{r}}}}}}}{-{{{r}} {{{ \Phi_{,{{r}}}}^{2}}}}} + {{{r}} {{ \Lambda_{,{{r}}}}} {{ \Phi_{,{{r}}}}}}}\right)}& 0& 0\\ 0& 0& \frac{{-{1}} + {\exp\left({{{2}} {{\Lambda}}}\right)} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}& 0\\ 0& 0& 0& \frac{{-{1}} + {{\cos\left( \theta\right)}^{2}} + {\exp\left({{{2}} {{\Lambda}}}\right)} + {{{r}} {{ \Lambda_{,{{r}}}}}}{-{{{r}} {{ \Phi_{,{{r}}}}}}}{-{{{\exp\left({{{2}} {{\Lambda}}}\right)}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Lambda_{,{{r}}}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}} {{ \Phi_{,{{r}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)}\end{array}\right]}}$
divergence: ${{{{ A} ^i} _{,i}} + {{{{{{ \Gamma} ^i} _j} _i}} {{{ A} ^j}}}} = {{{{2}} {{{A^{r}}}} \cdot {{\frac{1}{r}}}} + {{{{A^{\theta}}}} \cdot {{\cos\left( \theta\right)}} {{\frac{1}{\sin\left( \theta\right)}}}} + { {A^{\phi}}_{,{{\phi}}}} + { {A^{\theta}}_{,{{\theta}}}} + { {A^{r}}_{,{{r}}}} + { {A^{t}}_{,{{t}}}} + {{{{A^{r}}}} \cdot {{ \Lambda_{,{{r}}}}}} + {{{{A^{r}}}} \cdot {{ \Phi_{,{{r}}}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} {\ddot{t}} + {{{2}} {{\dot{r}}} \cdot {{\dot{t}}} \cdot {{ \Phi_{,{{r}}}}}} \\ \frac{{{{\ddot{r}}} \cdot {{\exp\left({{{2}} {{\Lambda}}}\right)}}}{-{{{r}} {{{\dot{\phi}}^{2}}}}}{-{{{r}} {{{\dot{\theta}}^{2}}}}} + {{{r}} {{{\dot{\phi}}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{\dot{r}}^{2}}} {{\exp\left({{{2}} {{\Lambda}}}\right)}} {{ \Lambda_{,{{r}}}}}} + {{{{\dot{t}}^{2}}} {{\exp\left({{{2}} {{\Phi}}}\right)}} {{ \Phi_{,{{r}}}}}}}{\exp\left({{{2}} {{\Lambda}}}\right)} \\ {\frac{1}{r}}{\left({{{{\ddot{\theta}}} \cdot {{r}}} + {{{2}} {{\dot{\theta}}} \cdot {{\dot{r}}}}{-{{{r}} {{{\dot{\phi}}^{2}}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}}}}\right)} \\ \frac{{{{\ddot{\phi}}} \cdot {{r}} {{\sin\left( \theta\right)}}} + {{{2}} {{\dot{\phi}}} \cdot {{\dot{r}}} \cdot {{\sin\left( \theta\right)}}} + {{{2}} {{\dot{\phi}}} \cdot {{\dot{\theta}}} \cdot {{r}} {{\cos\left( \theta\right)}}}}{{{r}} {{\sin\left( \theta\right)}}}\end{matrix}\right]}} = {\overset{u\downarrow}{\left[\begin{matrix} 0 \\ 0 \\ 0 \\ 0\end{matrix}\right]}}$