metric:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} {\frac{1}{r}}{\left({{R}{-{r}}}\right)}& 0& 0& 0\\ 0& \frac{r}{{-{R}} + {r}}& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& {{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}\end{array}\right]}}$
metric inverse:
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} \frac{r}{{R}{-{r}}}& 0& 0& 0\\ 0& {\frac{1}{r}}{\left({{-{R}} + {r}}\right)}& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& 0& 0& \frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}\end{array}\right]}}$
metric derivative:
${{{{ g} _a} _b} _{,c}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{R}{{r}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{R}{{\left({{R}{-{r}}}\right)}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{2}} {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
1st kind Christoffel:
${{{{ \Gamma} _a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{R}{{{2}} {{{r}^{2}}}}}& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{2}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} \frac{R}{{{2}} {{{r}^{2}}}}& 0& 0& 0\\ 0& -{\frac{R}{{{2}} {{\left({{{R}^{2}} + {{r}^{2}}{-{{{2}} {{R}} {{r}}}}}\right)}}}}& 0& 0\\ 0& 0& -{r}& 0\\ 0& 0& 0& -{{{r}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& r& 0\\ 0& r& 0& 0\\ 0& 0& 0& -{{{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {{r}} {{{\sin\left( \theta\right)}^{2}}}\\ 0& 0& 0& {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}\\ 0& {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{{r}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} 0& -{\frac{R}{{{2}} {{r}} {{\left({{-{r}} + {R}}\right)}}}}& 0& 0\\ -{\frac{R}{{{2}} {{r}} {{\left({{-{r}} + {R}}\right)}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{3}}}}& 0& 0& 0\\ 0& \frac{R}{{{2}} {{r}} {{\left({{-{r}} + {R}}\right)}}}& 0& 0\\ 0& 0& {R}{-{r}}& 0\\ 0& 0& 0& {R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& \frac{1}{r}& 0& 0\\ 0& 0& 0& -{{{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}\end{array}\right] \\ \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{1}{r}\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}\\ 0& \frac{1}{r}& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}& 0\end{array}\right]\end{matrix}\right]}}$
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& \frac{{{R}} {{\left({{R}{-{{{2}} {{r}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{{R}} {{\left({{R}{-{{{2}} {{r}}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\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& 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& \frac{{{{3}} {{{R}^{2}}} {{{r}^{2}}}}{-{{{2}} {{R}} {{{r}^{3}}}}}}{{{2}} {{{r}^{6}}}}& 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& \frac{{{R}} {{\left({{-{R}} + {{{2}} {{r}}}}\right)}}}{{{2}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 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& -{1}& 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& -{{\sin\left( \theta\right)}^{2}}& {{2}} {{\left({{R}{-{r}}}\right)}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\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} \frac{{R}^{2}}{{{4}} {{{r}^{4}}}}& 0& 0& 0\\ 0& \frac{{R}^{2}}{{{4}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{{R}^{2}}{{{4}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}}& 0& 0\\ \frac{{R}^{2}}{{{4}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\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& -{\frac{R}{{{2}} {{r}}}}& 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}}}}{{{2}} {{r}}}}\\ 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{{R}^{2}}{{{4}} {{{r}^{4}}}}& 0& 0\\ -{\frac{{R}^{2}}{{{4}} {{{r}^{4}}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} \frac{{R}^{2}}{{{4}} {{{r}^{4}}}}& 0& 0& 0\\ 0& \frac{{R}^{2}}{{{4}} {{\left({{{{{R}^{2}}} {{{r}^{2}}}} + {{r}^{4}}{-{{{2}} {{R}} {{{r}^{3}}}}}}\right)}}}& 0& 0\\ 0& 0& {\frac{1}{r}}{\left({{R}{-{r}}}\right)}& 0\\ 0& 0& 0& {\frac{1}{r}}{\left({{R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{R}{{{2}} {{r}}}& 0\\ 0& {\frac{1}{r}}{\left({{R}{-{r}}}\right)}& 0& 0\\ 0& 0& 0& {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\left({{-{r}} + {R}}\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}}}}{{{2}} {{r}}}\\ 0& 0& 0& {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}} {{\left({{-{R}} + {r}}\right)}}\\ 0& {\frac{1}{r}}{\left({{R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}}\right)}& {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}} {{\left({{-{r}} + {R}}\right)}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{4}}}}& 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{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}& 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}{r}}{\left({{R}{-{r}}}\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{1}{r}}{\left({{R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{r}} {{{\cos\left( \theta\right)}^{2}}}}}\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{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{4}}}}& 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{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}& \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}{r}}{\left({{R}{-{r}}}\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}{r}}{\left({{R}{-{r}}{-{{{R}} {{{\cos\left( \theta\right)}^{2}}}}}}\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& \frac{R}{{{{r}^{2}}} {{\left({{r}{-{R}}}\right)}}}& 0& 0\\ \frac{R}{{{{r}^{2}}} {{\left({{-{r}} + {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& -{\frac{R}{{{2}} {{r}}}}& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{r}}}& 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}}}}{{{2}} {{r}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{r}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& \frac{{-{{{{R}^{2}}} {{{r}^{2}}}}} + {{{R}} {{{r}^{3}}}}}{{r}^{6}}& 0& 0\\ \frac{{{{{R}^{2}}} {{{r}^{2}}}}{-{{{R}} {{{r}^{3}}}}}}{{r}^{6}}& 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}{{{2}} {{r}}}}& 0\\ 0& \frac{R}{{{2}} {{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& -{\frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{r}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{r}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{{{R}} {{\left({{R}{-{r}}}\right)}}}{{{2}} {{{r}^{4}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{4}}}}& 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}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}}& 0\\ 0& \frac{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\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& 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}} {{{R}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& -{{\frac{1}{r}} {{{R}} {{{\sin\left( \theta\right)}^{2}}}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{{{R}} {{\left({{R}{-{r}}}\right)}}}{{{2}} {{{r}^{4}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{4}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}}\\ 0& 0& 0& 0\\ 0& \frac{R}{{{2}} {{{r}^{2}}} {{\left({{-{r}} + {R}}\right)}}}& 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}} {R}}\\ 0& 0& {\frac{1}{r}} {R}& 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{R}{{r}^{3}}& 0& 0\\ -{\frac{R}{{r}^{3}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& -{\frac{R}{{r}^{3}}}& 0& 0\\ \frac{R}{{r}^{3}}& 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{R}{{{2}} {{{r}^{3}}}}}& 0\\ 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}\\ 0& 0& 0& 0\\ 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0\\ 0& 0& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0\\ 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 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}{{r}^{3}}\\ 0& 0& -{\frac{R}{{r}^{3}}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}\\ 0& 0& 0& 0\\ 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{\frac{R}{{r}^{3}}}\\ 0& 0& \frac{R}{{r}^{3}}& 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, $\sharp\flat$:
${{{ R} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Gaussian curvature:
${R} = {0}$
trace-free Ricci, $\sharp\flat$:
${{{ {(R^{TF})}} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${{{ G} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Schouten, $\sharp\flat$:
${{{ P} ^a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
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{R}{{r}^{3}}& 0& 0\\ -{\frac{R}{{r}^{3}}}& 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}{{{2}} {{{r}^{3}}}}}& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& -{\frac{R}{{r}^{3}}}& 0& 0\\ \frac{R}{{r}^{3}}& 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}{{{2}} {{{r}^{3}}}}}& 0\\ 0& \frac{R}{{{2}} {{{r}^{3}}}}& 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}{{{2}} {{{r}^{3}}}}}\\ 0& 0& 0& 0\\ 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}& 0\\ 0& 0& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{3}}}}}& 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}{{{2}} {{{r}^{3}}}}& 0\\ 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 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{R}{{r}^{3}}\\ 0& 0& -{\frac{R}{{r}^{3}}}& 0\end{array}\right]}\\ \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{\frac{R}{{{2}} {{{r}^{3}}}}}& 0& 0& 0\end{array}\right]}& \overset{c\downarrow d\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{R}{{{2}} {{{r}^{3}}}}\\ 0& 0& 0& 0\\ 0& -{\frac{R}{{{2}} {{{r}^{3}}}}}& 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{R}{{r}^{3}}}\\ 0& 0& \frac{R}{{r}^{3}}& 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{R}{{r}^{3}}}& 0& 0\\ \frac{R}{{r}^{3}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{R}{-{r}}}\right)}}}{{{2}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& \frac{{-{{R}^{2}}} + {{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{R}} {{r}}}{-{{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}}{{{2}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{{R}^{2}}{-{{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{R}} {{r}}}} + {{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}{{{2}} {{{r}^{2}}}}& 0& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& \frac{R}{{r}^{3}}& 0& 0\\ -{\frac{R}{{r}^{3}}}& 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{R}{{{2}} {{\left({{-{R}} + {r}}\right)}}}}& 0\\ 0& \frac{R}{{{2}} {{\left({{-{R}} + {r}}\right)}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{\frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{\left({{-{R}} + {r}}\right)}}}}\\ 0& 0& 0& 0\\ 0& \frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{\left({{-{R}} + {r}}\right)}}}& 0& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& \frac{{{R}} {{\left({{R}{-{r}}}\right)}}}{{{2}} {{{r}^{2}}}}& 0\\ 0& 0& 0& 0\\ \frac{{{R}} {{\left({{-{R}} + {r}}\right)}}}{{{2}} {{{r}^{2}}}}& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& \frac{R}{{{2}} {{\left({{-{R}} + {r}}\right)}}}& 0\\ 0& -{\frac{R}{{{2}} {{\left({{-{R}} + {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& {{R}} {{r}} {{{\sin\left( \theta\right)}^{2}}}\\ 0& 0& -{{{R}} {{r}} {{{\sin\left( \theta\right)}^{2}}}}& 0\end{array}\right]\\ \left[\begin{array}{cccc} 0& 0& 0& \frac{{{R}^{2}}{-{{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{-{{{R}} {{r}}}} + {{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}{{{2}} {{{r}^{2}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ \frac{{-{{R}^{2}}} + {{{{R}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{R}} {{r}}}{-{{{R}} {{r}} {{{\cos\left( \theta\right)}^{2}}}}}}{{{2}} {{{r}^{2}}}}& 0& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& \frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{\left({{-{R}} + {r}}\right)}}}\\ 0& 0& 0& 0\\ 0& -{\frac{{{R}} {{{\sin\left( \theta\right)}^{2}}}}{{{2}} {{\left({{-{R}} + {r}}\right)}}}}& 0& 0\end{array}\right]& \left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& -{{{R}} {{r}} {{{\sin\left( \theta\right)}^{2}}}}\\ 0& 0& {{R}} {{r}} {{{\sin\left( \theta\right)}^{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]}}$
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& 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& 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& 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& 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& 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& 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& 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& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
divergence: ${{{{ A} ^i} _{,i}} + {{{{{{ \Gamma} ^i} _j} _i}} {{{ A} ^j}}}} = {{{{-1}} {{{A^{r}}}} \cdot {{r}} {{\cos\left( \theta\right)}} {{\frac{1}{{{{R}} {{\sin\left( \theta\right)}}} + {{{-1}} {{r}} {{\sin\left( \theta\right)}}}}}}} + {{{-1}} {{r}} {{\partial_ {{\phi}}\left( {A^{\theta}}\right)}} {{\frac{1}{{R} + {{{-1}} {{r}}}}}}} + {{{-1}} {{r}} {{\partial_ {{\theta}}\left( {A^{r}}\right)}} {{\frac{1}{{R} + {{{-1}} {{r}}}}}}} + {{{-1}} {{r}} {{\partial_ {{r}}\left( {A^{t}}\right)}} {{\frac{1}{{R} + {{{-1}} {{r}}}}}}} + {{{5}} {{{A^{t}}}} \cdot {{R}} {{\frac{1}{{{{-2}} {{r}} {{r}}} + {{{2}} {{r}} {{R}}}}}}} + {{{-2}} {{{A^{t}}}} \cdot {{\frac{1}{{R} + {{{-1}} {{r}}}}}}} + {{{{A^{r}}}} \cdot {{R}} {{\cos\left( \theta\right)}} {{\frac{1}{{{{R}} {{\sin\left( \theta\right)}}} + {{{-1}} {{r}} {{\sin\left( \theta\right)}}}}}}} + {{{R}} {{\partial_ {{\phi}}\left( {A^{\theta}}\right)}} {{\frac{1}{{R} + {{{-1}} {{r}}}}}}} + {{{R}} {{\partial_ {{\theta}}\left( {A^{r}}\right)}} {{\frac{1}{{R} + {{{-1}} {{r}}}}}}} + {{{R}} {{\partial_ {{r}}\left( {A^{t}}\right)}} {{\frac{1}{{R} + {{{-1}} {{r}}}}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{t} \\ \ddot{r} \\ \ddot{\theta} \\ \ddot{\phi}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} \frac{{{R}} {{\dot{r}}} \cdot {{\dot{t}}}}{{{r}} {{\left({{-{r}} + {R}}\right)}}} \\ \frac{{-{{{R}} {{{\dot{r}}^{2}}} {{{r}^{2}}}}} + {{{R}} {{{\dot{t}}^{2}}} {{{r}^{2}}}} + {{{{R}^{3}}} {{{\dot{t}}^{2}}}}{-{{{2}} {{r}} {{{R}^{2}}} {{{\dot{t}}^{2}}}}}{-{{{2}} {{{\dot{\phi}}^{2}}} {{{r}^{5}}}}}{-{{{2}} {{{\dot{\theta}}^{2}}} {{{r}^{5}}}}}{-{{{2}} {{{R}^{2}}} {{{\dot{\phi}}^{2}}} {{{r}^{3}}}}}{-{{{2}} {{{R}^{2}}} {{{\dot{\theta}}^{2}}} {{{r}^{3}}}}} + {{{2}} {{{\dot{\phi}}^{2}}} {{{r}^{5}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{2}} {{{R}^{2}}} {{{\dot{\phi}}^{2}}} {{{r}^{3}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{4}} {{R}} {{{\dot{\phi}}^{2}}} {{{r}^{4}}}} + {{{4}} {{R}} {{{\dot{\theta}}^{2}}} {{{r}^{4}}}}{-{{{4}} {{R}} {{{\dot{\phi}}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}}}{{{2}} {{{r}^{3}}} {{\left({{-{r}} + {R}}\right)}}} \\ {\frac{1}{r}}{\left({{-{{{2}} {{\dot{\theta}}} \cdot {{\dot{r}}}}} + {{{r}} {{{\dot{\phi}}^{2}}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}}}\right)} \\ -{\frac{{{2}} {{\dot{\phi}}} \cdot {{\left({{{{\dot{r}}} \cdot {{\sin\left( \theta\right)}}} + {{{\dot{\theta}}} \cdot {{r}} {{\cos\left( \theta\right)}}}}\right)}}}{{{r}} {{\sin\left( \theta\right)}}}}\end{matrix}\right]}}$

ADM variables:
${\alpha} = {\sqrt{-{{{ g} ^t} ^t}}}$
${{ \beta} _i} = {{{ g} _t} _i}$
${{{ \gamma} _i} _j} = {{{ g} _i} _j}$

${\alpha} = {\frac{\sqrt{{R}{-{r}}}}{{{i}} {{\sqrt{r}}}}}$
${{{ \gamma} _i} _j} = {\overset{i\downarrow j\rightarrow}{\left[\begin{array}{ccc} \frac{r}{{-{R}} + {r}}& 0& 0\\ 0& {r}^{2}& 0\\ 0& 0& {{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}\end{array}\right]}}$
${{{ \gamma} ^i} ^j} = {\overset{i\downarrow j\rightarrow}{\left[\begin{array}{ccc} \frac{{-{R}} + {r} + {{{R}} {{{\cos\left( \theta\right)}^{2}}}}{-{{{r}} {{{\cos\left( \theta\right)}^{2}}}}}}{{{r}} {{{\sin\left( \theta\right)}^{2}}}}& 0& 0\\ 0& \frac{1}{{r}^{2}}& 0\\ 0& 0& \frac{1}{{{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}\end{array}\right]}}$
${{ \beta} _i} = {\overset{i\downarrow}{\left[\begin{matrix} 0 \\ 0 \\ 0\end{matrix}\right]}}$
${{ \beta} ^i} = {\overset{i\downarrow}{\left[\begin{matrix} 0 \\ 0 \\ 0\end{matrix}\right]}}$
${{ n} _a} = {\overset{a\downarrow}{\left[\begin{matrix} -{\frac{\sqrt{{R}{-{r}}}}{{{i}} {{\sqrt{r}}}}} \\ 0 \\ 0 \\ 0\end{matrix}\right]}}$
${{ n} ^a} = {\overset{a\downarrow}{\left[\begin{matrix} -{\frac{\sqrt{r}}{{{i}} {{\sqrt{{R}{-{r}}}}}}} \\ 0 \\ 0 \\ 0\end{matrix}\right]}}$
${{{{ n} ^a}} {{{ n} _a}}} = {-{1}}$
${{{{ n} _a}} {{{ n} _b}} {{{{ g} ^a} ^b}}} = {-{1}}$
${{{ \gamma} _a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{ K} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$