Schwarzschild, anholonomic
chart coordinates: $x^\tilde{\mu} = \{t, r, \theta, \phi\}$
chart coordinate basis: $e_\tilde{\mu} = \{e_{\tilde{t}}, e_{\tilde{r}}, e_{\tilde{\theta}}, e_{\tilde{\phi}}\}$
embedding coordinates: $u^I = \{t, x, y, z\}$
embedding basis $e_I = \{e_{t}, e_{x}, e_{y}, e_{z}\}$
flat metric:
${{{ \eta} _I} _J} = {\overset{I\downarrow J\rightarrow}{\left[\begin{array}{cccc} -1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
transform from basis to coordinate:
${{{ \tilde{e}} _A} ^a} = {\overset{a\downarrow A\rightarrow}{\left[\begin{array}{cccc} \alpha& 0& 0& 0\\ 0& \frac{1}{\alpha}& 0& 0\\ 0& 0& r& 0\\ 0& 0& 0& {{r}} {{\sin\left( \theta\right)}}\end{array}\right]}}$
transform from coorinate to basis:
${{{ \tilde{e}} ^a} _A} = {\overset{a\downarrow A\rightarrow}{\left[\begin{array}{cccc} \frac{1}{\alpha}& 0& 0& 0\\ 0& \alpha& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& \frac{1}{{{r}} {{\sin\left( \theta\right)}}}\end{array}\right]}}$
tensor index associated with coordinate $t$ has operator $e_{t}(\zeta) = $${\frac{1}{\alpha}} {\frac{\partial \zeta}{\partial t}}$
tensor index associated with coordinate $r$ has operator $e_{r}(\zeta) = $${{\alpha}} \cdot {{\frac{\partial \zeta}{\partial r}}}$
tensor index associated with coordinate $\theta$ has operator $e_{\theta}(\zeta) = $${\frac{1}{r}} {\frac{\partial \zeta}{\partial \theta}}$
tensor index associated with coordinate $\phi$ has operator $e_{\phi}(\zeta) = $$\frac{\frac{\partial \zeta}{\partial \phi}}{{{r}} {{\sin\left( \theta\right)}}}$
basis in embedded coordinates:
${{{ e} _u} ^I} = {\overset{u\downarrow I\rightarrow}{\left[\begin{array}{cccc} \alpha& 0& 0& 0\\ 0& \frac{1}{\alpha}& 0& 0\\ 0& 0& r& 0\\ 0& 0& 0& {{r}} {{\sin\left( \theta\right)}}\end{array}\right]}}$
${{{ e} ^u} _I} = {\overset{u\downarrow I\rightarrow}{\left[\begin{array}{cccc} \frac{1}{\alpha}& 0& 0& 0\\ 0& \alpha& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& \frac{1}{{{r}} {{\sin\left( \theta\right)}}}\end{array}\right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^v} _I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{{{ e} _u} ^I}} {{{{ e} ^u} _J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
basis determinant: ${det(e)} = {{{{r}^{2}}} {{\sin\left( \theta\right)}}}$
${{{{ c} _a} _b} ^c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ -{\frac{\partial \alpha}{\partial r}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} \frac{\partial \alpha}{\partial r}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& {\frac{1}{r}} {\alpha}& 0\\ 0& 0& 0& {\frac{1}{r}} {\alpha}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {\alpha}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {\alpha}}\\ 0& 0& 0& -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{{\alpha}^{2}}& 0& 0& 0\\ 0& \frac{1}{{\alpha}^{2}}& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& {{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}\end{array}\right]}}$
${{{ g} _u} _v} = {{{{{ e} _u} ^I}} {{{{ e} _v} ^J}} {{{{ \eta} _I} _J}}}$
metric determinant: ${det(g)} = {-{{{{r}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}}$
${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{{{ g} _a} _b} _{,c}} + {{{{ g} _a} _c} _{,b}}{-{{{{ g} _b} _c} _{,a}}} + {{{{ c} _a} _b} _c} + {{{{ c} _a} _c} _b}{-{{{{ c} _c} _b} _a}}}\right)}}}$
commutation coefficients:
${c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ -{\frac{\partial \alpha}{\partial r}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} \frac{\partial \alpha}{\partial r}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& {\frac{1}{r}} {\alpha}& 0\\ 0& 0& 0& {\frac{1}{r}} {\alpha}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& -{{\frac{1}{r}} {\alpha}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& -{{\frac{1}{r}} {\alpha}}\\ 0& 0& 0& -{\frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}}\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
metric:
${g} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{{\alpha}^{2}}& 0& 0& 0\\ 0& \frac{1}{{\alpha}^{2}}& 0& 0\\ 0& 0& {r}^{2}& 0\\ 0& 0& 0& {{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}\end{array}\right]}}$
metric inverse:
${g} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{\frac{1}{{\alpha}^{2}}}& 0& 0& 0\\ 0& {\alpha}^{2}& 0& 0\\ 0& 0& \frac{1}{{r}^{2}}& 0\\ 0& 0& 0& \frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}\end{array}\right]}}$
metric derivative:
${{\partial g}} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& -{{{2}} {{{\alpha}^{2}}} {{\frac{\partial \alpha}{\partial r}}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{{{2}} {{\frac{\partial \alpha}{\partial r}}}}{{\alpha}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{\alpha}} \cdot {{r}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& {{2}} {{\alpha}} \cdot {{r}} {{{\sin\left( \theta\right)}^{2}}}& {{2}} {{r}} {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}& 0\end{array}\right]}\end{matrix}\right]}}$
1st kind Christoffel:
${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ -{{{{\alpha}^{2}}} {{\frac{\partial \alpha}{\partial r}}}}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{\frac{\partial \alpha}{\partial r}}{{\alpha}^{2}}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {{\alpha}} \cdot {{r}}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {{\alpha}} \cdot {{r}} {{{\sin\left( \theta\right)}^{2}}}\\ 0& 0& 0& {{r}} {{\cos\left( \theta\right)}} {{\sin\left( \theta\right)}}\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients / 2nd kind Christoffel:
${\Gamma} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ \frac{\partial \alpha}{\partial r}& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& -{\frac{\partial \alpha}{\partial r}}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& {\frac{1}{r}} {\alpha}& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& {\frac{1}{r}} {\alpha}\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{{{r}} {{\sin\left( \theta\right)}}}\\ 0& 0& 0& 0\end{array}\right]}\end{matrix}\right]}}$
connection coefficients derivative:
${{\partial \Gamma}} = {\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& {{\alpha}} \cdot {{\frac{\partial^ 2 \alpha}{\partial r^ 2}}}& 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& -{{{\alpha}} \cdot {{\frac{\partial^ 2 \alpha}{\partial 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& 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& \frac{{{\alpha}} \cdot {{\left({{-{\alpha}} + {{{r}} {{\frac{\partial \alpha}{\partial r}}}}}\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& 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& \frac{{{\alpha}} \cdot {{\left({{-{\alpha}} + {{{r}} {{\frac{\partial \alpha}{\partial r}}}}}\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& 0\\ 0& -{\frac{{{\alpha}} \cdot {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}}& -{\frac{1}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\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]}}$
connection coefficients squared:
${{(\Gamma^2)}} = {\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{\partial \alpha}{\partial 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& 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& {\frac{\partial \alpha}{\partial 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& 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& \frac{{\alpha}^{2}}{{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& 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& \frac{{\alpha}^{2}}{{r}^{2}}& \frac{{{\alpha}} \cdot {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}& 0\\ 0& \frac{{{\alpha}} \cdot {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}& \frac{{\cos\left( \theta\right)}^{2}}{{{{r}^{2}}} {{\left({{1} + {\cos\left( \theta\right)}}\right)}} {{\left({{1}{-{\cos\left( \theta\right)}}}\right)}}}& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${R} = {\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& -{\frac{{{2}} {{\alpha}} \cdot {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}}& 0\\ 0& \frac{{{2}} {{\alpha}} \cdot {{\cos\left( \theta\right)}}}{{{{r}^{2}}} {{\sin\left( \theta\right)}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${R} = {\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& -{\frac{{{2}} {{\alpha}} \cdot {{\cos\left( \theta\right)}}}{{{{r}^{4}}} {{{\sin\left( \theta\right)}^{3}}}}}& 0\\ 0& \frac{{{2}} {{\alpha}} \cdot {{\cos\left( \theta\right)}}}{{{{r}^{4}}} {{{\sin\left( \theta\right)}^{3}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Ricci curvature, $\sharp\flat$:
${R} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Gaussian curvature:
$0$
trace-free Ricci, $\sharp\flat$:
${{(R^{TF})}} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Einstein / trace-reversed Ricci curvature, $\sharp\flat$:
${G} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Schouten, $\sharp\flat$:
${P} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}$
Weyl, $\sharp\sharp\flat\flat$:
${C} = {\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& -{\frac{{{2}} {{\alpha}} \cdot {{\cos\left( \theta\right)}}}{{{{r}^{4}}} {{{\sin\left( \theta\right)}^{3}}}}}& 0\\ 0& \frac{{{2}} {{\alpha}} \cdot {{\cos\left( \theta\right)}}}{{{{r}^{4}}} {{{\sin\left( \theta\right)}^{3}}}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Weyl, $\flat\flat\flat\flat$:
${C} = {\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& -{{{2}} {{\alpha}} \cdot {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}}& 0\\ 0& {{2}} {{\alpha}} \cdot {{\sin\left( \theta\right)}} {{\cos\left( \theta\right)}}& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\end{array}\right]}}$
Plebanski, $\sharp\sharp\flat\flat$:
${P} = {\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: ${{{{ A} ^i} _{,i}} + {{{{{{ \Gamma} ^i} _i} _j}} {{{ A} ^j}}}} = {{{{\frac{\partial {A^{\hat{\phi}}}}{\partial \phi}}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}} + {{{\frac{\partial {A^{\hat{\theta}}}}{\partial \theta}}} {{\frac{1}{r}}}} + {{{\frac{\partial {A^{\hat{t}}}}{\partial t}}} {{\frac{1}{\alpha}}}} + {{{\alpha}} \cdot {{\frac{\partial {A^{\hat{r}}}}{\partial r}}}} + {{{-1}} {{{A^{\hat{r}}}}} \cdot {{\frac{\partial \alpha}{\partial r}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{\hat{t}} \\ \ddot{\hat{r}} \\ \ddot{\hat{\theta}} \\ \ddot{\hat{\phi}}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} {{-1}} {{\dot{\hat{r}}}} \cdot {{\dot{\hat{t}}}} \cdot {{\frac{\partial \alpha}{\partial r}}} \\ {{{\dot{\hat{r}}}^{2}}} {{\frac{\partial \alpha}{\partial r}}} \\ {{-1}} {{\alpha}} \cdot {{\dot{\hat{\theta}}}} \cdot {{\dot{\hat{r}}}} \cdot {{\frac{1}{r}}} \\ {{{-1}} {{\dot{\hat{\phi}}}} \cdot {{\dot{\hat{\theta}}}} \cdot {{\cos\left( \theta\right)}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}} + {{{-1}} {{\dot{\hat{\phi}}}} \cdot {{\alpha}} \cdot {{\dot{\hat{r}}}} \cdot {{\sin\left( \theta\right)}} {{\frac{1}{r}}} {{\frac{1}{\sin\left( \theta\right)}}}}\end{matrix}\right]}}$
parallel propagators:
${{[\Gamma_t]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
$\int\limits_{{{t_L}}}^{{{t_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d t$
= $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
${ P} _t$ = $\exp\left( -{\int\limits_{{{t_L}}}^{{{t_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d t}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
${{ P} _t}^{-1}$ = $\exp\left({\int\limits_{{{t_L}}}^{{{t_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d t}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
${{[\Gamma_r]}} = {\left[\begin{array}{cccc} {\frac{1}{\alpha}} {\frac{\partial \alpha}{\partial r}}& 0& 0& 0\\ 0& -{{\frac{1}{\alpha}} {\frac{\partial \alpha}{\partial r}}}& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& \frac{1}{r}\end{array}\right]}$
$\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{cccc} {\frac{1}{\alpha}} {\frac{\partial \alpha}{\partial r}}& 0& 0& 0\\ 0& -{{\frac{1}{\alpha}} {\frac{\partial \alpha}{\partial r}}}& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& \frac{1}{r}\end{array}\right]}}d r$
= $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& \log\left( {{\frac{1}{{r_L}}} {{r_R}}}\right)& 0\\ 0& 0& 0& \log\left( {{\frac{1}{{r_L}}} {{r_R}}}\right)\end{array}\right]$
${ P} _r$ = $\exp\left( -{\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{cccc} {\frac{1}{\alpha}} {\frac{\partial \alpha}{\partial r}}& 0& 0& 0\\ 0& -{{\frac{1}{\alpha}} {\frac{\partial \alpha}{\partial r}}}& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& \frac{1}{r}\end{array}\right]}}d r}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]$
${{ P} _r}^{-1}$ = $\exp\left({\int\limits_{{{r_L}}}^{{{r_R}}}{{\left[\begin{array}{cccc} {\frac{1}{\alpha}} {\frac{\partial \alpha}{\partial r}}& 0& 0& 0\\ 0& -{{\frac{1}{\alpha}} {\frac{\partial \alpha}{\partial r}}}& 0& 0\\ 0& 0& \frac{1}{r}& 0\\ 0& 0& 0& \frac{1}{r}\end{array}\right]}}d r}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_L}}} {{r_R}}& 0\\ 0& 0& 0& {\frac{1}{{r_L}}} {{r_R}}\end{array}\right]$
${{[\Gamma_\theta]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}\end{array}\right]}$
$\int\limits_{{{\theta_L}}}^{{{\theta_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}\end{array}\right]}}d \theta$
= $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \log\left( {\frac{\left|{\sin\left( {\theta_R}\right)}\right|}{\left|{\sin\left( {\theta_L}\right)}\right|}}\right)\end{array}\right]$
${ P} _{\theta}$ = $\exp\left( -{\int\limits_{{{\theta_L}}}^{{{\theta_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}\end{array}\right]}}d \theta}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]$
${{ P} _{\theta}}^{-1}$ = $\exp\left({\int\limits_{{{\theta_L}}}^{{{\theta_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \frac{\cos\left( \theta\right)}{\sin\left( \theta\right)}\end{array}\right]}}d \theta}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_R}\right)}\right|}{\left|{\sin\left( {\theta_L}\right)}\right|}\end{array}\right]$
${{[\Gamma_\phi]}} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
$\int\limits_{{{\phi_L}}}^{{{\phi_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d \phi$
= $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
${ P} _{\phi}$ = $\exp\left( -{\int\limits_{{{\phi_L}}}^{{{\phi_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d \phi}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
${{ P} _{\phi}}^{-1}$ = $\exp\left({\int\limits_{{{\phi_L}}}^{{{\phi_R}}}{{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}}d \phi}\right)$
= $\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$
propagator commutation:
[ ${ P} _t$ , ${ P} _r$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _t$ , ${ P} _{\theta}$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _t$ , ${ P} _{\phi}$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _r$ , ${ P} _{\theta}$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _r$ , ${ P} _{\phi}$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
[ ${ P} _{\theta}$ , ${ P} _{\phi}$ ] = ${{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}{-{{{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}} {{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}}}}$ = $\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]$
propagator partials
${{\frac{\partial}{\partial t}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \theta}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \phi}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial t}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \theta}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \phi}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& {\frac{1}{{r_R}}} {{r_L}}& 0\\ 0& 0& 0& {\frac{1}{{r_R}}} {{r_L}}\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial t}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \theta}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \phi}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& \frac{\left|{\sin\left( {\theta_L}\right)}\right|}{\left|{\sin\left( {\theta_R}\right)}\right|}\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial t}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial r}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \theta}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
${{\frac{\partial}{\partial \phi}}\left({\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}\right)} = {\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}$
volume element: ${{i}} {{{r}^{2}}} {{\sin\left( \theta\right)}}$
volume integral: ${-{{\frac{1}{3}} {{{\Delta (cos(\theta))}} \cdot {{{\Delta (r^3)}}} \cdot {{\Delta t}} \cdot {{i}}}}} {{\Delta \phi}}$
finite volume (0,0)-form:
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{{\mathcal{V}(x_C)}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{ {-{{\frac{1}{3}} {{{\Delta (cos(\theta))}} \cdot {{{\Delta (r^3)}}} \cdot {{\Delta t}} \cdot {{i}}}}} {{\Delta \phi}}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{\Delta t}} \cdot {{\left({{{{\frac{1}{ {-{{\frac{1}{3}} {{{\Delta (cos(\theta))}} \cdot {{{\Delta (r^3)}}} \cdot {{\Delta t}} \cdot {{i}}}}} {{\Delta \phi}}}}} {{0}}} + {{S(x_C)}}}\right)}}}}$
${{u(x_C, t_R)}} = {{{u(x_C, t_L)}} + {{{{S(x_C)}}} \cdot {{\Delta t}}}}$