spherical and time, lapse varying in radial
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 $e_I = \{e_{t}, e_{x}, e_{y}, e_{z}\}$
transform from basis to coordinate:
${{{ \tilde{e}}_t}^t} = {1}$;
${{{ \tilde{e}}_r}^r} = {1}$;
${{{ \tilde{e}}_{\theta}}^{\theta}} = {1}$;
${{{ \tilde{e}}_{\phi}}^{\phi}} = {1}$
transform from coorinate to basis:
${{{ \tilde{e}}^t}_t} = {1}$;
${{{ \tilde{e}}^r}_r} = {1}$;
${{{ \tilde{e}}^{\theta}}_{\theta}} = {1}$;
${{{ \tilde{e}}^{\phi}}_{\phi}} = {1}$
tensor index associated with coordinate $t$ is index $t$ with operator $e_{t}(\zeta) = $${\partial \zeta}\over{\partial t}$
tensor index associated with coordinate $r$ is index $r$ with operator $e_{r}(\zeta) = $${\partial \zeta}\over{\partial r}$
tensor index associated with coordinate $\theta$ is index $\theta$ with operator $e_{\theta}(\zeta) = $${\partial \zeta}\over{\partial\theta}$
tensor index associated with coordinate $\phi$ is index $\phi$ with operator $e_{\phi}(\zeta) = $${\partial \zeta}\over{\partial\phi}$
flat metric:
${{{ \eta}_t}_t} = {-1}$;
${{{ \eta}_x}_x} = {1}$;
${{{ \eta}_y}_y} = {1}$;
${{{ \eta}_z}_z} = {1}$
chart in embedded coordinates:
${{ u}^t} = {{{t}} {{\alpha}}}$;
${{ u}^x} = {{{{{r}} {{sin\left( \theta\right)}}}} {{cos\left( \phi\right)}}}$;
${{ u}^y} = {{{{{r}} {{sin\left( \theta\right)}}}} {{sin\left( \phi\right)}}}$;
${{ u}^z} = {{{r}} {{cos\left( \theta\right)}}}$
basis operators applied to chart:
${{{ e}_u}^I} = {{{ u}^I}_{,u}}$
${{{ e}_t}^t} = {\alpha}$;
${{{ e}_r}^t} = {{{t}} {{{\partial \alpha}\over{\partial r}}}}$;
${{{ e}_r}^x} = {{{sin\left( \theta\right)}} {{cos\left( \phi\right)}}}$;
${{{ e}_r}^y} = {{{sin\left( \theta\right)}} {{sin\left( \phi\right)}}}$;
${{{ e}_r}^z} = {cos\left( \theta\right)}$;
${{{ e}_{\theta}}^x} = {{{r}} {{cos\left( \theta\right)}} {{cos\left( \phi\right)}}}$;
${{{ e}_{\theta}}^y} = {{{r}} {{cos\left( \theta\right)}} {{sin\left( \phi\right)}}}$;
${{{ e}_{\theta}}^z} = {-{{{r}} {{sin\left( \theta\right)}}}}$;
${{{ e}_{\phi}}^x} = {-{{{r}} {{sin\left( \theta\right)}} {{sin\left( \phi\right)}}}}$;
${{{ e}_{\phi}}^y} = {{{r}} {{sin\left( \theta\right)}} {{cos\left( \phi\right)}}}$
${{{ e}^t}_t} = {\frac{1}{\alpha}}$;
${{{ e}^t}_x} = {{\frac{1}{\alpha}}{({-{{{cos\left( \phi\right)}} {{t}} {{{\partial \alpha}\over{\partial r}}} {{sin\left( \theta\right)}}}})}}$;
${{{ e}^t}_y} = {{\frac{1}{\alpha}}{({-{{{sin\left( \phi\right)}} {{t}} {{{\partial \alpha}\over{\partial r}}} {{sin\left( \theta\right)}}}})}}$;
${{{ e}^t}_z} = {\frac{{{cos\left( \theta\right)}} {{t}} {{{\partial \alpha}\over{\partial r}}}}{-{\alpha}}}$;
${{{ e}^r}_x} = {{{cos\left( \phi\right)}} {{sin\left( \theta\right)}}}$;
${{{ e}^r}_y} = {{{sin\left( \phi\right)}} {{sin\left( \theta\right)}}}$;
${{{ e}^r}_z} = {cos\left( \theta\right)}$;
${{{ e}^{\theta}}_x} = {{\frac{1}{r}}{({{{cos\left( \theta\right)}} {{cos\left( \phi\right)}}})}}$;
${{{ e}^{\theta}}_y} = {{\frac{1}{r}}{({{{sin\left( \phi\right)}} {{cos\left( \theta\right)}}})}}$;
${{{ e}^{\theta}}_z} = {\frac{sin\left( \theta\right)}{-{r}}}$;
${{{ e}^{\phi}}_x} = {\frac{-{sin\left( \phi\right)}}{{{r}} {{sin\left( \theta\right)}}}}$;
${{{ e}^{\phi}}_y} = {\frac{cos\left( \phi\right)}{{{r}} {{sin\left( \theta\right)}}}}$
${{{{{ e}_u}^I}} {{{{ e}^v}_I}}} = {\overset{u\downarrow v\rightarrow}{\left[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{matrix}\right]}}$
${{{{{ e}_u}^I}} {{{{ e}^u}_J}}} = {\overset{I\downarrow J\rightarrow}{\left[\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{matrix}\right]}}$
basis determinant: ${det(e)} = {{{\alpha}} \cdot {{{r}^{2}}} {{sin\left( \theta\right)}}}$
${{{{ c}_a}_b}^c} = {0}$
${{{ g}_u}_v} = {{{{{{{ e}_u}^I}} {{{{ e}_v}^J}}}} {{{{ \eta}_I}_J}}}$
${{{ g}_t}_t} = {-{{\alpha}^{2}}}$;
${{{ g}_t}_r} = {-{{{\alpha}} \cdot {{t}} {{{\partial \alpha}\over{\partial r}}}}}$;
${{{ g}_r}_t} = {-{{{t}} {{{\partial \alpha}\over{\partial r}}} {{\alpha}}}}$;
${{{ g}_r}_r} = {{1} - {{{{t}^{2}}} {{{{\partial \alpha}\over{\partial r}}^{2}}}}}$;
${{{ g}_{\theta}}_{\theta}} = {{r}^{2}}$;
${{{ g}_{\phi}}_{\phi}} = {{{{r}^{2}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}$
${{{ g}_u}_v} = {{{{{{{ e}_u}^I}} {{{{ e}_v}^J}}}} {{{{ \eta}_I}_J}}}$
metric determinant: ${det(g)} = {-{{{{r}^{4}}} {{{\alpha}^{2}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}}$
${{{{ \Gamma}_a}_b}_c} = {{{\frac{1}{2}}} {{({{{{{{{{{ g}_a}_b}_{,c}} + {{{{ g}_a}_c}_{,b}}} - {{{{ g}_b}_c}_{,a}}} + {{{{ c}_a}_b}_c}} + {{{{ c}_a}_c}_b}} - {{{{ c}_c}_b}_a}})}}}$
commutation coefficients:
${{{{ c}_a}_b}^c} = {0}$
metric:
${{{ g}_t}_t} = {-{{\alpha}^{2}}}$;
${{{ g}_t}_r} = {-{{{\alpha}} \cdot {{t}} {{{\partial \alpha}\over{\partial r}}}}}$;
${{{ g}_r}_t} = {-{{{t}} {{{\partial \alpha}\over{\partial r}}} {{\alpha}}}}$;
${{{ g}_r}_r} = {{1} - {{{{t}^{2}}} {{{{\partial \alpha}\over{\partial r}}^{2}}}}}$;
${{{ g}_{\theta}}_{\theta}} = {{r}^{2}}$;
${{{ g}_{\phi}}_{\phi}} = {{{{r}^{2}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}$
metric inverse:
${{{ g}^t}^t} = {\frac{{1} - {{{{t}^{2}}} {{{{\partial \alpha}\over{\partial r}}^{2}}}}}{-{{\alpha}^{2}}}}$;
${{{ g}^t}^r} = {{\frac{1}{\alpha}}{({-{{{t}} {{{\partial \alpha}\over{\partial r}}}}})}}$;
${{{ g}^r}^t} = {{\frac{1}{\alpha}}{({-{{{t}} {{{\partial \alpha}\over{\partial r}}}}})}}$;
${{{ g}^r}^r} = {1}$;
${{{ g}^{\theta}}^{\theta}} = {\frac{1}{{r}^{2}}}$;
${{{ g}^{\phi}}^{\phi}} = {\frac{1}{{{{r}^{2}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}}$
metric derivative:
${{{{ {\partial g}}_t}_t}_r} = {-{{{2}} {{\alpha}} \cdot {{{\partial \alpha}\over{\partial r}}}}}$;
${{{{ {\partial g}}_t}_r}_t} = {-{{{\alpha}} \cdot {{{\partial \alpha}\over{\partial r}}}}}$;
${{{{ {\partial g}}_t}_r}_r} = {-{{{t}} {{({{{{\partial \alpha}\over{\partial r}}^{2}} + {{{\alpha}} \cdot {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}}})}}}}$;
${{{{ {\partial g}}_r}_t}_t} = {-{{{{\partial \alpha}\over{\partial r}}} {{\alpha}}}}$;
${{{{ {\partial g}}_r}_t}_r} = {-{{{t}} {{({{{{\partial \alpha}\over{\partial r}}^{2}} + {{{\alpha}} \cdot {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}}})}}}}$;
${{{{ {\partial g}}_r}_r}_t} = {-{{{2}} {{t}} {{{{\partial \alpha}\over{\partial r}}^{2}}}}}$;
${{{{ {\partial g}}_r}_r}_r} = {-{{{2}} {{{t}^{2}}} {{{\partial \alpha}\over{\partial r}}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}}}$;
${{{{ {\partial g}}_{\theta}}_{\theta}}_r} = {{{2}} {{r}}}$;
${{{{ {\partial g}}_{\phi}}_{\phi}}_r} = {{{2}} {{r}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}$;
${{{{ {\partial g}}_{\phi}}_{\phi}}_{\theta}} = {{{2}} {{{r}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}$
1st kind Christoffel:
${{{{ \Gamma}_t}_t}_r} = {-{{{\alpha}} \cdot {{{\partial \alpha}\over{\partial r}}}}}$;
${{{{ \Gamma}_t}_r}_t} = {-{{{\alpha}} \cdot {{{\partial \alpha}\over{\partial r}}}}}$;
${{{{ \Gamma}_t}_r}_r} = {-{{{t}} {{\alpha}} \cdot {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}}}$;
${{{{ \Gamma}_r}_t}_r} = {-{{{t}} {{{{\partial \alpha}\over{\partial r}}^{2}}}}}$;
${{{{ \Gamma}_r}_r}_t} = {-{{{t}} {{{{\partial \alpha}\over{\partial r}}^{2}}}}}$;
${{{{ \Gamma}_r}_r}_r} = {-{{{{t}^{2}}} {{{\partial \alpha}\over{\partial r}}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}}}$;
${{{{ \Gamma}_r}_{\theta}}_{\theta}} = {-{r}}$;
${{{{ \Gamma}_r}_{\phi}}_{\phi}} = {-{{{r}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}}$;
${{{{ \Gamma}_{\theta}}_r}_{\theta}} = {r}$;
${{{{ \Gamma}_{\theta}}_{\theta}}_r} = {r}$;
${{{{ \Gamma}_{\theta}}_{\phi}}_{\phi}} = {-{{{{r}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}}$;
${{{{ \Gamma}_{\phi}}_r}_{\phi}} = {{{r}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}$;
${{{{ \Gamma}_{\phi}}_{\theta}}_{\phi}} = {{{{r}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}$;
${{{{ \Gamma}_{\phi}}_{\phi}}_r} = {{{r}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}$;
${{{{ \Gamma}_{\phi}}_{\phi}}_{\theta}} = {{{{r}^{2}}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}$
connection coefficients / 2nd kind Christoffel:
${{{{ \Gamma}^t}_t}_r} = {{\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})}}$;
${{{{ \Gamma}^t}_r}_t} = {{\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})}}$;
${{{{ \Gamma}^t}_r}_r} = {{\frac{1}{\alpha}}{({{{t}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}})}}$;
${{{{ \Gamma}^t}_{\theta}}_{\theta}} = {{\frac{1}{\alpha}}{({{{t}} {{{\partial \alpha}\over{\partial r}}} {{r}}})}}$;
${{{{ \Gamma}^t}_{\phi}}_{\phi}} = {{\frac{1}{\alpha}}{({{{t}} {{{\partial \alpha}\over{\partial r}}} {{r}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}})}}$;
${{{{ \Gamma}^r}_{\theta}}_{\theta}} = {-{r}}$;
${{{{ \Gamma}^r}_{\phi}}_{\phi}} = {-{{{r}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}}$;
${{{{ \Gamma}^{\theta}}_r}_{\theta}} = {\frac{1}{r}}$;
${{{{ \Gamma}^{\theta}}_{\theta}}_r} = {\frac{1}{r}}$;
${{{{ \Gamma}^{\theta}}_{\phi}}_{\phi}} = {-{{{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}}$;
${{{{ \Gamma}^{\phi}}_r}_{\phi}} = {\frac{1}{r}}$;
${{{{ \Gamma}^{\phi}}_{\theta}}_{\phi}} = {\frac{cos\left( \theta\right)}{sin\left( \theta\right)}}$;
${{{{ \Gamma}^{\phi}}_{\phi}}_r} = {\frac{1}{r}}$;
${{{{ \Gamma}^{\phi}}_{\phi}}_{\theta}} = {\frac{cos\left( \theta\right)}{sin\left( \theta\right)}}$
connection coefficients derivative:
${{{{{ {\partial \Gamma}}^t}_t}_r}_r} = {\frac{{{{{\partial^ 2 \alpha}\over{\partial r^ 2}}} {{\alpha}}} - {{{\partial \alpha}\over{\partial r}}^{2}}}{{\alpha}^{2}}}$;
${{{{{ {\partial \Gamma}}^t}_r}_t}_r} = {\frac{{{{{\partial^ 2 \alpha}\over{\partial r^ 2}}} {{\alpha}}} - {{{\partial \alpha}\over{\partial r}}^{2}}}{{\alpha}^{2}}}$;
${{{{{ {\partial \Gamma}}^t}_r}_r}_t} = {{\frac{1}{\alpha}}{({{\partial^ 2 \alpha}\over{\partial r^ 2}})}}$;
${{{{{ {\partial \Gamma}}^t}_r}_r}_r} = {\frac{-{{{t}} {{({{{{{\partial \alpha}\over{\partial r}}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}} - {{{\alpha}} \cdot {{{\partial^ 3 \alpha}\over{\partial r^ 3}}}}})}}}}{{\alpha}^{2}}}$;
${{{{{ {\partial \Gamma}}^t}_{\theta}}_{\theta}}_t} = {{\frac{1}{\alpha}}{({{{{\partial \alpha}\over{\partial r}}} {{r}}})}}$;
${{{{{ {\partial \Gamma}}^t}_{\theta}}_{\theta}}_r} = {\frac{{{t}} {{({{{{{\alpha}} \cdot {{{\partial \alpha}\over{\partial r}}}} - {{{r}} {{{{\partial \alpha}\over{\partial r}}^{2}}}}} + {{{r}} {{\alpha}} \cdot {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}}})}}}{{\alpha}^{2}}}$;
${{{{{ {\partial \Gamma}}^t}_{\phi}}_{\phi}}_t} = {{\frac{1}{\alpha}}{({{{{\partial \alpha}\over{\partial r}}} {{r}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}})}}$;
${{{{{ {\partial \Gamma}}^t}_{\phi}}_{\phi}}_r} = {\frac{{{t}} {{({{{{{{\alpha}} \cdot {{{\partial \alpha}\over{\partial r}}}} - {{{\alpha}} \cdot {{{\partial \alpha}\over{\partial r}}} {{{cos\left( \theta\right)}^{2}}}}} - {{{r}} {{{{\partial \alpha}\over{\partial r}}^{2}}}}} + {{{r}} {{{cos\left( \theta\right)}^{2}}} {{{{\partial \alpha}\over{\partial r}}^{2}}}} + {{{{r}} {{\alpha}} \cdot {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}} - {{{r}} {{\alpha}} \cdot {{{\partial^ 2 \alpha}\over{\partial r^ 2}}} {{{cos\left( \theta\right)}^{2}}}}}})}}}{{\alpha}^{2}}}$;
${{{{{ {\partial \Gamma}}^t}_{\phi}}_{\phi}}_{\theta}} = {{\frac{1}{\alpha}}{({{{2}} {{t}} {{{\partial \alpha}\over{\partial r}}} {{r}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}})}}$;
${{{{{ {\partial \Gamma}}^r}_{\theta}}_{\theta}}_r} = {-{1}}$;
${{{{{ {\partial \Gamma}}^r}_{\phi}}_{\phi}}_r} = {-{({{1} - {{cos\left( \theta\right)}^{2}}})}}$;
${{{{{ {\partial \Gamma}}^r}_{\phi}}_{\phi}}_{\theta}} = {-{{{2}} {{r}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}}$;
${{{{{ {\partial \Gamma}}^{\theta}}_r}_{\theta}}_r} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {\partial \Gamma}}^{\theta}}_{\theta}}_r}_r} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {\partial \Gamma}}^{\theta}}_{\phi}}_{\phi}}_{\theta}} = {{{({{sin\left( \theta\right)} + {cos\left( \theta\right)}})}} {{({{sin\left( \theta\right)} - {cos\left( \theta\right)}})}}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_r}_{\phi}}_r} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_{\theta}}_{\phi}}_{\theta}} = {-{\frac{1}{{1} - {{cos\left( \theta\right)}^{2}}}}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_{\phi}}_r}_r} = {-{\frac{1}{{r}^{2}}}}$;
${{{{{ {\partial \Gamma}}^{\phi}}_{\phi}}_{\theta}}_{\theta}} = {-{\frac{1}{{1} - {{cos\left( \theta\right)}^{2}}}}}$
connection coefficients squared:
${{{{{ {(\Gamma^2)}}^t}_t}_r}_r} = {\frac{{{\partial \alpha}\over{\partial r}}^{2}}{{\alpha}^{2}}}$;
${{{{{ {(\Gamma^2)}}^t}_r}_r}_t} = {\frac{{{\partial \alpha}\over{\partial r}}^{2}}{{\alpha}^{2}}}$;
${{{{{ {(\Gamma^2)}}^t}_r}_r}_r} = {\frac{{{{\partial \alpha}\over{\partial r}}} {{t}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}}{{\alpha}^{2}}}$;
${{{{{ {(\Gamma^2)}}^t}_r}_{\theta}}_{\theta}} = {{\frac{1}{\alpha}}{({{{t}} {{{\partial \alpha}\over{\partial r}}}})}}$;
${{{{{ {(\Gamma^2)}}^t}_r}_{\phi}}_{\phi}} = {{\frac{1}{\alpha}}{({{{t}} {{{\partial \alpha}\over{\partial r}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}})}}$;
${{{{{ {(\Gamma^2)}}^t}_{\theta}}_t}_{\theta}} = {{\frac{1}{\alpha}}{({-{{{{\partial \alpha}\over{\partial r}}} {{r}}}})}}$;
${{{{{ {(\Gamma^2)}}^t}_{\theta}}_r}_{\theta}} = {\frac{{{t}} {{r}} {{({{{{\partial \alpha}\over{\partial r}}^{2}} - {{{\alpha}} \cdot {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}}})}}}{{\alpha}^{2}}}$;
${{{{{ {(\Gamma^2)}}^t}_{\theta}}_{\theta}}_r} = {{\frac{1}{\alpha}}{({{{t}} {{{\partial \alpha}\over{\partial r}}}})}}$;
${{{{{ {(\Gamma^2)}}^t}_{\theta}}_{\phi}}_{\phi}} = {{\frac{1}{\alpha}}{({{{t}} {{{\partial \alpha}\over{\partial r}}} {{r}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}})}}$;
${{{{{ {(\Gamma^2)}}^t}_{\phi}}_t}_{\phi}} = {{\frac{1}{\alpha}}{({-{{{{\partial \alpha}\over{\partial r}}} {{r}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}}})}}$;
${{{{{ {(\Gamma^2)}}^t}_{\phi}}_r}_{\phi}} = {\frac{{{t}} {{r}} {{({{{{{\partial \alpha}\over{\partial r}}^{2}} - {{{{cos\left( \theta\right)}^{2}}} {{{{\partial \alpha}\over{\partial r}}^{2}}}}} + {{{{\alpha}} \cdot {{{\partial^ 2 \alpha}\over{\partial r^ 2}}} {{{cos\left( \theta\right)}^{2}}}} - {{{\alpha}} \cdot {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}}}})}}}{{\alpha}^{2}}}$;
${{{{{ {(\Gamma^2)}}^t}_{\phi}}_{\theta}}_{\phi}} = {{\frac{1}{\alpha}}{({-{{{t}} {{{\partial \alpha}\over{\partial r}}} {{r}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})}}$;
${{{{{ {(\Gamma^2)}}^t}_{\phi}}_{\phi}}_r} = {{\frac{1}{\alpha}}{({{{t}} {{{\partial \alpha}\over{\partial r}}} {{({{1} - {{cos\left( \theta\right)}^{2}}})}}})}}$;
${{{{{ {(\Gamma^2)}}^t}_{\phi}}_{\phi}}_{\theta}} = {{\frac{1}{\alpha}}{({{{t}} {{{\partial \alpha}\over{\partial r}}} {{r}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}})}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_{\theta}}_{\theta}} = {-{1}}$;
${{{{{ {(\Gamma^2)}}^r}_r}_{\phi}}_{\phi}} = {-{({{1} - {{cos\left( \theta\right)}^{2}}})}}$;
${{{{{ {(\Gamma^2)}}^r}_{\theta}}_{\theta}}_r} = {-{1}}$;
${{{{{ {(\Gamma^2)}}^r}_{\theta}}_{\phi}}_{\phi}} = {-{{{r}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}}$;
${{{{{ {(\Gamma^2)}}^r}_{\phi}}_{\theta}}_{\phi}} = {{{r}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}$;
${{{{{ {(\Gamma^2)}}^r}_{\phi}}_{\phi}}_r} = {-{({{1} - {{cos\left( \theta\right)}^{2}}})}}$;
${{{{{ {(\Gamma^2)}}^r}_{\phi}}_{\phi}}_{\theta}} = {-{{{r}} {{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}}$;
${{{{{ {(\Gamma^2)}}^{\theta}}_r}_r}_{\theta}} = {\frac{1}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\theta}}_r}_{\phi}}_{\phi}} = {{\frac{1}{r}}{({-{{{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})}}$;
${{{{{ {(\Gamma^2)}}^{\theta}}_{\theta}}_r}_r} = {\frac{1}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\theta}}_{\theta}}_{\theta}}_{\theta}} = {-{1}}$;
${{{{{ {(\Gamma^2)}}^{\theta}}_{\theta}}_{\phi}}_{\phi}} = {-{{cos\left( \theta\right)}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\theta}}_{\phi}}_r}_{\phi}} = {{\frac{1}{r}}{({-{{{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})}}$;
${{{{{ {(\Gamma^2)}}^{\theta}}_{\phi}}_{\theta}}_{\phi}} = {-{({{1} - {{cos\left( \theta\right)}^{2}}})}}$;
${{{{{ {(\Gamma^2)}}^{\theta}}_{\phi}}_{\phi}}_r} = {{\frac{1}{r}}{({-{{{cos\left( \theta\right)}} {{sin\left( \theta\right)}}}})}}$;
${{{{{ {(\Gamma^2)}}^{\theta}}_{\phi}}_{\phi}}_{\theta}} = {-{{cos\left( \theta\right)}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_r}_r}_{\phi}} = {\frac{1}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_r}_{\theta}}_{\phi}} = {\frac{cos\left( \theta\right)}{{{sin\left( \theta\right)}} {{r}}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_r}_{\phi}}_{\theta}} = {\frac{cos\left( \theta\right)}{{{sin\left( \theta\right)}} {{r}}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\theta}}_r}_{\phi}} = {\frac{cos\left( \theta\right)}{{{r}} {{sin\left( \theta\right)}}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\theta}}_{\theta}}_{\phi}} = {\frac{{cos\left( \theta\right)}^{2}}{{1} - {{cos\left( \theta\right)}^{2}}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\theta}}_{\phi}}_r} = {\frac{cos\left( \theta\right)}{{{sin\left( \theta\right)}} {{r}}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\theta}}_{\phi}}_{\theta}} = {-{1}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_r}_r} = {\frac{1}{{r}^{2}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_r}_{\theta}} = {\frac{cos\left( \theta\right)}{{{r}} {{sin\left( \theta\right)}}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_{\theta}}_r} = {\frac{cos\left( \theta\right)}{{{sin\left( \theta\right)}} {{r}}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_{\theta}}_{\theta}} = {\frac{{cos\left( \theta\right)}^{2}}{{1} - {{cos\left( \theta\right)}^{2}}}}$;
${{{{{ {(\Gamma^2)}}^{\phi}}_{\phi}}_{\phi}}_{\phi}} = {-{1}}$
Riemann curvature, $\sharp\flat\flat\flat$:
${{{{{ R}^a}_b}_c}_d} = {0}$
Riemann curvature, $\sharp\sharp\flat\flat$:
${{{{{ R}^a}^b}_c}_d} = {0}$
Ricci curvature, $\sharp\flat$:
${{{ R}^a}_b} = {0}$
Gaussian curvature:
$0$
Einstein $\sharp\flat$ / trace-reversed Ricci curvature:
${{{ G}^a}_b} = {0}$
divergence: ${{{{ A}^i}_{,i}} + {{{{{{ \Gamma}^i}_i}_j}} {{{ A}^j}}}} = {{{\partial {A^{t}}}\over{\partial t}} + {{\partial {A^{r}}}\over{\partial r}} + {{\partial {A^{\theta}}}\over{\partial\theta}} + {{\partial {A^{\phi}}}\over{\partial\phi}} + {{{2}} {{{A^{r}}}} \cdot {{\frac{1}{r}}}} + {{{{\partial \alpha}\over{\partial r}}} {{{A^{r}}}} \cdot {{\frac{1}{\alpha}}}} + {{{cos\left( \theta\right)}} {{{A^{\theta}}}} \cdot {{\frac{1}{sin\left( \theta\right)}}}}}$
geodesic:
${\overset{a\downarrow}{\left[\begin{matrix} \ddot{t} \\ \ddot{r} \\ \ddot{\theta} \\ \ddot{\phi}\end{matrix}\right]}} = {\overset{a\downarrow}{\left[\begin{matrix} {{{-2}} {{\dot{r}}} \cdot {{\dot{t}}} \cdot {{{\partial \alpha}\over{\partial r}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{r}} {{t}} {{{\partial \alpha}\over{\partial r}}} {{{\dot{\phi}}^{2}}} {{\frac{1}{\alpha}}}} + {{{r}} {{t}} {{{\partial \alpha}\over{\partial r}}} {{{\dot{\phi}}^{2}}} {{{cos\left( \theta\right)}^{2}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{r}} {{t}} {{{\partial \alpha}\over{\partial r}}} {{{\dot{\theta}}^{2}}} {{\frac{1}{\alpha}}}} + {{{-1}} {{t}} {{{\dot{r}}^{2}}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}} {{\frac{1}{\alpha}}}} \\ {{{r}} {{{\dot{\phi}}^{2}}}} + {{{-1}} {{r}} {{{\dot{\phi}}^{2}}} {{{cos\left( \theta\right)}^{2}}}} + {{{r}} {{{\dot{\theta}}^{2}}}} \\ {{{cos\left( \theta\right)}} {{sin\left( \theta\right)}} {{{\dot{\phi}}^{2}}}} + {{{-2}} {{\dot{\theta}}} \cdot {{\dot{r}}} \cdot {{\frac{1}{r}}}} \\ {{{-2}} {{\dot{\phi}}} \cdot {{\dot{r}}} \cdot {{sin\left( \theta\right)}} {{\frac{1}{sin\left( \theta\right)}}} {{\frac{1}{r}}}} + {{{-2}} {{\dot{\phi}}} \cdot {{r}} {{\dot{\theta}}} \cdot {{cos\left( \theta\right)}} {{\frac{1}{sin\left( \theta\right)}}} {{\frac{1}{r}}}}\end{matrix}\right]}}$
parallel propagators:
${{[\Gamma_t]}} = {\left[\begin{matrix} 0 & {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix}\right]}$
$\int\limits_{{{t_L}}}^{{{t_R}}}d t\left( \left[\begin{matrix} 0 & {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix}\right]\right)$
= $\left[\begin{matrix} 0 & {\frac{1}{\alpha}}{({-{{{{\partial \alpha}\over{\partial r}}} {{({{{t_L}} - {{t_R}}})}}}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix}\right]$
${ P}_t$ = ${e}^{( -{({\int\limits_{{{t_L}}}^{{{t_R}}}d t\left( \left[\begin{matrix} 0 & {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix}\right]\right)})})}$
= $\left[\begin{matrix} 0 & {\frac{1}{\alpha}}{({{{{\partial \alpha}\over{\partial r}}} {{({{{t_L}} - {{t_R}}})}}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix}\right]$
${{ P}_t}^{-1}$ = ${e}^{({\int\limits_{{{t_L}}}^{{{t_R}}}d t\left( \left[\begin{matrix} 0 & {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix}\right]\right)})}$
= $\left[\begin{matrix} 0 & {\frac{1}{\alpha}}{({-{{{{\partial \alpha}\over{\partial r}}} {{({{{t_L}} - {{t_R}}})}}}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix}\right]$
${{[\Gamma_r]}} = {\left[\begin{matrix} {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & {\frac{1}{\alpha}}{({{{t}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & 0 & 0 & \frac{1}{r}\end{matrix}\right]}$
$\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & {\frac{1}{\alpha}}{({{{t}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & 0 & 0 & \frac{1}{r}\end{matrix}\right]\right)$
= $\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & {\frac{1}{\alpha}}{({{{t}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & 0 & 0 & \frac{1}{r}\end{matrix}\right]\right)$
${ P}_r$ = ${e}^{( -{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & {\frac{1}{\alpha}}{({{{t}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & 0 & 0 & \frac{1}{r}\end{matrix}\right]\right)})})}$
negIntConn $-{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & {\frac{1}{\alpha}}{({{{t}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & 0 & 0 & \frac{1}{r}\end{matrix}\right]\right)})}$
negIntConn $-{({\int\limits_{{{r_L}}}^{{{r_R}}}d r\left( \left[\begin{matrix} {\frac{1}{\alpha}}{({{\partial \alpha}\over{\partial r}})} & {\frac{1}{\alpha}}{({{{t}} {{{\partial^ 2 \alpha}\over{\partial r^ 2}}}})} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{1}{r} & 0 \\ 0 & 0 & 0 & \frac{1}{r}\end{matrix}\right]\right)})}$