${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{{\alpha}^{2}} & 0 & 0 & 0 \\ 0 & \frac{1}{{\alpha}^{2}} & 0 & 0 \\ 0 & 0 & {r}^{2} & 0 \\ 0 & 0 & 0 & {\left({{{r}} {{\sin\left( \theta\right)}}}\right)}^{2}\end{matrix} \right]}}$ ${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{-1}{{\alpha}^{2}} & 0 & 0 & 0 \\ 0 & {\alpha}^{2} & 0 & 0 \\ 0 & 0 & {r}^{-2} & 0 \\ 0 & 0 & 0 & {\left({{{r}} {{\sin\left( \theta\right)}}}\right)}^{-2}\end{matrix} \right]}}$
resting gravity:
${{{{ \Gamma} ^u} _t} _t} = {\overset{u\downarrow}{\left[ \begin{matrix} 0 \\ {{{\alpha}^{3}}} {{\frac{\partial \alpha}{\partial r}}} \\ 0 \\ 0\end{matrix} \right]}}$
${{{ G} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{{\alpha}^{2}}} {{\left({{{1} - {{\alpha}^{2}}} - {{{2}} {{\alpha}} \cdot {{r}} {{\frac{\partial \alpha}{\partial r}}}}}\right)}}}{{r}^{2}} & 0 & 0 & 0 \\ 0 & \frac{{-{1}} + {{{2}} {{\alpha}} \cdot {{r}} {{\frac{\partial \alpha}{\partial r}}}} + {{\alpha}^{2}}}{{{{\alpha}^{2}}} {{{r}^{2}}}} & 0 & 0 \\ 0 & 0 & {{{{r}^{2}}} {{{\frac{\partial \alpha}{\partial r}}^{2}}}} + {{{\alpha}} \cdot {{{r}^{2}}} {{\frac{\partial^ 2 \alpha}{\partial r^ 2}}}} + {{{2}} {{\alpha}} \cdot {{r}} {{\frac{\partial \alpha}{\partial r}}}} & 0 \\ 0 & 0 & 0 & {{{{{r}^{2}}} {{{\frac{\partial \alpha}{\partial r}}^{2}}}} - {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial \alpha}{\partial r}}^{2}}}}} + {{{{\alpha}} \cdot {{{r}^{2}}} {{\frac{\partial^ 2 \alpha}{\partial r^ 2}}}} - {{{\alpha}} \cdot {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 \alpha}{\partial r^ 2}}}}} + {{{{2}} {{\alpha}} \cdot {{r}} {{\frac{\partial \alpha}{\partial r}}}} - {{{2}} {{\alpha}} \cdot {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial \alpha}{\partial r}}}}}\end{matrix} \right]}}$ ${{{ T} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \rho & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}}$
${\frac{{{{\alpha}^{2}}} {{\left({{{1} - {{\alpha}^{2}}} - {{{2}} {{\alpha}} \cdot {{r}} {{\frac{\partial \alpha}{\partial r}}}}}\right)}}}{{r}^{2}}} = {{{8}} {{\rho}} \cdot {{\pi}}}$
${\frac{{-{1}} + {{{2}} {{\alpha}} \cdot {{r}} {{\frac{\partial \alpha}{\partial r}}}} + {{\alpha}^{2}}}{{{{\alpha}^{2}}} {{{r}^{2}}}}} = {0}$
${{{{{r}^{2}}} {{{\frac{\partial \alpha}{\partial r}}^{2}}}} + {{{\alpha}} \cdot {{{r}^{2}}} {{\frac{\partial^ 2 \alpha}{\partial r^ 2}}}} + {{{2}} {{\alpha}} \cdot {{r}} {{\frac{\partial \alpha}{\partial r}}}}} = {0}$
${{{{{{r}^{2}}} {{{\frac{\partial \alpha}{\partial r}}^{2}}}} - {{{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\frac{\partial \alpha}{\partial r}}^{2}}}}} + {{{{\alpha}} \cdot {{{r}^{2}}} {{\frac{\partial^ 2 \alpha}{\partial r^ 2}}}} - {{{\alpha}} \cdot {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial^ 2 \alpha}{\partial r^ 2}}}}} + {{{{2}} {{\alpha}} \cdot {{r}} {{\frac{\partial \alpha}{\partial r}}}} - {{{2}} {{\alpha}} \cdot {{r}} {{{\cos\left( \theta\right)}^{2}}} {{\frac{\partial \alpha}{\partial r}}}}}} = {0}$
Gnuplot Produced by GNUPLOT 5.4 patchlevel 1 0 2x10-23 4x10-23 6x10-23 8x10-23 1x10-22 1.2x10-22 0 2x106 4x106 6x106 8x106 1x107 1.2x107 1.4x107 8 pi Ttt 8 pi Ttt

solutions using explicit integration backwards


Gnuplot Produced by GNUPLOT 5.4 patchlevel 1 0.999999998 0.999999998 0.999999998 0.999999999 0.999999999 0.999999999 0.999999999 0.999999999 1.000000000 1.000000000 1.000000000 0 2x106 4x106 6x106 8x106 1x107 1.2x107 1.4x107 alpha alpha
Gnuplot Produced by GNUPLOT 5.4 patchlevel 1 0 1x10-16 2x10-16 3x10-16 4x10-16 5x10-16 6x10-16 7x10-16 0 2x106 4x106 6x106 8x106 1x107 1.2x107 1.4x107 d/dr alpha d/dr alpha
Gnuplot Produced by GNUPLOT 5.4 patchlevel 1 -1.2x10-22 -1x10-22 -8x10-23 -6x10-23 -4x10-23 -2x10-23 0 0 2x106 4x106 6x106 8x106 1x107 1.2x107 1.4x107 d2/dr2 alpha d2/dr2 alpha
Gnuplot Produced by GNUPLOT 5.4 patchlevel 1 -1.8x1010 -1.6x1010 -1.4x1010 -1.2x1010 -1x1010 -8x109 -6x109 -4x109 -2x109 0 0 2x106 4x106 6x106 8x106 1x107 1.2x107 1.4x107 gravity gravity
solutions using explicit integration forwards
Gnuplot Produced by GNUPLOT 5.4 patchlevel 1 0.09999993 0.09999994 0.09999995 0.09999996 0.09999997 0.09999998 0.09999999 0.10000000 0 2x106 4x106 6x106 8x106 1x107 1.2x107 1.4x107 alpha alpha
Gnuplot Produced by GNUPLOT 5.4 patchlevel 1 -7x10-15 -6x10-15 -5x10-15 -4x10-15 -3x10-15 -2x10-15 -1x10-15 0 0 2x106 4x106 6x106 8x106 1x107 1.2x107 1.4x107 d/dr alpha d/dr alpha
Gnuplot Produced by GNUPLOT 5.4 patchlevel 1 -1.2x10-21 -1x10-21 -8x10-22 -6x10-22 -4x10-22 -2x10-22 0 0 2x106 4x106 6x106 8x106 1x107 1.2x107 1.4x107 d2/dr2 alpha d2/dr2 alpha
Gnuplot Produced by GNUPLOT 5.4 patchlevel 1 0 2x109 4x109 6x109 8x109 1x1010 1.2x1010 1.4x1010 1.6x1010 1.8x1010 0 2x106 4x106 6x106 8x106 1x107 1.2x107 1.4x107 gravity gravity