$c$ = speed of light
$G$ = gravitational constant
${r_🜨}$ = Earth radius at equator
${m_🜨}$ = Earth mass
${{r_s}} = {\frac{{{2}} {{G}} {{{m_🜨}}}}{{c}^{2}}}$ = Schwarzschild radius of Earth
${{{f_🜨}} = {{\frac{1}{{a_{sph}}}}{\left({{{a_{sph}}} - {{b_{sph}}}}\right)}}} = {\frac{1}{298.257222101}}$ = flattening of Earth
${{I_🜨}} = {{{\frac{2}{5}}} {{{m_🜨}}} \cdot {{{{r_🜨}}^{2}}}}$ = moment of inertia
${{J_🜨}} = {{{{I_🜨}}} \cdot {{{\omega_🜨}}}}$ = angular momentum
${{J_🜨}} = {{{2}} \cdot {{\frac{1}{5}}} {{{\omega_🜨}}} \cdot {{{m_🜨}}} \cdot {{{{r_🜨}}^{2}}}}$
${a} = {\frac{{J_🜨}}{{{{m_🜨}}} \cdot {{c}}}}$
${a} = {\frac{{{2}} {{{\omega_🜨}}} \cdot {{{{r_🜨}}^{2}}}}{{{5}} {{c}}}}$
with numerical values:
${c} = {{{299792458}} \cdot {{{\frac{1}{s}} {m}}}}$
${G} = {{{6.67384\cdot{10^{-11}}}} \cdot {{\frac{{m}^{3}}{{{kg}} \cdot {{{s}^{2}}}}}}}$
${{r_🜨}} = {{{63781000}} {{m}}}$
${{m_🜨}} = {{{5.792\cdot{10^{24}}}} {{kg}}}$
${{r_s}} = {{{0.0086018711645876}} {{m}}}$
${{\omega_🜨}} = {\frac{{{2}} {{\pi}}}{{{86164}} {{s}}}}$ = sidereal rotation of Earth
${{\omega_🜨}} = {{{2.4323905830181\cdot{10^{-13}}}} \cdot {{\frac{1}{m}}}}$
${{I_🜨}} = {{{9.4247793784448\cdot{10^{39}}}} {{kg}} \cdot {{{m}^{2}}}}$
${{J_🜨}} = {{{2.2924744607152\cdot{10^{27}}}} {{kg}} \cdot {{m}}}$
${a} = {{{395.80014860414}} {{m}}}$
${\Sigma} = {{{r}^{2}} + {{{{a}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}$
${\Delta} = {{{{r}^{2}} - {{{{r_s}}} \cdot {{r}}}} + {{a}^{2}}}$
${A} = {{{\left({{{r}^{2}} + {{a}^{2}}}\right)}^{2}} - {{{{a}^{2}}} {{\Delta}} \cdot {{{\sin\left( \theta\right)}^{2}}}}}$
${{{ g} _{\phi}} _{\phi}} = {{{\left({{{r}^{2}} + {{a}^{2}} + {{\frac{1}{\Sigma}} {{{{r_s}}} \cdot {{r}} {{{a}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}}}\right)}} {{{\sin\left( \theta\right)}^{2}}}}$
${\frac{A}{{\Sigma}^{2}}} = {\frac{{{{2}} {{{a}^{2}}} {{{r}^{2}}}} + {{r}^{4}} + {{{a}^{4}} - {{{\Delta}} \cdot {{{a}^{2}}}}} + {{{\Delta}} \cdot {{{a}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{{{r}^{4}} + {{{2}} {{{a}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{a}^{4}}} {{{\cos\left( \theta\right)}^{4}}}}}}$
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{\left({{1} - {{\frac{1}{\Sigma}} {{{{r_s}}} \cdot {{r}}}}}\right)} & 0 & 0 & -{{\frac{1}{\Sigma}} {{{{r_s}}} \cdot {{r}} {{a}} {{{\sin\left( \theta\right)}^{2}}}}} \\ 0 & {\frac{1}{\Delta}} {\Sigma} & 0 & 0 \\ 0 & 0 & \Sigma & 0 \\ -{{\frac{1}{\Sigma}} {{{{r_s}}} \cdot {{r}} {{a}} {{{\sin\left( \theta\right)}^{2}}}}} & 0 & 0 & {{\left({{{r}^{2}} + {{a}^{2}} + {{\frac{1}{\Sigma}} {{{{r_s}}} \cdot {{r}} {{{a}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}}}\right)}} {{{\sin\left( \theta\right)}^{2}}}\end{matrix} \right]}}$
${{{ g} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{-{A}}{{{\Sigma}} \cdot {{\Delta}}} & 0 & 0 & \frac{-{{{{r_s}}} \cdot {{a}} {{r}}}}{{{\Sigma}} \cdot {{\Delta}}} \\ 0 & {\frac{1}{\Sigma}} {\Delta} & 0 & 0 \\ 0 & 0 & \frac{1}{\Sigma} & 0 \\ \frac{-{{{{r_s}}} \cdot {{a}} {{r}}}}{{{\Sigma}} \cdot {{\Delta}}} & 0 & 0 & \frac{{\Delta} - {{{{a}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}}{{{\Sigma}} \cdot {{\Delta}} \cdot {{{\sin\left( \theta\right)}^{2}}}}\end{matrix} \right]}}$
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} \frac{{{-{{r}^{2}}} - {{{156657.75763506}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{0.0086018711645876}} {{m}} {{r}}}}{{{r}^{2}} + {{{156657.75763506}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} & 0 & 0 & \frac{{{3.4046218852175}} {{r}} {{\left({{{{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{m}^{2}}}\right)}}}{{{r}^{2}} + {{{156657.75763506}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} \\ 0 & \frac{{{r}^{2}} + {{{156657.75763506}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}{{{{r}^{2}} - {{{0.0086018711645876}} {{m}} {{r}}}} + {{{156657.75763506}} {{{m}^{2}}}}} & 0 & 0 \\ 0 & 0 & {{r}^{2}} + {{{156657.75763506}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} & 0 \\ \frac{{{3.4046218852175}} {{r}} {{\left({{{{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{m}^{2}}}\right)}}}{{{r}^{2}} + {{{156657.75763506}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} & 0 & 0 & \frac{{{{r}^{4}} - {{{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{{313315.51527012}} {{{m}^{2}}} {{{r}^{2}}}} - {{{313315.51527012}} {{{m}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{156657.75763506}} {{{m}^{2}}} {{{r}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{156657.75763506}} {{{m}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\sin\left( \theta\right)}^{2}}}} + {{{{{24541653027.246}} {{{m}^{4}}}} - {{{24541653027.246}} {{{m}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{24541653027.246}} {{{m}^{4}}} {{{\sin\left( \theta\right)}^{2}}}}} + {{{24541653027.246}} {{{m}^{4}}} {{{\sin\left( \theta\right)}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} + {{{{1347.54984811}} {{r}} {{{m}^{3}}} {{{\sin\left( \theta\right)}^{2}}}} - {{{1347.54984811}} {{r}} {{{m}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}}}{{{r}^{2}} + {{{156657.75763506}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}\end{matrix} \right]}}$
at earth surface:
${{{ g} _a} _b} = {\overset{a\downarrow b\rightarrow}{\left[ \begin{matrix} -{0.99999999986513} & 0 & 0 & 0 \\ 0 & 1.0000000001349 & 0 & 0 \\ 0 & 0 & {{4.0680159611567\cdot{10^{15}}}} {{{m}^{2}}} & 0 \\ 0 & 0 & 0 & 0\end{matrix} \right]}}$
time vector = $\overset{a\downarrow}{\left[ \begin{matrix} dt \\ 0 \\ 0 \\ 0\end{matrix} \right]}$
timelike arclength
${{ds}^{2}} = {{\frac{1}{\Sigma}} {{{{dt}^{2}}} {{\left({{-{\Sigma}} + {{{r}} {{{r_s}}}}}\right)}}}}$
${{ds}^{2}} = {\frac{{{-{{{{dt}^{2}}} {{{r}^{2}}}}} - {{{156657.75763506}} {{{dt}^{2}}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{0.0086018711645876}} {{m}} {{r}} {{{dt}^{2}}}}}{{{r}^{2}} + {{{156657.75763506}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}$
time and rotating:
${{ds}^{2}} = {{\frac{1}{\Sigma}}{\left({{{{{\Sigma}} \cdot {{{d\phi}^{2}}} {{{r}^{2}}}} - {{{\Sigma}} \cdot {{{d\phi}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{\Sigma}} \cdot {{{a}^{2}}} {{{d\phi}^{2}}}} - {{{\Sigma}} \cdot {{{a}^{2}}} {{{d\phi}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{{{r}} {{{r_s}}} \cdot {{{a}^{2}}} {{{d\phi}^{2}}} {{{\sin\left( \theta\right)}^{2}}}} - {{{r}} {{{r_s}}} \cdot {{{a}^{2}}} {{{d\phi}^{2}}} {{{\cos\left( \theta\right)}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} - {{{2}} {{a}} {{d\phi}} \cdot {{dt}} \cdot {{r}} {{{r_s}}}}} + {{{{2}} {{a}} {{d\phi}} \cdot {{dt}} \cdot {{r}} {{{r_s}}} \cdot {{{\cos\left( \theta\right)}^{2}}}} - {{{\Sigma}} \cdot {{{dt}^{2}}}}} + {{{r}} {{{r_s}}} \cdot {{{dt}^{2}}}}}\right)}}$
${{ds}^{2}} = {\frac{{{{{{{d\phi}^{2}}} {{{r}^{4}}}} - {{{{d\phi}^{2}}} {{{r}^{4}}} {{{\cos\left( \theta\right)}^{2}}}}} - {{{156657.75763506}} {{{d\phi}^{2}}} {{{m}^{2}}} {{{r}^{2}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{156657.75763506}} {{{d\phi}^{2}}} {{{m}^{2}}} {{{r}^{2}}}} + {{{{24541653027.246}} {{{d\phi}^{2}}} {{{m}^{4}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{24541653027.246}} {{{d\phi}^{2}}} {{{m}^{4}}} {{{\cos\left( \theta\right)}^{4}}}}} + {{{{{1347.54984811}} {{r}} {{{d\phi}^{2}}} {{{m}^{3}}} {{{\sin\left( \theta\right)}^{2}}}} - {{{1347.54984811}} {{r}} {{{d\phi}^{2}}} {{{m}^{3}}} {{{\cos\left( \theta\right)}^{2}}} {{{\sin\left( \theta\right)}^{2}}}}} - {{{6.8092437704349}} {{d\phi}} \cdot {{dt}} \cdot {{r}} {{{m}^{2}}}}} + {{{{{6.8092437704349}} {{d\phi}} \cdot {{dt}} \cdot {{r}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}} - {{{{dt}^{2}}} {{{r}^{2}}}}} - {{{156657.75763506}} {{{dt}^{2}}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}} + {{{0.0086018711645876}} {{m}} {{r}} {{{dt}^{2}}}}}{{{r}^{2}} + {{{156657.75763506}} {{{m}^{2}}} {{{\cos\left( \theta\right)}^{2}}}}}}$