${{{ \delta} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{ \eta} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{ g} _u} _v} = {{{{ \eta} _u} _v}{-{{{2}} {{\Phi}} \cdot {{{{ \delta} _u} _v}}}}}$
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{\left({{1} + {{{2}} {{\Phi}}}}\right)}& 0& 0& 0\\ 0& {1}{-{{{2}} {{\Phi}}}}& 0& 0\\ 0& 0& {1}{-{{{2}} {{\Phi}}}}& 0\\ 0& 0& 0& {1}{-{{{2}} {{\Phi}}}}\end{array}\right]}}$
${{{ g} ^u} ^v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{\frac{1}{{1} + {{{2}} {{\Phi}}}}}& 0& 0& 0\\ 0& \frac{1}{{1}{-{{{2}} {{\Phi}}}}}& 0& 0\\ 0& 0& \frac{1}{{1}{-{{{2}} {{\Phi}}}}}& 0\\ 0& 0& 0& \frac{1}{{1}{-{{{2}} {{\Phi}}}}}\end{array}\right]}}$

${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{ {{ g} _a} _b} _{,c}} + {{ {{ g} _a} _c} _{,b}}{-{{ {{ g} _b} _c} _{,a}}}}\right)}}}$

${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{\left( {{{ \eta} _a} _b}{-{{{2}} {{\Phi}} \cdot {{{{ \delta} _a} _b}}}}\right)} _{,c}} + {{\left( {{{ \eta} _a} _c}{-{{{2}} {{\Phi}} \cdot {{{{ \delta} _a} _c}}}}\right)} _{,b}}{-{{\left( {{{ \eta} _b} _c}{-{{{2}} {{\Phi}} \cdot {{{{ \delta} _b} _c}}}}\right)} _{,a}}}}\right)}}}$

${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{{\left( {-{{{2}} {{\Phi}} \cdot {{\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}} + {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}\right)} _{,c}} + {{\left( {-{{{2}} {{\Phi}} \cdot {{\overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}} + {\overset{a\downarrow c\rightarrow}{\left[\begin{array}{cccc} -1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}\right)} _{,b}}{-{{\left( {-{{{2}} {{\Phi}} \cdot {{\overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} 1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}} + {\overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} -1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}\right)} _{,a}}}}\right)}}}$

${{{{ \Gamma} _a} _b} _c} = {{{\frac{1}{2}}} {{\left({{-{\overset{b\downarrow[{c\downarrow a\rightarrow}]}{\left[\begin{matrix} \overset{c\downarrow a\rightarrow}{\left[\begin{array}{cccc} -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{c\downarrow a\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{c\downarrow a\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{c\downarrow a\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\end{array}\right]}\end{matrix}\right]}}} + {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \overset{b\downarrow c\rightarrow}{\left[\begin{array}{cccc} -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\\ 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\\ -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\\ 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\\ -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\\ 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\\ -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\end{array}\right]}\end{matrix}\right]}} + {\overset{a\downarrow[{c\downarrow b\rightarrow}]}{\left[\begin{matrix} \overset{c\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{c\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{c\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\\ 0& 0& 0& 0\end{array}\right]} \\ \overset{c\downarrow b\rightarrow}{\left[\begin{array}{cccc} 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -{{{2}} {{ \Phi_{,{{t}}}}}}& -{{{2}} {{ \Phi_{,{{x}}}}}}& -{{{2}} {{ \Phi_{,{{y}}}}}}& -{{{2}} {{ \Phi_{,{{z}}}}}}\end{array}\right]}\end{matrix}\right]}}}\right)}}}$

${{{{ \Gamma} _a} _b} _c} = {\overset{b\downarrow[{c\downarrow a\rightarrow}]}{\left[\begin{matrix} \overset{c\downarrow a\rightarrow}{\left[\begin{array}{cccc} -{ \Phi_{,{{t}}}}& \Phi_{,{{x}}}& \Phi_{,{{y}}}& \Phi_{,{{z}}}\\ -{ \Phi_{,{{x}}}}& -{ \Phi_{,{{t}}}}& 0& 0\\ -{ \Phi_{,{{y}}}}& 0& -{ \Phi_{,{{t}}}}& 0\\ -{ \Phi_{,{{z}}}}& 0& 0& -{ \Phi_{,{{t}}}}\end{array}\right]} \\ \overset{c\downarrow a\rightarrow}{\left[\begin{array}{cccc} -{ \Phi_{,{{x}}}}& -{ \Phi_{,{{t}}}}& 0& 0\\ \Phi_{,{{t}}}& -{ \Phi_{,{{x}}}}& \Phi_{,{{y}}}& \Phi_{,{{z}}}\\ 0& -{ \Phi_{,{{y}}}}& -{ \Phi_{,{{x}}}}& 0\\ 0& -{ \Phi_{,{{z}}}}& 0& -{ \Phi_{,{{x}}}}\end{array}\right]} \\ \overset{c\downarrow a\rightarrow}{\left[\begin{array}{cccc} -{ \Phi_{,{{y}}}}& 0& -{ \Phi_{,{{t}}}}& 0\\ 0& -{ \Phi_{,{{y}}}}& -{ \Phi_{,{{x}}}}& 0\\ \Phi_{,{{t}}}& \Phi_{,{{x}}}& -{ \Phi_{,{{y}}}}& \Phi_{,{{z}}}\\ 0& 0& -{ \Phi_{,{{z}}}}& -{ \Phi_{,{{y}}}}\end{array}\right]} \\ \overset{c\downarrow a\rightarrow}{\left[\begin{array}{cccc} -{ \Phi_{,{{z}}}}& 0& 0& -{ \Phi_{,{{t}}}}\\ 0& -{ \Phi_{,{{z}}}}& 0& -{ \Phi_{,{{x}}}}\\ 0& 0& -{ \Phi_{,{{z}}}}& -{ \Phi_{,{{y}}}}\\ \Phi_{,{{t}}}& \Phi_{,{{x}}}& \Phi_{,{{y}}}& -{ \Phi_{,{{z}}}}\end{array}\right]}\end{matrix}\right]}}$

${{{{ \Gamma} ^a} _b} _c} = {{{{{ g} ^a} ^d}} {{{{{ \Gamma} _d} _b} _c}}}$

${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} \frac{ \Phi_{,{{t}}}}{{1} + {{{2}} {{\Phi}}}}& -{\frac{ \Phi_{,{{x}}}}{{1} + {{{2}} {{\Phi}}}}}& -{\frac{ \Phi_{,{{y}}}}{{1} + {{{2}} {{\Phi}}}}}& -{\frac{ \Phi_{,{{z}}}}{{1} + {{{2}} {{\Phi}}}}}\\ \frac{ \Phi_{,{{x}}}}{{1} + {{{2}} {{\Phi}}}}& \frac{ \Phi_{,{{t}}}}{{1} + {{{2}} {{\Phi}}}}& 0& 0\\ \frac{ \Phi_{,{{y}}}}{{1} + {{{2}} {{\Phi}}}}& 0& \frac{ \Phi_{,{{t}}}}{{1} + {{{2}} {{\Phi}}}}& 0\\ \frac{ \Phi_{,{{z}}}}{{1} + {{{2}} {{\Phi}}}}& 0& 0& \frac{ \Phi_{,{{t}}}}{{1} + {{{2}} {{\Phi}}}}\end{array}\right] \\ \left[\begin{array}{cccc} -{\frac{ \Phi_{,{{x}}}}{{1}{-{{{2}} {{\Phi}}}}}}& -{\frac{ \Phi_{,{{t}}}}{{1}{-{{{2}} {{\Phi}}}}}}& 0& 0\\ \frac{ \Phi_{,{{t}}}}{{1}{-{{{2}} {{\Phi}}}}}& -{\frac{ \Phi_{,{{x}}}}{{1}{-{{{2}} {{\Phi}}}}}}& \frac{ \Phi_{,{{y}}}}{{1}{-{{{2}} {{\Phi}}}}}& \frac{ \Phi_{,{{z}}}}{{1}{-{{{2}} {{\Phi}}}}}\\ 0& -{\frac{ \Phi_{,{{y}}}}{{1}{-{{{2}} {{\Phi}}}}}}& -{\frac{ \Phi_{,{{x}}}}{{1}{-{{{2}} {{\Phi}}}}}}& 0\\ 0& -{\frac{ \Phi_{,{{z}}}}{{1}{-{{{2}} {{\Phi}}}}}}& 0& -{\frac{ \Phi_{,{{x}}}}{{1}{-{{{2}} {{\Phi}}}}}}\end{array}\right] \\ \left[\begin{array}{cccc} -{\frac{ \Phi_{,{{y}}}}{{1}{-{{{2}} {{\Phi}}}}}}& 0& -{\frac{ \Phi_{,{{t}}}}{{1}{-{{{2}} {{\Phi}}}}}}& 0\\ 0& -{\frac{ \Phi_{,{{y}}}}{{1}{-{{{2}} {{\Phi}}}}}}& -{\frac{ \Phi_{,{{x}}}}{{1}{-{{{2}} {{\Phi}}}}}}& 0\\ \frac{ \Phi_{,{{t}}}}{{1}{-{{{2}} {{\Phi}}}}}& \frac{ \Phi_{,{{x}}}}{{1}{-{{{2}} {{\Phi}}}}}& -{\frac{ \Phi_{,{{y}}}}{{1}{-{{{2}} {{\Phi}}}}}}& \frac{ \Phi_{,{{z}}}}{{1}{-{{{2}} {{\Phi}}}}}\\ 0& 0& -{\frac{ \Phi_{,{{z}}}}{{1}{-{{{2}} {{\Phi}}}}}}& -{\frac{ \Phi_{,{{y}}}}{{1}{-{{{2}} {{\Phi}}}}}}\end{array}\right] \\ \left[\begin{array}{cccc} -{\frac{ \Phi_{,{{z}}}}{{1}{-{{{2}} {{\Phi}}}}}}& 0& 0& -{\frac{ \Phi_{,{{t}}}}{{1}{-{{{2}} {{\Phi}}}}}}\\ 0& -{\frac{ \Phi_{,{{z}}}}{{1}{-{{{2}} {{\Phi}}}}}}& 0& -{\frac{ \Phi_{,{{x}}}}{{1}{-{{{2}} {{\Phi}}}}}}\\ 0& 0& -{\frac{ \Phi_{,{{z}}}}{{1}{-{{{2}} {{\Phi}}}}}}& -{\frac{ \Phi_{,{{y}}}}{{1}{-{{{2}} {{\Phi}}}}}}\\ \frac{ \Phi_{,{{t}}}}{{1}{-{{{2}} {{\Phi}}}}}& \frac{ \Phi_{,{{x}}}}{{1}{-{{{2}} {{\Phi}}}}}& \frac{ \Phi_{,{{y}}}}{{1}{-{{{2}} {{\Phi}}}}}& -{\frac{ \Phi_{,{{z}}}}{{1}{-{{{2}} {{\Phi}}}}}}\end{array}\right]\end{matrix}\right]}}$


let $\Phi$ ~ 0, but keep $\partial\Phi$ to find:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} \Phi_{,{{t}}}& -{ \Phi_{,{{x}}}}& -{ \Phi_{,{{y}}}}& -{ \Phi_{,{{z}}}}\\ \Phi_{,{{x}}}& \Phi_{,{{t}}}& 0& 0\\ \Phi_{,{{y}}}& 0& \Phi_{,{{t}}}& 0\\ \Phi_{,{{z}}}& 0& 0& \Phi_{,{{t}}}\end{array}\right] \\ \left[\begin{array}{cccc} -{ \Phi_{,{{x}}}}& -{ \Phi_{,{{t}}}}& 0& 0\\ \Phi_{,{{t}}}& -{ \Phi_{,{{x}}}}& \Phi_{,{{y}}}& \Phi_{,{{z}}}\\ 0& -{ \Phi_{,{{y}}}}& -{ \Phi_{,{{x}}}}& 0\\ 0& -{ \Phi_{,{{z}}}}& 0& -{ \Phi_{,{{x}}}}\end{array}\right] \\ \left[\begin{array}{cccc} -{ \Phi_{,{{y}}}}& 0& -{ \Phi_{,{{t}}}}& 0\\ 0& -{ \Phi_{,{{y}}}}& -{ \Phi_{,{{x}}}}& 0\\ \Phi_{,{{t}}}& \Phi_{,{{x}}}& -{ \Phi_{,{{y}}}}& \Phi_{,{{z}}}\\ 0& 0& -{ \Phi_{,{{z}}}}& -{ \Phi_{,{{y}}}}\end{array}\right] \\ \left[\begin{array}{cccc} -{ \Phi_{,{{z}}}}& 0& 0& -{ \Phi_{,{{t}}}}\\ 0& -{ \Phi_{,{{z}}}}& 0& -{ \Phi_{,{{x}}}}\\ 0& 0& -{ \Phi_{,{{z}}}}& -{ \Phi_{,{{y}}}}\\ \Phi_{,{{t}}}& \Phi_{,{{x}}}& \Phi_{,{{y}}}& -{ \Phi_{,{{z}}}}\end{array}\right]\end{matrix}\right]}}$
${{{ g} _u} _v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$
${{{ g} ^u} ^v} = {\overset{u\downarrow v\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}$

let ${ \Phi_{,{{t}}}} = {0}$ to find:
${{{{ \Gamma} ^a} _b} _c} = {\overset{a\downarrow[{b\downarrow c\rightarrow}]}{\left[\begin{matrix} \left[\begin{array}{cccc} 0& -{ \Phi_{,{{x}}}}& -{ \Phi_{,{{y}}}}& -{ \Phi_{,{{z}}}}\\ \Phi_{,{{x}}}& 0& 0& 0\\ \Phi_{,{{y}}}& 0& 0& 0\\ \Phi_{,{{z}}}& 0& 0& 0\end{array}\right] \\ \left[\begin{array}{cccc} -{ \Phi_{,{{x}}}}& 0& 0& 0\\ 0& -{ \Phi_{,{{x}}}}& \Phi_{,{{y}}}& \Phi_{,{{z}}}\\ 0& -{ \Phi_{,{{y}}}}& -{ \Phi_{,{{x}}}}& 0\\ 0& -{ \Phi_{,{{z}}}}& 0& -{ \Phi_{,{{x}}}}\end{array}\right] \\ \left[\begin{array}{cccc} -{ \Phi_{,{{y}}}}& 0& 0& 0\\ 0& -{ \Phi_{,{{y}}}}& -{ \Phi_{,{{x}}}}& 0\\ 0& \Phi_{,{{x}}}& -{ \Phi_{,{{y}}}}& \Phi_{,{{z}}}\\ 0& 0& -{ \Phi_{,{{z}}}}& -{ \Phi_{,{{y}}}}\end{array}\right] \\ \left[\begin{array}{cccc} -{ \Phi_{,{{z}}}}& 0& 0& 0\\ 0& -{ \Phi_{,{{z}}}}& 0& -{ \Phi_{,{{x}}}}\\ 0& 0& -{ \Phi_{,{{z}}}}& -{ \Phi_{,{{y}}}}\\ 0& \Phi_{,{{x}}}& \Phi_{,{{y}}}& -{ \Phi_{,{{z}}}}\end{array}\right]\end{matrix}\right]}}$

let
${{ u} ^a} = {\overset{a\downarrow}{\left[\begin{matrix} {u^t} \\ {u^x} \\ {u^y} \\ {u^z}\end{matrix}\right]}}$

matter stress-energy tensor:
${{{ T} ^a} ^b} = {{{{\left({{\rho} + {P}}\right)}} {{{ u} ^a}} {{{ u} ^b}}} + {{{P}} {{{{ g} ^a} ^b}}}}$

${{{ T} ^a} ^b} = {{{{\left({{\rho} + {P}}\right)}} {{\overset{a\downarrow}{\left[\begin{matrix} {u^t} \\ {u^x} \\ {u^y} \\ {u^z}\end{matrix}\right]}}} {{\overset{b\downarrow}{\left[\begin{matrix} {u^t} \\ {u^x} \\ {u^y} \\ {u^z}\end{matrix}\right]}}}} + {{{P}} {{\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} -{1}& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}}}}}$
${{{ T} ^a} ^b} = {\overset{a\downarrow b\rightarrow}{\left[\begin{array}{cccc} {-{P}} + {{{P}} {{{{u^t}}^{2}}}} + {{{\rho}} \cdot {{{{u^t}}^{2}}}}& {{{u^t}}} \cdot {{{u^x}}} \cdot {{\left({{P} + {\rho}}\right)}}& {{{u^t}}} \cdot {{{u^y}}} \cdot {{\left({{P} + {\rho}}\right)}}& {{{u^t}}} \cdot {{{u^z}}} \cdot {{\left({{P} + {\rho}}\right)}}\\ {{{u^t}}} \cdot {{{u^x}}} \cdot {{\left({{P} + {\rho}}\right)}}& {P} + {{{P}} {{{{u^x}}^{2}}}} + {{{\rho}} \cdot {{{{u^x}}^{2}}}}& {{{u^x}}} \cdot {{{u^y}}} \cdot {{\left({{P} + {\rho}}\right)}}& {{{u^x}}} \cdot {{{u^z}}} \cdot {{\left({{P} + {\rho}}\right)}}\\ {{{u^t}}} \cdot {{{u^y}}} \cdot {{\left({{P} + {\rho}}\right)}}& {{{u^x}}} \cdot {{{u^y}}} \cdot {{\left({{P} + {\rho}}\right)}}& {P} + {{{P}} {{{{u^y}}^{2}}}} + {{{\rho}} \cdot {{{{u^y}}^{2}}}}& {{{u^y}}} \cdot {{{u^z}}} \cdot {{\left({{P} + {\rho}}\right)}}\\ {{{u^t}}} \cdot {{{u^z}}} \cdot {{\left({{P} + {\rho}}\right)}}& {{{u^x}}} \cdot {{{u^z}}} \cdot {{\left({{P} + {\rho}}\right)}}& {{{u^y}}} \cdot {{{u^z}}} \cdot {{\left({{P} + {\rho}}\right)}}& {P} + {{{P}} {{{{u^z}}^{2}}}} + {{{\rho}} \cdot {{{{u^z}}^{2}}}}\end{array}\right]}}$


${\nabla \cdot T} = {0}$
${{-{ P_{,{{t}}}}} + {{{{{u^t}}^{2}}} {{ P_{,{{t}}}}}} + {{{{{u^t}}^{2}}} {{ \rho_{,{{t}}}}}} + {{{P}} {{{u^t}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{P}} {{{u^t}}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{P}} {{{u^t}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^t}_{,{{x}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^t}_{,{{y}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^t}_{,{{z}}}}}} + {{{\rho}} \cdot {{{u^t}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^t}}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^t}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^t}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^t}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^t}_{,{{z}}}}}} + {{{{u^t}}} \cdot {{{u^x}}} \cdot {{ P_{,{{x}}}}}} + {{{{u^t}}} \cdot {{{u^x}}} \cdot {{ \rho_{,{{x}}}}}} + {{{{u^t}}} \cdot {{{u^y}}} \cdot {{ P_{,{{y}}}}}} + {{{{u^t}}} \cdot {{{u^y}}} \cdot {{ \rho_{,{{y}}}}}} + {{{{u^t}}} \cdot {{{u^z}}} \cdot {{ P_{,{{z}}}}}} + {{{{u^t}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{z}}}}}} + {{{2}} {{P}} {{{u^t}}} \cdot {{ {u^t}_{,{{t}}}}}} + {{{2}} {{\rho}} \cdot {{{u^t}}} \cdot {{ {u^t}_{,{{t}}}}}}{-{{{2}} {{P}} {{{u^t}}} \cdot {{{u^x}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{P}} {{{u^t}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{P}} {{{u^t}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^t}}} \cdot {{{u^x}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^t}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^t}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}} = {0}$
${{ P_{,{{x}}}} + {{{{{u^x}}^{2}}} {{ P_{,{{x}}}}}} + {{{{{u^x}}^{2}}} {{ \rho_{,{{x}}}}}} + {{{P}} {{{u^t}}} \cdot {{ {u^x}_{,{{t}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^t}_{,{{t}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^x}_{,{{y}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^x}_{,{{z}}}}}}{-{{{P}} {{{{u^t}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{P}} {{{{u^y}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{P}} {{{{u^z}}^{2}}} {{ \Phi_{,{{x}}}}}}} + {{{\rho}} \cdot {{{u^t}}} \cdot {{ {u^x}_{,{{t}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^t}_{,{{t}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^x}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^x}_{,{{z}}}}}}{-{{{\rho}} \cdot {{{{u^t}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{\rho}} \cdot {{{{u^y}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{\rho}} \cdot {{{{u^z}}^{2}}} {{ \Phi_{,{{x}}}}}}} + {{{{u^t}}} \cdot {{{u^x}}} \cdot {{ P_{,{{t}}}}}} + {{{{u^t}}} \cdot {{{u^x}}} \cdot {{ \rho_{,{{t}}}}}} + {{{{u^x}}} \cdot {{{u^y}}} \cdot {{ P_{,{{y}}}}}} + {{{{u^x}}} \cdot {{{u^y}}} \cdot {{ \rho_{,{{y}}}}}} + {{{{u^x}}} \cdot {{{u^z}}} \cdot {{ P_{,{{z}}}}}} + {{{{u^x}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{z}}}}}}{-{{{4}} {{P}} {{ \Phi_{,{{x}}}}}}} + {{{2}} {{P}} {{{u^x}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{ {u^x}_{,{{x}}}}}}{-{{{3}} {{P}} {{{{u^x}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{3}} {{\rho}} \cdot {{{{u^x}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}} = {0}$
${{ P_{,{{y}}}} + {{{{{u^y}}^{2}}} {{ P_{,{{y}}}}}} + {{{{{u^y}}^{2}}} {{ \rho_{,{{y}}}}}} + {{{P}} {{{u^t}}} \cdot {{ {u^y}_{,{{t}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^y}_{,{{x}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^t}_{,{{t}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^y}_{,{{z}}}}}}{-{{{P}} {{{{u^t}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{P}} {{{{u^x}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{P}} {{{{u^z}}^{2}}} {{ \Phi_{,{{y}}}}}}} + {{{\rho}} \cdot {{{u^t}}} \cdot {{ {u^y}_{,{{t}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^y}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^t}_{,{{t}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^y}_{,{{z}}}}}}{-{{{\rho}} \cdot {{{{u^t}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{\rho}} \cdot {{{{u^x}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{\rho}} \cdot {{{{u^z}}^{2}}} {{ \Phi_{,{{y}}}}}}} + {{{{u^t}}} \cdot {{{u^y}}} \cdot {{ P_{,{{t}}}}}} + {{{{u^t}}} \cdot {{{u^y}}} \cdot {{ \rho_{,{{t}}}}}} + {{{{u^x}}} \cdot {{{u^y}}} \cdot {{ P_{,{{x}}}}}} + {{{{u^x}}} \cdot {{{u^y}}} \cdot {{ \rho_{,{{x}}}}}} + {{{{u^y}}} \cdot {{{u^z}}} \cdot {{ P_{,{{z}}}}}} + {{{{u^y}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{z}}}}}}{-{{{4}} {{P}} {{ \Phi_{,{{y}}}}}}} + {{{2}} {{P}} {{{u^y}}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{2}} {{\rho}} \cdot {{{u^y}}} \cdot {{ {u^y}_{,{{y}}}}}}{-{{{3}} {{P}} {{{{u^y}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{3}} {{\rho}} \cdot {{{{u^y}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{P}} {{{u^y}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^y}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}} = {0}$
${{ P_{,{{z}}}} + {{{{{u^z}}^{2}}} {{ P_{,{{z}}}}}} + {{{{{u^z}}^{2}}} {{ \rho_{,{{z}}}}}} + {{{P}} {{{u^t}}} \cdot {{ {u^z}_{,{{t}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^z}_{,{{x}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^z}_{,{{y}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^t}_{,{{t}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^y}_{,{{y}}}}}}{-{{{P}} {{{{u^t}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{P}} {{{{u^x}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{P}} {{{{u^y}}^{2}}} {{ \Phi_{,{{z}}}}}}} + {{{\rho}} \cdot {{{u^t}}} \cdot {{ {u^z}_{,{{t}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^z}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^z}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^t}_{,{{t}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^y}_{,{{y}}}}}}{-{{{\rho}} \cdot {{{{u^t}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{\rho}} \cdot {{{{u^x}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{\rho}} \cdot {{{{u^y}}^{2}}} {{ \Phi_{,{{z}}}}}}} + {{{{u^t}}} \cdot {{{u^z}}} \cdot {{ P_{,{{t}}}}}} + {{{{u^t}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{t}}}}}} + {{{{u^x}}} \cdot {{{u^z}}} \cdot {{ P_{,{{x}}}}}} + {{{{u^x}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{x}}}}}} + {{{{u^y}}} \cdot {{{u^z}}} \cdot {{ P_{,{{y}}}}}} + {{{{u^y}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{y}}}}}}{-{{{4}} {{P}} {{ \Phi_{,{{z}}}}}}} + {{{2}} {{P}} {{{u^z}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{2}} {{\rho}} \cdot {{{u^z}}} \cdot {{ {u^z}_{,{{z}}}}}}{-{{{3}} {{P}} {{{{u^z}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{3}} {{\rho}} \cdot {{{{u^z}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{P}} {{{u^y}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^y}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{y}}}}}}}} = {0}$

low velocity relativistic approximations:
${{u^t}} = {1}$

${\nabla \cdot T} = {0}$ becomes:
${{ \rho_{,{{t}}}} + {{{P}} {{ {u^x}_{,{{x}}}}}} + {{{P}} {{ {u^y}_{,{{y}}}}}} + {{{P}} {{ {u^z}_{,{{z}}}}}} + {{{\rho}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{\rho}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{\rho}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{{u^x}}} \cdot {{ P_{,{{x}}}}}} + {{{{u^x}}} \cdot {{ \rho_{,{{x}}}}}} + {{{{u^y}}} \cdot {{ P_{,{{y}}}}}} + {{{{u^y}}} \cdot {{ \rho_{,{{y}}}}}} + {{{{u^z}}} \cdot {{ P_{,{{z}}}}}} + {{{{u^z}}} \cdot {{ \rho_{,{{z}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{P}} {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{P}} {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}} = {0}$
${{ P_{,{{x}}}} + {{{P}} {{ {u^x}_{,{{t}}}}}} + {{{\rho}} \cdot {{ {u^x}_{,{{t}}}}}}{-{{{\rho}} \cdot {{ \Phi_{,{{x}}}}}}} + {{{{u^x}}} \cdot {{ P_{,{{t}}}}}} + {{{{u^x}}} \cdot {{ \rho_{,{{t}}}}}} + {{{{{u^x}}^{2}}} {{ P_{,{{x}}}}}} + {{{{{u^x}}^{2}}} {{ \rho_{,{{x}}}}}}{-{{{5}} {{P}} {{ \Phi_{,{{x}}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^x}_{,{{y}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^x}_{,{{z}}}}}}{-{{{P}} {{{{u^y}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{P}} {{{{u^z}}^{2}}} {{ \Phi_{,{{x}}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^x}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^x}_{,{{z}}}}}}{-{{{\rho}} \cdot {{{{u^y}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{\rho}} \cdot {{{{u^z}}^{2}}} {{ \Phi_{,{{x}}}}}}} + {{{{u^x}}} \cdot {{{u^y}}} \cdot {{ P_{,{{y}}}}}} + {{{{u^x}}} \cdot {{{u^y}}} \cdot {{ \rho_{,{{y}}}}}} + {{{{u^x}}} \cdot {{{u^z}}} \cdot {{ P_{,{{z}}}}}} + {{{{u^x}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{z}}}}}} + {{{2}} {{P}} {{{u^x}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{ {u^x}_{,{{x}}}}}}{-{{{3}} {{P}} {{{{u^x}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{3}} {{\rho}} \cdot {{{{u^x}}^{2}}} {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}} = {0}$
${{ P_{,{{y}}}} + {{{P}} {{ {u^y}_{,{{t}}}}}} + {{{\rho}} \cdot {{ {u^y}_{,{{t}}}}}}{-{{{\rho}} \cdot {{ \Phi_{,{{y}}}}}}} + {{{{u^y}}} \cdot {{ P_{,{{t}}}}}} + {{{{u^y}}} \cdot {{ \rho_{,{{t}}}}}} + {{{{{u^y}}^{2}}} {{ P_{,{{y}}}}}} + {{{{{u^y}}^{2}}} {{ \rho_{,{{y}}}}}}{-{{{5}} {{P}} {{ \Phi_{,{{y}}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^y}_{,{{x}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^y}_{,{{z}}}}}}{-{{{P}} {{{{u^x}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{P}} {{{{u^z}}^{2}}} {{ \Phi_{,{{y}}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^y}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^y}_{,{{z}}}}}}{-{{{\rho}} \cdot {{{{u^x}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{\rho}} \cdot {{{{u^z}}^{2}}} {{ \Phi_{,{{y}}}}}}} + {{{{u^x}}} \cdot {{{u^y}}} \cdot {{ P_{,{{x}}}}}} + {{{{u^x}}} \cdot {{{u^y}}} \cdot {{ \rho_{,{{x}}}}}} + {{{{u^y}}} \cdot {{{u^z}}} \cdot {{ P_{,{{z}}}}}} + {{{{u^y}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{z}}}}}} + {{{2}} {{P}} {{{u^y}}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{2}} {{\rho}} \cdot {{{u^y}}} \cdot {{ {u^y}_{,{{y}}}}}}{-{{{3}} {{P}} {{{{u^y}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{3}} {{\rho}} \cdot {{{{u^y}}^{2}}} {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{P}} {{{u^y}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^y}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}} = {0}$
${{ P_{,{{z}}}} + {{{P}} {{ {u^z}_{,{{t}}}}}} + {{{\rho}} \cdot {{ {u^z}_{,{{t}}}}}}{-{{{\rho}} \cdot {{ \Phi_{,{{z}}}}}}} + {{{{u^z}}} \cdot {{ P_{,{{t}}}}}} + {{{{u^z}}} \cdot {{ \rho_{,{{t}}}}}} + {{{{{u^z}}^{2}}} {{ P_{,{{z}}}}}} + {{{{{u^z}}^{2}}} {{ \rho_{,{{z}}}}}}{-{{{5}} {{P}} {{ \Phi_{,{{z}}}}}}} + {{{P}} {{{u^x}}} \cdot {{ {u^z}_{,{{x}}}}}} + {{{P}} {{{u^y}}} \cdot {{ {u^z}_{,{{y}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{P}} {{{u^z}}} \cdot {{ {u^y}_{,{{y}}}}}}{-{{{P}} {{{{u^x}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{P}} {{{{u^y}}^{2}}} {{ \Phi_{,{{z}}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^z}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^z}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^y}_{,{{y}}}}}}{-{{{\rho}} \cdot {{{{u^x}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{\rho}} \cdot {{{{u^y}}^{2}}} {{ \Phi_{,{{z}}}}}}} + {{{{u^x}}} \cdot {{{u^z}}} \cdot {{ P_{,{{x}}}}}} + {{{{u^x}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{x}}}}}} + {{{{u^y}}} \cdot {{{u^z}}} \cdot {{ P_{,{{y}}}}}} + {{{{u^y}}} \cdot {{{u^z}}} \cdot {{ \rho_{,{{y}}}}}} + {{{2}} {{P}} {{{u^z}}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{2}} {{\rho}} \cdot {{{u^z}}} \cdot {{ {u^z}_{,{{z}}}}}}{-{{{3}} {{P}} {{{{u^z}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{3}} {{\rho}} \cdot {{{{u^z}}^{2}}} {{ \Phi_{,{{z}}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{P}} {{{u^y}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^y}}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{y}}}}}}}} = {0}$
${{{{P}} {{ {u^x}_{,{{x}}}}}} + {{{P}} {{ {u^y}_{,{{y}}}}}} + {{{P}} {{ {u^z}_{,{{z}}}}}} + {{{{u^x}}} \cdot {{ P_{,{{x}}}}}} + {{{{u^y}}} \cdot {{ P_{,{{y}}}}}} + {{{{u^z}}} \cdot {{ P_{,{{z}}}}}}} = {0}$

first equation in terms of $\partial_t \rho$
${ \rho_{,{{t}}}} = {{-{{{\rho}} \cdot {{ {u^x}_{,{{x}}}}}}}{-{{{\rho}} \cdot {{ {u^y}_{,{{y}}}}}}}{-{{{\rho}} \cdot {{ {u^z}_{,{{z}}}}}}}{-{{{{u^x}}} \cdot {{ \rho_{,{{x}}}}}}}{-{{{{u^y}}} \cdot {{ \rho_{,{{y}}}}}}}{-{{{{u^z}}} \cdot {{ \rho_{,{{z}}}}}}} + {{{2}} {{P}} {{{u^x}}} \cdot {{ \Phi_{,{{x}}}}}} + {{{2}} {{P}} {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}} + {{{2}} {{P}} {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}} + {{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{ \Phi_{,{{x}}}}}} + {{{2}} {{\rho}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}} + {{{2}} {{\rho}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}$

spatial equations neglect $P_{,t}$, $(P u^j)_{,j}$, $P$, and $\Phi_{,i} u_j$ and substitutes the definition of $\partial_t \rho$

${\nabla \cdot T} = {0}$ becomes:
${{ \rho_{,{{t}}}} + {{{\rho}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{\rho}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{\rho}} \cdot {{ {u^z}_{,{{z}}}}}} + {{{{u^x}}} \cdot {{ \rho_{,{{x}}}}}} + {{{{u^y}}} \cdot {{ \rho_{,{{y}}}}}} + {{{{u^z}}} \cdot {{ \rho_{,{{z}}}}}}{-{{{2}} {{P}} {{{u^x}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{P}} {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{P}} {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^x}}} \cdot {{ \Phi_{,{{x}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^y}}} \cdot {{ \Phi_{,{{y}}}}}}}{-{{{2}} {{\rho}} \cdot {{{u^z}}} \cdot {{ \Phi_{,{{z}}}}}}}} = {0}$
${{ P_{,{{x}}}} + {{{\rho}} \cdot {{ {u^x}_{,{{t}}}}}}{-{{{\rho}} \cdot {{ \Phi_{,{{x}}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^x}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^x}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^x}_{,{{z}}}}}}} = {0}$
${{ P_{,{{y}}}} + {{{\rho}} \cdot {{ {u^y}_{,{{t}}}}}}{-{{{\rho}} \cdot {{ \Phi_{,{{y}}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^y}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^y}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^y}_{,{{z}}}}}}} = {0}$
${{ P_{,{{z}}}} + {{{\rho}} \cdot {{ {u^z}_{,{{t}}}}}}{-{{{\rho}} \cdot {{ \Phi_{,{{z}}}}}}} + {{{\rho}} \cdot {{{u^x}}} \cdot {{ {u^z}_{,{{x}}}}}} + {{{\rho}} \cdot {{{u^y}}} \cdot {{ {u^z}_{,{{y}}}}}} + {{{\rho}} \cdot {{{u^z}}} \cdot {{ {u^z}_{,{{z}}}}}}} = {0}$