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

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

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

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

${{{\mathrm{\Gamma }}_a}_b}_c=\frac{1}{2}\left(-\stackrel{b\downarrow [c\downarrow a\to ]}{\left[\begin{array}{c}\stackrel{c\downarrow a\to }{\left[\begin{array}{cccc}-2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \stackrel{c\downarrow a\to }{\left[\begin{array}{cccc}0& 0& 0& 0\\ -2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \stackrel{c\downarrow a\to }{\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ -2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\\ 0& 0& 0& 0\end{array}\right]}\\ \stackrel{c\downarrow a\to }{\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\end{array}\right]}\end{array}\right]}+\stackrel{a\downarrow [b\downarrow c\to ]}{\left[\begin{array}{c}\stackrel{b\downarrow c\to }{\left[\begin{array}{cccc}-2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \stackrel{b\downarrow c\to }{\left[\begin{array}{cccc}0& 0& 0& 0\\ -2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \stackrel{b\downarrow c\to }{\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ -2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\\ 0& 0& 0& 0\end{array}\right]}\\ \stackrel{b\downarrow c\to }{\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\end{array}\right]}\end{array}\right]}+\stackrel{a\downarrow [c\downarrow b\to ]}{\left[\begin{array}{c}\stackrel{c\downarrow b\to }{\left[\begin{array}{cccc}-2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \stackrel{c\downarrow b\to }{\left[\begin{array}{cccc}0& 0& 0& 0\\ -2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]}\\ \stackrel{c\downarrow b\to }{\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ -2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\\ 0& 0& 0& 0\end{array}\right]}\\ \stackrel{c\downarrow b\to }{\left[\begin{array}{cccc}0& 0& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ -2{\mathrm{\Phi }}_{,t}& -2{\mathrm{\Phi }}_{,x}& -2{\mathrm{\Phi }}_{,y}& -2{\mathrm{\Phi }}_{,z}\end{array}\right]}\end{array}\right]}\right)$

${{{\mathrm{\Gamma }}_a}_b}_c=\stackrel{b\downarrow [c\downarrow a\to ]}{\left[\begin{array}{c}\stackrel{c\downarrow a\to }{\left[\begin{array}{cccc}-{\mathrm{\Phi }}_{,t}& {\mathrm{\Phi }}_{,x}& {\mathrm{\Phi }}_{,y}& {\mathrm{\Phi }}_{,z}\\ -{\mathrm{\Phi }}_{,x}& -{\mathrm{\Phi }}_{,t}& 0& 0\\ -{\mathrm{\Phi }}_{,y}& 0& -{\mathrm{\Phi }}_{,t}& 0\\ -{\mathrm{\Phi }}_{,z}& 0& 0& -{\mathrm{\Phi }}_{,t}\end{array}\right]}\\ \stackrel{c\downarrow a\to }{\left[\begin{array}{cccc}-{\mathrm{\Phi }}_{,x}& -{\mathrm{\Phi }}_{,t}& 0& 0\\ {\mathrm{\Phi }}_{,t}& -{\mathrm{\Phi }}_{,x}& {\mathrm{\Phi }}_{,y}& {\mathrm{\Phi }}_{,z}\\ 0& -{\mathrm{\Phi }}_{,y}& -{\mathrm{\Phi }}_{,x}& 0\\ 0& -{\mathrm{\Phi }}_{,z}& 0& -{\mathrm{\Phi }}_{,x}\end{array}\right]}\\ \stackrel{c\downarrow a\to }{\left[\begin{array}{cccc}-{\mathrm{\Phi }}_{,y}& 0& -{\mathrm{\Phi }}_{,t}& 0\\ 0& -{\mathrm{\Phi }}_{,y}& -{\mathrm{\Phi }}_{,x}& 0\\ {\mathrm{\Phi }}_{,t}& {\mathrm{\Phi }}_{,x}& -{\mathrm{\Phi }}_{,y}& {\mathrm{\Phi }}_{,z}\\ 0& 0& -{\mathrm{\Phi }}_{,z}& -{\mathrm{\Phi }}_{,y}\end{array}\right]}\\ \stackrel{c\downarrow a\to }{\left[\begin{array}{cccc}-{\mathrm{\Phi }}_{,z}& 0& 0& -{\mathrm{\Phi }}_{,t}\\ 0& -{\mathrm{\Phi }}_{,z}& 0& -{\mathrm{\Phi }}_{,x}\\ 0& 0& -{\mathrm{\Phi }}_{,z}& -{\mathrm{\Phi }}_{,y}\\ {\mathrm{\Phi }}_{,t}& {\mathrm{\Phi }}_{,x}& {\mathrm{\Phi }}_{,y}& -{\mathrm{\Phi }}_{,z}\end{array}\right]}\end{array}\right]}$

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

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


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

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

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

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

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


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

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

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

first equation in terms of ${\partial }_t\rho$
${\rho }_{,t}=-\rho \cdot {u^x}_{,x}-\rho \cdot {u^y}_{,y}-\rho \cdot {u^z}_{,z}-u^x\cdot {\rho }_{,x}-u^y\cdot {\rho }_{,y}-u^z\cdot {\rho }_{,z}+2P{u}^x\cdot {\mathrm{\Phi }}_{,x}+2P{u}^y\cdot {\mathrm{\Phi }}_{,y}+2P{u}^z\cdot {\mathrm{\Phi }}_{,z}+2\rho \cdot u^x\cdot {\mathrm{\Phi }}_{,x}+2\rho \cdot u^y\cdot {\mathrm{\Phi }}_{,y}+2\rho \cdot u^z\cdot {\mathrm{\Phi }}_{,z}$

spatial equations neglect $P_{,t}$, $(P{u}^j{)}_{,j}$, $P$, and ${\mathrm{\Phi }}_{,i}{u}_j$ and substitutes the definition of ${\partial }_t\rho$

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