properties of hypersurface normal:
$n^a{n}_a=-1$ = unit hypersurface normal
$n_a{n^a}_{;b}+n^a{n_a}_{;b}=0$
$n^a{n_a}_{;b}=0$
$n_a{n^a}_{;b}=0$

matter hypersurface normal definition:
$\rho =n^a{n}^b{T_a}_b$ = density

projection tensor:
${{\gamma }_a}_b={g_a}_b+n_a{n}_b$ = projection operator
${{\gamma }_a}^b={{\delta }_a}^b+n_a{n}^b$
${{\delta }_a}^b={{\gamma }_a}^b-n_a{n}^b$
$n_a{n}^b={{\gamma }_a}^b-{{\delta }_a}^b$

${{\gamma }_a}^c{{\gamma }_c}^b=\left({{\delta }_a}^c+n_a{n}^c\right)\left({{\delta }_c}^b+n_c{n}^b\right)$
${{\gamma }_a}^c{{\gamma }_c}^b={{\delta }_a}^b+n_a{n}^b$
${{\gamma }_a}^c{{\gamma }_c}^b={{\gamma }_a}^b$

$n^a{{\gamma }_a}^b=n^a\left({{\delta }_a}^b+n_a{n}^b\right)$
$n^a{{\gamma }_a}^b=n^b+n^a{n}_a{n}^b$
$n^b{{\gamma }_b}^a=0$

$n_b{{\gamma }_a}^b=n_b\left({{\delta }_a}^b+n_a{n}^b\right)$
$n_b{{\gamma }_a}^b=n_a+n_a{n}_b{n}^b$
$n_b{{\gamma }_a}^b=0$

properties of spatial vectors:
$v^a{n}_a=0$
$n_a{v^a}_{;b}+v^a{n_a}_{;b}=0$
$n_a{v^a}_{;b}=-v^a{n_a}_{;b}$

$v_a{n}^a=0$
$n^a{v_a}_{;b}=-v_a{n^a}_{;b}$

${{\gamma }_b}^a{v}^b=v^a$

extrinsic curvature
${K_a}_b=-{{\gamma }_a}^c{{\gamma }_b}^d{n_c}_{;d}$
${K_a}_b=-{{\gamma }_a}^c{{\gamma }_b}^d{n_c}_{;d}$
${K_a}_b=-\left({{\delta }_a}^c+n_a{n}^c\right)\left({{\delta }_b}^d+n_b{n}^d\right){n_c}_{;d}$
${K_a}_b=-1{n_a}_{;b}+-1n_a{n}^c{n_c}_{;b}+-1n_b{n}^d{n_a}_{;d}+-1n_a{n}_b{n}^c{n}^d{n_c}_{;d}$
${K_a}_b=-\left({n_a}_{;b}+n_b{n}^d{n_a}_{;d}\right)$
${n_a}_{;b}=-\left({K_a}_b+n_b{n}^d{n_a}_{;d}\right)$

$n^a{K_a}_b=0$ because $n\cdot \gamma =0$ and $K=\perp \mathrm{\nabla }n$

projection of covariant derivative on a spatial vector
${v^a}_{|b}={{\gamma }_c}^a{{\gamma }_b}^d{v^c}_{;d}$
using ${{\gamma }_a}^c={{\delta }_a}^c+n_a{n}^c$
${v^a}_{|b}={{\gamma }_b}^d{v^c}_{;d}\left({{\delta }_c}^a+n_c{n}^a\right)$
${v^a}_{|b}={{\gamma }_b}^d{v^a}_{;d}+n^a{n}_c{{\gamma }_b}^d{v^c}_{;d}$
${v^a}_{|b}={{\gamma }_b}^d\left({v^a}_{;d}-n^a{v}^c{n_c}_{;d}\right)$
${v^a}_{|b}={{\gamma }_b}^d\left({v^a}_{;d}+n^a{v}^c{K_c}_d+n^a{n}_d{n}^e{v}^c{n_c}_{;e}\right)$
${v^a}_{|b}=\left({{\delta }_b}^d+n_b{n}^d\right)\left({v^a}_{;d}+n^a{v}^c{K_c}_d+n^a{n}_d{n}^e{v}^c{n_c}_{;e}\right)$
${v^a}_{|b}={v^a}_{;b}+n_b{n}^d{v^a}_{;d}+n^a{v}^c{K_c}_b+n^a{n}_b{n}^d{v}^c{K_c}_d+n^a{n}_b{n}^e{v}^c{n_c}_{;e}+n^a{n}_b{n}_d{n}^d{n}^e{v}^c{n_c}_{;e}$
${v^a}_{|b}={v^a}_{;b}+n^a{v}^c{K_c}_b+n_b{n}^d{v^a}_{;d}+n^a{n}_b{n}^d{v}^c{K_c}_d$
using $n^d{K_c}_d=0$
${v^a}_{|b}={v^a}_{;b}+n^a{v}^c{K_c}_b+n_b{n}^d{v^a}_{;d}$
${v^a}_{|b}={{\gamma }_b}^c{v^a}_{;c}+n^a{v}^c{K_c}_b$

2nd projection covariant derivative of a spatial vector
${{v^a}_{|b}}_{|c}={({{\gamma }_b}^d{v^a}_{;d}+n^a{v}^d{K_d}_b)}_{|c}$
${{v^a}_{|b}}_{|c}={({{\gamma }_e}^d{v^g}_{;d}+n^g{v}^d{K_d}_e)}_{;f}{{\gamma }_g}^a{{\gamma }_b}^e{{\gamma }_c}^f$
${{v^a}_{|b}}_{|c}={{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_e}^d{{\gamma }_g}^a{{v^g}_{;d}}_{;f}+{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{v^g}_{;d}{{{\gamma }_e}^d}_{;f}+n^g{v}^d{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{{K_d}_e}_{;f}+n^g{K_d}_e{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{v^d}_{;f}+v^d{K_d}_e{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{n^g}_{;f}$
${{v^a}_{|b}}_{|c}={{\gamma }_c}^f{{\gamma }_g}^a{{v^g}_{;d}}_{;f}{{\gamma }_b}^d+{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{v^g}_{;d}{{{\gamma }_e}^d}_{;f}+n^g{v}^d{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{{K_d}_e}_{;f}+n^g{K_d}_e{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{v^d}_{;f}+v^d{K_d}_e{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{n^g}_{;f}$
${{v^a}_{|b}}_{|c}={{\gamma }_b}^d{{\gamma }_c}^f{{\gamma }_g}^a{{v^g}_{;d}}_{;f}+n^g{K_d}_b{{\gamma }_c}^f{{\gamma }_g}^a{v^d}_{;f}+v^d{K_d}_b{{\gamma }_c}^f{{\gamma }_g}^a{n^g}_{;f}+{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{v^g}_{;d}{{{\gamma }_e}^d}_{;f}+n^g{v}^d{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{{K_d}_e}_{;f}$
${{v^a}_{|b}}_{|c}={{\gamma }_b}^d{{\gamma }_c}^f{{\gamma }_g}^a{{v^g}_{;d}}_{;f}+v^d{K_d}_b{{\gamma }_c}^f{{\gamma }_g}^a{n^g}_{;f}+{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{v^g}_{;d}{{{\gamma }_e}^d}_{;f}$
${{v^a}_{|b}}_{|c}={{\gamma }_b}^d{{\gamma }_c}^f{{\gamma }_g}^a{{v^g}_{;d}}_{;f}+v^d{K_d}_b{{\gamma }_c}^f{{\gamma }_g}^a{n^g}_{;f}+{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{v^g}_{;d}\left({{{\delta }_e}^d}_{;f}+n_e{n^d}_{;f}+n^d{n_e}_{;f}\right)$
${{v^a}_{|b}}_{|c}={{\gamma }_b}^d{{\gamma }_c}^f{{\gamma }_g}^a{{v^g}_{;d}}_{;f}+v^d{K_d}_b{{\gamma }_c}^f{{\gamma }_g}^a{n^g}_{;f}+{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{v^g}_{;d}{{{\delta }_e}^d}_{;f}+n_e{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{n^d}_{;f}{v^g}_{;d}+n^d{{\gamma }_b}^e{{\gamma }_c}^f{{\gamma }_g}^a{n_e}_{;f}{v^g}_{;d}$
${{v^a}_{|b}}_{|c}=-1n^d{K_b}_c{{\gamma }_e}^a{v^e}_{;d}+-1v^d{K_c}_e{K_d}_b{{\gamma }^a}^e+{{\gamma }_b}^d{{\gamma }_c}^e{{\gamma }_f}^a{{v^f}_{;d}}_{;e}+{{\gamma }_b}^d{{\gamma }_c}^e{{\gamma }_f}^a{v^f}_{;g}{{{\delta }_d}^g}_{;e}+n_d{{\gamma }_b}^d{{\gamma }_c}^e{{\gamma }_f}^a{n^g}_{;e}{v^f}_{;g}$

Riemann curvature of covariant derivative
${{{R^a}_b}_c}_d{v}^b={{v^a}_{;d}}_{;c}-{{v^a}_{;c}}_{;d}$

Riemann curvature of projection covariant derivative
${{{{(R^{\perp })}^a}_b}_c}_d{v}^b={{v^a}_{|d}}_{|c}-{{v^a}_{|c}}_{|d}$
${{{{(R^{\perp })}^a}_b}_c}_d{v}^b=-\left(-1n^g{K_c}_d{{\gamma }_e}^a{v^e}_{;g}+-1v^g{K_d}_e{K_g}_c{{\gamma }^a}^e+{{\gamma }_c}^g{{\gamma }_d}^e{{\gamma }_f}^a{{v^f}_{;g}}_{;e}+{{\gamma }_c}^g{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;g}{{{\delta }_g}^g}_{;e}+n_g{{\gamma }_c}^g{{\gamma }_d}^e{{\gamma }_f}^a{n^g}_{;e}{v^f}_{;g}\right)+-1n^g{K_d}_c{{\gamma }_e}^a{v^e}_{;g}+-1v^g{K_c}_e{K_g}_d{{\gamma }^a}^e+{{\gamma }_d}^g{{\gamma }_c}^e{{\gamma }_f}^a{{v^f}_{;g}}_{;e}+{{\gamma }_d}^g{{\gamma }_c}^e{{\gamma }_f}^a{v^f}_{;g}{{{\delta }_g}^g}_{;e}+n_g{{\gamma }_d}^g{{\gamma }_c}^e{{\gamma }_f}^a{n^g}_{;e}{v^f}_{;g}$
using ${{v^f}_{;g}}_{;e}{{\gamma }_f}^a{{\gamma }_c}^g{{\gamma }_d}^e=\left({{v^f}_{;e}}_{;g}+v^b{{{R^f}_b}_e}_g\right){{\gamma }_f}^a{{\gamma }_c}^g{{\gamma }_d}^e$
$v^b{{{{(R^{\perp })}^a}_b}_c}_d=n^g{K_c}_d{{\gamma }_e}^a{v^e}_{;g}+-1n^g{K_d}_c{{\gamma }_e}^a{v^e}_{;g}+-1v^g{K_c}_e{K_g}_d{{\gamma }^a}^e+v^g{K_d}_e{K_g}_c{{\gamma }^a}^e+-1\left({{v^f}_{;e}}_{;g}+v^b{{{R^f}_b}_e}_g\right){{\gamma }_f}^a{{\gamma }_c}^g{{\gamma }_d}^e+{{\gamma }_c}^e{{\gamma }_d}^g{{\gamma }_f}^a{{v^f}_{;g}}_{;e}+{{\gamma }_c}^e{{\gamma }_d}^g{{\gamma }_f}^a{v^f}_{;g}{{{\delta }_g}^g}_{;e}+-1{{\gamma }_c}^g{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;g}{{{\delta }_g}^g}_{;e}+-1n_g{{\gamma }_c}^g{{\gamma }_d}^e{{\gamma }_f}^a{n^g}_{;e}{v^f}_{;g}+n_g{{\gamma }_c}^e{{\gamma }_d}^g{{\gamma }_f}^a{n^g}_{;e}{v^f}_{;g}$
$v^b{{{{(R^{\perp })}^a}_b}_c}_d=n^b{K_c}_d{{\gamma }_e}^a{v^e}_{;b}+-1n^b{K_d}_c{{\gamma }_e}^a{v^e}_{;b}+v^b{K_b}_c{K_d}_e{{\gamma }^a}^e+-1v^b{K_b}_d{K_c}_e{{\gamma }^a}^e+-1v^b{{\gamma }_c}^e{{\gamma }_d}^f{{\gamma }_g}^a{{{R^g}_b}_f}_e+{{\gamma }_c}^b{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;e}{{{\delta }_e}^e}_{;b}+-1{{\gamma }_c}^b{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;b}{{{\delta }_b}^b}_{;e}+-1n_b{{\gamma }_c}^b{{\gamma }_d}^e{{\gamma }_f}^a{n^b}_{;e}{v^f}_{;b}+n_b{{\gamma }_c}^e{{\gamma }_d}^b{{\gamma }_f}^a{n^b}_{;e}{v^f}_{;b}$
$v^b{{{{(R^{\perp })}^a}_b}_c}_d=v^b{K_b}_c{K_d}_e{{\gamma }^a}^e+-1v^b{K_b}_d{K_c}_e{{\gamma }^a}^e+-1v^b{{\gamma }_c}^e{{\gamma }_d}^f{{\gamma }_g}^a{{{R^g}_b}_f}_e+-1{{\gamma }_c}^b{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;b}{{{\delta }_b}^b}_{;e}+{{\gamma }_c}^b{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;e}{{{\delta }_e}^e}_{;b}+-1n_b{{\gamma }_c}^b{{\gamma }_d}^e{{\gamma }_f}^a{n^b}_{;e}{v^f}_{;b}+n_b{{\gamma }_c}^e{{\gamma }_d}^b{{\gamma }_f}^a{n^b}_{;e}{v^f}_{;b}$
${{\gamma }_b}^h{{{{(R^{\perp })}^a}_h}_c}_d={K_d}_e{K_h}_c{{\gamma }^a}^e{{\gamma }_b}^h+-1{K_c}_e{K_h}_d{{\gamma }^a}^e{{\gamma }_b}^h+{{\gamma }_c}^h{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;e}{{{\delta }_e}^e}_{;h}+-1{{\gamma }_c}^h{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;h}{{{\delta }_h}^h}_{;e}+-1{{\gamma }_b}^h{{\gamma }_c}^e{{\gamma }_d}^f{{\gamma }_g}^a{{{R^g}_h}_f}_e+-1n_h{{\gamma }_c}^h{{\gamma }_d}^e{{\gamma }_f}^a{n^h}_{;e}{v^f}_{;h}+n_h{{\gamma }_c}^e{{\gamma }_d}^h{{\gamma }_f}^a{n^h}_{;e}{v^f}_{;h}$
${{{{(R^{\perp })}^a}_b}_c}_d={K_d}^a{K_b}_c+-1{K_c}^a{K_b}_d+-1{{\gamma }_b}^h{{\gamma }_c}^e{{\gamma }_d}^f{{\gamma }_g}^a{{{R^g}_h}_f}_e+-1{{\gamma }_c}^h{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;h}0+{{\gamma }_c}^h{{\gamma }_d}^e{{\gamma }_f}^a{v^f}_{;e}0+-1n_h{{\gamma }_c}^h{{\gamma }_d}^e{{\gamma }_f}^a{n^h}_{;e}{v^f}_{;h}+n_h{{\gamma }_c}^e{{\gamma }_d}^h{{\gamma }_f}^a{n^h}_{;e}{v^f}_{;h}$
${{{{(R^{\perp })}^k}_b}_c}_d{{\gamma }_a}_k=\left({K_d}^k{K_b}_c+-1{K_c}^k{K_b}_d+-1{{\gamma }_b}^h{{\gamma }_c}^e{{\gamma }_d}^f{{\gamma }_g}^k{{{R^g}_h}_f}_e+-1{{\gamma }_c}^h{{\gamma }_d}^e{{\gamma }_f}^k{v^f}_{;h}0+{{\gamma }_c}^h{{\gamma }_d}^e{{\gamma }_f}^k{v^f}_{;e}0+-1n_h{{\gamma }_c}^h{{\gamma }_d}^e{{\gamma }_f}^k{n^h}_{;e}{v^f}_{;h}+n_h{{\gamma }_c}^e{{\gamma }_d}^h{{\gamma }_f}^k{n^h}_{;e}{v^f}_{;h}\right){{\gamma }_a}_k$
${{{{(R^{\perp })}_a}_b}_c}_d={K_a}_d{K_b}_c+-1{K_a}_c{K_b}_d+-1{{\gamma }_a}^e{{\gamma }_b}^f{{\gamma }_c}^g{{\gamma }_d}^h{{{R_e}_f}_h}_g+-1n_f{{\gamma }_a}_k{{\gamma }_c}^f{{\gamma }_d}^g{{\gamma }_h}^k{n^f}_{;g}{v^h}_{;f}+n_f{{\gamma }_a}_k{{\gamma }_c}^g{{\gamma }_d}^f{{\gamma }_h}^k{n^f}_{;g}{v^h}_{;f}$
${{{{(R^{\perp })}_a}_b}_c}_d={K_a}_d{K_b}_c-{K_a}_c{K_b}_d-{{\gamma }_b}^f{{\gamma }_a}^e{{\gamma }_c}^g{{\gamma }_d}^h{{{R_e}_f}_h}_g-n_b{{\gamma }_a}_e{{\gamma }_c}^b{{\gamma }_d}^f{{\gamma }_g}^e{n^b}_{;f}{v^g}_{;b}+n_b{{\gamma }_c}^f{{\gamma }_a}_e{{\gamma }_d}^b{{\gamma }_g}^e{n^b}_{;f}{v^g}_{;b}$
Gauss' equation

Einstein Field Equations
${G_a}_b=8\pi \cdot {T_a}_b$
using ${G_a}_b={R_a}_b-\frac{1}{2}R{g_a}_b$
${R_a}_b-\frac{1}{2}R{g_a}_b=8\pi \cdot {T_a}_b$

time-by-time
$\frac{1}{2}{n}^a{n}^b\left(2{R_a}_b-R{g_a}_b\right)=8\pi \cdot n^a{n}^b{T_a}_b$
$n^a{n}^b{R_a}_b+-1\cdot \frac{1}{2}R{n}_b{n}^b=8\pi \cdot n^a{n}^b{T_a}_b$
using $n^a{n}_a=-1$ , $\rho =n^a{n}^b{T_a}_b$
$\frac{1}{2}R+n^a{n}^b{R_a}_b=8\rho \cdot \pi$

time-by-space
$\frac{1}{2}{n}^a{{\gamma }_i}^b\left(2{R_a}_b-R{g_a}_b\right)=8\pi \cdot n^a{T_a}_b{{\gamma }_i}^b$
$\frac{1}{2}{n}^a\left({{\delta }_i}^b+n_i{n}^b\right)\left(2{R_a}_b-R{g_a}_b\right)=8\pi \cdot n^a{T_a}_b\left({{\delta }_i}^b+n_i{n}^b\right)$
$\frac{1}{2}\left(-R{n}_i+2n^a{R_a}_i-R{n}_b{n}^b{n}_i+2n^a{n}^b{n}_i{R_a}_b\right)=8\pi \cdot n^a\left({T_a}_i+n^b{n}_i{T_a}_b\right)$
$n^a\left({R_a}_i+n^b{n}_i{R_a}_b\right)=8\pi \cdot n^a\left({T_a}_i+n^b{n}_i{T_a}_b\right)$

space-by-space
$\frac{1}{2}{{\gamma }_i}^a{{\gamma }_j}^b\left(2{R_a}_b-R{g_a}_b\right)=8\pi \cdot {T_a}_b{{\gamma }_i}^a{{\gamma }_j}^b$
$\frac{1}{2}\left({{\delta }_i}^a+n_i{n}^a\right)\left({{\delta }_j}^b+n_j{n}^b\right)\left(2{R_a}_b-R{g_a}_b\right)=8\pi \cdot {T_a}_b\left({{\delta }_i}^a+n_i{n}^a\right)\left({{\delta }_j}^b+n_j{n}^b\right)$
$\frac{1}{2}\left(2{R_i}_j-R{g_i}_j-2R{n}_i{n}_j+2n^b{n}_j{R_i}_b+2n^a{n}_i{R_a}_j-R{n}_b{n}^b{n}_i{n}_j+2n^a{n}^b{n}_i{n}_j{R_a}_b\right)=8\pi \cdot \left({T_i}_j+n^a{n}_i{T_a}_j+n^b{n}_j{T_i}_b+n^a{n}^b{n}_i{n}_j{T_a}_b\right)$
$\frac{1}{2}\left(2{R_i}_j-R{g_i}_j-R{n}_i{n}_j+2n^b{n}_j{R_i}_b+2n^a{n}_i{R_a}_j+2n^a{n}^b{n}_i{n}_j{R_a}_b\right)=8\pi \cdot \left({T_i}_j+\rho \cdot n_i{n}_j+n^b{n}_j{T_i}_b+n^a{n}_i{T_a}_j\right)$