gauge vars
$Q = {{\alpha \cdot f} {{K}^i}_i}$
primitive $\partial_t$ defs
${\alpha}_{,t} = {{{\alpha}_{,i} {\beta}^i} - {\alpha \cdot Q}}$
${\alpha}_{,t} = {{{\alpha}_{,i} {\beta}^i} - {{{\alpha}^{2}} f {{K}^i}_i}}$
${{{\gamma}_i}_j}_{,t} = {{{{{{-2} \alpha} {{K}_i}_j} + {{{{\gamma}_i}_j}_{,k} {\beta}^k}} + {{{\gamma}_k}_j {{\beta}^k}_{,i}}} + {{{\gamma}_i}_k {{\beta}^k}_{,j}}}$
${{{K}_i}_j}_{,t} = {{{{{{{{{{K}_i}_j}_{,k} {\beta}^k} + {{{K}_k}_i {{\beta}^k}_{,j}}} + {{{K}_k}_j {{\beta}^k}_{,i}}} - {{\alpha}_{,i}}_{,j}} + {{{{\Gamma}^k}_i}_j {\alpha}_{,k}}} + {\alpha \cdot {({{{{{R}_i}_j + 0} + {{({{{K}^k}_k + 0})} {{K}_i}_j}} - {{2 {{K}_i}_k} {{K}^k}_j}})}}} + {{{4 \pi} \alpha} {({{{{\gamma}_i}_j {({S - \rho})}} - {2 {{S}_i}_j}})}}}$
lapse vars
${f}_{,k} = {{f' \cdot \alpha} {a}_k}$
hyperbolic state variables
${a}_k = {\left(log\left(\alpha\right)\right)}_{,k}$
${a}_k = {{\frac1\alpha}{({{\alpha}_{,k}})}}$
${\alpha}_{,k} = {{a}_k \alpha}$
${{{d}_k}_i}_j = {{\frac{1}{2}} {{{\gamma}_i}_j}_{,k}}$
${{{\gamma}_i}_j}_{,k} = {2 {{{d}_k}_i}_j}$
${\gamma^{ij}}_{,k}$ wrt aux vars
${{{\gamma}^i}^j}_{,k} = {{{-{{\gamma}^i}^l} {{{\gamma}_l}_m}_{,k}} {{\gamma}^m}^j}$
${{{\gamma}^i}^j}_{,k} = {-{2 {{\gamma}^i}^l {{{d}_k}_l}_m {{\gamma}^m}^j}}$
connections wrt aux vars
${{{\Gamma}_i}_j}_k = {{\frac{1}{2}} {({{{{{\gamma}_i}_j}_{,k} + {{{\gamma}_i}_k}_{,j}} - {{{\gamma}_j}_k}_{,i}})}}$
${{{\Gamma}_i}_j}_k = {-{({{{{{d}_i}_j}_k - {{{d}_j}_i}_k} - {{{d}_k}_i}_j})}}$
${{{\Gamma}^i}_j}_k = {{{\gamma}^i}^l {({{{{d}_j}_l}_k + {{{{d}_k}_l}_j - {{{d}_l}_j}_k}})}}$
Ricci wrt aux vars
${{R}_i}_j = {{{{{{{\Gamma}^k}_i}_j}_{,k} - {{{{\Gamma}^k}_i}_k}_{,j}} + {{{{\Gamma}^k}_l}_k {{{\Gamma}^l}_i}_j}} - {{{{\Gamma}^k}_l}_j {{{\Gamma}^l}_i}_k}}$
${{R}_i}_j = {{{{\left({{\gamma}^k}^a {({{{{d}_i}_a}_j + {{{{d}_j}_a}_i - {{{d}_a}_i}_j}})}\right)}_{,k} - {\left({{\gamma}^k}^a {({{{{d}_i}_a}_k + {{{{d}_k}_a}_i - {{{d}_a}_i}_k}})}\right)}_{,j}} + {{{{\gamma}^k}^a {({{{{d}_l}_a}_k + {{{{d}_k}_a}_l - {{{d}_a}_l}_k}})}} {{{\gamma}^l}^b {({{{{d}_i}_b}_j + {{{{d}_j}_b}_i - {{{d}_b}_i}_j}})}}}} - {{{{\gamma}^k}^a {({{{{d}_l}_a}_j + {{{{d}_j}_a}_l - {{{d}_a}_l}_j}})}} {{{\gamma}^l}^b {({{{{d}_i}_b}_k + {{{{d}_k}_b}_i - {{{d}_b}_i}_k}})}}}}$
${{R}_i}_j = {{{{{{{{\gamma}^k}^a}_{,j} {{{d}_a}_i}_k} - {{{{\gamma}^k}^a}_{,j} {{{d}_i}_a}_k}} - {{{{\gamma}^k}^a}_{,j} {{{d}_k}_a}_i}} - {{{{\gamma}^k}^a}_{,k} {{{d}_a}_i}_j}} + {{{{\gamma}^k}^a}_{,k} {{{d}_i}_a}_j} + {{{{{\gamma}^k}^a}_{,k} {{{d}_j}_a}_i} - {{{\gamma}^k}^a {{{{d}_a}_i}_j}_{,k}}} + {{{{\gamma}^k}^a {{{{d}_a}_i}_k}_{,j}} - {{{\gamma}^k}^a {{{d}_a}_l}_j {{\gamma}^l}^b {{{d}_b}_i}_k}} + {{{\gamma}^k}^a {{{d}_a}_l}_j {{\gamma}^l}^b {{{d}_i}_b}_k} + {{{\gamma}^k}^a {{{d}_a}_l}_j {{\gamma}^l}^b {{{d}_k}_b}_i} + {{{{{\gamma}^k}^a {{{d}_a}_l}_k {{\gamma}^l}^b {{{d}_b}_i}_j} - {{{\gamma}^k}^a {{{d}_a}_l}_k {{\gamma}^l}^b {{{d}_i}_b}_j}} - {{{\gamma}^k}^a {{{d}_a}_l}_k {{\gamma}^l}^b {{{d}_j}_b}_i}} + {{{{\gamma}^k}^a {{{{d}_i}_a}_j}_{,k}} - {{{\gamma}^k}^a {{{{d}_i}_a}_k}_{,j}}} + {{{\gamma}^k}^a {{{{d}_j}_a}_i}_{,k}} + {{{{{{{\gamma}^k}^a {{{d}_j}_a}_l {{\gamma}^l}^b {{{d}_b}_i}_k} - {{{\gamma}^k}^a {{{d}_j}_a}_l {{\gamma}^l}^b {{{d}_i}_b}_k}} - {{{\gamma}^k}^a {{{d}_j}_a}_l {{\gamma}^l}^b {{{d}_k}_b}_i}} - {{{\gamma}^k}^a {{{{d}_k}_a}_i}_{,j}}} - {{{\gamma}^k}^a {{{d}_k}_a}_l {{\gamma}^l}^b {{{d}_b}_i}_j}} + {{{\gamma}^k}^a {{{d}_k}_a}_l {{\gamma}^l}^b {{{d}_i}_b}_j} + {{{\gamma}^k}^a {{{d}_k}_a}_l {{\gamma}^l}^b {{{d}_j}_b}_i} + {{{{{{\gamma}^k}^a {{{d}_l}_a}_j {{\gamma}^l}^b {{{d}_b}_i}_k} - {{{\gamma}^k}^a {{{d}_l}_a}_j {{\gamma}^l}^b {{{d}_i}_b}_k}} - {{{\gamma}^k}^a {{{d}_l}_a}_j {{\gamma}^l}^b {{{d}_k}_b}_i}} - {{{\gamma}^k}^a {{{d}_l}_a}_k {{\gamma}^l}^b {{{d}_b}_i}_j}} + {{{\gamma}^k}^a {{{d}_l}_a}_k {{\gamma}^l}^b {{{d}_i}_b}_j} + {{{\gamma}^k}^a {{{d}_l}_a}_k {{\gamma}^l}^b {{{d}_j}_b}_i}}$
${{R}_i}_j = {-{({{{{{\gamma}^k}^a {{{{d}_a}_i}_j}_{,k}} - {{{\gamma}^k}^a {{{{d}_a}_i}_k}_{,j}}} + {{{{{{\gamma}^k}^a {{{d}_a}_l}_j {{\gamma}^l}^b {{{d}_b}_i}_k} - {{{\gamma}^k}^a {{{d}_a}_l}_j {{\gamma}^l}^b {{{d}_i}_b}_k}} - {{{\gamma}^k}^a {{{d}_a}_l}_j {{\gamma}^l}^b {{{d}_k}_b}_i}} - {{{\gamma}^k}^a {{{d}_a}_l}_k {{\gamma}^l}^b {{{d}_b}_i}_j}} + {{{\gamma}^k}^a {{{d}_a}_l}_k {{\gamma}^l}^b {{{d}_i}_b}_j} + {{{{\gamma}^k}^a {{{d}_a}_l}_k {{\gamma}^l}^b {{{d}_j}_b}_i} - {{{\gamma}^k}^a {{{{d}_i}_a}_j}_{,k}}} + {{{{{\gamma}^k}^a {{{{d}_i}_a}_k}_{,j}} - {{{\gamma}^k}^a {{{{d}_j}_a}_i}_{,k}}} - {{{\gamma}^k}^a {{{d}_j}_a}_l {{\gamma}^l}^b {{{d}_b}_i}_k}} + {{{\gamma}^k}^a {{{d}_j}_a}_l {{\gamma}^l}^b {{{d}_i}_b}_k} + {{{\gamma}^k}^a {{{d}_j}_a}_l {{\gamma}^l}^b {{{d}_k}_b}_i} + {{{\gamma}^k}^a {{{{d}_k}_a}_i}_{,j}} + {{{{{{\gamma}^k}^a {{{d}_k}_a}_l {{\gamma}^l}^b {{{d}_b}_i}_j} - {{{\gamma}^k}^a {{{d}_k}_a}_l {{\gamma}^l}^b {{{d}_i}_b}_j}} - {{{\gamma}^k}^a {{{d}_k}_a}_l {{\gamma}^l}^b {{{d}_j}_b}_i}} - {{{\gamma}^k}^a {{{d}_l}_a}_j {{\gamma}^l}^b {{{d}_b}_i}_k}} + {{{\gamma}^k}^a {{{d}_l}_a}_j {{\gamma}^l}^b {{{d}_i}_b}_k} + {{{\gamma}^k}^a {{{d}_l}_a}_j {{\gamma}^l}^b {{{d}_k}_b}_i} + {{{{{\gamma}^k}^a {{{d}_l}_a}_k {{\gamma}^l}^b {{{d}_b}_i}_j} - {{{\gamma}^k}^a {{{d}_l}_a}_k {{\gamma}^l}^b {{{d}_i}_b}_j}} - {{{\gamma}^k}^a {{{d}_l}_a}_k {{\gamma}^l}^b {{{d}_j}_b}_i}} + {{{{2 {{\gamma}^k}^d {{{d}_j}_d}_c {{\gamma}^c}^a {{{d}_a}_i}_k} - {2 {{\gamma}^k}^d {{{d}_j}_d}_c {{\gamma}^c}^a {{{d}_i}_a}_k}} - {2 {{\gamma}^k}^d {{{d}_j}_d}_c {{\gamma}^c}^a {{{d}_k}_a}_i}} - {2 {{\gamma}^k}^d {{{d}_k}_d}_c {{\gamma}^c}^a {{{d}_a}_i}_j}} + {2 {{\gamma}^k}^d {{{d}_k}_d}_c {{\gamma}^c}^a {{{d}_i}_a}_j} + {2 {{\gamma}^k}^d {{{d}_k}_d}_c {{\gamma}^c}^a {{{d}_j}_a}_i}})}}$
symmetrizing
${{R}_i}_j = {-{({{{{\gamma}^a}^k {{{{d}_a}_i}_j}_{,k}} + {{{{{{\gamma}^a}^k {{{d}_a}_j}_l {{\gamma}^b}^l {{{d}_b}_i}_k} - {{{\gamma}^a}^k {{{d}_a}_j}_l {{\gamma}^b}^l {{{d}_i}_b}_k}} - {{{\gamma}^a}^k {{{d}_a}_j}_l {{\gamma}^b}^l {{{d}_k}_b}_i}} - {{{\gamma}^a}^k {{{{d}_i}_a}_j}_{,k}}} + {{{{{\gamma}^a}^k {{{{d}_i}_a}_k}_{,j}} - {{{\gamma}^a}^k {{{{d}_j}_a}_i}_{,k}}} - {{{\gamma}^a}^k {{{d}_j}_a}_l {{\gamma}^b}^l {{{d}_b}_i}_k}} + {{{\gamma}^a}^k {{{d}_j}_a}_l {{\gamma}^b}^l {{{d}_i}_b}_k} + {{{{\gamma}^a}^k {{{d}_j}_a}_l {{\gamma}^b}^l {{{d}_k}_b}_i} - {{{\gamma}^a}^k {{{d}_l}_a}_j {{\gamma}^b}^l {{{d}_b}_i}_k}} + {{{\gamma}^a}^k {{{d}_l}_a}_j {{\gamma}^b}^l {{{d}_i}_b}_k} + {{{\gamma}^a}^k {{{d}_l}_a}_j {{\gamma}^b}^l {{{d}_k}_b}_i} + {{{{{{\gamma}^a}^k {{{d}_l}_a}_k {{\gamma}^b}^l {{{d}_b}_i}_j} - {{{\gamma}^a}^k {{{d}_l}_a}_k {{\gamma}^b}^l {{{d}_i}_b}_j}} - {{{\gamma}^a}^k {{{d}_l}_a}_k {{\gamma}^b}^l {{{d}_j}_b}_i}} - {2 {{\gamma}^d}^k {{{d}_d}_c}_k {{\gamma}^a}^c {{{d}_a}_i}_j}} + {2 {{\gamma}^d}^k {{{d}_d}_c}_k {{\gamma}^a}^c {{{d}_i}_a}_j} + {2 {{\gamma}^d}^k {{{d}_d}_c}_k {{\gamma}^a}^c {{{d}_j}_a}_i} + {{{2 {{\gamma}^d}^k {{{d}_j}_c}_d {{\gamma}^a}^c {{{d}_a}_i}_k} - {2 {{\gamma}^d}^k {{{d}_j}_c}_d {{\gamma}^a}^c {{{d}_i}_a}_k}} - {2 {{\gamma}^d}^k {{{d}_j}_c}_d {{\gamma}^a}^c {{{d}_k}_a}_i}}})}}$
${{R}_i}_j = {{{\gamma}^e}^f {({{{{2 {{{d}_e}_f}_g {{\gamma}^g}^h {{{d}_h}_i}_j} - {2 {{{d}_e}_f}_g {{\gamma}^g}^h {{{d}_i}_h}_j}} - {2 {{{d}_e}_f}_g {{\gamma}^g}^h {{{d}_j}_h}_i}} + {{{{{d}_e}_g}_j {{\gamma}^g}^h {{{d}_f}_h}_i} - {{{{d}_e}_g}_j {{\gamma}^g}^h {{{d}_h}_f}_i}} + {{{{{d}_e}_g}_j {{\gamma}^g}^h {{{d}_i}_f}_h} - {{{{d}_e}_i}_j}_{,f}} + {{{{d}_f}_e}_i}_{,j} + {{{{{d}_f}_e}_j}_{,i} - {{{{d}_g}_e}_f {{\gamma}^g}^h {{{d}_h}_i}_j}} + {{{{d}_g}_e}_f {{\gamma}^g}^h {{{d}_i}_h}_j} + {{{{{d}_g}_e}_f {{\gamma}^g}^h {{{d}_j}_h}_i} - {{{{d}_g}_e}_j {{\gamma}^g}^h {{{d}_f}_h}_i}} + {{{{{{d}_g}_e}_j {{\gamma}^g}^h {{{d}_h}_f}_i} - {{{{d}_g}_e}_j {{\gamma}^g}^h {{{d}_i}_f}_h}} - {{{{d}_i}_e}_f}_{,j}} + {{{{{d}_j}_e}_g {{\gamma}^g}^h {{{d}_f}_h}_i} - {{{{d}_j}_e}_g {{\gamma}^g}^h {{{d}_h}_f}_i}} + {{{{d}_j}_e}_g {{\gamma}^g}^h {{{d}_i}_f}_h}})}}$
time derivative of $\alpha_{,t}$
${\alpha}_{,t} = {{{\beta}^i {{a}_i \alpha}} - {{{\alpha}^{2}} f {{K}^i}_i}}$
time derivative of $\gamma_{ij,t}$
${{{\gamma}_i}_j}_{,t} = {{{{{{-2} \alpha} {{K}_i}_j} + {{{{\gamma}_i}_j}_{,k} {\beta}^k}} + {{{\gamma}_k}_j {{\beta}^k}_{,i}}} + {{{\gamma}_i}_k {{\beta}^k}_{,j}}}$
${{{\gamma}_i}_j}_{,t} = {{{{{{-2} \alpha} {{K}_i}_j} + {{\beta}^k {2 {{{d}_k}_i}_j}}} + {{{\gamma}_k}_j {{\beta}^k}_{,i}}} + {{{\gamma}_i}_k {{\beta}^k}_{,j}}}$
time derivative of $a_{k,t}$
${{a}_k}_{,t} = {\frac{{{{\alpha}_{,k}}_{,t} \alpha} - {{\alpha}_{,k} {\alpha}_{,t}}}{{\alpha}^{2}}}$
${{a}_k}_{,t} = {\frac{{\alpha \cdot {\left({{\beta}^i {{a}_i \alpha}} - {{{\alpha}^{2}} f {{K}^i}_i}\right)}_{,k}} - {{\alpha}_{,k} {({{{\beta}^i {{a}_i \alpha}} - {{{\alpha}^{2}} f {{K}^i}_i}})}}}{{\alpha}^{2}}}$
${{a}_k}_{,t} = {-{({{{{{\alpha}_{,k} f {{K}^i}_i} - {{{\beta}^i}_{,k} {a}_i}} - {{\beta}^i {{a}_i}_{,k}}} + {\alpha \cdot {f}_{,k} {{K}^i}_i} + {\alpha \cdot f {{{K}^i}_i}_{,k}}})}}$
${{a}_k}_{,t} = {-{({{{{{\alpha}_{,k} f {{K}^i}_i} - {{{\beta}^i}_{,k} {a}_i}} - {{\beta}^i {{a}_i}_{,k}}} + {\alpha \cdot {f}_{,k} {{K}^i}_i} + {\alpha \cdot f {\left({{\gamma}^i}^j {{K}_i}_j\right)}_{,k}}})}}$
${{a}_k}_{,t} = {-{({{{{{{a}_k \alpha} f {{K}^i}_i} - {{{\beta}^i}_{,k} {a}_i}} - {{\beta}^i {{a}_i}_{,k}}} + {\alpha \cdot {{f' \cdot \alpha} {a}_k} {{K}^i}_i} + {\alpha \cdot f {\left({{\gamma}^i}^j {{K}_i}_j\right)}_{,k}}})}}$
${{a}_k}_{,t} = {{{{\beta}^i}_{,k} {a}_i} + {{{{{{\beta}^i {{a}_i}_{,k}} - {{a}_k \alpha \cdot f {{K}^i}_i}} - {{{\alpha}^{2}} f' \cdot {a}_k {{K}^i}_i}} - {\alpha \cdot f {{{\gamma}^i}^j}_{,k} {{K}_i}_j}} - {\alpha \cdot f {{\gamma}^i}^j {{{K}_i}_j}_{,k}}}}$
${{a}_k}_{,t} = {{{{\beta}^i}_{,k} {a}_i} + {{{{\beta}^i {{a}_i}_{,k}} - {{a}_k \alpha \cdot f {{K}^i}_i}} - {{{\alpha}^{2}} f' \cdot {a}_k {{K}^i}_i}} + {{2 \alpha \cdot f {{\gamma}^i}^b {{{d}_k}_b}_a {{\gamma}^a}^j {{K}_i}_j} - {\alpha \cdot f {{\gamma}^i}^j {{{K}_i}_j}_{,k}}}}$
${{a}_k}_{,t} = {{{{\beta}^i}_{,k} {a}_i} + {{{{\beta}^i {{a}_i}_{,k}} - {{a}_k \alpha \cdot f {{K}^i}_i}} - {{{\alpha}^{2}} f' \cdot {a}_k {{K}^i}_i}} + {{2 \alpha \cdot f {{\gamma}^i}^b {{{d}_k}_b}_a {{\gamma}^a}^j {{K}_i}_j} - {\alpha \cdot f {{\gamma}^i}^j {{{K}_i}_j}_{,k}}}}$
time derivative of $d_{kij,t}$
${{{{d}_k}_i}_j}_{,t} = {{\frac12}{({{{{{\gamma}_i}_j}_{,k}}_{,t}})}}$
${{{{d}_k}_i}_j}_{,t} = {{\frac12}{({{\left({{{{{-2} \alpha} {{K}_i}_j} + {{\beta}^l {2 {{{d}_l}_i}_j}}} + {{{\gamma}_l}_j {{\beta}^l}_{,i}}} + {{{\gamma}_i}_l {{\beta}^l}_{,j}}\right)}_{,k}})}}$
${{{{d}_k}_i}_j}_{,t} = {{\frac12}{({-{({{{{{{{{2 {\alpha}_{,k} {{K}_i}_j} - {2 {{\beta}^l}_{,k} {{{d}_l}_i}_j}} - {2 {\beta}^l {{{{d}_l}_i}_j}_{,k}}} - {{{{\gamma}_i}_l}_{,k} {{\beta}^l}_{,j}}} - {{{\gamma}_i}_l {{{\beta}^l}_{,j}}_{,k}}} - {{{{\gamma}_l}_j}_{,k} {{\beta}^l}_{,i}}} - {{{\gamma}_l}_j {{{\beta}^l}_{,i}}_{,k}}} + {2 \alpha \cdot {{{K}_i}_j}_{,k}}})}})}}$
${{{{d}_k}_i}_j}_{,t} = {{\frac12}{({-{({{{{{{{{2 {{K}_i}_j {a}_k \alpha} - {2 {{\beta}^l}_{,i} {{{d}_k}_l}_j}} - {2 {{\beta}^l}_{,j} {{{d}_k}_i}_l}} - {2 {{\beta}^l}_{,k} {{{d}_l}_i}_j}} - {2 {\beta}^l {{{{d}_l}_i}_j}_{,k}}} - {{{\gamma}_i}_l {{{\beta}^l}_{,j}}_{,k}}} - {{{\gamma}_l}_j {{{\beta}^l}_{,i}}_{,k}}} + {2 \alpha \cdot {{{K}_i}_j}_{,k}}})}})}}$
$K_{ij,t}$ with hyperbolic terms
${{{K}_i}_j}_{,t} = {{{{{{{{{{K}_i}_j}_{,k} {\beta}^k} + {{{K}_k}_i {{\beta}^k}_{,j}}} + {{{K}_k}_j {{\beta}^k}_{,i}}} - {{\alpha}_{,i}}_{,j}} + {{{{\Gamma}^k}_i}_j {\alpha}_{,k}}} + {\alpha \cdot {({{{{{R}_i}_j + 0} + {{({{{K}^k}_k + 0})} {{K}_i}_j}} - {{2 {{K}_i}_k} {{K}^k}_j}})}}} + {{{4 \pi} \alpha} {({{{{\gamma}_i}_j {({S - \rho})}} - {2 {{S}_i}_j}})}}}$
${{{K}_i}_j}_{,t} = {{{{{{{{{{K}_i}_j}_{,k} {\beta}^k} + {{{K}_k}_i {{\beta}^k}_{,j}}} + {{{K}_k}_j {{\beta}^k}_{,i}}} - {{\frac{1}{2}} {({{\left({a}_i \alpha\right)}_{,j} + {\left({a}_j \alpha\right)}_{,i}})}}} + {{{{\Gamma}^k}_i}_j {{a}_k \alpha}}} + {\alpha \cdot {({{{{{R}_i}_j + 0} + {{({{{K}^k}_k + 0})} {{K}_i}_j}} - {{2 {{K}_i}_k} {{K}^k}_j}})}}} + {{{4 \pi} \alpha} {({{{{\gamma}_i}_j {({S - \rho})}} - {2 {{S}_i}_j}})}}}$
${{{K}_i}_j}_{,t} = {{\frac12}{({{2 {{{K}_i}_j}_{,k} {\beta}^k} + {2 {{K}_k}_i {{\beta}^k}_{,j}} + {2 {{K}_k}_j {{\beta}^k}_{,i}} + {{{{{2 {{{\Gamma}^k}_i}_j {a}_k \alpha} - {{{a}_i}_{,j} \alpha}} - {{a}_i {\alpha}_{,j}}} - {{{a}_j}_{,i} \alpha}} - {{a}_j {\alpha}_{,i}}} + {{2 \alpha \cdot {{K}^k}_k {{K}_i}_j} - {4 \alpha \cdot {{K}_i}_k {{K}^k}_j}} + {{2 \alpha \cdot {{R}_i}_j} - {{16} \pi \cdot \alpha \cdot {{S}_i}_j}} + {{8 \pi \cdot \alpha \cdot {{\gamma}_i}_j S} - {8 \pi \cdot \alpha \cdot {{\gamma}_i}_j \rho}}})}}$
${{{K}_i}_j}_{,t} = {{\frac12}{({{2 {{{K}_i}_j}_{,k} {\beta}^k} + {2 {{K}_k}_i {{\beta}^k}_{,j}} + {2 {{K}_k}_j {{\beta}^k}_{,i}} + {{{{2 {{{\Gamma}^k}_i}_j {a}_k \alpha} - {{{a}_i}_{,j} \alpha}} - {2 {a}_i {a}_j \alpha}} - {{{a}_j}_{,i} \alpha}} + {{2 \alpha \cdot {{K}^k}_k {{K}_i}_j} - {4 \alpha \cdot {{K}_i}_k {{K}^k}_j}} + {{2 \alpha \cdot {{R}_i}_j} - {{16} \pi \cdot \alpha \cdot {{S}_i}_j}} + {{8 \pi \cdot \alpha \cdot {{\gamma}_i}_j S} - {8 \pi \cdot \alpha \cdot {{\gamma}_i}_j \rho}}})}}$
${{{K}_i}_j}_{,t} = {{\frac12}{({{2 {{{K}_i}_j}_{,k} {\beta}^k} + {2 {{K}_k}_i {{\beta}^k}_{,j}} + {2 {{K}_k}_j {{\beta}^k}_{,i}} + {{{{2 {a}_k \alpha \cdot {{{\gamma}^k}^l {({{{{d}_i}_l}_j + {{{{d}_j}_l}_i - {{{d}_l}_i}_j}})}}} - {{{a}_i}_{,j} \alpha}} - {2 {a}_i {a}_j \alpha}} - {{{a}_j}_{,i} \alpha}} + {{2 \alpha \cdot {{K}^k}_k {{K}_i}_j} - {4 \alpha \cdot {{K}_i}_k {{K}^k}_j}} + {{2 \alpha \cdot {{R}_i}_j} - {{16} \pi \cdot \alpha \cdot {{S}_i}_j}} + {{8 \pi \cdot \alpha \cdot {{\gamma}_i}_j S} - {8 \pi \cdot \alpha \cdot {{\gamma}_i}_j \rho}}})}}$
${{{K}_i}_j}_{,t} = {{\frac12}{({{2 {{{K}_i}_j}_{,k} {\beta}^k} + {2 {{K}_k}_i {{\beta}^k}_{,j}} + {2 {{K}_k}_j {{\beta}^k}_{,i}} + {{{{2 {a}_k \alpha \cdot {{{\gamma}^k}^l {({{{{d}_i}_l}_j + {{{{d}_j}_l}_i - {{{d}_l}_i}_j}})}}} - {{{a}_i}_{,j} \alpha}} - {2 {a}_i {a}_j \alpha}} - {{{a}_j}_{,i} \alpha}} + {{2 \alpha \cdot {{K}^k}_k {{K}_i}_j} - {4 \alpha \cdot {{K}_i}_k {{K}^k}_j}} + {{2 \alpha \cdot {{{\gamma}^e}^f {({{{{2 {{{d}_e}_f}_g {{\gamma}^g}^h {{{d}_h}_i}_j} - {2 {{{d}_e}_f}_g {{\gamma}^g}^h {{{d}_i}_h}_j}} - {2 {{{d}_e}_f}_g {{\gamma}^g}^h {{{d}_j}_h}_i}} + {{{{{d}_e}_g}_j {{\gamma}^g}^h {{{d}_f}_h}_i} - {{{{d}_e}_g}_j {{\gamma}^g}^h {{{d}_h}_f}_i}} + {{{{{d}_e}_g}_j {{\gamma}^g}^h {{{d}_i}_f}_h} - {{{{d}_e}_i}_j}_{,f}} + {{{{d}_f}_e}_i}_{,j} + {{{{{d}_f}_e}_j}_{,i} - {{{{d}_g}_e}_f {{\gamma}^g}^h {{{d}_h}_i}_j}} + {{{{d}_g}_e}_f {{\gamma}^g}^h {{{d}_i}_h}_j} + {{{{{d}_g}_e}_f {{\gamma}^g}^h {{{d}_j}_h}_i} - {{{{d}_g}_e}_j {{\gamma}^g}^h {{{d}_f}_h}_i}} + {{{{{{d}_g}_e}_j {{\gamma}^g}^h {{{d}_h}_f}_i} - {{{{d}_g}_e}_j {{\gamma}^g}^h {{{d}_i}_f}_h}} - {{{{d}_i}_e}_f}_{,j}} + {{{{{d}_j}_e}_g {{\gamma}^g}^h {{{d}_f}_h}_i} - {{{{d}_j}_e}_g {{\gamma}^g}^h {{{d}_h}_f}_i}} + {{{{d}_j}_e}_g {{\gamma}^g}^h {{{d}_i}_f}_h}})}}} - {{16} \pi \cdot \alpha \cdot {{S}_i}_j}} + {{8 \pi \cdot \alpha \cdot {{\gamma}_i}_j S} - {8 \pi \cdot \alpha \cdot {{\gamma}_i}_j \rho}}})}}$
${{{K}_i}_j}_{,t} = {{\frac12}{({{2 {{{K}_i}_j}_{,k} {\beta}^k} + {2 {{K}_k}_i {{\beta}^k}_{,j}} + {{{{2 {{K}_k}_j {{\beta}^k}_{,i}} - {{{a}_i}_{,j} \alpha}} - {2 {a}_i {a}_j \alpha}} - {{{a}_j}_{,i} \alpha}} + {2 {a}_k \alpha \cdot {{\gamma}^k}^l {{{d}_i}_l}_j} + {{2 {a}_k \alpha \cdot {{\gamma}^k}^l {{{d}_j}_l}_i} - {2 {a}_k \alpha \cdot {{\gamma}^k}^l {{{d}_l}_i}_j}} + {{2 \alpha \cdot {{K}^k}_k {{K}_i}_j} - {4 \alpha \cdot {{K}_i}_k {{K}^k}_j}} + {{{4 \alpha \cdot {{\gamma}^e}^f {{{d}_e}_f}_g {{\gamma}^g}^h {{{d}_h}_i}_j} - {4 \alpha \cdot {{\gamma}^e}^f {{{d}_e}_f}_g {{\gamma}^g}^h {{{d}_i}_h}_j}} - {4 \alpha \cdot {{\gamma}^e}^f {{{d}_e}_f}_g {{\gamma}^g}^h {{{d}_j}_h}_i}} + {{2 \alpha \cdot {{\gamma}^e}^f {{{d}_e}_g}_j {{\gamma}^g}^h {{{d}_f}_h}_i} - {2 \alpha \cdot {{\gamma}^e}^f {{{d}_e}_g}_j {{\gamma}^g}^h {{{d}_h}_f}_i}} + {{2 \alpha \cdot {{\gamma}^e}^f {{{d}_e}_g}_j {{\gamma}^g}^h {{{d}_i}_f}_h} - {2 \alpha \cdot {{\gamma}^e}^f {{{{d}_e}_i}_j}_{,f}}} + {2 \alpha \cdot {{\gamma}^e}^f {{{{d}_f}_e}_i}_{,j}} + {{2 \alpha \cdot {{\gamma}^e}^f {{{{d}_f}_e}_j}_{,i}} - {2 \alpha \cdot {{\gamma}^e}^f {{{d}_g}_e}_f {{\gamma}^g}^h {{{d}_h}_i}_j}} + {2 \alpha \cdot {{\gamma}^e}^f {{{d}_g}_e}_f {{\gamma}^g}^h {{{d}_i}_h}_j} + {{2 \alpha \cdot {{\gamma}^e}^f {{{d}_g}_e}_f {{\gamma}^g}^h {{{d}_j}_h}_i} - {2 \alpha \cdot {{\gamma}^e}^f {{{d}_g}_e}_j {{\gamma}^g}^h {{{d}_f}_h}_i}} + {{{2 \alpha \cdot {{\gamma}^e}^f {{{d}_g}_e}_j {{\gamma}^g}^h {{{d}_h}_f}_i} - {2 \alpha \cdot {{\gamma}^e}^f {{{d}_g}_e}_j {{\gamma}^g}^h {{{d}_i}_f}_h}} - {2 \alpha \cdot {{\gamma}^e}^f {{{{d}_i}_e}_f}_{,j}}} + {{2 \alpha \cdot {{\gamma}^e}^f {{{d}_j}_e}_g {{\gamma}^g}^h {{{d}_f}_h}_i} - {2 \alpha \cdot {{\gamma}^e}^f {{{d}_j}_e}_g {{\gamma}^g}^h {{{d}_h}_f}_i}} + {{2 \alpha \cdot {{\gamma}^e}^f {{{d}_j}_e}_g {{\gamma}^g}^h {{{d}_i}_f}_h} - {{16} \pi \cdot \alpha \cdot {{S}_i}_j}} + {{8 \pi \cdot \alpha \cdot {{\gamma}_i}_j S} - {8 \pi \cdot \alpha \cdot {{\gamma}_i}_j \rho}}})}}$
${{{K}_i}_j}_{,t} = {{\frac12}{({{2 {{K}_a}_i {{\beta}^a}_{,j}} + {2 {{K}_a}_j {{\beta}^a}_{,i}} + {{2 {{{K}_i}_j}_{,a} {\beta}^a} - {2 {a}_a \alpha \cdot {{\gamma}^a}^b {{{d}_b}_i}_j}} + {2 {a}_a \alpha \cdot {{\gamma}^a}^b {{{d}_i}_b}_j} + {{{{2 {a}_a \alpha \cdot {{\gamma}^a}^b {{{d}_j}_b}_i} - {{{a}_i}_{,j} \alpha}} - {2 {a}_i {a}_j \alpha}} - {{{a}_j}_{,i} \alpha}} + {{2 \alpha \cdot {{K}^a}_a {{K}_i}_j} - {4 \alpha \cdot {{K}_i}_a {{K}^a}_j}} + {{{4 \alpha \cdot {{\gamma}^a}^b {{{d}_a}_b}_c {{\gamma}^c}^d {{{d}_d}_i}_j} - {4 \alpha \cdot {{\gamma}^a}^b {{{d}_a}_b}_c {{\gamma}^c}^d {{{d}_i}_d}_j}} - {4 \alpha \cdot {{\gamma}^a}^b {{{d}_a}_b}_c {{\gamma}^c}^d {{{d}_j}_d}_i}} + {2 \alpha \cdot {{\gamma}^a}^b {{{{d}_a}_b}_i}_{,j}} + {2 \alpha \cdot {{\gamma}^a}^b {{{{d}_a}_b}_j}_{,i}} + {{2 \alpha \cdot {{\gamma}^a}^b {{{d}_a}_c}_j {{\gamma}^c}^d {{{d}_b}_d}_i} - {2 \alpha \cdot {{\gamma}^a}^b {{{d}_a}_c}_j {{\gamma}^c}^d {{{d}_d}_b}_i}} + {{{2 \alpha \cdot {{\gamma}^a}^b {{{d}_a}_c}_j {{\gamma}^c}^d {{{d}_i}_b}_d} - {2 \alpha \cdot {{\gamma}^a}^b {{{{d}_a}_i}_j}_{,b}}} - {2 \alpha \cdot {{\gamma}^a}^b {{{d}_c}_a}_b {{\gamma}^c}^d {{{d}_d}_i}_j}} + {2 \alpha \cdot {{\gamma}^a}^b {{{d}_c}_a}_b {{\gamma}^c}^d {{{d}_i}_d}_j} + {{2 \alpha \cdot {{\gamma}^a}^b {{{d}_c}_a}_b {{\gamma}^c}^d {{{d}_j}_d}_i} - {2 \alpha \cdot {{\gamma}^a}^b {{{d}_c}_a}_j {{\gamma}^c}^d {{{d}_b}_d}_i}} + {{{2 \alpha \cdot {{\gamma}^a}^b {{{d}_c}_a}_j {{\gamma}^c}^d {{{d}_d}_b}_i} - {2 \alpha \cdot {{\gamma}^a}^b {{{d}_c}_a}_j {{\gamma}^c}^d {{{d}_i}_b}_d}} - {2 \alpha \cdot {{\gamma}^a}^b {{{{d}_i}_a}_b}_{,j}}} + {{2 \alpha \cdot {{\gamma}^a}^b {{{d}_j}_a}_c {{\gamma}^c}^d {{{d}_b}_d}_i} - {2 \alpha \cdot {{\gamma}^a}^b {{{d}_j}_a}_c {{\gamma}^c}^d {{{d}_d}_b}_i}} + {{2 \alpha \cdot {{\gamma}^a}^b {{{d}_j}_a}_c {{\gamma}^c}^d {{{d}_i}_b}_d} - {{16} \pi \cdot \alpha \cdot {{S}_i}_j}} + {{8 \pi \cdot \alpha \cdot {{\gamma}_i}_j S} - {8 \pi \cdot \alpha \cdot {{\gamma}_i}_j \rho}}})}}$
partial derivatives
${\alpha}_{,t} = {{{\beta}^i {{a}_i \alpha}} - {{{\alpha}^{2}} f {{K}^i}_i}}$
${{{\gamma}_i}_j}_{,t} = {{{{{{-2} \alpha} {{K}_i}_j} + {{\beta}^k {2 {{{d}_k}_i}_j}}} + {{{\gamma}_k}_j {{\beta}^k}_{,i}}} + {{{\gamma}_i}_k {{\beta}^k}_{,j}}}$
${{a}_k}_{,t} = {{{{\beta}^i}_{,k} {a}_i} + {{{{\beta}^i {{a}_i}_{,k}} - {{a}_k \alpha \cdot f {{K}^i}_i}} - {{{\alpha}^{2}} f' \cdot {a}_k {{K}^i}_i}} + {{2 \alpha \cdot f {{\gamma}^i}^b {{{d}_k}_b}_a {{\gamma}^a}^j {{K}_i}_j} - {\alpha \cdot f {{\gamma}^i}^j {{{K}_i}_j}_{,k}}}}$
${{{{d}_k}_i}_j}_{,t} = {{\frac12}{({-{({{{{{{{{2 {{K}_i}_j {a}_k \alpha} - {2 {{\beta}^l}_{,i} {{{d}_k}_l}_j}} - {2 {{\beta}^l}_{,j} {{{d}_k}_i}_l}} - {2 {{\beta}^l}_{,k} {{{d}_l}_i}_j}} - {2 {\beta}^l {{{{d}_l}_i}_j}_{,k}}} - {{{\gamma}_i}_l {{{\beta}^l}_{,j}}_{,k}}} - {{{\gamma}_l}_j {{{\beta}^l}_{,i}}_{,k}}} + {2 \alpha \cdot {{{K}_i}_j}_{,k}}})}})}}$
${{{K}_i}_j}_{,t} = {{{{{{{{{{{-{\frac{1}{2}}} \alpha} {{a}_i}_{,j}} - {{{\frac{1}{2}} \alpha} {{a}_j}_{,i}}} + {\alpha \cdot {({{{{\frac{1}{2}} {{\gamma}^p}^q} {({{{{{d}_i}_p}_q}_{,j} + {{{{d}_j}_p}_q}_{,i}})}} - {{{\frac{1}{2}} {{\gamma}^m}^r} {({{{{{d}_m}_i}_j}_{,r} + {{{{d}_m}_j}_i}_{,r}})}}})}}} - {\alpha \cdot {{V}_j}_{,i}}} - {\alpha \cdot {{V}_i}_{,j}}} + {\alpha \cdot {({{{{{{{{{{-{a}_i} {a}_j} + {{({{{{{d}_j}_i}^k + {{{d}_i}_j}^k} - {{{d}^k}_i}_j})} {({{{a}_k + {V}_k} - {{{d}^l}_l}_k})}}} + {{2 {({{{{d}^k}^l}_j - {{{d}^l}^k}_j})}} {{{d}_k}_l}_i}} + {{2 {{{d}_i}^k}^l} {{{d}_k}_l}_j}} + {{2 {{{d}_j}^k}^l} {{{d}_k}_l}_i}} - {{3 {{{d}_i}^k}^l} {{{d}_j}_k}_l}} + {{{K}^k}_k {{K}_i}_j}} - {{{K}_i}^k {{K}_k}_j}} - {{{K}_j}^k {{K}_k}_i}})}}} + {{{{K}_i}_j}_{,k} {\beta}^k}} + {{{K}_k}_j {{\beta}^k}_{,i}}} + {{{K}_i}_k {{\beta}^k}_{,j}}}$
${{V}_k}_{,t} = 0$
neglecting shift
${\alpha}_{,t} = {{-1} {{\alpha}^{2}} f {{K}^i}_i}$
${{{\gamma}_i}_j}_{,t} = {{-2} \alpha \cdot {{K}_i}_j}$
${{a}_k}_{,t} = {{-{\alpha \cdot {a}_k f {{K}^i}_i}} + {-{{{\alpha}^{2}} f' \cdot {a}_k {{K}^i}_i}} + {2 \alpha \cdot f {{\gamma}^i}^b {{{d}_k}_b}_a {{\gamma}^a}^j {{K}_i}_j} + {-{\alpha \cdot f {{\gamma}^i}^j {{{K}_i}_j}_{,k}}}}$
${{{{d}_k}_i}_j}_{,t} = {{-{\alpha \cdot {{{K}_i}_j}_{,k}}} + {-{\alpha \cdot {{K}_i}_j {a}_k}}}$
${{{K}_i}_j}_{,t} = {{\alpha \cdot {{K}^k}_k {{K}_i}_j} + {-{\alpha \cdot {{K}_i}^k {{K}_k}_j}} + {-{\alpha \cdot {{K}_j}^k {{K}_k}_i}} + {-{\alpha \cdot {{V}_i}_{,j}}} + {-{\alpha \cdot {{V}_j}_{,i}}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_i}_j}_{,r}}})}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_j}_i}_{,r}}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_i}_p}_q}_{,j}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_j}_p}_q}_{,i}})}} + {{\frac12}{({-{\alpha \cdot {{a}_i}_{,j}}})}} + {-{\alpha \cdot {a}_i {a}_j}} + {{\frac12}{({-{\alpha \cdot {{a}_j}_{,i}}})}} + {2 \alpha \cdot {{{d}^k}^l}_j {{{d}_k}_l}_i} + {-{\alpha \cdot {{{d}^k}_i}_j {V}_k}} + {-{\alpha \cdot {{{d}^k}_i}_j {a}_k}} + {\alpha \cdot {{{d}^k}_i}_j {{{d}^l}_l}_k} + {-{2 \alpha \cdot {{{d}^l}^k}_j {{{d}_k}_l}_i}} + {-{3 \alpha \cdot {{{d}_i}^k}^l {{{d}_j}_k}_l}} + {2 \alpha \cdot {{{d}_i}^k}^l {{{d}_k}_l}_j} + {\alpha \cdot {{{d}_i}_j}^k {V}_k} + {\alpha \cdot {{{d}_i}_j}^k {a}_k} + {-{\alpha \cdot {{{d}_i}_j}^k {{{d}^l}_l}_k}} + {2 \alpha \cdot {{{d}_j}^k}^l {{{d}_k}_l}_i} + {\alpha \cdot {{{d}_j}_i}^k {V}_k} + {\alpha \cdot {{{d}_j}_i}^k {a}_k} + {-{\alpha \cdot {{{d}_j}_i}^k {{{d}^l}_l}_k}}}$
${{V}_k}_{,t} = 0$
neglecting source terms
${\alpha}_{,t} = 0$
${{{\gamma}_i}_j}_{,t} = 0$
${{a}_k}_{,t} = {{-1} \alpha \cdot f {{\gamma}^i}^j {{{K}_i}_j}_{,k}}$
${{{{d}_k}_i}_j}_{,t} = {{-1} \alpha \cdot {{{K}_i}_j}_{,k}}$
${{{K}_i}_j}_{,t} = {{-{\alpha \cdot {{V}_i}_{,j}}} + {-{\alpha \cdot {{V}_j}_{,i}}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_i}_j}_{,r}}})}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_j}_i}_{,r}}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_i}_p}_q}_{,j}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_j}_p}_q}_{,i}})}} + {{\frac12}{({-{\alpha \cdot {{a}_i}_{,j}}})}} + {{\frac12}{({-{\alpha \cdot {{a}_j}_{,i}}})}}}$
${{V}_k}_{,t} = 0$
...and those source terms are...
${\alpha}_{,t}$$ + \dots = $$-{{{\alpha}^{2}} f {{K}^i}_i}$
${{{\gamma}_i}_j}_{,t}$$ + \dots = $$-{2 \alpha \cdot {{K}_i}_j}$
${{a}_k}_{,t}$$ + \dots = $$-{\alpha \cdot {({{{a}_k f {{K}^i}_i} + {{\alpha \cdot f' \cdot {a}_k {{K}^i}_i} - {2 f {{\gamma}^i}^b {{{d}_k}_b}_a {{\gamma}^a}^j {{K}_i}_j}}})}}$
${{{{d}_k}_i}_j}_{,t}$$ + \dots = $$-{\alpha \cdot {{K}_i}_j {a}_k}$
${{{K}_i}_j}_{,t}$$ + \dots = $$\alpha \cdot {({{{{{{{K}^k}_k {{K}_i}_j} - {{{K}_i}^k {{K}_k}_j}} - {{{K}_j}^k {{K}_k}_i}} - {{a}_i {a}_j}} + {{{2 {{{d}^k}^l}_j {{{d}_k}_l}_i} - {{{{d}^k}_i}_j {V}_k}} - {{{{d}^k}_i}_j {a}_k}} + {{{{{{d}^k}_i}_j {{{d}^l}_l}_k} - {2 {{{d}^l}^k}_j {{{d}_k}_l}_i}} - {3 {{{d}_i}^k}^l {{{d}_j}_k}_l}} + {2 {{{d}_i}^k}^l {{{d}_k}_l}_j} + {{{{d}_i}_j}^k {V}_k} + {{{{{d}_i}_j}^k {a}_k} - {{{{d}_i}_j}^k {{{d}^l}_l}_k}} + {2 {{{d}_j}^k}^l {{{d}_k}_l}_i} + {{{{d}_j}_i}^k {V}_k} + {{{{{d}_j}_i}^k {a}_k} - {{{{d}_j}_i}^k {{{d}^l}_l}_k}}})}$
${{V}_k}_{,t}$$ + \dots = $$0$
separating x from other dimensions:
${{{\gamma}_x}_x}_{,t} = 0$
${{{\gamma}_i}_x}_{,t} = 0$
${{{\gamma}_x}_j}_{,t} = 0$
${{{\gamma}_i}_j}_{,t} = 0$
${{a}_x}_{,t} = {{-1} \alpha \cdot f {{\gamma}^i}^j {{{K}_i}_j}_{,x}}$
${{a}_k}_{,t} = {{-1} \alpha \cdot f {{\gamma}^i}^j {{{K}_i}_j}_{,k}}$
${{{{d}_x}_x}_x}_{,t} = {{-1} \alpha \cdot {{{K}_x}_x}_{,x}}$
${{{{d}_k}_x}_x}_{,t} = {{-1} \alpha \cdot {{{K}_x}_x}_{,k}}$
${{{{d}_x}_i}_x}_{,t} = {{-1} \alpha \cdot {{{K}_i}_x}_{,x}}$
${{{{d}_k}_i}_x}_{,t} = {{-1} \alpha \cdot {{{K}_i}_x}_{,k}}$
${{{{d}_x}_x}_j}_{,t} = {{-1} \alpha \cdot {{{K}_x}_j}_{,x}}$
${{{{d}_k}_x}_j}_{,t} = {{-1} \alpha \cdot {{{K}_x}_j}_{,k}}$
${{{{d}_x}_i}_j}_{,t} = {{-1} \alpha \cdot {{{K}_i}_j}_{,x}}$
${{{{d}_k}_i}_j}_{,t} = {{-1} \alpha \cdot {{{K}_i}_j}_{,k}}$
${{{K}_x}_x}_{,t} = {{-{\alpha \cdot {{V}_x}_{,x}}} + {-{\alpha \cdot {{V}_x}_{,x}}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_x}_x}_{,r}}})}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_x}_x}_{,r}}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_x}_p}_q}_{,x}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_x}_p}_q}_{,x}})}} + {{\frac12}{({-{\alpha \cdot {{a}_x}_{,x}}})}} + {{\frac12}{({-{\alpha \cdot {{a}_x}_{,x}}})}}}$
${{{K}_i}_x}_{,t} = {{-{\alpha \cdot {{V}_i}_{,x}}} + {-{\alpha \cdot {{V}_x}_{,i}}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_i}_x}_{,r}}})}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_x}_i}_{,r}}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_i}_p}_q}_{,x}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_x}_p}_q}_{,i}})}} + {{\frac12}{({-{\alpha \cdot {{a}_i}_{,x}}})}} + {{\frac12}{({-{\alpha \cdot {{a}_x}_{,i}}})}}}$
${{{K}_x}_j}_{,t} = {{-{\alpha \cdot {{V}_x}_{,j}}} + {-{\alpha \cdot {{V}_j}_{,x}}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_x}_j}_{,r}}})}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_j}_x}_{,r}}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_x}_p}_q}_{,j}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_j}_p}_q}_{,x}})}} + {{\frac12}{({-{\alpha \cdot {{a}_x}_{,j}}})}} + {{\frac12}{({-{\alpha \cdot {{a}_j}_{,x}}})}}}$
${{{K}_i}_j}_{,t} = {{-{\alpha \cdot {{V}_i}_{,j}}} + {-{\alpha \cdot {{V}_j}_{,i}}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_i}_j}_{,r}}})}} + {{\frac12}{({-{\alpha \cdot {{\gamma}^m}^r {{{{d}_m}_j}_i}_{,r}}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_i}_p}_q}_{,j}})}} + {{\frac12}{({\alpha \cdot {{\gamma}^p}^q {{{{d}_j}_p}_q}_{,i}})}} + {{\frac12}{({-{\alpha \cdot {{a}_i}_{,j}}})}} + {{\frac12}{({-{\alpha \cdot {{a}_j}_{,i}}})}}}$
${{V}_x}_{,t} = 0$
${{V}_k}_{,t} = 0$
spelled out
${\partial_{{t}}({a_x})} = {{-{\alpha \cdot f {\gamma^{xx}} \cdot {\partial_{{x}}({K_{xx}})}}} + {-{2 \alpha \cdot f {\gamma^{xy}} \cdot {\partial_{{x}}({K_{xy}})}}} + {-{2 \alpha \cdot f {\gamma^{xz}} \cdot {\partial_{{x}}({K_{xz}})}}} + {-{\alpha \cdot f {\gamma^{yy}} \cdot {\partial_{{x}}({K_{yy}})}}} + {-{2 \alpha \cdot f {\gamma^{yz}} \cdot {\partial_{{x}}({K_{yz}})}}} + {-{\alpha \cdot f {\gamma^{zz}} \cdot {\partial_{{x}}({K_{zz}})}}}}$
${\partial_{{t}}({a_y})} = 0$
${\partial_{{t}}({a_z})} = 0$
${\partial_{{t}}({d_{xxx}})} = {{-1} \alpha \cdot {\partial_{{x}}({K_{xx}})}}$
${\partial_{{t}}({d_{xxy}})} = {{-1} \alpha \cdot {\partial_{{x}}({K_{xy}})}}$
${\partial_{{t}}({d_{xxz}})} = {{-1} \alpha \cdot {\partial_{{x}}({K_{xz}})}}$
${\partial_{{t}}({d_{xyy}})} = {{-1} \alpha \cdot {\partial_{{x}}({K_{yy}})}}$
${\partial_{{t}}({d_{xyz}})} = {{-1} \alpha \cdot {\partial_{{x}}({K_{yz}})}}$
${\partial_{{t}}({d_{xzz}})} = {{-1} \alpha \cdot {\partial_{{x}}({K_{zz}})}}$
${\partial_{{t}}({d_{yxx}})} = 0$
${\partial_{{t}}({d_{yxy}})} = 0$
${\partial_{{t}}({d_{yxz}})} = 0$
${\partial_{{t}}({d_{yyy}})} = 0$
${\partial_{{t}}({d_{yyz}})} = 0$
${\partial_{{t}}({d_{yzz}})} = 0$
${\partial_{{t}}({d_{zxx}})} = 0$
${\partial_{{t}}({d_{zxy}})} = 0$
${\partial_{{t}}({d_{zxz}})} = 0$
${\partial_{{t}}({d_{zyy}})} = 0$
${\partial_{{t}}({d_{zyz}})} = 0$
${\partial_{{t}}({d_{zzz}})} = 0$
${\partial_{{t}}({K_{xx}})} = {{-{2 \alpha \cdot {\partial_{{x}}({V_x})}}} + {-{\alpha \cdot {\partial_{{x}}({a_x})}}} + {2 \alpha \cdot {\gamma^{xy}} \cdot {\partial_{{x}}({d_{xxy}})}} + {-{\alpha \cdot {\gamma^{xy}} \cdot {\partial_{{x}}({d_{yxx}})}}} + {2 \alpha \cdot {\gamma^{xz}} \cdot {\partial_{{x}}({d_{xxz}})}} + {-{\alpha \cdot {\gamma^{xz}} \cdot {\partial_{{x}}({d_{zxx}})}}} + {\alpha \cdot {\gamma^{yy}} \cdot {\partial_{{x}}({d_{xyy}})}} + {2 \alpha \cdot {\gamma^{yz}} \cdot {\partial_{{x}}({d_{xyz}})}} + {\alpha \cdot {\gamma^{zz}} \cdot {\partial_{{x}}({d_{xzz}})}}}$
${\partial_{{t}}({K_{xy}})} = {{-{\alpha \cdot {\partial_{{x}}({V_y})}}} + {{\frac12}{({-{\alpha \cdot {\partial_{{x}}({a_y})}}})}} + {-{\alpha \cdot {\gamma^{xx}} \cdot {\partial_{{x}}({d_{xxy}})}}} + {{\frac12}{({\alpha \cdot {\gamma^{xx}} \cdot {\partial_{{x}}({d_{yxx}})}})}} + {\alpha \cdot {\gamma^{xz}} \cdot {\partial_{{x}}({d_{yxz}})}} + {-{\alpha \cdot {\gamma^{xz}} \cdot {\partial_{{x}}({d_{zxy}})}}} + {{\frac12}{({\alpha \cdot {\gamma^{yy}} \cdot {\partial_{{x}}({d_{yyy}})}})}} + {\alpha \cdot {\gamma^{yz}} \cdot {\partial_{{x}}({d_{yyz}})}} + {{\frac12}{({\alpha \cdot {\gamma^{zz}} \cdot {\partial_{{x}}({d_{yzz}})}})}}}$
${\partial_{{t}}({K_{xz}})} = {{-{\alpha \cdot {\partial_{{x}}({V_z})}}} + {{\frac12}{({-{\alpha \cdot {\partial_{{x}}({a_z})}}})}} + {-{\alpha \cdot {\gamma^{xx}} \cdot {\partial_{{x}}({d_{xxz}})}}} + {{\frac12}{({\alpha \cdot {\gamma^{xx}} \cdot {\partial_{{x}}({d_{zxx}})}})}} + {-{\alpha \cdot {\gamma^{xy}} \cdot {\partial_{{x}}({d_{yxz}})}}} + {\alpha \cdot {\gamma^{xy}} \cdot {\partial_{{x}}({d_{zxy}})}} + {{\frac12}{({\alpha \cdot {\gamma^{yy}} \cdot {\partial_{{x}}({d_{zyy}})}})}} + {\alpha \cdot {\gamma^{yz}} \cdot {\partial_{{x}}({d_{zyz}})}} + {{\frac12}{({\alpha \cdot {\gamma^{zz}} \cdot {\partial_{{x}}({d_{zzz}})}})}}}$
${\partial_{{t}}({K_{yy}})} = {{-{\alpha \cdot {\gamma^{xx}} \cdot {\partial_{{x}}({d_{xyy}})}}} + {-{\alpha \cdot {\gamma^{xy}} \cdot {\partial_{{x}}({d_{yyy}})}}} + {-{\alpha \cdot {\gamma^{xz}} \cdot {\partial_{{x}}({d_{zyy}})}}}}$
${\partial_{{t}}({K_{yz}})} = {{-{\alpha \cdot {\gamma^{xx}} \cdot {\partial_{{x}}({d_{xyz}})}}} + {-{\alpha \cdot {\gamma^{xy}} \cdot {\partial_{{x}}({d_{yyz}})}}} + {-{\alpha \cdot {\gamma^{xz}} \cdot {\partial_{{x}}({d_{zyz}})}}}}$
${\partial_{{t}}({K_{zz}})} = {{-{\alpha \cdot {\gamma^{xx}} \cdot {\partial_{{x}}({d_{xzz}})}}} + {-{\alpha \cdot {\gamma^{xy}} \cdot {\partial_{{x}}({d_{yzz}})}}} + {-{\alpha \cdot {\gamma^{xz}} \cdot {\partial_{{x}}({d_{zzz}})}}}}$
${\partial_{{t}}({V_x})} = 0$
${\partial_{{t}}({V_y})} = 0$
${\partial_{{t}}({V_z})} = 0$
${{\left[\matrix{{\partial_{{t}}({a_x})}\\{\partial_{{t}}({a_y})}\\{\partial_{{t}}({a_z})}\\{\partial_{{t}}({d_{xxx}})}\\{\partial_{{t}}({d_{xxy}})}\\{\partial_{{t}}({d_{xxz}})}\\{\partial_{{t}}({d_{xyy}})}\\{\partial_{{t}}({d_{xyz}})}\\{\partial_{{t}}({d_{xzz}})}\\{\partial_{{t}}({d_{yxx}})}\\{\partial_{{t}}({d_{yxy}})}\\{\partial_{{t}}({d_{yxz}})}\\{\partial_{{t}}({d_{yyy}})}\\{\partial_{{t}}({d_{yyz}})}\\{\partial_{{t}}({d_{yzz}})}\\{\partial_{{t}}({d_{zxx}})}\\{\partial_{{t}}({d_{zxy}})}\\{\partial_{{t}}({d_{zxz}})}\\{\partial_{{t}}({d_{zyy}})}\\{\partial_{{t}}({d_{zyz}})}\\{\partial_{{t}}({d_{zzz}})}\\{\partial_{{t}}({K_{xx}})}\\{\partial_{{t}}({K_{xy}})}\\{\partial_{{t}}({K_{xz}})}\\{\partial_{{t}}({K_{yy}})}\\{\partial_{{t}}({K_{yz}})}\\{\partial_{{t}}({K_{zz}})}\\{\partial_{{t}}({V_x})}\\{\partial_{{t}}({V_y})}\\{\partial_{{t}}({V_z})}}\right]} + {{\left[\matrix{0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&{\alpha \cdot f {\gamma^{xx}}}&{2 \alpha \cdot f {\gamma^{xy}}}&{2 \alpha \cdot f {\gamma^{xz}}}&{\alpha \cdot f {\gamma^{yy}}}&{2 \alpha \cdot f {\gamma^{yz}}}&{\alpha \cdot f {\gamma^{zz}}}&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\alpha&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\alpha&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\alpha&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\alpha&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\alpha&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&\alpha&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\\alpha&0&0&0&{-{2 \alpha \cdot {\gamma^{xy}}}}&{-{2 \alpha \cdot {\gamma^{xz}}}}&{-{\alpha \cdot {\gamma^{yy}}}}&{-{2 \alpha \cdot {\gamma^{yz}}}}&{-{\alpha \cdot {\gamma^{zz}}}}&{\alpha \cdot {\gamma^{xy}}}&0&0&0&0&0&{\alpha \cdot {\gamma^{xz}}}&0&0&0&0&0&0&0&0&0&0&0&{2 \alpha}&0&0\\0&{{\frac12}{({\alpha})}}&0&0&{\alpha \cdot {\gamma^{xx}}}&0&0&0&0&{{\frac12}{({-{\alpha \cdot {\gamma^{xx}}}})}}&0&{-{\alpha \cdot {\gamma^{xz}}}}&{{\frac12}{({-{\alpha \cdot {\gamma^{yy}}}})}}&{-{\alpha \cdot {\gamma^{yz}}}}&{{\frac12}{({-{\alpha \cdot {\gamma^{zz}}}})}}&0&{\alpha \cdot {\gamma^{xz}}}&0&0&0&0&0&0&0&0&0&0&0&\alpha&0\\0&0&{{\frac12}{({\alpha})}}&0&0&{\alpha \cdot {\gamma^{xx}}}&0&0&0&0&0&{\alpha \cdot {\gamma^{xy}}}&0&0&0&{{\frac12}{({-{\alpha \cdot {\gamma^{xx}}}})}}&{-{\alpha \cdot {\gamma^{xy}}}}&0&{{\frac12}{({-{\alpha \cdot {\gamma^{yy}}}})}}&{-{\alpha \cdot {\gamma^{yz}}}}&{{\frac12}{({-{\alpha \cdot {\gamma^{zz}}}})}}&0&0&0&0&0&0&0&0&\alpha\\0&0&0&0&0&0&{\alpha \cdot {\gamma^{xx}}}&0&0&0&0&0&{\alpha \cdot {\gamma^{xy}}}&0&0&0&0&0&{\alpha \cdot {\gamma^{xz}}}&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&{\alpha \cdot {\gamma^{xx}}}&0&0&0&0&0&{\alpha \cdot {\gamma^{xy}}}&0&0&0&0&0&{\alpha \cdot {\gamma^{xz}}}&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&{\alpha \cdot {\gamma^{xx}}}&0&0&0&0&0&{\alpha \cdot {\gamma^{xy}}}&0&0&0&0&0&{\alpha \cdot {\gamma^{xz}}}&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0}\right]} {\left[\matrix{{\partial_{{x}}({a_x})}\\{\partial_{{x}}({a_y})}\\{\partial_{{x}}({a_z})}\\{\partial_{{x}}({d_{xxx}})}\\{\partial_{{x}}({d_{xxy}})}\\{\partial_{{x}}({d_{xxz}})}\\{\partial_{{x}}({d_{xyy}})}\\{\partial_{{x}}({d_{xyz}})}\\{\partial_{{x}}({d_{xzz}})}\\{\partial_{{x}}({d_{yxx}})}\\{\partial_{{x}}({d_{yxy}})}\\{\partial_{{x}}({d_{yxz}})}\\{\partial_{{x}}({d_{yyy}})}\\{\partial_{{x}}({d_{yyz}})}\\{\partial_{{x}}({d_{yzz}})}\\{\partial_{{x}}({d_{zxx}})}\\{\partial_{{x}}({d_{zxy}})}\\{\partial_{{x}}({d_{zxz}})}\\{\partial_{{x}}({d_{zyy}})}\\{\partial_{{x}}({d_{zyz}})}\\{\partial_{{x}}({d_{zzz}})}\\{\partial_{{x}}({K_{xx}})}\\{\partial_{{x}}({K_{xy}})}\\{\partial_{{x}}({K_{xz}})}\\{\partial_{{x}}({K_{yy}})}\\{\partial_{{x}}({K_{yz}})}\\{\partial_{{x}}({K_{zz}})}\\{\partial_{{x}}({V_x})}\\{\partial_{{x}}({V_y})}\\{\partial_{{x}}({V_z})}}\right]}}} = {\left[\matrix{0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0\\0}\right]}$
characteristic polynomial:
${{\lambda}^{{18}}} {({{{{\lambda}^{{12}}} - {{{\lambda}^{2}} {{\alpha}^{{10}}} {{{\gamma^{xx}}}^{5}}}} + {{{10} {{\lambda}^{8}} {{\alpha}^{4}} {{{\gamma^{xx}}}^{2}}} - {{10} {{\lambda}^{6}} {{\alpha}^{6}} {{{\gamma^{xx}}}^{3}}}} + {{5 {{\lambda}^{4}} {{\alpha}^{8}} {{{\gamma^{xx}}}^{4}}} - {5 {{\lambda}^{{10}}} {{\alpha}^{2}} {\gamma^{xx}}}} + {{{{{\alpha}^{{12}}} {{{\gamma^{xx}}}^{6}} f} - {{10} {{\lambda}^{6}} {{\alpha}^{6}} {{{\gamma^{xx}}}^{3}} f}} - {5 {{\lambda}^{2}} {{\alpha}^{{10}}} {{{\gamma^{xx}}}^{5}} f}} + {{10} {{\lambda}^{4}} {{\alpha}^{8}} {{{\gamma^{xx}}}^{4}} f} + {{5 {{\lambda}^{8}} {{\alpha}^{4}} {{{\gamma^{xx}}}^{2}} f} - {{{\lambda}^{{10}}} {{\alpha}^{2}} f {\gamma^{xx}}}}})}$