This is supposed to generalize perturbative metric math. Typical perturbative metric math is done using $g_{ab} = \eta_{ab} + h_{ab}$, where $\eta_{ab}$ must be $\pm diag\{-1, 1, 1, 1\}$ so that its derivative will be zero (or else a lot of other terms pop up) and $h_{ab}$ must be small. This is shit-tier because most the physicists who use this metric themselves only do math in non-Cartesian background metrics (like spherical, cylindrical, etc). This is also shit-tier because the metric is quadratic in nature ($ds = \int \sqrt{ g_{uv} \frac{\partial x^u}{\partial \lambda} \frac{\partial x^v}{\partial \lambda}} d\lambda$) and represents the inner product of basis vectors ($g_{uv} = e_u \cdot e_v$), such that, if you perturb the basis then you find $|g'_{ab}| = |g_{ab}|(1 + 2 |{\epsilon_a}^u| + |{\epsilon_a}^u|^2)$), and this $h_{ab}$ is only accounting for the $|{\epsilon_a}^u|^2$ term. If you use this model then you are essentially throwing away the linear error term and only looking exclusively at quadratic error.

With that disclaimer aside, I was asked once to extend gravitoelectrmagnetics to non-Cartesian backgrounds. GEM relies on perturbative metrics and de-Donder gauge. So let's see what a perturbed metric would look like with a non-Cartesian background metric.



${{ \hat{g}}_a}_b$ = first metric
${{ \tilde{g}}_a}_b$ = second metric

metric
${{{ g}_a}_b} = {{{{ \hat{g}}_a}_b} + {{{ \tilde{g}}_a}_b}}$

metric inverse
${{g_{12}}} = {{{{{ \hat{g}}^u}^v}} {{{{ \tilde{g}}_u}_v}}}$
${{{ g}^a}^b} = {{{{ \hat{g}}^a}^b} - {\frac{{{{{ \hat{g}}^a}^c}} {{{{ \tilde{g}}_c}_d}} {{{{ \hat{g}}^d}^b}}}{{1} + {{g_{12}}}}}}$

useful identity:
${{ \delta}_i}^j$ = 1 for i=j, 0 otherwise
${{{{ \delta}_i}^j}_{,k}} = {0}$
${{\left( {{{{ g}_l}_i}} {{{{ g}^i}^j}}\right)}_{,k}} = {0}$
${{{{{{ g}_l}_i}} {{{{{ g}^i}^j}_{,k}}}} + {{{{{ g}^i}^j}} {{{{{ g}_l}_i}_{,k}}}}} = {0}$
${{{{{{ g}^l}^m}} {{{{ g}^i}^j}} {{{{{ g}_l}_i}_{,k}}}} + {{{{{ g}^l}^m}} {{{{ g}_l}_i}} {{{{{ g}^i}^j}_{,k}}}}} = {0}$
${{{{ g}^i}^j}_{,k}} = {-{{{{{ g}^l}^i}} {{{{ g}^m}^j}} {{{{{ g}_l}_m}_{,k}}}}}$

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

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

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

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

${{{{ \Gamma}^a}_b}_c} = {{{{{ g}^a}^d}} {{{{{ \Gamma}_d}_b}_c}}}$
${{{{ \Gamma}^a}_b}_c} = {{{{{ g}^a}^d}} {{({{{{{ \hat{\Gamma}}_d}_b}_c} + {{{{ \tilde{\Gamma}}_d}_b}_c}})}}}$
${{{{ \Gamma}^a}_b}_c} = {{\frac{1}{2}}{({-{{{{{ g}^a}^d}} {{({{{{{{{ \hat{g}}_b}_c}_{,d}} - {{{{ \hat{g}}_d}_b}_{,c}}} - {{{{ \hat{g}}_d}_c}_{,b}}} + {{{{{{ \tilde{g}}_b}_c}_{,d}} - {{{{ \tilde{g}}_d}_b}_{,c}}} - {{{{ \tilde{g}}_d}_c}_{,b}}}})}}}})}}$

connection partial
${\left( {{{{ \Gamma}_a}_b}_c} = {{{{{ \hat{\Gamma}}_a}_b}_c} + {{{{ \tilde{\Gamma}}_a}_b}_c}}\right)}_{,d}$
${{{{{ \Gamma}_a}_b}_c}_{,d}} = {{{{{{ \hat{\Gamma}}_a}_b}_c}_{,d}} + {{{{{ \tilde{\Gamma}}_a}_b}_c}_{,d}}}$

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

${\left( {{{{ \Gamma}^a}_b}_c} = {{{{{ g}^a}^e}} {{{{{ \Gamma}_e}_b}_c}}}\right)}_{,d}$
${{{{{ \Gamma}^a}_b}_c}_{,d}} = {{{{{{{ \Gamma}_e}_b}_c}} {{{{{ g}^a}^e}_{,d}}}} + {{{{{ g}^a}^e}} {{{{{{ \Gamma}_e}_b}_c}_{,d}}}}}$
${{{{{ \Gamma}^a}_b}_c}_{,d}} = {{-{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \Gamma}_e}_b}_c}} {{{{{ g}_f}_g}_{,d}}}}} + {{{{{ g}^a}^e}} {{{{{{ \Gamma}_e}_b}_c}_{,d}}}}}$

${{{{{ \Gamma}^a}_b}_c}_{,d}} = {{{-{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{\Gamma}}_e}_b}_c}} {{{{{ g}_f}_g}_{,d}}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \tilde{\Gamma}}_e}_b}_c}} {{{{{ g}_f}_g}_{,d}}}}} + {{{{{ g}^a}^e}} {{{{{{ \hat{\Gamma}}_e}_b}_c}_{,d}}}} + {{{{{ g}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_c}_{,d}}}}}$
${{{{{ \Gamma}^a}_b}_c}_{,d}} = {{{{{-{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{\Gamma}}_e}_b}_c}} {{{{{ \hat{g}}_f}_g}_{,d}}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{\Gamma}}_e}_b}_c}} {{{{{ \tilde{g}}_f}_g}_{,d}}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_f}_g}_{,d}}} {{{{{ \tilde{\Gamma}}_e}_b}_c}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \tilde{\Gamma}}_e}_b}_c}} {{{{{ \tilde{g}}_f}_g}_{,d}}}}} + {{{{{ g}^a}^e}} {{{{{{ \hat{\Gamma}}_e}_b}_c}_{,d}}}} + {{{{{ g}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_c}_{,d}}}}}$
${{{{{ \Gamma}^a}_b}_c}_{,d}} = {{\frac{1}{2}}{({{{{2}} {{{{ g}^a}^e}} {{{{{{ \hat{\Gamma}}_e}_b}_c}_{,d}}}} + {{{{{2}} {{{{ g}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_c}_{,d}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \tilde{g}}_e}_b}_{,c}}} {{{{{ \tilde{g}}_f}_g}_{,d}}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \tilde{g}}_e}_c}_{,b}}} {{{{{ \tilde{g}}_f}_g}_{,d}}}}} + {{{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \tilde{g}}_b}_c}_{,e}}} {{{{{ \tilde{g}}_f}_g}_{,d}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_f}_g}_{,d}}} {{{{{ \tilde{g}}_e}_b}_{,c}}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_f}_g}_{,d}}} {{{{{ \tilde{g}}_e}_c}_{,b}}}}} + {{{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_f}_g}_{,d}}} {{{{{ \tilde{g}}_b}_c}_{,e}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_e}_b}_{,c}}} {{{{{ \tilde{g}}_f}_g}_{,d}}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_e}_c}_{,b}}} {{{{{ \tilde{g}}_f}_g}_{,d}}}}} + {{{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_b}_c}_{,e}}} {{{{{ \tilde{g}}_f}_g}_{,d}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_e}_b}_{,c}}} {{{{{ \hat{g}}_f}_g}_{,d}}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_e}_c}_{,b}}} {{{{{ \hat{g}}_f}_g}_{,d}}}}} + {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{g}}_b}_c}_{,e}}} {{{{{ \hat{g}}_f}_g}_{,d}}}}})}}$

Riemann curvature
${{{{{ R}^a}_b}_c}_d} = {{{{{{{{ \Gamma}^a}_b}_d}_{,c}} - {{{{{ \Gamma}^a}_b}_c}_{,d}}} + {{{{{{ \Gamma}^a}_e}_c}} {{{{{ \Gamma}^e}_b}_d}}}} - {{{{{{ \Gamma}^a}_e}_d}} {{{{{ \Gamma}^e}_b}_c}}}}$
${{{{{ R}^a}_b}_c}_d} = {{-{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \Gamma}_e}_b}_d}} {{{{{ g}_f}_g}_{,c}}}}} + {{{{{ g}^a}^e}} {{{{{{ \Gamma}_e}_b}_d}_{,c}}}} + {{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \Gamma}_e}_b}_c}} {{{{{ g}_f}_g}_{,d}}}} - {{{{{ g}^a}^e}} {{{{{{ \Gamma}_e}_b}_c}_{,d}}}}} + {{{{{{{ \Gamma}^a}_e}_c}} {{{{{ \Gamma}^e}_b}_d}}} - {{{{{{ \Gamma}^a}_e}_d}} {{{{{ \Gamma}^e}_b}_c}}}}}$
${{{{{ R}^a}_b}_c}_d} = {{{{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_c}_{,g}}} {{{{{ g}_d}_e}_{,f}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_c}_{,g}}} {{{{{ g}_d}_f}_{,e}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_c}_{,g}}} {{{{{ g}_e}_f}_{,d}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_d}_{,g}}} {{{{{ g}_c}_e}_{,f}}}} + {{{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_d}_{,g}}} {{{{{ g}_c}_f}_{,e}}}} + {{{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_d}_{,g}}} {{{{{ g}_e}_f}_{,c}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_g}_{,c}}} {{{{{ g}_d}_e}_{,f}}}} + {{{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_g}_{,c}}} {{{{{ g}_d}_f}_{,e}}}} + {{{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_g}_{,c}}} {{{{{ g}_e}_f}_{,d}}}} + {{{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_g}_{,d}}} {{{{{ g}_c}_e}_{,f}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_g}_{,d}}} {{{{{ g}_c}_f}_{,e}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_b}_g}_{,d}}} {{{{{ g}_e}_f}_{,c}}}} + {{{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_c}_e}_{,f}}} {{{{{ g}_d}_g}_{,b}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_c}_f}_{,e}}} {{{{{ g}_d}_g}_{,b}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_c}_g}_{,b}}} {{{{{ g}_d}_e}_{,f}}}} + {{{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_c}_g}_{,b}}} {{{{{ g}_d}_f}_{,e}}}} + {{{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_c}_g}_{,b}}} {{{{{ g}_e}_f}_{,d}}}} + {{{-1}} \cdot {{\frac{1}{4}}} {{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ g}_d}_g}_{,b}}} {{{{{ g}_e}_f}_{,c}}}} + {{{\frac{1}{2}}} {{{{ g}^a}^e}} {{{{{{ g}_b}_c}_{,d}}_{,e}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ g}^a}^e}} {{{{{{ g}_b}_d}_{,c}}_{,e}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ g}^a}^e}} {{{{{{ g}_c}_e}_{,b}}_{,d}}}} + {{{\frac{1}{2}}} {{{{ g}^a}^e}} {{{{{{ g}_d}_e}_{,b}}_{,c}}}}}$

Riemann curvature if the ${{{{ g}_a}_b}_{,c}}^{2}$ terms vanish:
${{{{{ R}^a}_b}_c}_d} = {{\frac{1}{2}}{({{{{{ g}^a}^e}} {{({{{{{{{{ g}_b}_c}_{,d}}_{,e}} - {{{{{ g}_b}_d}_{,c}}_{,e}}} - {{{{{ g}_c}_e}_{,b}}_{,d}}} + {{{{{ g}_d}_e}_{,b}}_{,c}}})}}})}}$

Riemann curvature of a sum of two metrics
${{{{{ R}^a}_b}_c}_d} = {{{-{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{\Gamma}}_e}_b}_d}} {{{{{ g}_f}_g}_{,c}}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \tilde{\Gamma}}_e}_b}_d}} {{{{{ g}_f}_g}_{,c}}}}} + {{{{{ g}^a}^e}} {{{{{{ \hat{\Gamma}}_e}_b}_d}_{,c}}}} + {{{{{ g}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_d}_{,c}}}} + {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{\Gamma}}_e}_b}_c}} {{{{{ g}_f}_g}_{,d}}}} + {{{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \tilde{\Gamma}}_e}_b}_c}} {{{{{ g}_f}_g}_{,d}}}} - {{{{{ g}^a}^e}} {{{{{{ \hat{\Gamma}}_e}_b}_c}_{,d}}}}} - {{{{{ g}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_c}_{,d}}}}} + {{{{{{{ \Gamma}^a}_e}_c}} {{{{{ \Gamma}^e}_b}_d}}} - {{{{{{ \Gamma}^a}_e}_d}} {{{{{ \Gamma}^e}_b}_c}}}}}$
${{{{{ R}^a}_b}_c}_d} = {-{({{{{{{{ g}^a}^e}} {{{{{{ \hat{\Gamma}}_e}_b}_c}_{,d}}}} - {{{{{ g}^a}^e}} {{{{{{ \hat{\Gamma}}_e}_b}_d}_{,c}}}}} + {{{{{{{{ g}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_c}_{,d}}}} - {{{{{ g}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_d}_{,c}}}}} - {{{{{ g}^a}^f}} {{{{ g}^e}^g}} {{{{{ \hat{\Gamma}}_f}_e}_c}} {{{{{ \hat{\Gamma}}_g}_b}_d}}}} - {{{{{ g}^a}^f}} {{{{ g}^e}^g}} {{{{{ \hat{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_g}_b}_d}}}} + {{{{{ g}^a}^f}} {{{{ g}^e}^g}} {{{{{ \hat{\Gamma}}_f}_e}_d}} {{{{{ \hat{\Gamma}}_g}_b}_c}}} + {{{{{ g}^a}^f}} {{{{ g}^e}^g}} {{{{{ \hat{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_g}_b}_c}}} + {{{{{{{ g}^a}^f}} {{{{ g}^e}^g}} {{{{{ \hat{\Gamma}}_g}_b}_c}} {{{{{ \tilde{\Gamma}}_f}_e}_d}}} - {{{{{ g}^a}^f}} {{{{ g}^e}^g}} {{{{{ \hat{\Gamma}}_g}_b}_d}} {{{{{ \tilde{\Gamma}}_f}_e}_c}}}} - {{{{{ g}^a}^f}} {{{{ g}^e}^g}} {{{{{ \tilde{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_g}_b}_d}}}} + {{{{{{ g}^a}^f}} {{{{ g}^e}^g}} {{{{{ \tilde{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_g}_b}_c}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{\Gamma}}_e}_b}_c}} {{{{{ g}_f}_g}_{,d}}}}} + {{{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \hat{\Gamma}}_e}_b}_d}} {{{{{ g}_f}_g}_{,c}}}} - {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \tilde{\Gamma}}_e}_b}_c}} {{{{{ g}_f}_g}_{,d}}}}} + {{{{{ g}^f}^a}} {{{{ g}^g}^e}} {{{{{ \tilde{\Gamma}}_e}_b}_d}} {{{{{ g}_f}_g}_{,c}}}}})}}$
${{{{{ R}^a}_b}_c}_d} = {{\frac{1}{4}}{({{{{{ g}^a}^e}} {{({{{{{{2}} {{{{{{ \hat{g}}_b}_c}_{,d}}_{,e}}}} - {{{2}} {{{{{{ \hat{g}}_b}_d}_{,c}}_{,e}}}}} - {{{2}} {{{{{{ \hat{g}}_c}_e}_{,b}}_{,d}}}}} + {{{2}} {{{{{{ \hat{g}}_d}_e}_{,b}}_{,c}}}} + {{{{{2}} {{{{{{ \tilde{g}}_b}_c}_{,d}}_{,e}}}} - {{{2}} {{{{{{ \tilde{g}}_b}_d}_{,c}}_{,e}}}}} - {{{2}} {{{{{{ \tilde{g}}_c}_e}_{,b}}_{,d}}}}} + {{{2}} {{{{{{ \tilde{g}}_d}_e}_{,b}}_{,c}}}} + {{{{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_c}_{,g}}} {{{{{ \hat{g}}_d}_e}_{,f}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_c}_{,g}}} {{{{{ \hat{g}}_d}_f}_{,e}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_c}_{,g}}} {{{{{ \hat{g}}_e}_f}_{,d}}}}} + {{{{{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_c}_{,g}}} {{{{{ \tilde{g}}_d}_e}_{,f}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_c}_{,g}}} {{{{{ \tilde{g}}_d}_f}_{,e}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_c}_{,g}}} {{{{{ \tilde{g}}_e}_f}_{,d}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_d}_{,g}}} {{{{{ \hat{g}}_c}_e}_{,f}}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_d}_{,g}}} {{{{{ \hat{g}}_c}_f}_{,e}}}} + {{{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_d}_{,g}}} {{{{{ \hat{g}}_e}_f}_{,c}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_d}_{,g}}} {{{{{ \tilde{g}}_c}_e}_{,f}}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_d}_{,g}}} {{{{{ \tilde{g}}_c}_f}_{,e}}}} + {{{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_d}_{,g}}} {{{{{ \tilde{g}}_e}_f}_{,c}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,c}}} {{{{{ \hat{g}}_d}_e}_{,f}}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,c}}} {{{{{ \hat{g}}_d}_f}_{,e}}}} + {{{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,c}}} {{{{{ \hat{g}}_e}_f}_{,d}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,c}}} {{{{{ \tilde{g}}_d}_e}_{,f}}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,c}}} {{{{{ \tilde{g}}_d}_f}_{,e}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,c}}} {{{{{ \tilde{g}}_e}_f}_{,d}}}} + {{{{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,d}}} {{{{{ \hat{g}}_c}_e}_{,f}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,d}}} {{{{{ \hat{g}}_c}_f}_{,e}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,d}}} {{{{{ \hat{g}}_e}_f}_{,c}}}}} + {{{{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,d}}} {{{{{ \tilde{g}}_c}_e}_{,f}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,d}}} {{{{{ \tilde{g}}_c}_f}_{,e}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,d}}} {{{{{ \tilde{g}}_e}_f}_{,c}}}}} + {{{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_e}_{,f}}} {{{{{ \hat{g}}_d}_g}_{,b}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_e}_{,f}}} {{{{{ \tilde{g}}_b}_d}_{,g}}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_e}_{,f}}} {{{{{ \tilde{g}}_b}_g}_{,d}}}} + {{{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_e}_{,f}}} {{{{{ \tilde{g}}_d}_g}_{,b}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_f}_{,e}}} {{{{{ \hat{g}}_d}_g}_{,b}}}}} + {{{{{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_f}_{,e}}} {{{{{ \tilde{g}}_b}_d}_{,g}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_f}_{,e}}} {{{{{ \tilde{g}}_b}_g}_{,d}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_f}_{,e}}} {{{{{ \tilde{g}}_d}_g}_{,b}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_g}_{,b}}} {{{{{ \hat{g}}_d}_e}_{,f}}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_g}_{,b}}} {{{{{ \hat{g}}_d}_f}_{,e}}}} + {{{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_g}_{,b}}} {{{{{ \hat{g}}_e}_f}_{,d}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_g}_{,b}}} {{{{{ \tilde{g}}_d}_e}_{,f}}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_g}_{,b}}} {{{{{ \tilde{g}}_d}_f}_{,e}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_g}_{,b}}} {{{{{ \tilde{g}}_e}_f}_{,d}}}} + {{{{{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_e}_{,f}}} {{{{{ \tilde{g}}_b}_c}_{,g}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_e}_{,f}}} {{{{{ \tilde{g}}_b}_g}_{,c}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_e}_{,f}}} {{{{{ \tilde{g}}_c}_g}_{,b}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_f}_{,e}}} {{{{{ \tilde{g}}_b}_c}_{,g}}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_f}_{,e}}} {{{{{ \tilde{g}}_b}_g}_{,c}}}} + {{{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_f}_{,e}}} {{{{{ \tilde{g}}_c}_g}_{,b}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_g}_{,b}}} {{{{{ \hat{g}}_e}_f}_{,c}}}}} + {{{{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_g}_{,b}}} {{{{{ \tilde{g}}_c}_e}_{,f}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_g}_{,b}}} {{{{{ \tilde{g}}_c}_f}_{,e}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_g}_{,b}}} {{{{{ \tilde{g}}_e}_f}_{,c}}}}} + {{{{{{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,c}}} {{{{{ \tilde{g}}_b}_d}_{,g}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,c}}} {{{{{ \tilde{g}}_b}_g}_{,d}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,c}}} {{{{{ \tilde{g}}_d}_g}_{,b}}}}} - {{{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,d}}} {{{{{ \tilde{g}}_b}_c}_{,g}}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,d}}} {{{{{ \tilde{g}}_b}_g}_{,c}}}} + {{{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,d}}} {{{{{ \tilde{g}}_c}_g}_{,b}}}} + {{{{{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_c}_{,g}}} {{{{{ \tilde{g}}_d}_e}_{,f}}}} - {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_c}_{,g}}} {{{{{ \tilde{g}}_d}_f}_{,e}}}}} - {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_c}_{,g}}} {{{{{ \tilde{g}}_e}_f}_{,d}}}}} - {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_d}_{,g}}} {{{{{ \tilde{g}}_c}_e}_{,f}}}}} + {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_d}_{,g}}} {{{{{ \tilde{g}}_c}_f}_{,e}}}} + {{{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_d}_{,g}}} {{{{{ \tilde{g}}_e}_f}_{,c}}}} - {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_g}_{,c}}} {{{{{ \tilde{g}}_d}_e}_{,f}}}}} + {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_g}_{,c}}} {{{{{ \tilde{g}}_d}_f}_{,e}}}} + {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_g}_{,c}}} {{{{{ \tilde{g}}_e}_f}_{,d}}}} + {{{{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_g}_{,d}}} {{{{{ \tilde{g}}_c}_e}_{,f}}}} - {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_g}_{,d}}} {{{{{ \tilde{g}}_c}_f}_{,e}}}}} - {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_b}_g}_{,d}}} {{{{{ \tilde{g}}_e}_f}_{,c}}}}} + {{{{{{{ g}^f}^g}} {{{{{ \tilde{g}}_c}_e}_{,f}}} {{{{{ \tilde{g}}_d}_g}_{,b}}}} - {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_c}_f}_{,e}}} {{{{{ \tilde{g}}_d}_g}_{,b}}}}} - {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_c}_g}_{,b}}} {{{{{ \tilde{g}}_d}_e}_{,f}}}}} + {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_c}_g}_{,b}}} {{{{{ \tilde{g}}_d}_f}_{,e}}}} + {{{{{{ g}^f}^g}} {{{{{ \tilde{g}}_c}_g}_{,b}}} {{{{{ \tilde{g}}_e}_f}_{,d}}}} - {{{{{ g}^f}^g}} {{{{{ \tilde{g}}_d}_g}_{,b}}} {{{{{ \tilde{g}}_e}_f}_{,c}}}}}})}}})}}$

What if we assume ${{{{ g}^a}^b}\approx{{{ \hat{g}}^a}^b}}\approx{{{ \tilde{g}}^a}^b}$

${{{{{ R}^a}_b}_c}_d} = {{\frac{1}{4}}{({{{{4}} {{{{{{ \tilde{R}}^a}_b}_c}_d}}} + {{{4}} {{{{{{ \hat{R}}^a}_b}_c}_d}}} + {{{{{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_c}_{,g}}} {{{{{ \tilde{g}}_d}_e}_{,f}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_c}_{,g}}} {{{{{ \tilde{g}}_d}_f}_{,e}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_c}_{,g}}} {{{{{ \tilde{g}}_e}_f}_{,d}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_d}_{,g}}} {{{{{ \tilde{g}}_c}_e}_{,f}}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_d}_{,g}}} {{{{{ \tilde{g}}_c}_f}_{,e}}}} + {{{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_d}_{,g}}} {{{{{ \tilde{g}}_e}_f}_{,c}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,c}}} {{{{{ \tilde{g}}_d}_e}_{,f}}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,c}}} {{{{{ \tilde{g}}_d}_f}_{,e}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,c}}} {{{{{ \tilde{g}}_e}_f}_{,d}}}} + {{{{{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,d}}} {{{{{ \tilde{g}}_c}_e}_{,f}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,d}}} {{{{{ \tilde{g}}_c}_f}_{,e}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_b}_g}_{,d}}} {{{{{ \tilde{g}}_e}_f}_{,c}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_e}_{,f}}} {{{{{ \tilde{g}}_b}_d}_{,g}}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_e}_{,f}}} {{{{{ \tilde{g}}_b}_g}_{,d}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_e}_{,f}}} {{{{{ \tilde{g}}_d}_g}_{,b}}}} + {{{{{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_f}_{,e}}} {{{{{ \tilde{g}}_b}_d}_{,g}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_f}_{,e}}} {{{{{ \tilde{g}}_b}_g}_{,d}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_f}_{,e}}} {{{{{ \tilde{g}}_d}_g}_{,b}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_g}_{,b}}} {{{{{ \tilde{g}}_d}_e}_{,f}}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_g}_{,b}}} {{{{{ \tilde{g}}_d}_f}_{,e}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_c}_g}_{,b}}} {{{{{ \tilde{g}}_e}_f}_{,d}}}} + {{{{{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_e}_{,f}}} {{{{{ \tilde{g}}_b}_c}_{,g}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_e}_{,f}}} {{{{{ \tilde{g}}_b}_g}_{,c}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_e}_{,f}}} {{{{{ \tilde{g}}_c}_g}_{,b}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_f}_{,e}}} {{{{{ \tilde{g}}_b}_c}_{,g}}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_f}_{,e}}} {{{{{ \tilde{g}}_b}_g}_{,c}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_f}_{,e}}} {{{{{ \tilde{g}}_c}_g}_{,b}}}} + {{{{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_g}_{,b}}} {{{{{ \tilde{g}}_c}_e}_{,f}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_g}_{,b}}} {{{{{ \tilde{g}}_c}_f}_{,e}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_d}_g}_{,b}}} {{{{{ \tilde{g}}_e}_f}_{,c}}}}} + {{{{{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,c}}} {{{{{ \tilde{g}}_b}_d}_{,g}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,c}}} {{{{{ \tilde{g}}_b}_g}_{,d}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,c}}} {{{{{ \tilde{g}}_d}_g}_{,b}}}}} - {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,d}}} {{{{{ \tilde{g}}_b}_c}_{,g}}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,d}}} {{{{{ \tilde{g}}_b}_g}_{,c}}}} + {{{{{ g}^a}^e}} {{{{ g}^f}^g}} {{{{{ \hat{g}}_e}_f}_{,d}}} {{{{{ \tilde{g}}_c}_g}_{,b}}}}})}}$

What if instead we assume ${{{ g}^a}^b}\approx{{{ \hat{g}}^a}^b}$ , ${{ \tilde{g}}_a}_b$ $ << 1$ s.t. ${{{{{ \tilde{g}}_a}_b}_{,c}}^{2}}\approx{0}$ ?
${{{{{ R}^a}_b}_c}_d} = {{{{{{{{ \Gamma}^a}_b}_d}_{,c}} - {{{{{ \Gamma}^a}_b}_c}_{,d}}} + {{{{{{ \Gamma}^a}_e}_c}} {{{{{ \Gamma}^e}_b}_d}}}} - {{{{{{ \Gamma}^a}_e}_d}} {{{{{ \Gamma}^e}_b}_c}}}}$
${{{{{ R}^a}_b}_c}_d} = {{{{-1}} {{{{{ \Gamma}_e}_b}_c}} {{{{{ g}^a}^e}_{,d}}}} + {{{{{{ \Gamma}_e}_b}_d}} {{{{{ g}^a}^e}_{,c}}}} + {{{-1}} {{{{ g}^a}^e}} {{{{{{ \Gamma}_e}_b}_c}_{,d}}}} + {{{{{ g}^a}^e}} {{{{{{ \Gamma}_e}_b}_d}_{,c}}}} + {{{{{ g}^a}^f}} {{{{ g}^e}^h}} {{{{{ \Gamma}_f}_e}_c}} {{{{{ \Gamma}_h}_b}_d}}} + {{{-1}} {{{{ g}^a}^f}} {{{{ g}^e}^h}} {{{{{ \Gamma}_f}_e}_d}} {{{{{ \Gamma}_h}_b}_c}}}}$
using ${{{ g}^a}^b}\approx{{{ \hat{g}}^a}^b}$
${{{{{ R}^a}_b}_c}_d} = {{{{-1}} {{{{{ \Gamma}_e}_b}_c}} {{{{{ \hat{g}}^a}^e}_{,d}}}} + {{{{{{ \Gamma}_e}_b}_d}} {{{{{ \hat{g}}^a}^e}_{,c}}}} + {{{-1}} {{{{ \hat{g}}^a}^e}} {{{{{{ \Gamma}_e}_b}_c}_{,d}}}} + {{{{{ \hat{g}}^a}^e}} {{{{{{ \Gamma}_e}_b}_d}_{,c}}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \Gamma}_f}_e}_c}} {{{{{ \Gamma}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \Gamma}_f}_e}_d}} {{{{{ \Gamma}_h}_b}_c}}}}$
using ${{{{ \Gamma}_a}_b}_c} = {{{{{ \hat{\Gamma}}_a}_b}_c} + {{{{ \tilde{\Gamma}}_a}_b}_c}}$
${{{{{ R}^a}_b}_c}_d} = {{{{-1}} {{{{{ \hat{\Gamma}}_e}_b}_c}} {{{{{ \hat{g}}^a}^e}_{,d}}}} + {{{{{{ \hat{\Gamma}}_e}_b}_d}} {{{{{ \hat{g}}^a}^e}_{,c}}}} + {{{{{{ \hat{g}}^a}^e}_{,c}}} {{{{{ \tilde{\Gamma}}_e}_b}_d}}} + {{{-1}} {{{{{ \hat{g}}^a}^e}_{,d}}} {{{{{ \tilde{\Gamma}}_e}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^e}} {{{{{{ \hat{\Gamma}}_e}_b}_c}_{,d}}}} + {{{{{ \hat{g}}^a}^e}} {{{{{{ \hat{\Gamma}}_e}_b}_d}_{,c}}}} + {{{-1}} {{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_c}_{,d}}}} + {{{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_d}_{,c}}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_c}} {{{{{ \hat{\Gamma}}_h}_b}_d}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_d}} {{{{{ \hat{\Gamma}}_h}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_h}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_c}} {{{{{ \tilde{\Gamma}}_f}_e}_d}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_d}} {{{{{ \tilde{\Gamma}}_f}_e}_c}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \tilde{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \tilde{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_h}_b}_c}}}}$
recombining terms of ${{{{ \hat{R}}^a}_b}_c}_d$
${{{{{ R}^a}_b}_c}_d} = {{{{-1}} {{{{{{ \hat{\Gamma}}^a}_b}_c}_{,d}}}} + {{{{{ \hat{\Gamma}}^a}_b}_d}_{,c}} + {{{{{{ \hat{\Gamma}}^a}_e}_c}} {{{{{ \hat{\Gamma}}^e}_b}_d}}} + {{{-1}} {{{{{ \hat{\Gamma}}^a}_e}_d}} {{{{{ \hat{\Gamma}}^e}_b}_c}}} + {{{{{{ \hat{g}}^a}^e}_{,c}}} {{{{{ \tilde{\Gamma}}_e}_b}_d}}} + {{{-1}} {{{{{ \hat{g}}^a}^e}_{,d}}} {{{{{ \tilde{\Gamma}}_e}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_c}_{,d}}}} + {{{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_d}_{,c}}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_h}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_c}} {{{{{ \tilde{\Gamma}}_f}_e}_d}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_d}} {{{{{ \tilde{\Gamma}}_f}_e}_c}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \tilde{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \tilde{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_h}_b}_c}}}}$
${{{{{ R}^a}_b}_c}_d} = {{{{{{{ \hat{g}}^a}^e}_{,c}}} {{{{{ \tilde{\Gamma}}_e}_b}_d}}} + {{{-1}} {{{{{ \hat{g}}^a}^e}_{,d}}} {{{{{ \tilde{\Gamma}}_e}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_c}_{,d}}}} + {{{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_d}_{,c}}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_h}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_c}} {{{{{ \tilde{\Gamma}}_f}_e}_d}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_d}} {{{{{ \tilde{\Gamma}}_f}_e}_c}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \tilde{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \tilde{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_h}_b}_c}}} + {{{{{ \hat{R}}^a}_b}_c}_d}}$
Take note that I'm not touching ${{ \tilde{g}}^a}^b$ specifically because ${{ \tilde{g}}_a}_b$ << 1 implies ${{ \tilde{g}}^a}^b$ >> 1. Another detail I think everyone who ever did perturbative metric / weak field GR ignores.
Next comes ${{{{{ \tilde{g}}_a}_b}_{,c}}^{2}} = {0}$ and ${{{{{{ \tilde{g}}_a}_b}_{,c}}} {{{{{ \tilde{\Gamma}}_a}_b}_c}}} = {0}$ . I don't know why. It's not like $|y|<<1$ ever implied $|\frac{\partial y}{\partial x}|<<1$ ever in the history of math. But this is part of the weak-field process. Surprisingly this only removes two terms.
${{{{{ R}^a}_b}_c}_d} = {{{{{{{ \hat{g}}^a}^e}_{,c}}} {{{{{ \tilde{\Gamma}}_e}_b}_d}}} + {{{-1}} {{{{{ \hat{g}}^a}^e}_{,d}}} {{{{{ \tilde{\Gamma}}_e}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_c}_{,d}}}} + {{{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{\Gamma}}_e}_b}_d}_{,c}}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_h}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_c}} {{{{{ \tilde{\Gamma}}_f}_e}_d}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_d}} {{{{{ \tilde{\Gamma}}_f}_e}_c}}} + {{{{{ \hat{R}}^a}_b}_c}_d}}$
So since that just removes two terms, I'm not going to remove them from here on. I'll let you do that yourself if you want.
Now expand the ${{{{ \tilde{\Gamma}}_a}_b}_c}_{,d}$ terms
${{{{{ R}^a}_b}_c}_d} = {{{{{{{ \hat{g}}^a}^e}_{,c}}} {{{{{ \tilde{\Gamma}}_e}_b}_d}}} + {{{-1}} {{{{{ \hat{g}}^a}^e}_{,d}}} {{{{{ \tilde{\Gamma}}_e}_b}_c}}} + {{{{{ \hat{g}}^a}^e}} {{{{ \hat{g}}^f}^g}} {{{{{ \hat{\Gamma}}_e}_f}_c}} {{{{{ \tilde{\Gamma}}_g}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^e}} {{{{ \hat{g}}^f}^g}} {{{{{ \hat{\Gamma}}_e}_f}_d}} {{{{{ \tilde{\Gamma}}_g}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^e}} {{{{ \hat{g}}^f}^g}} {{{{{ \hat{\Gamma}}_g}_b}_c}} {{{{{ \tilde{\Gamma}}_e}_f}_d}}} + {{{{{ \hat{g}}^a}^e}} {{{{ \hat{g}}^f}^g}} {{{{{ \hat{\Gamma}}_g}_b}_d}} {{{{{ \tilde{\Gamma}}_e}_f}_c}}} + {{{{{ \hat{g}}^a}^e}} {{{{ \hat{g}}^f}^g}} {{{{{ \tilde{\Gamma}}_e}_f}_c}} {{{{{ \tilde{\Gamma}}_g}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^e}} {{{{ \hat{g}}^f}^g}} {{{{{ \tilde{\Gamma}}_e}_f}_d}} {{{{{ \tilde{\Gamma}}_g}_b}_c}}} + {{{{{ \hat{R}}^a}_b}_c}_d} + {{{\frac{1}{2}}} {{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{g}}_b}_c}_{,d}}_{,e}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{g}}_e}_c}_{,b}}_{,d}}}} + {{{-1}} \cdot {{\frac{1}{2}}} {{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{g}}_b}_d}_{,c}}_{,e}}}} + {{{\frac{1}{2}}} {{{{ \hat{g}}^a}^e}} {{{{{{ \tilde{g}}_e}_d}_{,b}}_{,c}}}}}$
Now from here most perturbative literature assumes the background metric uses constant components, i.e. ${{{ \hat{g}}_a}_b} = {{{ \eta}_a}_b}$ and ${{{{ \eta}_a}_b}_{,c}} = {0}$ , and that removes all the other terms except for the 4 ${{{{ \tilde{g}}_a}_b}_{,c}}_{,d}$ perturbation terms.
Mind you, if you don't expand the perturbation connection to produce those 4 ${{{{ \tilde{g}}_a}_b}_{,c}}_{,d}$ terms then instead you can recombine the partial to see it is the antisymmetric portion of ${\left( {{{{ \hat{g}}^a}^e}} {{{{{ \tilde{\Gamma}}_e}_b}_c}}\right)}_{,d}$
${{{{{ R}^a}_b}_c}_d} = {{{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_h}_b}_c}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_c}} {{{{{ \tilde{\Gamma}}_f}_e}_d}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \hat{\Gamma}}_h}_b}_d}} {{{{{ \tilde{\Gamma}}_f}_e}_c}}} + {{{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \tilde{\Gamma}}_f}_e}_c}} {{{{{ \tilde{\Gamma}}_h}_b}_d}}} + {{{-1}} {{{{ \hat{g}}^a}^f}} {{{{ \hat{g}}^e}^h}} {{{{{ \tilde{\Gamma}}_f}_e}_d}} {{{{{ \tilde{\Gamma}}_h}_b}_c}}} + {{{{{ \hat{R}}^a}_b}_c}_d} + {{\left( {{{{ \hat{g}}^a}^e}} {{{{{ \tilde{\Gamma}}_e}_b}_d}}\right)}_{,c}} + {{\left( {-{{{ \hat{g}}^a}^e}} {{{{{ \tilde{\Gamma}}_e}_b}_c}}\right)}_{,d}}}$