Numerical Relativity - ADM from EFE

Derivations:

I've got the whole thing in notebooks progressing the other way, from the EFE to the ADM, following Baumgarte & Shapiro, so TODO copy it here.

ADM

γ_{i j, t} - L_{\vec{β}} γ_{i j} = - 2 α K_{i j}

K_{i j, t} - L_{\vec{β}} K_{i j} = - D_{i} D_{j} α + α (R_{i j} + K K_{i j} - 2 K_{i k} {K^{k}}_{j})

expanding Lie derivatives and projection covariant derivatives:

γ_{i j, t} - γ_{i j, k} β^{k} - γ_{i k} {β^{k}}_{, j} - γ_{k j} {β^{k}}_{, i} = - 2 α K_{i j}

K_{i j, t} - β^{k} K_{i j, k} - K_{k j} {β^{k}}_{, i} - K_{i k} {β^{k}}_{, j} = - {⊥_{i}}^{a} \nabla_{a} ({⊥_{j}}^{b} \nabla_{b} α) + α (R_{i j} + K K_{i j} - 2 K_{i k} {K^{k}}_{j})

= - {⊥_{i}}^{a} \nabla_{a} (α_{, j} - n_{j} n^{b} α_{, b}) + α (R_{i j} + K K_{i j} - 2 K_{i k} {K^{k}}_{j})

= - {⊥_{i}}^{a} (\nabla_{a} α_{, j} - \nabla_{a} (n_{j} n^{b} α_{, b})) + α (R_{i j} + K K_{i j} - 2 K_{i k} {K^{k}}_{j})

= - {⊥_{i}}^{a} (\nabla_{a} α_{, j} - \nabla_{a} n_{j} n^{b} α_{, b} - n_{j} \nabla_{a} n^{b} α_{, b} - n_{j} n^{b} \nabla_{a} α_{, b}) + α (R_{i j} + K K_{i j} - 2 K_{i k} {K^{k}}_{j})

= - (δ_{i}^{a} - n_{i} n^{a}) (α_{, j a} - {^{4} Γ^{b}}_{j a} α_{, b} - (n_{j, a} - {^{4} Γ^{c}}_{j a} n_{c}) n^{b} α_{, b} - n_{j} ({n^{b}}_{, a} + {^{4} Γ^{b}}_{c a} n^{c}) α_{, b} - n_{j} n^{b} (α_{, b a} - {^{4} Γ^{c}}_{b a} α_{, c})) + α (R_{i j} + K K_{i j} - 2 K_{i k} {K^{k}}_{j})

...is

D_{i}

the same as applying the spatial connections?

K_{i j, t} - β^{k} K_{i j, k} - K_{k j} {β^{k}}_{, i} - K_{i k} {β^{k}}_{, j}

= - D_{i} α_{, j} + α (R_{i j} + K K_{i j} - 2 K_{i k} {K^{k}}_{j})

= - α_{, i j} + {Γ^{k}}_{j i} α_{, k} + α (R_{i j} + K K_{i j} - 2 K_{i k} {K^{k}}_{j})