Initial Value Equations out of ADM formalism

[TerryW]

Homework Statement

Use the ADM formalism to derive dynamic and initial value equations

Relevant Equations

MTW (21.115) - See attachment

In my earlier post, I demonstrated a way to derive MTW's Equation (21.90),

$16π\mathfrak{L}_{geom} = - \dot π^{ij} γ_{ij} - N\mathcal{H} -N_i\mathcal{H^i}-2(π^{ij}N_j-½N^iTrπ+γ^½N^{|i})_{,i}$ MTW (21.90)

I received my first two 'likes' for this which made me really happy - Thanks guys!

I've moved on a bit now and have reached page 525, (four whole pages - wow!) and I am now trying to produce MTW (21.115) - see attachment.

MTW suggests that this can be derived from:

$I = \frac{1}{16π}\int[ \dot π^{ij} γ_{ij} - N\mathcal{H} -N_i\mathcal{H^i}]d^4x + \int\mathfrak{L}_{Field}d^4x$ MTW (21.95)

I've managed to produce nearly all of (21.115) but I cannot find a way to produce the terms I've underlined.

Instead of the term $(π^{ij}N^m)_{|m}$ I have $(N^iπ^{jk})_{|k}$

I can't see any way to produce terms with $N^{|ij}$ or $N^{|m}{}_{|m}$ from any of the terms in 21.95, but if I go back to (21.90), I can use the term $[N^{|i}(\gamma)^{\frac{1}{2}}]_{,i}$, which nearly gets me home, but I end up with

$(\gamma)^{\frac{1}{2}}N^{|ij} - \frac{1}{2}(\gamma)^{\frac{1}{2}}g^{ij}N^{|m}{}_{|m}$

and a bonus term$-(-g)^{\frac{1}{2}}\frac{1}{N^2}N^{|i}N_{|i}$

(I have a copy of the original ADM paper and interestingly, they do use the full (21.90) to derive (21.115), ie without dropping the divergence!)

Is there something I have missed that would enable me to show that the bonus term somehow resolves my problems, or is there another route altogether?
Regards

Terry W

Attachment:

 

[Nate810]

MTW (21.115): $16π\mathfrak{L}_{geom} = - \dot π^{ij} γ_{ij} - N\mathcal{H} -N_i\mathcal{H^i}-2(π^{ij}N_j-½N^iTrπ+γ^½N^{|i})_{,i} + (π^{ij}N^m)_{|m} - \frac{1}{2}N^{|ij} - \frac{1}{2}N^{|m}{}_{|m}$