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Advanced Physics Homework Help
Initial Value Equations out of ADM formalism
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[QUOTE="TerryW, post: 6176650, member: 119371"] [B]Homework Statement:[/B] Use the ADM formalism to derive dynamic and initial value equations [B]Relevant Equations:[/B] 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: [ATTACH type="full"]243235[/ATTACH] [/QUOTE]
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Initial Value Equations out of ADM formalism
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