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Advanced Physics Homework Help
Problems deriving the conjugate momenta for the ADM formalism
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[QUOTE="TerryW, post: 6039022, member: 119371"] [h2]Homework Statement [/h2]If you look in Wikipedia for ADM formalism, you are given a Derivation, which starts from the Lagrangian: ##\mathfrak {L}## = [SUP](4)[/SUP]R ##\sqrt{^{(4)}g}## and moves rapidly to... The conjugate momenta can then be computed as ##\pi^{ij} = \sqrt {^{(4)}g} (^{(4)} \Gamma ^0 \,_{pq} - g_{pq} ^{(4)} \Gamma ^0\, _{rs} g^{rs}) g^{pq} g^{jq}## ...(1) [I]using standard techniques and definitions [/I](my italics) From MTW chapter 21 I have: ##\pi^{ij} = \frac {\delta (action) }{\delta (g_{ij}) }## ...... (21.91) and from D'Inverno Chapter 11, I have ##\frac {\delta (action) }{\delta (g_{\alpha\beta}) } = (\frac {\partial (action) }{\partial (g_{\alpha\beta}) }) - (\frac {\partial (action) }{\partial (g_{\alpha\beta,\gamma}) }) _{,\gamma}## + ##(\frac {\partial (action) }{\partial (g_{\alpha\beta,\gamma\delta})})_{,\gamma\delta} ##...(11.25) I have tried to produce ##\pi^{ij}## as above starting with work on the more general ##(\frac {\partial}{\partial (g_{\alpha\beta ,\gamma}) })g^{\mu\nu}g^{\gamma\alpha}g^{\beta\delta}(\Gamma_{\gamma\beta\alpha}\Gamma_{\delta\mu\nu} - \Gamma_{\gamma\beta\nu}\Gamma_{\delta\mu\alpha})## ie just taking the derivative wrt ##g_{\alpha\beta,\gamma}## and ignoring the ## \sqrt {^{(4)}g}## and the second derivative wrt ##\gamma## for now. After much fiddling with indices, I have ended up with (see more complete workings below) ##=\frac{1}{2}g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu} - \frac{1}{2}g^{\mu\beta}g^{\nu\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}+\frac{1}{2}g^{\beta\gamma}g^{\mu\delta}g^{\nu\alpha}\Gamma_{\delta\mu\nu} - \frac{1}{2}g^{\beta\mu}g^{\delta\gamma}g^{\nu\alpha}\Gamma_{\delta\mu\nu}## ##=\frac{1}{2}g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu} +\frac{1}{2}g^{\beta\gamma}g^{\mu\delta}g^{\nu\alpha}\Gamma_{\delta\mu\nu} -g^{\beta\mu}g^{\alpha\nu}g^{\gamma\delta}\Gamma_{\delta\mu\nu}## I've also reached the same result by working on the 18 terms obtained when you multiply out ##\frac{1}{2}(g_{γβ,α} + g_{γα,β} - g_{αβ,γ})\frac{1}{2}(g_{δμ,ν} + g_{δν,μ} - g_{νμ,δ}) - \frac{1}{2}(g_{γβ,ν} + g_{γν,β} - g_{νβ,γ})\frac{1}{2}(g_{δμ,α} + g_{δα,μ} - g_{αμ,δ})## I can manipulate ##\frac{1}{2}g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}## into ##\frac{1}{2}g_{μν} \Gamma ^γ\, _{ρσ} g^{ρσ} g^{αμ} g^{βν}## and I can manipulate ##g^{\beta\mu}g^{\alpha\nu}g^{\gamma\delta}\Gamma_{\delta\mu\nu}## into ##\Gamma^\gamma_{\mu\nu}g^{α\nu} g^{βμ}## I'd like to be able to transform ##\frac{1}{2}g^{\beta\gamma}g^{\mu\delta}g^{\nu\alpha}\Gamma_{\delta\mu\nu}## into ##\frac{1}{2}g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}## but I can't see a way of doing this. This would then give me: ##g_{μν} \Gamma ^γ\, _{ρσ} g^{ρσ} g^{αμ} g^{βν} - \Gamma^\gamma_{\mu\nu}g^{αμ} g^{β\nu}## which would become ##g_{μν} \Gamma ^γ\, _{ρσ} g^{ρσ} g^{iμ} g^{jν} - \Gamma^\gamma_{\mu\nu}g^{i\mu} g^{j\nu}## So my problems are: 1. Am I missing something which would enable me to transform ##\frac{1}{2}g^{\beta\gamma}g^{\mu\delta}g^{\nu\alpha}\Gamma_{\delta\mu\nu}## into ##\frac{1}{2}g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}##? 2. How do I justify the selection of '0' (and only '0') for γ ? 3. How to I justify limiting the other indices to just the 3D indices. ? 4. Why is the further differentiation wrt γ being ignored? [h2]Homework Equations[/h2][h2]The Attempt at a Solution[/h2] [/B] ##(\frac {\partial}{\partial (g_{\alpha\beta ,\gamma}) })g^{\mu\nu}g^{\gamma\alpha}g^{\beta\delta}(\Gamma_{\gamma\beta\alpha}\Gamma_{\delta\mu\nu} - \Gamma_{\gamma\beta\nu}\Gamma_{\delta\mu\alpha})## ##= (\frac {\partial }{\partial (g_{\alpha\beta ,\gamma})}g^{\mu\nu}g^{\gamma\alpha}g^{\beta\delta}\Gamma_{\gamma\beta\alpha}\Gamma_{\delta\mu\nu} - g^{\mu\alpha}g^{\gamma\nu}g^{\beta\delta}\Gamma_{\gamma\beta\alpha}\Gamma_{\delta\mu\nu})## ##= (\frac {\partial }{\partial (g_{\alpha\beta ,\gamma}) }g^{\mu\nu}g^{\gamma\alpha}g^{\beta\delta}\Gamma_{\alpha\beta\gamma}\Gamma_{\delta\mu\nu} - g^{\mu\gamma}g^{\alpha\nu}g^{\beta\delta}\Gamma_{\alpha\beta\gamma}\Gamma_{\delta\mu\nu})## ##= \frac{1}{2}(g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}) + \Gamma_{\alpha\beta\gamma}\frac {\partial }{\partial (g_{\alpha\beta ,\gamma})}(g^{\mu\nu}g^{\alpha\gamma}g^{\beta\delta}\Gamma_{\delta\mu\nu}) - \frac{1}{2}(g^{\mu\beta}g^{\nu\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}) - \Gamma_{\alpha\beta\gamma}\frac {\partial}{\partial (g_{\alpha\beta ,\gamma}) }(g^{\mu\gamma}g^{\alpha\nu}g^{\beta\delta}\Gamma_{\delta\mu\nu})## ##=\frac{1}{2}(g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}) - \frac{1}{2}(g^{\mu\beta}g^{\nu\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}) + \Gamma_{\delta\mu\nu}\frac {\partial }{\partial (g_{\alpha\beta ,\gamma})}(g^{\beta\gamma}g^{\nu\delta}g^{\mu\alpha}\Gamma_{\alpha\beta\gamma}) - \Gamma_{\delta\mu\nu}\frac {\partial}{\partial (g_{\alpha\beta ,\gamma}) }(g^{\beta\nu}g^{\gamma\delta}g^{\mu\alpha}\Gamma_{\alpha\beta\gamma})## ##=\frac{1}{2}g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu} - \frac{1}{2}g^{\mu\beta}g^{\nu\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu} + \frac{1}{2}(g^{\beta\gamma}g^{\nu\delta}g^{\mu\alpha}+g^{\beta\gamma}g^{\nu\delta}g^{\mu\alpha}-g^{\beta\alpha}g^{\nu\delta}g^{\mu\gamma})\Gamma_{\delta\mu\nu}- \frac{1}{2}(g^{\beta\nu}g^{\delta\gamma}g^{\mu\alpha}+g^{\gamma\nu}g^{\beta\delta}g^{\mu\alpha}-g^{\beta\nu}g^{\delta\alpha}g^{\mu\gamma})\Gamma_{\delta\mu\nu}## I reckon I can interchange β and γ because ##\Gamma_{αβγ} = \Gamma_{αγβ}## and I can interchange μ and ν because ##\Gamma_{δμν} = \Gamma_{δνμ}## Also I can interchange α and β because ##\pi^{αβ} = \pi^{βα}## Using these two tools, I can reduce ##\frac{1}{2}(g^{\beta\gamma}g^{\nu\delta}g^{\mu\alpha}+g^{\beta\gamma}g^{\nu\delta}g^{\mu\alpha}-g^{\beta\alpha}g^{\nu\delta}g^{\mu\gamma})\Gamma_{\delta\mu\nu}- \frac{1}{2}(g^{\beta\nu}g^{\delta\gamma}g^{\mu\alpha}+g^{\gamma\nu}g^{\beta\delta}g^{\mu\alpha}-g^{\beta\nu}g^{\delta\alpha}g^{\mu\gamma})\Gamma_{\delta\mu\nu}## to ##\frac{1}{2}(g^{\beta\gamma}g^{\mu\delta}g^{\nu\alpha} - g^{\beta\nu}g^{\delta\gamma}g^{\mu\alpha})\Gamma_{\delta\mu\nu}## Giving me the result: ##\frac{1}{2}g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu} - \frac{1}{2}g^{\mu\beta}g^{\nu\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}+\frac{1}{2}g^{\beta\gamma}g^{\mu\delta}g^{\nu\alpha}\Gamma_{\delta\mu\nu} - \frac{1}{2}g^{\beta\mu}g^{\delta\gamma}g^{\nu\alpha}\Gamma_{\delta\mu\nu}## [/QUOTE]
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