Problems deriving the conjugate momenta for the ADM formalism

[TerryW]

Homework Statement

If you look in Wikipedia for ADM formalism, you are given a Derivation, which starts from the Lagrangian:

$\mathfrak {L}$ = ⁽⁴⁾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)
using standard techniques and definitions (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?

Homework Equations

The Attempt at a Solution

[/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:

$\frac{1}{2}g^{\mu\nu}g^{\beta\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu} - \frac{1}{2}g^{\mu\beta}g^{\nu\alpha}g^{\gamma\delta}\Gamma_{\delta\mu\nu}+\frac{1}{2}g^{\beta\gamma}g^{\mu\delta}g^{\nu\alpha}\Gamma_{\delta\mu\nu} - \frac{1}{2}g^{\beta\mu}g^{\delta\gamma}g^{\nu\alpha}\Gamma_{\delta\mu\nu}$