Problems deriving the conjugate momenta for the ADM formalism

  • Thread starter TerryW
  • Start date
  • #1
TerryW
Gold Member
168
6

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}##
 

Answers and Replies

Related Threads on Problems deriving the conjugate momenta for the ADM formalism

Replies
0
Views
755
Replies
0
Views
309
Replies
0
Views
463
Replies
7
Views
1K
Replies
0
Views
621
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
2K
Replies
4
Views
7K
  • Last Post
Replies
3
Views
2K
Replies
0
Views
1K
Top