Dynamic Equations of the ADM Formalism

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TerryW
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Homework Statement:
I have been trying to find my way to reproducing MTW's equation 21.115. I've identified a couple of errors in my earlier postings on this and I've worked on these to get me closer to the answer but I'm still not quite there.
Relevant Equations:
MTW's equation 21.115
The main error in my earlier work was forgetting that to obtain ##\delta X## you have to find not only ##\frac {\partial X}{\partial g_{ij}}##, but also ##\big(-\frac {\partial X}{\partial g_{ij,k}}\big)_{,k}## and ##\big(\frac {\partial X}{\partial g_{ij,kl}}\big)_{,kl}##. I also missed a trick when I worked on ##\delta (N\gamma^\frac{1}{2}R)##.

So the results of my reworking are as follows (There are many pages of work producing the results for ##\big(\frac {\partial (-N\mathcal{H})}{\partial g_{ij}}\big)## ,##\big(-\frac {\partial (-N\mathcal{H})}{\partial g_{ij,k}}\big)_{,k}## , ##\big(\frac {\partial (-N\mathcal{H})}{\partial g_{ij,kl}}\big)_{,kl}## and ##\big(\frac {\partial (-N_i\mathcal{H}^i)}{\partial g_{ij}}\big)## ,##\big(-\frac {\partial (-N_i\mathcal{H}^i)}{\partial g_{ij,kl}}\big)_{,kl}## , ##\big(\frac {\partial (-N_i\mathcal{H}^i)}{\partial g_{ij,kl}}\big)_{,kl}##) I'm happy to share these if anyone is interested!

The results for ##\delta (-N\mathcal{H})## are:

A. ##\delta (N\gamma^\frac{1}{2}R) = -N(\gamma^\frac{1}{2})(R^{ij} - \frac{1}{2}g^{ij}R)##

(The Palatini method is a short cut to this result - saves a lot of work)

(i) ##\frac {\partial }{\partial g_{ij}}(-N\gamma^\frac{1}{2}(Tr\pi^2 - \frac{1}{2}(Tr\pi)^2))## produces:

B. ## \frac{1}{2}g^{ij}(-N\gamma^\frac{1}{2}(Tr\pi^2 - \frac{1}{2}(Tr\pi)^2))## and

C. ##-2N\gamma^\frac{1}{2}(Tr|pi^2 - \frac{1}{2}(\pi^{im}\pi_m{}^j - \frac{1}{2}\pi^{ij}Tr\pi)##

(ii) ##\big(-\frac {\partial }{\partial g_{ij,k}}(-N\gamma^\frac{1}{2}(Tr\pi^2 - \frac{1}{2}(Tr\pi)^2))\big)_{,k}## produces:

##(-2N^i\pi^{jk} +N^k\pi^{ij})_{,k}##

This can then be turned into:

E. ##(-2N^i\pi^{jk})_{|k}## which is unwanted plus
more unwanted terms ##+2N^m\Gamma^i{}_{mk}\pi^{jk} +2N^i\Gamma^j{}_{mk}\pi^{mk} +2N^i\Gamma^k{}_{mk}\pi^{mj} ## plus

F. ##(N^k\pi^{ij}))_{|k}## which we do want plus
more unwanted terms ## - N^m\Gamma^i{}_{mk}\pi^{ij} - N^k\Gamma^j{}_{mk}\pi^{im} - N^k\Gamma^k{}_{mk}\pi^{mj} ##

##-N\mathcal{H}## contains no terms in ##g_{ij,kl}##

The results for ##\delta (-N_i\mathcal{H^i})## are:

(i) ##\frac {\partial (-N_i\mathcal{H}^i)}{\partial g_{ij}}## gives:

D. ##-N^j_{|k}\pi^{ik} - N^i_{|k}\pi^{jk}## plus

-E. ##(2N^i\pi^{jk})_{|k}## which cancels E. above plus an unwanted term ##g^{ij}N_l(\pi^{lk})_{|k}##

(ii) ##\big(-\frac {\partial (-N_i\mathcal{H}^i)}{\partial g_{ij,k}}\big)_{,k}## , gives

G. ##(-4N_{|m}\gamma^{\frac{1}{2}}(g^{il}g^{jk} - g^{ij}g^{kl})_{,k}##and a whole raft of unwanted terms

(iii) ##\big(\frac {\partial (-N_i\mathcal{H}^i)}{\partial g_{ij,kl}}\big)_{,kl}## gives

H. ##(-2N\gamma^{\frac{1}{2}}(g^{il}g^{jk} - g^{ij}g^{kl})_{,kl}##

I can work on G and H to produce a term like ## \gamma^{\frac{1}{2}}(N^{ij} - g^{ij}N^{|m}{}_{|m})## but it has an unwanted factor of '6' in it plus another load of unwanted terms.

In summary, I have been able to produce all the terms in MTW 21.115 without the need to mine into the divergence, but I am left with a rogue factor 6 and a whole load of unwanted bits and pieces which do not appear to cancel out in any way.

If there is anyone out there who would be willing to check through any of my workings to help identify where I am going wrong, it would be much appreciated.


Regards


TerryW
 

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