Dynamic Equations of the ADM Formalism

[TerryW]

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.RegardsTerryW

 

[mark726]

Dear TerryW,

Thank you for sharing your results and asking for feedback. From your description, it seems like you have put a lot of effort into reworking your previous work and have made some progress. However, there are a few points that I would like to address in order to help identify where you may have gone wrong.

Firstly, it is important to remember that in order to obtain $\delta X$, you need to find not only $\frac {\partial X}{\partial g_{ij}}$, but also $\big(-\frac {\partial X}{\partial g_{ij,k}}\big)_{,k}$ and $\big(\frac {\partial X}{\partial g_{ij,kl}}\big)_{,kl}$. It seems like you have correctly identified this as the main error in your earlier work.

Secondly, I would suggest double checking your calculations for the terms $\big(-\frac {\partial }{\partial g_{ij,k}}(-N\gamma^\frac{1}{2}(Tr\pi^2 - \frac{1}{2}(Tr\pi)^2))\big)_{,k}$ and $\big(\frac {\partial }{\partial g_{ij,kl}}(-N\gamma^\frac{1}{2}(Tr\pi^2 - \frac{1}{2}(Tr\pi)^2))\big)_{,kl}$. It seems like you have encountered some unwanted terms, and it is possible that these could be the result of a mistake in your calculations.

Thirdly, it is important to note that the Palatini method is a shortcut and may not always give you the most accurate results. It is possible that the factor of 6 that you are encountering is a result of using this method. I would suggest revisiting your calculations using a more rigorous approach to see if this factor can be eliminated.

Lastly, it may be helpful to have a colleague or another scientist review your work and provide feedback. Sometimes, a fresh set of eyes can help identify mistakes or offer new insights. I hope this helps and I wish you the best of luck in resolving any remaining issues with your work.