Question on Variational Techniques in General Relativity

[Freeman]

OK, so I had no clue where to put this, so here goes nothing. In general relativity, the variational methods used, I need to figure out the variation of the (three) christoffel symbol with respect to the (three) metric tensor:

[math]\frac{\delta\Gamma^{i}_{bc}}{\delta g_{bc}} = ?[/math]

the reason I ask is because I'm really looking for the variation of the Ricci tensor with respect to the metric (all of this is going on in three dimensions too, only the spatial ones)

[math]\frac{\delta R_{ab}}{\delta g_{ab}} = \frac{\delta\Gamma^{c}_{ab;c}}{\delta g_{ab}} - \frac{\delta\Gamma^{c}_{ac;b}}{\delta g_{ab}}[/math]

that's how I would figure it to be, so I'm wondering how would I go about this?

 

Any help would be greatly appreciated!

[ajb]

't Hooft goes through such a calculation in his lecture notes on general relativity. (Any other good account of GR will do something similar)

 

http://www.phys.uu.nl/~thooft/lectures/genrel.pdf

 

Hope that helps. If not, let us know.

[Freeman]

Cheers! One last question, I can't think straight at the moment but is the following acceptable:

[math]\frac{\delta \partial_{t}X}{\delta X} = \partial_{t}\frac{\delta X}{\delta X}[/math]

or is it more complicated than that?

[ajb]

You can move [math]\delta[/math] through a derivative provided the variation is not with respect to the argument of the derivative.

 

If it is then you do pick up an extra term as you have to consider how the derivative changes. The same is true of the measure for an integral.

[Freeman]

Cheers! Thanks for all the help everyone, it's really helped me better understand the use of variational techniques (and not just in general relativity )!