# Hilbert action according to robert wald

1. Jun 22, 2012

### naughtyeskimo

hi everyone
i'm brand new to forums and i'm holdin a seminar on a variation of the hilbert action as described in wald's book general relativity. if anyone knows that book and topic pretty well maybe you can help me, my question is this:

for the variation $\delta R_{ab}$ with respect to parameter $\lambda$ of the ricci tensor $R_{ab}$ wald uses a result from the end of chapter 7, where he assumes $g_0^{ab}$ to be a solution of einstein's equation, so in the calculation of $\delta R_{ab}$ he assumes already that $R_{ab} = 0$ at $\lambda = 0$ so he works it out in terms of the tensor $C^c_{ab}$ (the tensor difference between two covariant derivatives).

my problem is: he uses einstein's equation on the unperturbed metric to fish out $\delta R_{ab}$, then uses that to derive einstein's equation with the hilbert action.

is there a good explanation for using this equation to derive this very same equation? i hope i explained myself clearly enough.

Last edited: Jun 22, 2012
2. Jun 22, 2012

### nicksauce

Carroll's derivation seems pretty straight forward (see pg. 114): http://preposterousuniverse.com/grnotes/grnotes-four.pdf [Broken]

Last edited by a moderator: May 6, 2017
3. Jun 25, 2012

### naughtyeskimo

thanks very much, that is an easier method. good work.

4. Jun 25, 2012

### haushofer

That's one reason I've never really liked Wald's book. Some things are being made much more complicated than necessary, unless you're a die-hard mathematician. Indeed, the easiest way is to apply the variation directly to the definition of the Ricci tensor and use covariance arguments (or do the whole calculation).