hilbert action according to robert wald

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

my problem is: he uses einstein's equation on the unperturbed metric to fish out [itex]\delta R_{ab}[/itex], 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.

thanks in advance

nicksauce

Carroll's derivation seems pretty straight forward (see pg. 114): http://preposterousuniverse.com/grno...notes-four.pdf

naughtyeskimo

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

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).