My earlier question about up-dated accounts of Emmy Noether's own version(s) of the Noether conservation theorem(s) gets a thorough answer (with appreciative historical remarks on Lie and Noether) in Peter Olver's book Applications of Lie Groups to Differential Equations. But I am still struggling with the relation to GR.
As I understand it, one approach to GR is to define a Lagrangian function of the metric tensor, called the action of the metric. Then the Einstein field equation is derived by finding a metric field that minimizes the action. Is that right, so far as it goes? Anyway, it is vague.
Where is the integral taken? Some formulations seem to say that this only works for aymptotically flat regions in space time, and then you integrate over any region which is flat all around the boundary.
I suppose that in a compact space-time you could integrate over all of space-time. Or have I completely stopped making sense when I say that?
I would appreciate any answers or references. I have not mastered GR but I have read the first 4 chapters of Wald, General Relativity, and have read Misner Wheeler and Thorne through "How mass-energy generates curvature (chap 17)." I hope I need not actually read either one of those books all the way up to where they do Lagrangian formulations.