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I am not sure if the first one can give the geodesic equation. It also fails to have reparametrization invariance.

\phi^\ast g(x,y) = g(\phi_\ast x, \phi_\ast y) ≠ g(x,y) Hence it is not an isometry necessarily.

Cause the certainly larger set of RS(M)=RM(M)/Diff(M) corresponds to the same physical situation.

For instance the classes are, as you would already know, [M,g], [M,\Omega^2 g] etc.. To me that's the essence with which Beger is talking. And i don't think that we require anything else.

The relation is "Diffeomorphism". [a] contains all x, which are related to a by a diffeomorphism. Does that make sense?

Hmm. Let us try to understand that. An arbitrary diffeo would change the Metric. That would affect, say, the Mass distribution as well. But that again would correspond to the same physical situation much in the vein of what Nanaki was talking about. For once I thought,(using an example of...

I don't understand. Are you quotient-ing with the Diffeos that are Isometries or arbritary Diffeos? Or is it just the way he defines it?

Indeed :-)

How is it twisting words? :rolleyes: I said whatever you wrote makes sense.

Might be too late to reply. But yes, they make no mention of the geometry. Hence for example, transformations like g_{\mu\nu} \rightarrow \Omega^2 g_{\mu\nu} are not diffeos :-)

Oh, its not that simple. The above example is of the type when I make some kind of scaling, but then that would also change the Jacobian and the total mass would still remain M It still puzzles me.

Exactly! I believe it does use the EFE. If you are using the online notes, while going from 5.34 to 5.35, he uses the fact that the variation of the Hilbert action is zero to the first order.

Yeah. Thats what I was worried about. To which Nanaki has to say "They're not physically inequivalent. Suppose my mass distribution is two point particles. If I change my metric so that the particles are now twice as far apart, but I also act on the particles, moving them closer together, I...

That helped. :-) Thanks a lot.

The mass distribution for instance. Hence the statement " Two descriptions related by a diffeo can correspond to different physically inequivalent situations " should hold good.