Recognitions:

## energy in GR

 Quote by Ben Niehoff By the way, here is how to identify what transformations preserve Einstein's equations. $$\Box h_{\mu\nu} + 2 R_\mu{}^\rho{}_\nu{}^\sigma h_{\rho\sigma} = 0.$$ Any solutions to this equation give infinitesimal perturbations that preserve Einstein's equation. However, the gauge condition on h does not fully gauge-fix, so some of the solutions will just be infinitesimal coordinate transformations; you have to throw those ones out. Anything left will give you a non-trivial invariance of GR. These will be local diffeomorphisms which are not isometries.
Correct! There is also, as you mentioned a sublety with so called large diffeomorphisms (diffeormorphisms which are not continously connected to the identity), as well as with so called boundary diffeomorphisms (which change the asymptotic structure in some way). The linearization procedure obscures these facts, and they have to be added by hand.

Recognitions:
 Quote by Haelfix A conformal isometry is a diffeomorphism: Psi: M --> M such that (Psi*G)uv = Omega^2 Guv provided omega is everwhere real and positive. The case Omega = 1 is just a regular isometry.
Can you give an example? I'm having a hard time imagining such a map. Is it possible to have a continuous family of such maps?

 A conformal transformation (or Weyl rescaling): Guv' = Omega^2 Guv is NOT in general a diffeomorphism! If you don't take the pullback, then it wont be invariant.
But I'm looking for things which are not invariant. Is a round 2-sphere of radius A diffeomorphic to a round 2-sphere of radius B, or not?

I say it is. In patches, $\varphi: (\theta, \phi) \mapsto (\theta, \phi)$, which is clearly differentiable, and poses no problems with the transition functions. But $g_A$ is not the pullback of $g_B$ along $\varphi$.

One does have $\varphi^*(g_B) = (b^2/a^2) \, g_A$, is that what you mean above? In this case, $\varphi : A \rightarrow B$, not $\varphi : A \rightarrow A$.

 Quote by Haelfix So I agree with most of your post, except the last part. I am a little uneasy with the terminology. Following Wald and Nakahara: A conformal isometry is a diffeomorphism: Psi: M --> M such that (Psi*G)uv = Omega^2 Guv provided omega is everwhere real and positive. The case Omega = 1 is just a regular isometry. A conformal transformation (or Weyl rescaling): Guv' = Omega^2 Guv is NOT in general a diffeomorphism! If you don't take the pullback, then it wont be invariant.
You seem to be contradicting yourself here.
A Weyl rescaling is a conformal transformation of the metric, and all conformal transformations are diffeomorphisms (they are defined as the subgroup of diffeomorphisms that preserve the metric up to a scale, the conformal factor).

Recognitions: