How GR Resolves the Conservation of Momentum and Energy

[Ben Niehoff]

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 won't 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 $\varphi : A \rightarrow A$.

 

[TrickyDicky]

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 won't 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).

 

[Haelfix]

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).

Real quick, b/c I have to go. Yes. I did just change conventions from a few posts back, but that's b/c the definition of a conformal transformation differs between the texts I'm consulting and I just switched to Wald's convention. (Nakahara calls the former definition a conformal transformation, Wald calls the latter a conformal transformation). What's important is that they are distinct mathematical concepts. (scroll through a few pages in Wald as well)

http://books.google.com/books?id=9S...A#v=onepage&q=wald conformal isometry&f=false

http://books.google.com/books?id=cH...QHf78HeAw&ved=0CDcQ6AEwAA#v=onepage&q&f=false

(See the example from the latter).