How Do Conformal Maps Preserve Angles in Different Geometric Contexts?

[bcrowell]

I'm trying to make sure I thoroughly understand Penrose diagrams, and I'm finding that it gets a little confusing because there are so many different ways of talking about conformal maps. This post summarizes how I think the definition of a conformal map works. I'd be grateful for comments on whether I have this right.

In ordinary plane geometry, conformal maps are defined as transformations that preserve angles, implying that small shapes are preserved up to a scale factor.

In GR, we normally think of a coordinate transformation as changing both the vectors and the metric, so that a quantity like $g^{ab}u_av_b$ is a scalar that remains unchanged by any transformation whatsoever. Within this formalism, it becomes difficult to talk about the equivalent of preserving angles. *All* diffeomorphisms preserve inner products, i.e., not just angles but lengths as well.

To get a nontrivial distinction between conformal and nonconformal transformations, you have to either transform the vectors while keeping the metric the same, or transform the metric while keeping the vectors the same.

When people talk about conformal transformations in the complex plane, they are basically keeping the metric (i.e., definition of multiplication and the complex conjugate) the same.

On a manifold, I think it becomes awkward to talk about transforming vectors while keeping the metric same. There will be cases where your transformation takes a point in one chart to a point in another chart, and the metric may not even be formally well defined on the coordinates of that chart. I think this is why one instead uses the approach of defining a conformal transformation as one that rescales the metric by some factor $\Omega^2$, where $\Omega$ is nonzero.

Is this right?

 

[George Jones]

In GR, we normally think of a coordinate transformation as changing both the vectors and the metric

I don't think of a coordinate transformation as changing either vectors or the metric.

*All* diffeomorphisms preserve inner products

This isn't true. A diffeoemorphism that "preserves" the metric is called an isometry, i.e., all isometries are diffeomorphisms, but not all diffeomorphisms are isometries.

 

[bcrowell]

Thanks for the reply, George, although I have to admit it mystifies me completely.

I don't think of a coordinate transformation as changing either vectors or the metric.

Meaning that you normally use coordinate-independent notation? That's fine, but I'm posing my question in the context of "index gymnastics" notation.

This isn't true. A diffeoemorphism that "preserves" the metric is called an isometry, i.e., all isometries are diffeomorphisms, but not all diffeomorphisms are isometries.

This confuses me, because in your first post you say, "I don't think of a coordinate transformation as changing either vectors or the metric," but in your second post you say, 'A diffeoemorphism that "preserves" the metric is called an isometry,[...]' So do you not think of a coordinate transformation as a diffeomorphism?

All I'm saying is that *in the index-gymnastics notation,* the tensor g_ab does change under a change of coordinates, while the inner product g_abu^av^b does not.

 

[arkajad]

Penrose diagram deals with the conformal compactification. Conformal transformations that are related to it may be of two kinds - either conformal transformations in the conformally compactified tangent space or "conformal transformations of the metric", that is trasformations of the form

$$g(x)\mapsto e^{\sigma(x)}g(x)$$

Equivalence class of such metrics is called a conformal structure. Conformally related metrics have the same null geodesics.