This is a very very simple question and I am sure it will look dumb because I won't be using the correct terminology but here I go.(adsbygoogle = window.adsbygoogle || []).push({});

Consider the points in a manifold. Now we assign coordinates to those points.

ne thing that I find confusing about any type of transformation is whether

a) it is the points themselves which are moved about (in which case the manifold is being deformed in some way) with the coordinates "following" the points so that each actual point remains at the same coordinate or

b) it is only the coordinate chart that is being changed, without the points in the surface being actually moved (so the surface is not deformed in any way)

c) or if the surface is being deformed without the chart being changed so that the points acquire new coordinates.

I am not even totally sure if one makes a distinction between cases b) and c) in that in both cases the points change coordinates so maybe it's a "active vs passive" type of thing. It does seem that there is a physical difefrence but maybe mathematically it's really considered to be the same. I don't know.

To have specific examples in mind, consider a Weyl rescaling. If I understand correctly (I may be wrong on that), this is different from a conformal mapping in that there is no actual change of coordinates [itex] x \rightarrow x'(x) [/itex] involved. Only the metric is rescaled.

On the other hand, a conformal transformation does involve a tranformation of the coordinates [itex] x \rightarrow x'(x) [/itex] which leads to a rescaling of the metric also.

So the question is: what is the difference between the two cases? Which of the three cases a, b or c described above does each correspond to?

Here is what my gut feeling is but it's probably completely incorrect.

First there is the question of what a mapping [itex] x \rightarrow x'(x) [/itex] actually means. I woudl think that it means that the actual points are assigned new coordinates. So this corresponds to a change of the map which does not involve deforming the manifold itself. Now, the physical "distance" (as given by [itex] g_{\mu \nu} dx^\mu dx^\nu [/itex]) between two actual points has not changed, so the metric has to change in order to give the same distance between two given points (which now have different coordinates). A confomal transformation is simply one such that the metric is rescaled. But the key point is that the manifold itself (if I understand correctly) is not changed. we are talking about a pure change of the coordinate grid.

On the other hand, what would be a Weyl rescaling? It seems to me that since there is no transformation [itex] x \rightarrow x'(x) [/itex] , any actual point remains at the same coordinate. But the fact that the metric is rescaled implies that the distance between two given coordinates (and therefore two given actual points since they retain the same coordinate) is changed. That would imply that the manifold is actually being deformed here, with the coordinate grid "following" the points.

Two questions: is that correct? And if so, what would be the "mathspeak" way of conveying what I am talking about (so that I coudl recognize it in a math book)?

Thanks!

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# Simple questions about weyl rescaling/Conformal transfo

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