B How to select the smooth atlas to use for spacetime?

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Thread starter Shirish
Start date Jul 18, 2020
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Atlas Smooth Spacetime

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Selecting a smooth atlas for spacetime involves understanding that differentiability of curves is generally independent of the choice of atlas, contrary to some claims that suggest otherwise. The discussion highlights that while differentiability can depend on the atlas used in specific pathological cases, these situations are not commonly encountered in practical applications of general relativity. The consensus is that physicists typically rely on intuitive choices of atlases that work well without delving into complex mathematical nuances. Additionally, a referenced paper suggests that certain types of manifolds allow for a well-defined notion of differentiability across atlases, reinforcing the idea that the choice of atlas should not affect physical outcomes. Ultimately, the resolution emphasizes the importance of focusing on practical applications rather than getting lost in theoretical complexities.

Aug 4, 2020

cianfa72

martinbn said:

He wasn't being careful. He probably meant that the two are the same topological space with two different smooth structures. But in dimension 2 the to different structures are diffeomorphic.

Watching it again, I believe as follows:

for both the regular and the sphere with 'edge' he takes the subspace topology inherited from ##\mathbb R^3##
for the regular sphere all the smooth charts in ##\mathbb R^3## 'restricted' to the sphere are compatible and thus define a maximal atlas for it
however for the sphere with 'edge' we cannot take all those charts because when 'restricted' to it are not longer mutually compatible

Thus the sphere with 'edge' and the regular one have not the same smooth structure (actually the regular sphere is a submanifold of ##\mathbb R^3## whereas the other is not) nevertheless are diffeomorphic as manifolds of dimension 2

Last edited: Aug 4, 2020