- #1
Shirish
- 244
- 32
I'm studying differential geometry basics for general relativity (no specific source, just googling around). I know that spacetime is modeled as a ##4##-dimensional smooth manifold. Smooth manifold means that we consider a restriction of the maximal atlas such that all charts in it are compatible.
From what I've read, it seems like the differentiability of curves depends on the atlas we're using (I could be wrong about this, please correct me in that case). So a curve may be differentiable w.r.t. one atlas and not to another.
What if there are multiple non-equivalent restrictions of the maximal atlas such that their charts are compatible? I tried searching for a result that conveniently tells us, for example, that there's a unique such restriction. On the contrary, I found that there can be uncountably many such atlases!
So then which one do we select? The differentiability of a curve ("existence of velocities") in spacetime shouldn't depend on our choice, right? Do we make any additional assumptions to make this notion well-defined w.r.t. choice of atlas?
From what I've read, it seems like the differentiability of curves depends on the atlas we're using (I could be wrong about this, please correct me in that case). So a curve may be differentiable w.r.t. one atlas and not to another.
What if there are multiple non-equivalent restrictions of the maximal atlas such that their charts are compatible? I tried searching for a result that conveniently tells us, for example, that there's a unique such restriction. On the contrary, I found that there can be uncountably many such atlases!
So then which one do we select? The differentiability of a curve ("existence of velocities") in spacetime shouldn't depend on our choice, right? Do we make any additional assumptions to make this notion well-defined w.r.t. choice of atlas?
Last edited: