Chart(s) of space-time as a smooth manifold

So we all know that space-time in general relativity is modeled as a smooth (pseudoRiemannian) manifold.

Each point (event) on space-time is labeled with a unique coordinate $$(t,x,y,z)$$ in a specific reference frame.

Am I correct to think the map

$$spacetime \rightarrow \mathbb{R}^4$$

$$event \rightarrow (t,x,y,z)$$

as a chart on the entirety of the manifold?

Hence did we just assumed that space-time is homeomorphic to $$\mathbb{R}^4$$ so that a single chart can cover the entirety of the manifold?

What if space-time is homeomorphic to for example $$S^4$$ so that one single chart isn't enough?

• Spacetime is covered by a single metric-preserving chart if and only if you are studying special relativity. – WillO Feb 17 at 5:25
• Thanks, I understand now, indeed I was carrying stereotypes from special relatively – pig2000 Feb 17 at 5:49
• As an example I found atlas of de sitter space: arxiv.org/abs/1211.2363 – pig2000 Feb 17 at 5:55
• @WillO: Spacetime is covered by a single metric-preserving chart if and only if you are studying special relativity. The part about "metric-preserving" isn't relevant to the OP. – Ben Crowell Feb 17 at 15:22
• @BenCrowell: I agree that it is not relevant to the question that the OP actually asked, but I am at least 70% confident that it is relevant to the underlying confusion. – WillO Feb 17 at 16:38

1 Answer

One of the properties of a manifold is that you can map open subsets of the manifold to open sunsets of R4 with a one-to-one smooth mapping. This mapping can be done in the neighborhood of any event, but there is no guarantee that it can be done globally. Indeed, as you point out there are some manifolds where the mapping cannot be done globally.

Each mapping is called a coordinate chart. Thus, coordinate charts need not be global. The set of all charts on a manifold is called an atlas. Two charts need not overlap, but if they do then within the region that they overlap since there is a one-to-one mapping from each chart to events in the manifold there is also a one to one mapping from one chart to the other. This is the coordinate transform.