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  • $\begingroup$ On the level of smooth manifolds, one can not distinguish, for example, a plane and a wavy surface. What makes them different is the metric, and the metric is the primary object to be determined in GR. The metric is defined locally and can be studied locally. Therefore, one does not care what the chart is, but rather what the metric is in the coordinate system. (in fact in the context of abstract manifold, a single chart does not carry much information, but the transition functions between the charts do. ) If it is still not clear, I can make it a more rigorous answer. $\endgroup$ Commented Mar 24, 2017 at 2:49
  • $\begingroup$ user1620696: "The trouble is that we don't know what $M$ is!" -- Perhaps a bit more ... stringently: Considering a set of events ("as such") we don't know its topology (which subsets to call "open", or "neighborhood") to begin with; and considering any non-empty subset $U_a \subseteq M$ together with two (invertible) assignments (of coordinates, without any regard to topology) $$ \phi_j : U \rightarrow \mathbb R^n, \qquad \phi_k : U \rightarrow \mathbb R^n,$$ such that $$ \phi_j \circ \phi_k^{-1} : \mathbb R^n \rightarrow \mathbb R^n \text{ is NOT a Homeomorphism},$$ [... contd.] $\endgroup$ Commented Mar 24, 2017 at 6:23
  • $\begingroup$ ... (i.e. not a homeomorphisms wrt. the "natural topology of $\mathbb R^n$"), then we can't say whether $\phi_j$ is not a homeomorphism, or whether $\phi_k$ is not a homeomorphism, or whether neither is a homeomorphism; much less which assignment of coordinates is a homeom. wrt. $M$. "given a reference frame $$e_{\mu} : M \rightarrow TM$$ we can [...]" -- Being already in trouble since "we don't know what $M$ is!", how could we possibly "know what $TM$ is" ?? (IMHO, that's asking for even more trouble; and a misuse of "reference frame".) $\endgroup$ Commented Mar 24, 2017 at 6:24
  • $\begingroup$ Possible duplicate of Global Properties of Spacetime Manifolds $\endgroup$ Commented Mar 24, 2017 at 15:17
  • $\begingroup$ @JamalS, although related I don't believe this is a duplicate. I explain the reason: if I understood well, the OP is asking there about topological properties of spacetime and how they relate to the solutions to Einstein's equations. Here the question is another: how does one construct coordinate systems and give meaning to coordinate functions in the context of GR when $M$ isn't known even as a set, let alone as a topological space/manifold? I also am trying to discuss whether or not all GR coordinate systems are construct adapted to some observer or frame of reference. $\endgroup$ Commented Mar 24, 2017 at 15:54