general relativity - Coordinates vs. Geometries: How can we know two coordinate systems describe the same geometry?

Specifically, I'm asking this because I'm taking a class on General Relativity, and in Hartle's book Gravity, in Ch. 12, after having spent some time using Schwarzschild coordinates, we are introduced to two new coordinate systems, the Eddington-Finkelstein coordinates and the Kruskal-Szekeres coordinates.

The book claims that the three of these coordinates are just different coordinate representations of the same spacetime geometry. How does one prove this claim, particularly since the coordinates do not even remotely behave identically (e.g.: when $r=2M$ in Schwarzschild coordinates, the system appears to blow up, but really the issue is with its representation in Schwarzschild coordinates)? Obviously, there would be no point having 3 different coordinate systems if they all behave identically everywhere, but I'm having difficulty reconciling these different behaviours as all being different descriptions of the same spacetime geometry.

Answer

In a sense it is obvious that there are different descriptions of the same phenomena; suppose you drop an apple in front of you, then move to the left - nothing, as far as the physics of the falling apple is concerned, has changed

If you describe the two situations in coordinates then the two descriptions will be the different - but they describe the same physics; thus we think there is some way of relating the two descriptions.

Mathematically, this is a coordinate change; and this is how vectors and tensors were classically described.

For example, contravariant transforms in such and such a manner; and so on for coviant vectors, tensors and densities; mathematically we say this is the local description.

In differential geometry one defines tangent, cotangent, tensor and form bundles globally; then by taking sections we get the corresponding field ie a (tangent) vector field, or a symmetric 2-tensor field, aka a metric.

If one then takes two overlapping sections, say of a tensor, and look at their transformation properties; these will be the same as the classical description of a tensor, i.e. a tensor of type p,q transforms in such and such a manner.

Finally, we cannot tell from one single transformation in a single patch the whole geometry, but the usual description of such implicitly gives us an atlas of patches, and we can glue them together to get the manifold and bundle structure. Usually this is left implicit, but the introductory part of a differential geometry text usually spells this out in detail. See for example Michors text Natural Operations on manifold and bundle atlases.