# How one pair of Schwarzschild coordinates can specify 2 different points on a manifold?

My question is regarding Schwarzschild solution. I always heard that coordinate chart is a one to one map from a manifold to real numbers. But when we look inside the black hole using Kruskal coordinates we see that each pair (r,t) in Schwarzschild coordinates appears in 2 distinct places on the diagram. Each point in the interior of black hole with a certain value of (r,t) has its counterpart inside the white hole. How can a ceratin point (r,t,$$\phi$$,$$\theta$$ ) specify 2 different places on a manifold? Is this a result of coordinate singularities?

• This is an abuse of notation. They are different charts, defined in different open sets, which one for simplicity decides to use the same notation for.
– Gold
Jan 23, 2020 at 19:03
• There is a thing worse than that. We have an entire line in Kruskal diagram that is t=infinity and r=2GM. So entire line corresponds to a single coordinate. Jan 23, 2020 at 20:12

Coordinate charts are always one-to-one from the manifold to some open subset to $$\mathbb{R}^n$$. This is because we use charts as local homeomorphisms to define what a manifold is (it can be defined in other, more abstract ways, e.g. locally ringed spaces, but let's not get into that).