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General relativity is defined on a base manifold which, viewed as a topological space, is simply connected (which means there's no holes). However, we know that inside a black hole there's a singularity, a point of infinite density and curvature.

The way this was explained to me was thinking of it as a literal hole on the fabric of space-time, a point in which nothing is defined. This, however yields a serious problem. How is it possible that a theory defined on a simply connected space predicts holes on it? Wouldn't the prediction of holes in a space assumed to be simply connected prove the theory conceptually wrong?

paulina
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Tomás
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1 Answers1

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For physical reasons we usually assume that the spacetime manifold is connected, but it is not necessary to assume that it is simply connected. However, simply connected manifolds do have some nice properties that allow certain proofs or theorems which fail in non-simply connected manifolds.

One common class is regarding the existence of geodesics connecting two nearby points, meaning points close enough that the effects of curvature are negligible. In non-simply connected manifolds it is possible that there is no geodesic path connecting two such events because all paths must turn to avoid a small hole.

The Einstein field equations, the core of general relativity, can be well defined on a manifold with a hole. So despite the problems with geodesics, the curvature, the metric, and the stress energy tensor can still work.

However, having said all the above, the Schwarzschild spacetime is simply connected. The singularity in the Schwarzschild spacetime is spacelike, meaning that it is a moment in time, specifically the end of time.

The topology of the Schwarzschild spacetime is $R^2 \times S^2$. Both $R^2$ and $S^2$ are simply connected. So their product is simply connected also.

Dale
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