I plan to study general relativity in the next few months and for now I keep gathering information about it. I read here and there that Penrose and Hawking proved that general relativity as we know it entails the existence of a singularity inside a black hole. I would like to know if this follows from an assumption of a Riemaniann structure where spacetime may lose it. Laurent Nottale in his books explains quantum effects by lack of differentiability of spacetime, so are there models in the literature where the supposed singularity is described as a "smooth fractal" (loose translation of French "fractale lisse")? Like in the visualization of Nash isometric embedding theorem, that is a $C^{1}$ (and not $C^2$) manifold with infinitely many corrugations (see http://hevea-project.fr/ENPageToreDossierDePresse.html)?
1 Answer
In general relativity the space manifold has a metric described by the Einstein equation, and there is no need for an embedding space. When we say there is a singularity in the metric this usually means a curvature singularity, where scalar invariants diverge as you approach it, and the point is usually taken to not be a part of the manifold.
The corrugated embeddings of manifolds in higher spaces do not affect their differentiability or their internal curvature and metric. So the problem near a black hole singularity is not due to that. It is due to the curvatures diverging, but in a standard Schwarzschild black hole the divergence is smooth, spherically symmetric and has a fairly simple form. It is only at $r=0$ where things become undefined.
