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It's easy to show that a singularity of the riemanian tensor scalar $R_{\alpha \beta \mu \nu} R^{\alpha \beta \mu \nu}$ leads to a singularity of the spacetime. But what about the other way round? Is it possible to proof that each singularity of spacetime is also a singularity of $R_{\alpha \beta \mu \nu} R^{\alpha \beta \mu \nu}$ ?

Alpha001
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    Obvious counterexample: Conical singularities. – ACuriousMind Nov 09 '15 at 15:02
  • Or just any singularity that is just a point removed from the manifold – Slereah Nov 09 '15 at 15:11
  • But is the conical singularity a real singularity of spacetime or only a coordinate singularity? Since it is possible to choos coordinates where the metric tensor is finite? – Alpha001 Nov 09 '15 at 15:13
  • @ACuriousMind Perhaps an added condition that the manifold is smooth everywhere? However already there are counterexamples. There is a coordinate singularity at $r=r_s$ where $r_s=\frac{2Gm}{c^2}$ in Schwarzschild's coordinates. Such singularities are called coordinate singularities. – Horus Nov 09 '15 at 15:27
  • You'll have to define "singularity of spacetime", but the conical singularity isn't a coordinate singularity because you can't even define the differentiable structure at the tip of the cone - there are no coordinates in which the tip is just another point in a manifold. You really have to say what you mean by "singularity", because usually they just will not be points on the spacetime manifold itself, but "limit points" which are not actually part of it. – ACuriousMind Nov 09 '15 at 15:52
  • Ok, I don't thought about coordinate singularities. Since the riemanian tensor scalar should be finte at coordinate singularities? I thought about real singularities, which can not be removed by a change of coordinats. Such like the point singularity in Schwarzschild or the ring singularity in Kerr. – Alpha001 Nov 09 '15 at 16:02
  • @Slereah Often (e.g. in Hawking and Ellis) a spacetime is defined as a manifold and metric that has no regular points removed. So to show something is a singularity by that (most common) definition you have to show that it is impossible to add points to the manifold and have it still be regular at all points. – Timaeus Nov 10 '15 at 18:24

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