# Topologically, is a curvature singularity just a hole?

Topologically speaking, a hole can be introduced into a manifold and it will still be a manifold, e.g. remove points within a 2-sphere of some radius from the cartesian plane and you'll still have a manifold.

Penrose's singularity theorems prove the existence (mathematically) of incomplete null geodesics, but not of curvature singularities, per say. So Im wondering if, in my naive view, maybe a topological perspective is better suited to describe curvature singularities? IF so, I'm not sure how to view curvature singularities topologically.

My question: is a curvature singularity, e.g. a black hole, in general relativity simply a topological hole of the spacetime manifold, or is it more topologically complicated? Or instead, is the singular structure not a part of the manifold, as suggested in @benrg's answer to this question, and thus is not a topological hole? Or is it not this simple, and there's some nuance(s) that I'm missing?

EDIT: I suppose I can phrase my confusion like this: how is it logically consistent to say that the physical singularity is not a part of the spacetime manifold (like how $$\infty$$ is not a point on the real line) AND that we can have a description of the singular structure from the metric itself (e.g., the Kerr metric has a ring singularity as can be shown from the metric)? Or do we avoid this confusion if the singularity is a topological hole?

• Thank you for the reference! Those other SE questions are helpful. benrg is correct, a singularity is a coordinate region that is not a part of the spacetime manifold. I suppose my question is then, what does it mean to "not be a part of the spacetime manifold" if geodesics terminate at that point? This is why I framed my question around topological holes. Jul 13 '21 at 21:59
• Okay, I see. Thanks for the clarifications! I added an edited question which hopefully better approximates my confusion. Jul 14 '21 at 0:15
• Interestingly, the hawking's area theorem also applies to a hole - that it cannot shrink? Jul 14 '21 at 0:29
• There's some relevant info here: physics.stackexchange.com/q/144447/123208 Jul 14 '21 at 1:34
• I posted these questions on two different posts on Math SE: math.stackexchange.com/questions/4199439/… and math.stackexchange.com/questions/4198452/… since I realized they are really math questions. Jul 15 '21 at 19:58

Or instead, is the singular structure not a part of the manifold, as suggested in @benrg's answer to this question, and thus is not a topological hole?

These are not two different interpretations. A topological hole is precisely something that is not part of the manifold.

Im wondering if, in my naive view, maybe a topological perspective is better suited to describe curvature singularities?

Again, this is not an "or."

Hawking’s theorem (S. W. Hawking, CMP 25, 152-166 (1972)) states that Surfaces of the event horizon in (3+1)D asymptotically flat stationary black hole spacetimes obeying the dominant energy condition are topologically 2-spheres.

In addition, if you consider an Euclidean integral of Gauss-Bonnet scalar of a BH ($$r$$ from horizon to infinity and Euclidean time from $$0$$ to $$1/T$$), you will get Euler characteristics $$\chi =2$$, which is indeed a sphere.

• Well, I"m asking about the gravitational singularity, not the event horizon. But for my curiosity, does one have to specify an energy condition in order to determine the topology of an event horizon? And so does the topology of the event horizon depend on the energy condition? Jul 16 '21 at 12:29

"When Schwarzschild set out to determine the gravitational field of a point source placed at the origin of $$R^{3}$$. His solution should therefore be differentiable on R × ($$R^{3}$$ - {(0, 0, 0)}) and singular on R × {(0, 0, 0)} without transgressing the notion of a mathematical point or more precisely, the notion of a line of universe of a point, yet he switches to the edge variety R × [0, +∞[×S² which has singularities on the edge R × {0} × S² yet the spatial metric α²dω², induced on the edge, clearly restores the topology of S². Therefore the spatial metric restores the topology of [0, +∞[×S². The space of the observer is thus identified with a real three-dimensional semicylinder, i.e., with a space different from the space $$R^{3}$$ which appears in the formulation of the problem. The point mass was supposed to be placed at the origin of $$R^{3}$$ which would be by hypothesis a singular point for the the metric. Now we do not have a singular point, but an infinity of singular points constituting the edge {0} × S² which has the power of the continu. The very content of the problem is deeply modified. However, one continues to speak of "the origin r = 0" as if it were a point."

summary or extract of a long document in French.

From what I understand, the problem lies in the use of change of variables (explicit or implicit) without taking into account the mathematical conditions that give them a logical meaning.

• Care to give a source for this "long document in French?" Dec 13 '21 at 17:01
• Dec 13 '21 at 18:29
• Thanks. What page did you quote from? And just FYI, it's best practice on this site to include the sources of your answer in your actual answer. Dec 13 '21 at 18:35

The best way to answer this question is to study the interior Schwarzschild solution. What happens as the compactness parameter $$\alpha=r_s/R$$ reaches value $$8/9$$? Metric components becomes $$g_{00}=0$$ and $$g_{rr}=1$$. Energy density $$\varepsilon$$ remains finite and pressure diverges as $$p~\sim r^{-2}$$. It is the "birth" of the central singularity. However, these words are physically meaningless. What we have for sure is the initial event horizon, defined clearly by spacetime points where time ceases, i.e. $$g_{00}=0$$. It is also clear that curvature invariant $$R$$, due to divergence of pressure goes to infinity. But in spherically symmetric spacetime curvature is intrinsic property of a 2-sphere and not of a non-dimensional point. Thus, one should interpret $$r=0$$ as a 2-sphere having zero surface area.