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?

  • $\begingroup$ 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. $\endgroup$ Jul 13 at 21:59
  • $\begingroup$ Okay, I see. Thanks for the clarifications! I added an edited question which hopefully better approximates my confusion. $\endgroup$ Jul 14 at 0:15
  • $\begingroup$ Interestingly, the hawking's area theorem also applies to a hole - that it cannot shrink? $\endgroup$ Jul 14 at 0:29
  • $\begingroup$ There's some relevant info here: physics.stackexchange.com/q/144447/123208 $\endgroup$
    – PM 2Ring
    Jul 14 at 1:34
  • $\begingroup$ 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. $\endgroup$ Jul 15 at 19:58

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.

  • $\begingroup$ 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? $\endgroup$ Jul 16 at 12:29

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