This is a repost of a question I saw here:
Could the spacetime manifold itself end at the event horizon?
which was closed because it apparently didn't seem clear as to what the poster there was asking. However when I saw it I think I had a fairly immediate idea of what the asker is supposed to be asking about and I suspect there is a valid question here - however if I'm wrong, you can close this one too.
Namely, what I'm asking (and believe the OP of that Q was asking as well and I'm more asking it to resurrect the question in a better form) is the series of these questions, each of which builds on top of each other and it is asking in particular about the topological restrictions on spacetime manifolds in general relativity:
is it permissible for a space-time manifold in general relativity to have an edge in the same sense that a piece of paper (a 2D manifold) has an edge? (That is, the manifold has boundary points in the sense in topology, with the set of boundary points being of dimension one less than its own dimension i.e. if $M$ is the manifold then $\partial M \ne \emptyset$ and $\mathrm{dim}\ \partial M = (\mathrm{dim}\ M) - 1$)
if so, is it permissible for it to contain a "hole" in the same sense as if you punched a hole in said sheet using a hole punch (this is another 2D boundary, but we can enclose it with a loop and perhaps higher-dimensional enclosures in its higher dimensional analogue - another way to say this may be that the higher-dimensional boundary results in the manifold being not simply connected)?
if that is so, could the event horizon of a black hole be just such a boundary (or connected set of boundary points) of the spacetime manifold?
I note that it appears the regular singularity of an ordinary black hole is (at least from what I gather in readings) a boundary of dimension 1 - so the question basically is if you can have a boundary of dimension 3, and thus the manifold simply stops - as the sheet of paper does at its edge or better yet a hole punched in it - at the black hole's event horizon, so that the black hole is literally a hole in spacetime. One simplistic model of what is being asked would be just take the Schwarzschild spacetime and delete the interior part. No observer outside could tell the difference, right, so this would be consistent empirically, no? And anything falling in simply ceases to exist when it hits the horizon as that represents the termination of its worldline, same as with the singularity but here the "singularity fills the entire volume of the horizon"?
The context seems to be quantum-gravitational theories involving a black hole "firewall" and the idea is that at least on a cursory reading of some of the papers put forward suggested the firewall was just this kind of abrupt termination (mathematical boundary) of the manifold.
I also note however this question is fairly old - from 2013 - so I'd also be curious in knowing if that progress on the firewall problem since then has definitively been able to rule out the idea of such a space-time hole and if so, how exactly it did so (provided that was a proper characterization in the first place). Even modulo any connection to the firewall theories, what exactly would prevent a black hole from being such a "literal hole in space" as outlined in points (1)-(3) above?
Answer
is it permissible for a space-time manifold in general relativity to have an edge in the same sense that a piece of paper (a 2D manifold) has an edge?
Not really. The Einstein field equations only make sense at a point that has an open neighborhood of spacetime surrounding it, so we can only apply them on a manifold, not on a manifold-with-boundary. We do sometimes talk about a manifold-with-boundary in GR, but usually the context is that we're describing idealized points and surfaces that have been added to the spacetime, such as $\mathscr{I}^+$ or $i^0$. These are like vanishing points in perspective art. They are not actually part of the spacetime.
The fundamental reason that we do relativity on a manifold, not a manifold-with-boundary, is the equivalence principle. One way of stating the e.p. is that every region of spacetime is locally describable by special relativity. That is baked into the structure of GR and the Einstein field equations, and it would be violated at a boundary.
if so, is it permissible for it to contain a "hole" in the same sense as if you punched a hole in said sheet using a hole punch
GR doesn't impose any constraints on the topology of spacetime, so you can have holes. However, a hole does not imply a boundary in topology. If you take the Cartesian plane and remove the closed unit circle $r\le1$, you get a manifold, not a manifold-with-boundary.
if that is so, could the event horizon of a black hole be just such a boundary (or connected set of boundary points) of the spacetime manifold?
There is no physical motivation for doing this in classical GR. Nothing special happens, locally, at the event horizon. The event horizon is a set of points defined only in relation to distant points.
Historically, the misbehavior of the Schwarzschild metric, expressed in Schwarzschild coordinates, was not clearly understood at first. Later people realized that it was only a coordinate singularity. In GR, we aren't normally interested in spacetimes that are not maximally extended. When a spacetime has a proper extension, that is usually interpreted as meaning that something has just been artificially deleted from it. For example, you can take Minkowski space and delete a point, or delete everything at $t\ge 0$, but this is considered the kind of silly, artificial example that we want to rule out. We only want to talk about incomplete geodesics if the geodesics end at a singularity (a real singularity, not a coordinate singularity).
The reason that proposals such as firewalls are so radical is that they violate the equivalence principle. When, for example, people attempt to do semiclassical gravity and wind up with a prediction that something diverges at the event horizon of a black hole, it's a sign that their technique for doing semiclassical gravity isn't working properly. They can try to do things like renormalizations in order to get rid of this unphysical behavior. The basic problem is that semiclassical gravity lacks any clearly defined foundational principles. We have no reason to think that the techniques people use are valid approximation schemes.
I note that it appears the regular singularity of an ordinary black hole is (at least from what I gather in readings) a boundary of dimension 1
Not true. There is no standard way to define its dimensionality. See Is a black hole singularity a single point?
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