Does spacetime in general relativity contain holes?

Question

Are there physical models of spacetimes, which have bounded (four dimensional) holes in them?

And do the Einstein equations give restrictions to such phenomena?

Here by holes I mean constructions which are bounded in size in spacetime and which for example might be characterized by nontrivial higher fundamental groups.

On a related note:

Could relatively localized and special configurations of (classical) spacetime be interpreted as matter?

I.e. can field-like behavior emerge in characteristic structures of spacetime itself, like for example holes in the above sense, or localized areas of wild curvature? Gravitational waves certainly go in this direction, although they act on such a huge scale, that in their case we probably wouldn't recognize an organized, maybe even life-like behavior as such.

Answer 1

The spacetime in general relativity does not contain "holes" in the sense of excized regions because of a physical argument--- if you can shoot a particle at the region, it should continue into the region. This is the reason that geodesic completeness is used instead of completeness in GR. The condition of geodesic completeness says that the manifold must not have places where geodesics stop for no reason.

Of course, the singularity theorems guarantee that geodesic completeness fails inside a black hole. But the failure in the case of time-like singularities is mild--- the singularity is only reachable by light rays.

The closest thing to an excized region is a black hole. The interior is excized in the sense that it is disconnected causally from the exterior. You can remove the interior and simulate the exterior only (classically) and you don't expect to run into too many troubles. Whether this is completely true in the quantum version is not clear to me.

As for other topological quantities, you can put them in by hand, but it is not clear if they can appear dynamically. There is the topological censorship conjecture, which states that you won't be able to see a topological transition in classical general relativity. I do not know the status (or even the precise statement) of this conjecture.

Answer 2

Does spacetime in general relativity contain holes?

The Max Planck Institute for Gravitational Physics says: "the most drastic consequence of Einstein's description of gravity ... is the possibility that space and time may exhibit 'holes' or 'edges' ...". https://www.einstein-online.info/spotlights/singularities

GR clearly allows spacetime to contain holes. Geodesic incompleteness is built into the system. The question remains, though, as to whether or not such holes actually exist.

I believe they do, and exactly in the sense of excised regions of the manifold. The excised regions are what we call black holes.

The radial component of the Schwarzschild metric tells us that the metric stretching of space goes to infinity at the event horizon. If the vacuum has intrinsic mass, this means that space itself thins out and disappears completely at the horizon. The region inside the event horizon, then, becomes a cutout in the manifold, i.e., a cavitation bubble, or hole, in spacetime.

The infinity at the horizon is not just an unfortunate artifact of the Schwarzschild coordinate system. A coordinate transformation to a constantly accelerating frame in flat, Minkowski spacetime (the "free-fall" coordinate system) only makes the infinity analytically removable. It does not actually remove it or change the effect of metric stretching.

So, assuming an intrinsic mass of the vacuum, it looks to me like an event horizon marks the edge of the spacetime manifold and the degeneration of the metric.

Such cavitation would likely involve a topological change as well as degeneration of the metric but it may not be a problem. Here is a paper entitled Topology Change in General Relativity by Gary T. Horowitz, Department of Physics, University of California, that appears to turn the question on its head: "The question is not whether topology change can occur, but rather how do we stop topology from changing? Why doesn’t the space around us suddenly split into disconnected pieces?" https://arxiv.org/abs/hep-th/9109030

And here is another paper on metric degeneration and topology change in general relativity that I ran across recently: " ... even in standard general relativity (couched in first-order language) Horowitz [20] has shown that it is possible to construct reasonable topology-changing spacetimes if degenerate metrics are allowed. So, this might well prove to be the correct approach to describing topology change." https://arxiv.org/abs/gr-qc/9406053

The other main issue is the question of intrinsic mass. Does the vacuum have its own mass? Is the vacuum a substance or is it not? ... the old debate of abstract relationalism vs. substantivalism. As far as I know, the issue is still undecided, so, cutouts in the manifold must remain a possibility.

In my humble opinion, this is the most interesting and most important question going in physics today. If such cutouts exist, it would force a paradigm shift in our thinking about the nature of reality.

The external metric associated with a cavity in the manifold would be (by Birkhoff's theorem) the same as that for a normal massive object or a black hole. So a cavity in the spacetime manifold would be gravitationally indistinguishable from either. This, in answer to your related question, is the way localized and special configurations of (classical) spacetime could be interpreted as matter.

Answer 3

Are there physical models of spacetimes, which have bounded (four dimensional) holes in them? [...] And do the Einstein equations give restrictions to such phenomena?

The notion of a hole or the size of a hole doesn't automatically make sense unless you form the hole by cutting something out of some larger manifold that you already had in mind. For example, if you cut a point out of a 2-sphere, you get something that has the topology of the Euclidean plane. In that sense, you could consider the Euclidean plane to have a hole in it.

The Einstein field equations are differential equations, and since derivatives are local things, the field equations don't "see" global features such as topology. If you start with any spacetime that is a solution to the field equations, and cut part of it out, the only condition for the field equations to remain defined and satisfied is that what's left is still a manifold. Manifolds don't have boundaries, so you just have to make sure that what you cut out is a closed set, so that what's left over has the topology of an open set.

Here by holes I mean constructions which are bounded in size in spacetime and which for example might be characterized by nontrivial higher fundamental groups.

There is no restriction on the size of the hole or its topological features such as whether it's knotted, etc. The only condition is that what's left over after cutting is still a manifold.

Could relatively localized and special configurations of (classical) spacetime be interpreted as matter? [...] I.e. can field-like behavior emerge in characteristic structures of spacetime itself, like for example holes in the above sense, or localized areas of wild curvature?

There are some cases like this that are of physical interest, and others that are not.

A naked singularity is an example that's of physical interest. The singularity is not considered part of the spacetime manifold, so it could be considered a "hole" in the topological sense.

As an example that's not of physical interest, we could take Minkowski space and remove one point. Because the field equations are local, the missing point is completely undetectable from any finite distance.

We can fill in missing points by extending the spacetime, and if we continue doing so as much as possible, we get what's called the maximally extended version of the spacetime. The maximally extended version may or may not be more physically realistic/interesting than the original. For example, the maximal extension of Minkowski space is actually the Einstein universe, which is a whole different creature, and may or may not be what you wanted to study. The maximal extension of the Schwarzschild spacetime contains lots of funky stuff like a white hole and a second copy of the exterior region; these features are not physically present in a black hole that forms by gravitational collapse.