# Does spacetime in general relativity contain holes?

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.

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"bounded in size in spacetime" what do you mean by size? If there is a hole, you don't have a metric on the hole, and how to you measure the spacetime size of the hole? Note that topology doesn't generally concern sizes. –  Willie Wong Mar 12 '12 at 14:48
In any case, please see if this question on topology of GR answers your question already. –  Willie Wong Mar 12 '12 at 14:50
In numerical relativity one often excises black holes and replace them with isolated horizons or dynamical horizons. One can show that these are precisely the kind of "holes" necessary to maintain sympletic structure of the entire field theory (and one also gets extra terms like Chern-Simons restricted to the boundary). –  genneth Mar 12 '12 at 15:06
@WillieWong: Thanks for the link, I'll check it out and see if my question is considered in the answer. I don't know if my size-formulation was unclear maybe, but the idea is that the hole should be bounded in the sense that you can find a sphere of finite measure and wrap it around it. @ genneth: Mhm, unless it first evolves and then dissolves again, I think a black hole isn't a hole in 4-dimensional spacetime. –  NikolajK Mar 12 '12 at 15:16
For your second question see en.wikipedia.org/wiki/Geon_(physics). I don't think the existance of stable geons has ever been proved. –  John Rennie Mar 12 '12 at 16:58

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.

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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 black hole 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. The Schwarzschild metric is a vacuum spacetime, so the mass that it has (as, e.g., measured by a distant observer) is not made out of matter fields that are living in the spacetime itself.

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.

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You say "Manifolds don't have boundaries." Not true: en.wikipedia.org/wiki/Manifold#Manifold_with_boundary I'm sure Stokes is rolling over in his grave Ben. –  joshphysics Sep 2 '13 at 5:58
Manifolds don't have boundaries, Manifolds-with-boundaries have boundaries. –  MBN Sep 2 '13 at 11:10
@joshphysics: In addition to what MBN said, you can apply Stokes' theorem to a region within a manifold that has a boundary, but that doesn't mean that the whole manifold is a manifold-with-boundary. When you see manifolds-with-boundaries used in GR, usually it's in a context where the boundary is an idealized surface at infinity like $\mathscr{I}^+$; that surface is there for mathematical convenience, and doesn't represent a part of the spacetime that one could actually observe. It's similar to notations like $\int^\infty$, which isn't meant to imply that $\infty$ is a real number. –  Ben Crowell Sep 2 '13 at 16:11
@BenCrowell Ok sure, the strict mathematical definition of the term "Manifold" as a topological space that's locally homeomorphic to $\mathbb R^n$ disallows boundaries. I nonetheless think it's pedagogically unfavorable to make that statement; left unqualified, I think it will confuse people with a weaker mathematical background than yourself. Otherwise, I agree with everything you've said. –  joshphysics Sep 2 '13 at 17:09
@Ben Crowell. "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." So, am I understanding it correctly that, given the above conditions, you think holes or cutouts in the spacetime manifold are possible? Do I have that right? –  dcgeorge Sep 17 '13 at 17:21

Does spacetime in general relativity contain holes?

The Max Planck Institute for Gravitational Physic says: "the most drastic consequence of Einstein's description of gravity ... is the possibility that space and time may exhibit 'holes' or 'edges' ...". http://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?" http://arxiv.org/pdf/hep-th/9109030v1.pdf

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." http://arxiv.org/pdf/gr-qc/9406053v1.pdf

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.

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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. This is wrong. The infinity at the horizon is not just an unfortunate artifact of the Schwarzschild coordinate system. This puts you at odds with every relativist since 1960. –  Ben Crowell Sep 2 '13 at 16:14
it looks to me like an event horizon marks the edge of the spacetime manifold and the degeneration of the metric No, the metric is not degenerate at the horizon. It has the same signature there as everywhere else. 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. This is all nonsense. There is no such controversy. –  Ben Crowell Sep 2 '13 at 16:17
@Ben Crowell (The infinity at the horizon) Here's my source: "It's customary to assert that the Schwarzschild solution is unequivocally non-singular at r = 2m, and that the intrinsic curvature and proper time of a free-falling object are finite and well-behaved at that radius. ... However, ... with respect to the proper frame of an infalling test particle, we found that there remains a formal singularity at r = 2m. ... The free-falling coordinate system does not remove the singularity, but it makes the singularity analytically removable." mathpages.com/rr/s8-07/8-07.htm –  dcgeorge Sep 5 '13 at 15:47
@Ben Crowell (the metric is not degenerate at the horizon) That's true from the presently accepted perspective where spacetime is assumed to continue inside the horizon, but viewed in the context of the OP's question about "holes" in the spacetime manifold ... if such cutouts could exist ... would the metric not have to end and become degenerate at the edge of the hole? Black holes might not be such cutouts but the two would have the same vacuum metric and would therefore be indistinguishable. –  dcgeorge Sep 5 '13 at 15:51
@Ben Crowell (There is no such controversy.) Perhaps you could elaborate? I googled the phrase and got over a hundred thousand results. Here's one: "With the general theory of relativity, the traditional debate between absolutism and relationalism has been shifted to whether or not spacetime is a substance, ... " en.wikipedia.org/wiki/Philosophy_of_space_and_time –  dcgeorge Sep 5 '13 at 15:53