Is it possible for metric expansion to create holes, or cavities in the fabric of spacetime?

According to the Schwarzschild metric, the metric expansion of space around a black hole goes to infinity at the event horizon. Normally, when something expands, it thins out, so, this metric stretching seems to imply that the energy density of the vacuum goes to zero at the event horizon. In other words the implication is that the spacetime manifold comes to an end there; making the black hole an actual hole, or cutout, in the spacetime manifold.

I'm definitely not an expert in these matters but I can imagine the fabric of space being pulled apart by the explosive expansion of the early universe; empty bubbles opening up, imbedded regions devoid of manifold, devoid of vacuum energy, ... true voids, like what, presumably, lies beyond our universe.

  • $\begingroup$ I too will be interested in the answer on this, though I suspect it will be tied up with "quantum" . The expansion you are envisaging is classical GR and the early universe is primarily "quantum". $\endgroup$ – anna v Dec 19 '12 at 4:59
  • $\begingroup$ Subsequent question by OP: physics.stackexchange.com/q/51845/2451 $\endgroup$ – Qmechanic Jan 23 '13 at 17:34

The problem is the phrase "fabric of spacetime" conjures up images of some material sitting in some background space. However, this is fundamentally antithetical to the way standard GR is understood. Spacetime is all there is - there is nothing outside of it. Yes, we say that it is "curved," and we show pictures of rubber sheets floating in "space," but the math behind GR - differential geometry - is all about intrinsic curvature. You don't need spacetime to sit in anything.

So when the metric changes, test particles may very well find themselves further apart, but spacetime exists between them as much as before. As a thought experiment, imagine two test particles flanking the "hole" you want to form. Before allowing space to expand, make a line of very closely spaced test particles between these two. Between which of these many new test particles does the hole form? Why should two particles, placed arbitrarily close to one another, find themselves all of a sudden arbitrarily far apart?

This is all trying to convey a picture that reflects the math. On the more mathy side, spacetime is defined to be a manifold, which is essentially a set of points, the neighborhood of each of which looks something like $\mathbb{R}^4$. Just as there is a continuum of points in $\mathbb{R}^4$, there is a continuum of spacetime.

Now there may be singularities inside black holes, and these cause problems, but we generally don't worry, given that they are conjectured to be always hidden from us behind event horizons. The event horizons themselves do not present a problem either. While it is true that spacetime may seem to have stretched by an infinite amount as an event horizon formed, we are free to adopt different coordinate systems, and appropriate choices of coordinates remove all the infinities one encounters in these cases. Moreover, these are sensible reparameterizations, in the sense that the new coordinates are smooth (infinitely differentiable) functions of the old and vice versa.

Of course, the preceding was a four-paragraph crash course in both general relativity and differential geometry, so it doesn't do much justice to either. I'd be happy to elaborate on any points that are particularly unclear.

  • $\begingroup$ Next, as you say "... test particles ... find themselves moving apart ..." as space expands, together with the "fact?" that the vacuum has a mass of it makes the argument that the manifold has to actually thin out as it expands. The test particles would separate at the point where the manifold breaks apart to form the cavity and find themselves on opposite sides of the hole. $\endgroup$ – dcgeorge Dec 19 '12 at 15:44
  • $\begingroup$ (This first comment somehow got erased, sorry) Thanks Chris but I do have more questions. First: You say "Spacetime is all there is ..." and I have no problem with that. But, supposedly, the universe was once tiny on it way to getting bigger which implies that it has an outer boundary. The holes I'm thinking about are the same sort of thing: the edges of the holes, likewise, mark the edges of the spacetime manifold. The "insides" of the holes are outside of our universe. The holes, therefore, have no volume, only surfaces. The spacetime manifold would still be continuous and self contained. $\endgroup$ – dcgeorge Dec 19 '12 at 16:32
  • $\begingroup$ "The Universe was tiny" is sort of an odd way of putting it. I think it's better to think about it as "the Universe was more dense, now it is less dense". $\endgroup$ – Kyle Oman Dec 19 '12 at 16:39
  • $\begingroup$ Finally, the coordinate system you refer to is, I presume, the "free-fall" coordinate system. Its distinctive feature is that space, itself, must be considered to be flowing radially inward at the Newtonian escape velocity where it reaches the speed of light at the horizon and infinity at the central singularity. It makes the infinity problem at the horizon “analytically removable” but, as I understand it, that's not the same as it being completely eliminated. It also assumes that the spacetime manifold continues inside the hole which would make it inapplicable in this case. $\endgroup$ – dcgeorge Dec 19 '12 at 17:36
  • $\begingroup$ @Kyle. Yes, the universe was more dense and now it is less dense but only because it was smaller earlier on. I think it exploded from a highly compressed state, expanded through it's equilibrium point (because of it's momentum) at which time the pressure instantly turned negative. At that point, the negative pressure caused it to "boil" forming vast numbers of cavitation bubbles. I have fun speculating about these things. $\endgroup$ – dcgeorge Dec 19 '12 at 18:03

A footnote to Chris' answer: there is sort a meaning to "a hole in spacetime" as it could reflect the topology of spacetime. For example the surface of a doughnut is smooth and continuous, but there is a sense in which the doughnut has a hole in it because the genus of the surface is one rather than zero.

I believe string theory allows for topology change of a manifold, but this is far outside my area.

  • $\begingroup$ Thanks John. If the spacetime manifold had only one hole in it, it would be a torus structure. If it had more than one hole, as I assume, I'm not sure what its structure would be called. Anybody? $\endgroup$ – dcgeorge Dec 19 '12 at 18:15
  • $\begingroup$ Two holes is a double torus math.ucla.edu/~bon/multitorus.html $\endgroup$ – John Rennie Dec 19 '12 at 19:26
  • $\begingroup$ A manifold with an isolated cavity inside it wouldn't be a torus, though. To make it a torus, the hole (or tunnel) has to intersect the surface in two places. So, what, topologically speaking, is a solid that has a disconnected cavity inside, a cavity that doesn't intersect the surface of the solid? $\endgroup$ – dcgeorge Dec 20 '12 at 4:57
  • $\begingroup$ This is possibly better moved to MathSE. Remember that a (2D) torus is just the surface. If you're considering 2D manifolds, a solid with a hole in is two disconnected surfaces. $\endgroup$ – John Rennie Dec 20 '12 at 6:45
  • $\begingroup$ Yes, this is getting a bit off the main topic and should be moved or just discontinued but I'm a new to the site and don't know how to do either. $\endgroup$ – dcgeorge Dec 31 '12 at 17:16

My answer to the question of whether or not metric stretching could create holes or cavities in the spacetime manifold is:


At least, the possibility seems not to have been eliminated. Here's an illustration I made that sums up the basic idea: http://dcgeorge.com/images/ThePhysicsOFACavityInSpace.jpg

Whether or not this cavitation phenomenon is possible depends on a number of fundamental and interrelated issues, each of which is arguable in its own right.

Chris White makes the following points:

  • you don't need spacetime to sit in anything. ... (I agree)

  • the math is all about "intrinsic" curvature. ... (I agree)

  • when the metric changes, test particles may very well find themselves further apart, but spacetime exists between them as much as before. ... (I don't agree, at least in part)

This last point doesn't take into account the thinning-out effect of metric stretching. If you stretch something, it's still there but only up to a point. If you stretch it infinitely, it isn't there any more.

Chris again:

  • the infinity at the event horizon can be "removed" by a change of coordinates.

Yes, the free-fall coordinate system that does this makes the infinity “analytically removable” but this is not the same as being completely eliminated. In any case, the purpose of "removing" the infinity is to extend the math on through the event horizon on the assumption that the manifold continues into the interior region. This, in spite of what the math of general relativity appears to be telling us.

Even worse, this coordinate system assumes space itself to be flowing into the black hole like water down a drain, reaching the speed of light at the horizon and infinite speed at the central singularity. I find this unsatisfactory and unconvincing for a number of reasons but mainly because manifolds, as Chris says, are defined intrinsically, i.e., without reference to anything "outside". So, if the manifold were to end at the event horizon, would the coordinate system not, simply end with it?

The main issue, then, seems to come down to whether or not the spacetime manifold can be broken open.

Chris says:

  • You can no more tear spacetime than you can tear the abstract notion of the x,y-plane.

Oh? Is there some hard and fast rule that manifolds can't be broken? Just last week I read that Grigori Perelman (in order to prove the Poincaré conjecture) studied the nature of singularities in a 3-manifold and found that certain, simple kinds could exist inside the manifold (all were variations on stretching a sphere along a line). He then cut the manifold apart at the singularities and morphed the pieces (using Ricci flow) back into simple manifolds.

What's especially noteworthy here is that the degenerate form of these singularities, an undistorted, spheroidal singularity, is exactly the picture presented by the radial component of the Schwarzschild metric.

Perleman's singularities and his method of removing them demonstrates that manifolds can be broken up and at least suggests this as a possibility for the spacetime manifold. Given this possibility, metric stretching ending in cavitation would seem to be a clean, natural way to do that.

And, least we forget, mixed in with all of this is the question of:

  • whether of not the spacetime manifold has its own intrinsic mass.

The intrinsic mass of the vacuum has been an ongoing debate for decades but the general consensus appears to be moving in favor of it having a mass which is proportional to the cosmological constant Lambda.

The bottom line, it looks to me, is that the possibility of metric expansion creating cavities in the spacetime manifold hasn't yet been eliminated. Considering the implications, would it not be worthwhile for the question to be rigorously explored by experts in the field?


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