Is it possible for metric expansion to create holes, or cavities in the fabric o

Question

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.

Explanation / Answer

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 R4. Just as there is a continuum of points in R4, 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.

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