### Re: Is there a simple way of proving that the universe is not infinitely large?

Date: Sun Apr 22 19:53:48 2001
Posted By: Steve Furlanetto, Grad student, Astronomy, Harvard-Smithsonian Center for Astrophysics
Area of science: Astronomy
ID: 987244419.As
Message:

Good question! In fact, this does not prove that the universe is finite, but it is a technique that astronomers have tried to use to map the distribution of matter in the universe.

The reason that this doesn’t prove that the universe is finite is fairly straightforward, at least until you start to think about it hard. Imagine that you have a shell of matter of a uniform density (like a basketball). It turns out (and this was first shown by Isaac Newton, although it took him quite a while) that this shell exerts no gravitational force at any point interior to the shell. This is easy to see if you are sitting at the exact center of the sphere. Each point on the sphere pulls you toward it, but each point has one exactly opposite it pulling you in exactly the opposite direction, with the same amount of force. So the force due to each point cancels with that opposite it, and there is no net force.

The same is true at any point inside the sphere, though it is less obvious. The mathematical proof can be found in any calculus-based physics book, and I imagine you will get to it in your physics class one of these days. The proof works because the gravitational force decreases with distance squared, while the amount of material in any direction increases with distance squared, so the cancellation still works (check out a physics book for details).

As long as the universe is spherically symmetric (which is very close to being true), we can divide it into shells like the ones above. All the shells outside of us don’t contribute any gravitational force, because each piece cancels with one opposite it. In this view, the material in the shells inside of us is what slows down the expanding universe (the “Hubble flow”).

There are a couple of subtleties in this. The first is that, if the universe is infinite, how do we define the centers of all the shells so that we know which ones are external to us? This can be fixed by using general relativity, Einstein’s theory that replaces Newtonian gravity in a self-consistent way. In general relativity it turns out that the choice of a center won’t matter. But that brings up the second subtlety – is an analogous result to the cancellation even true in general relativity? Fortunately, it is, thanks to a result called “Birkhoff’s Theorem” that again lets us ignore all the shells external to us.

However, this is only exactly true if the universe is really spherically symmetric. While that’s true on the largest scales, it obviously isn’t true locally – after all, the sun is in one particular direction! So we’d expect some small deviations from the exact cancellation. For example, suppose that there’s a large chunk of matter in some direction. We’d expect to feel a force accelerating us in that direction. Correspondingly, if there’s an absence of matter in some particular direction, we’d expect to feel a force in the opposite direction.

These forces would of course make us move, and astronomers have tried to measure our motion to determine in which direction we are moving. (The first step in doing this is to subtract off the Hubble expansion velocity that all galaxies experience; the extra motions will be on top of this). There turn out to be several special motions. First, our galaxy is falling into the Virgo cluster, a large agglomeration of matter several million light years away toward the constellation Virgo. On top of that, the entire Virgo cluster (and us) are moving toward an enormous amount of matter (that we can’t see) known as the Great Attractor (in the direction of the Hydra and Centaurus constellations). We aren’t exactly sure what makes up the Great Attractor, but it is at least in part a cluster of clusters of galaxies, or a “supercluster.” However, there seems to be even more mass in that direction than we can see directly. So an extension of the method you’ve thought of allows us to learn something pretty fundamental about the universe that would otherwise be inaccessible to us! You can learn more about this by searching our site for
Great Attractor

Astronomers have also measured the motions of other galaxies and tried to turn those into maps of the matter densities. It is a very clever method, but unfortunately the observations are very difficult so it hasn’t been too successful yet.

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