# A center everywhere

Posted by on Apr 7, 2013 in Blog, Science | 0 comments

In a few posts now, I’ve introduced the shell theorem and derived some differential equations for the scale factor. One feature of these derivations is that, although they begin with a coordinate system with some given origin, and some given proper distance to some specific mass (say, a galaxy), in the end, the details of the proper distance, the sample mass, and even the origin of coordinates all disappear.

This is consistent with an assumed symmetry of the problem; namely, that on a grand cosmological scale the density of stuff is uniform everywhere. In this way, it doesn’t matter what location is chosen for the center of mass in the initial problem. Anywhere is as good as anywhere else. If we did have a problem with a specific center of mass, then our test mass (our chosen galaxy) would be attracted towards that center. But in the case of this uniform distribution of matter, what does it mean to say that our galaxy is attracted towards the center?

Of course, the answer is quite simple. It means just what it says. If anywhere is as good a center of mass as any other place, then everything is collapsing together. Or expanding apart; that depends on the conditions of the problem, as we’ll see.

There are a few different ways to think about this. One way is to postulate, perhaps as in general relativity, that space is a manifold that has properties such as curvature, a metric, and so on. In this view, the metric of space is dependent upon the matter and energy within it; and the notion that space would have a “scale factor” that would depend upon the density of stuff within it is not at all surprising. After all, that’s just what Einstein’s Field Equations are about: matter and energy curve space-time, and space-time supports matter and energy.

Another way to think about this is to assume that space is nothing but measurable relationships between matter and energy. To say that space is expanding (or contracting) is just another way of saying that the measurable distances between objects that are not bound by strong forces are increasing (or decreasing).

Either way that floats your balloon is equally valid, as far as I can tell you. I personally like the model that space and space-time have certain properties derived from a metric that is itself an inherent attribute of the manifold. But that’s just me. It makes thinking about certain objective computations more natural.

You should understand that this cosmological expansion (or contraction) does not mean that our galaxy is getting bigger or that our solar system is getting bigger or that our Earth is getting bigger or that we are getting bigger or that protons are getting bigger or whatever. All of these elements are bound by forces that are much greater than whatever is driving the expansion of the universe, which kicks in only at scales above about $0.002 H_d$, where $H_d$ is the Hubble distance (about 14 billion light-years). At scales below this value, material is collapsing, galactic clusters are aggregating, stars are forming, galaxies are colliding, dinner is cooking, neutrons are decaying, and so on.

Now, on to weirder forms of matter and energy…