Geometry, Topology and Destiny

By Mark Trodden | April 8, 2012 1:56 pm

I’ve reached the cosmology part of my General Relativity (GR) course, and one of the early points that comes up is my traditional rant against confusing three very distinct concepts when thinking about the universe. Roughly stated, these are; What is the shape of the universe? Is the universe finite or infinite? and Will the universe expand forever or recollapse.

When we apply GR to cosmology, we make use of the simplifying assumptions, backed up by observations, that there exists a definition of time such that at a fixed value of time, the universe is spatially homogeneous (looks the same wherever the observer is) and isotropic (looks the same in all directions around a point). We then specialize to the most general metric compatible with these assumptions, and write down the resulting Einstein equations with appropriate sources (regular matter, dark matter, radiation, a cosmological constant, etc.). The solutions to these equations are the famous Friedmann, Robertson-Walker spacetimes, describing the expansion (or contraction) of the universe.

It is important to take a moment to emphasize what we have done here. GR is indeed a beautiful geometric theory describing curved spacetime. But practically, we are solving differential equations, subject to (in this case) the condition that the universe look the way it does today. Differential equations describe the local behavior of a system and so, in GR, they describe the local geometry in the neighborhood of a spacetime point.

Because homogeneity and isotropy are quite restrictive assumptions, there are only three possible answers for the local geometry of space at any fixed point in time – it can be spatially positively curved (locally like a 3-dimensional sphere), flat (locally like a 3-dimensional version of a flat plane) or negatively spatially curved (locally like a 3-dimensional hyperboloid). A given cosmological solution to GR tells you one of these answers around a spacetime point, and homogeneity then tells you that this is the same answer around every spacetime point. This is what we mean when we say that GR tells us about geometry – the shape of the universe – as depicted in the NASA graphic below.

This raises a very different question that is often confused with the one above. If our solution tells us that the universe is locally a 3-sphere (or flat space, or a hyperboloid) around every point, then does that mean it is a 3-sphere, or an infinite flat 3-dimensional space, or an infinite hyperboloid. This is really a question of topology – how is it connected up – which also answers the question of whether the universe is finite or infinite. To illustrate the point, suppose we have solved the cosmological equations of GR, and discovered that at every spacetime point, the universe is locally a flat 3-dimensional space. This is, by the way, what observations actually indicate our universe is like. Then, just off the top of your head, you can think of many different spaces with precisely this same property. One example is, of course, that the universe is indeed a flat, infinite 3-dimensional space. Another is that the universe is a 3-torus, in which if you were to fix time and trace out a line away from any point along the x, y or z-axis, you traverse a circle and come right back to where you started. This is a finite volume space, that is connected up in a very specific way, but which is everywhere flat, just like the infinite example. In two dimensions, one might visualize it as

Of course, I could have only made one or two directions into circles (leaving it still infinite in some directions), or made the space into a finite one with more than one hole, or any number of other possibilities.

This is the beauty of topology, but it is not something that solving the equations of GR tells us. Rather it is an extra input into our solutions. It is, however, something we can test, most precisely through measurements of the Cosmic Microwave Background radiation, as I may discuss in a later post.

Completely independent of questions of topology, the geometry of a given cosmological solution raises another issue that is often mixed up with those of geometry and topology. Suppose that the universe contains only conventional matter sources (regular matter, dark matter and radiation, say), and suppose you know (you might question whether this is truly possible) that this is all it will ever contain. Then the equations easily predict that, in the case of positive spatial curvature, an expanding universe will ultimately reach a maximum size and recollapse in a big crunch, whereas flat or negatively curved universes will expand forever. These are predictions of the destiny of the universe, and often lead to the following connection

However, as I made clear, there are some assumptions that go into the connection between geometry and destiny, and although these may have seemed reasonable ones at one time, we know today that the accelerated expansion of the universe seems to point to the existence of some kind of dark energy (a cosmological constant, for example), that behaves in a way quite different from conventional mass-energy sources. In fact, we know that for sources like this, once acceleration begins, it is easily possible for a positively curved universe, for example, to expand forever. Indeed, in the case of a cosmological constant, this is precisely what happens.

So the universe may be positively or negatively curved, or flat, and our solutions to GR tell us this. They may be finite or infinite, and connected up in interesting ways, but GR does not tell us why this is the case. And the universe may expand forever or recollapse, but this depends on detailed properties of the cosmic energy budget, and not just on geometry. Cosmological spacetimes are some of the simplest solutions to GR that we know, and even they admit all kinds of potential complexities, beyond the most obvious possibilities. Wonderful, isn’t it?

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About Mark Trodden

Mark Trodden holds the Fay R. and Eugene L. Langberg Endowed Chair in Physics and is co-director of the Center for Particle Cosmology at the University of Pennsylvania. He is a theoretical physicist working on particle physics and gravity— in particular on the roles they play in the evolution and structure of the universe. When asked for a short phrase to describe his research area, he says he is a particle cosmologist.

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