Wednesday, January 16, 2008

The Balloon Blowup Analogy

Jim Kelnhofer, underwater videographer extraordinaire, asked the following nontrivial question (about cosmology, when it comes to underwater video, the questions flow in the other direction:)
ok, so I've been thinking about space and time a bit and the fact that there is no real center to the universe, or so everyone says. The analogy that I kind of get is that points in three dimensional space are kind of like points on the surface of a balloon. And, as you blow up the balloon all the points recede from one another as the universe, represented by the balloon, grows. I can sort of conceptually get this, though I don't completely comprehend all the math involved in the 3D/2D transformations. My question would be, is the expansion of the balloon representative of time, and does that translate to the center of the universe being time t=0? Is that how the math works out or what the math means?


Here's my less than satisfying response.

One way to understand a weird space is to formulate it as a subspace of a simpler space. For example, we can understand a balloon, the surface of a sphere, which is a slightly tricky space, by thinking of its embedding in good old flat three dimensional space.

General Relativity, on the other hand, is formulated in terms of Differential Geometry. In Differential Geometry, the spaces are called Manifolds. By definition an n-dimensional manifold can be completely covered (nicely) by n-dimensional Euclidean slabs (called charts in the literature).

By the way, if you think about it, the surface of a sphere cannot be modeled so nicely by just one chart (you end up leaving out at least one or two points, like the North and South poles). It takes at least two charts.

So there are these two ways to bootstrap our knowledge and intuition about Euclidean space to handle more general spaces: embeddings and coverings.

Can a given n-dimensional manifold (defined by coverings) always be embedded in a "nice" way in some higher-dimensional Euclidean space? That's an interesting question. John Nash, the dude depicted in the movie and book "A Beautiful Mind" found some impressive stuff along those lines.

General Relativity doesn't start off by worrying much about embeddings, it's fundamentally about the local properties of the space - that was the way Einstein went about it. Local properties can usually be addressed nicely by a covering formulation, you don't even need an embedding as it turns out.

There are more complications, in addition to Differential Geometry, once Time and Causality are involved.

There are however various ways to try to visualize a global picture of different space/time models. One popular way is the Penrose diagram.

Unfortunately it's not very easy to get from the local situation to a more global picture. It fact it is damn hard.

For example, Einstein came up with the local formulation, but he never got very far himself with global solutions (the ones he found are pretty trivial really).
One of Stephen Hawking's claims to fame is a book called "The Large Scale Structure of Space/Time" which is full of very hairy math addressing these global questions.

So the way mathematical physics often works is, people try to get the local formulation right, then they do some very hairy stuff to see what sorts of global implications follow.

What's been going on in the mathematics community, getting back to the time of Gauss in the 1800's, is that they've been able to come up with all kinds of bizarre "geometries" that are quite different than the Euclidean geometry that seems to be intuitive for most of us human types. However, the mathematical work tends to have a purely spatial flavor about it. Einstein happened to have a mathematician buddy who showed him one of these formalisms called Riemannian Geometry and Einstein basically kludged Time in. The nature of the kludge is that, locally, everything looks like Special Relativity, and the space-time geometry of that had already been worked out, by Einstein and other dudes at that time. The mathematical consequences of this have been slow to unfold, even though it's now been over 90 years.

Mathematicians have been very creative, they have given us powerful tools for formulating things with lots of dimensions and weird shapes. Sometimes there's even originally a physical motivation for some of that, but often they're just spinning out lots of crazy new junk for their own amusement. Every once and a while, someone with a more practical bent grovels through the math literature and finds something inadvertently useful, to the horror of Pure Mathematicians everywhere.

The interesting thing now is, we have these observations (red shifts, the cosmic microwave background, etc.) that seem to give some hints about what the global picture might be like. After the astronomers observe something new and cool, the physicists have to see if they can reproduce those results in their models. Lately they've been having a hard time of it, hence all the buzz about Dark Matter and Dark Energy.

So after all that did I even manage to address Jim's question about the Balloon Analogy? Not really! I think that model might have been specifically designed to show the universe could be finite, but still not have a center, at least at t>0. Even so there is apparently a singularity at t=0 when, if you extrapolate backward, the balloon seems to just materialize out of nowhere: a Big Bang.

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