Cosmology and Geometry

A Two-Dimensional Analogy

As it expands, there is no dot that is the "center", but if you stand on any dot you will see all other dots moving away from you. Furthermore, the rate at which they move away will be proportional to the distance: Dots close to you will be moving away slower than those further away. The expansion of the Universe appears to be like this, but in three rather than two space dimensions. That makes it much harder to visualize, but it is possible to describe it mathematically. Here is an animation that uses the balloon analogy to explain why light is redshifted in an expanding universe.

Notice some other important features of the balloon analogy that also carry over to the realistic case of our 3-dimensional expanding space.

The speed of a dot on the surface of the balloon relative to its neighbors is not created by anything that the dot is doing. All motion is caused by the expansion of the balloon (if the dots did move independent of the expansion of the balloon, this would correspond to peculiar motion above and beyond that implied by the expansion of the space).

The space (surface of the balloon) is not expanding "into" something (remember, hypothetical creatures living on the balloon see only the 2-dimensional surface of the balloon, not the third dimension in which we see it expanding). Instead, the space for the expansion is being "created" by the expansion itself because it is being stretched out of existing balloon.

As we note in the box below, the balloon analogy is just an analogy, and it is not a perfect one. There are some potentially misleading aspects of the balloon example if one takes it too literally.

Shortcomings of the Balloon Analogy The balloon analogy is very useful, but it is potentially misleading in several respects if we take it too literally. Spacetime is 4-Dimensional: The first is that the surface of the balloon has only two dimensions but space is 3-dimensional (and spacetime is 4-dimensional). We Can't Get Outside our Spacetime to Watch: The second is that in the balloon analogy we have the luxury of watching a 2-dimensional surface expanding in a 3-dimensional space. But since spacetime is 4-dimensional, the expansion of 4-dimensional spacetime should not be viewed as an expansion in a larger-dimensional space. That is, there is no physical sense in which we can get "outside" our spacetime to watch it expand like we can sit outside the balloon and watch it expand. The best analogy with the balloon is that we must imagine ourselves to be completely flat creatures confined to the surface of the balloon with no third dimension. Then we could still infer that our 2-dimensional space was expanding by doing geometry tests in our two dimensions, but we would have no sense of what it meant to get "outside" our 2-dimensional world to watch it expand because the third dimension would not exist for us. The Universe Is Not Being Expanded by an Internal Pressure: Finally, let us note that our observation of the 2-dimensional balloon in three dimensions can lead to another fallacy. Since it is increased pressure that blows the balloon up, we may be tempted to assume that there is a pressure associated with the Universe that causes the Hubble expansion. This is incorrect. The Hubble expansion is not generated by the pressure of gas or radiation in the Universe. In fact, if these are present they actually slow the expansion! (For more inquisitive students wishing an explanation of this counterintuitive result: First, expansion of the balloon is not caused by pressure but by the difference in pressure inside and outside the balloon. But the cosmological principle described later in this module implies that the pressure in one large part of the Universe must be the same as that in any other part, so there can be no pressure differences on cosmological scales. Second, in general relativity pressure, like mass, can act as a source of the gravitational field. Therefore, normal gas or radiation pressure actually increases the gravitational attraction within the Universe and slows the expansion.)