# Why does a flat universe imply an infinite universe?

This article claims that because the universe appears to be flat, it must be infinite. I've heard this idea mentioned in a few other places, but they never explain the reasoning at all.

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## 4 Answers

We need to be precise about the phrase the size of the universe. Specifically I'm going to take it to mean the maximum possible separation between any two points. In an infinite universe two points can be separated by an arbitrarily large distance, so if the maximum distance between two points is finite this means the universe must not be infinite.

The point of all this is that the distance between any two points is calculated using the metric. For a Friedmann universe like ours (at least we believe our universe to be a Friedmann universe) the metric is (in polar coordinates):

$$ds^2 = -dt^2 + a^2(t) \left[ \frac{dr^2}{1 - kr^2} + r^2d\Omega^2 \right]$$

The value of the parameter $k$ determines whether the universe is closed, flat or open. Specifically $k > 0$ is a closed universe, $k = 0$ is a flat universe and $k < 0$ is an open universe. The variable $s$ is the proper distance.

Now, suppose we choose an origin at some starting point, choose a fixed time, and calculate the proper distance, $s$ as we move radially away from the starting point. The question is whether $s$ can reach infinity or not. Because only $r$ is changing $dt = d\Omega = 0$, so the expression for the proper distance simplifies to:

$$ds^2 = a^2(t) \frac{dr^2}{1 - kr^2}$$

We'll choose our units of distance to make $a = 1$, and we'll consider only closed or flat space, $k \ge 0$, in which case we can integrate the above equation to give:

$$s(r) = \frac{\sin^{-1}(\sqrt{k}r)}{\sqrt{k}}$$

So the maximum possible value for $s(r)$ is when $\sqrt{k}r = 1$, in which case:

$$s_{max} = \frac{\pi}{2\sqrt{k}}$$

And there's the result we want. For a closed universe $k > 0$ and therefore the maximum possible distance between two points is finite. However as $k \rightarrow 0$ the maximum possible distance $s_{max} \rightarrow\infty$. That's why a flat universe is infinite.

However we should note that, As Rexcirus points out in his answer, even a flat universe can be finite is if has a non-trivial topology.

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For example the surface of a torus is flat but finite slight correction: the surface of a torus can be flat, but the generic 2-dimensional torus embedded in $\mathbb R^3$ isn't; Wikipedia informs me that it cannot be flat if the embedding is at least $\mathcal C^2$, but an explicit $\mathcal C^1$ embedding has been found rather recently (April 2012); see eg gipsa-lab.fr/~francis.lazarus/Hevea/Presse/index-en.html for some pictures –  Christoph Jul 8 at 17:18
I think general relativity can supply some topological information (see my old bounty question). If we know the metric, in certain cases we may apply the Gauss-Bonnet-Chern theorem to compute the Euler characteristic, which specifies topological information, namely the number of handles. So 'general relativity cannot tell us anything about the global topology' is inaccurate for certain cases. Nevertheless, +1, a good answer. –  JamalS Jul 8 at 17:49
For a non-expert, mentioning the torus may be confusing as usually people imagine the partially negative-curved and partially positive-curve one in $\mathbb{R}^3$ (as touched upon by Christoph). I would maybe edit the post to say it's like in the game Snake, where you can go forward forever on a plane, but appearing at the same place because the edges of the screen are "topologically glued together". But anyways, also plused. –  Void Jul 8 at 18:49
@Void A Snake fan? I always thought of tori in terms of Asteroids myself :) –  Chris White Jul 8 at 19:08
+1: "In an infinite universe two points can be separated by an infinite distance" - FWIW, I think this should be "arbitrarily large distance", or put another way, there is no maximum distance just as there is no largest number. Infinity is not a number and all that.. –  Alfred Centauri Jul 9 at 3:01
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This claim is simply wrong. The flat hyperplane is of course infinite, but non trivial topologies can be flat and still finite. The simplest example is the 3-torus, but there are even the Klein bottle and the Hantzsche-Wendt manifold.

See for example page 27 of Janna Levin - Topology and the Cosmic Microwave Background, which show you ten different closed flat 3-manifolds.

For further reading I suggest: William Thurston, Three-Dimensional Geometry and Topology, Princeton University press (1997).

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I think that it is important to note that (almost) everyone doing cosmology works within the framework of the FLRW universe.

This implies that we assume that the universe is spatially homogeneous and isotropic, i.e. 'every place is the same (at least on large scales)'. Now, think of a flat, finite universe: Is it possible to maintain that all places are the same?

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I suppose the universe being "infinite" may simply mean that it his infinite room for expansion, the area outside of what is currently filled with matter being an infinite void, correct? –  Sintrastes Jul 8 at 15:20
@Sintrastes No, it is not like that. The expansion of the universe is a whole other matter. As far as we know, there is NOTHING outside of the universe: There is also nothing to be filled. This is not in contradiction with our universe expanding, although it is hard/impossible for us to visualize. –  Danu Jul 8 at 15:25
So all of infinite space is necessarily filled with matter, thus meaning that the information content of the universe is infinite? I'm not saying that there is nothing outside of the universe, I'm saying that there is a void (still part of the universe) outside of the matter that has expanded from the big bang. That would make the information in the universe finite, but the size infinite, which I think is easier to comprehend. –  Sintrastes Jul 8 at 15:39
This all being hypothetical, of course. –  Sintrastes Jul 8 at 15:40
@Sintrastes Statements about 'the information content of the universe' are speculative at best, since they are hard to properly define. As for the 'void' you speak of: It is clear to physicists that most of the universe is not filled 'to the brim' with matter. There are, however, other things that we know as fields which are believed to permeate all of space. In fact, matter is viewed as excitations of these fields. –  Danu Jul 8 at 15:43
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Other answers have made clear the 'flat' only implies infinite given additional assumptions around the topology.

In short: A universe which is the same everywhere but not simply connected can be finite.

It's worth mentioning that whilst the main working model assumes that the universe is simply connected, the actual topology is an open and serious question.

Consequently there are ongoing studies on firstly the topological possibilities and secondly looking for them.

For example, the next simplest space would be a 3-torus. With that shape, and a sufficiently small universe, you might be able to see the same galaxies by looking in opposite directions in the sky.

[ As far as I am aware ] There is no hard evidence for such galaxy mirroring.

As a jumping off point, see Wikipedia Doughnut Universe, but there are also a load of technical papers on the subject.

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