# 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.

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 if it has a non-trivial topology.

• 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 '14 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 '14 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 '14 at 18:49
• @Void A Snake fan? I always thought of tori in terms of Asteroids myself :) – user10851 Jul 8 '14 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 '14 at 3:01

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).

• Question is about "universe" and a standard requirement on universe is "isotropic". Torus is not isotropic, there is a specific direction to go around shorter diameter, there is a specific direction to go around larger diameter. I believe "flat universe implies infinite plane" is correct. – F. Jatpil May 9 '19 at 13:37
• Isotropic is an additional property that you may or may not require. For instance is it not valid in our universe at small scales. Anyway the point of my answer is that a big enough torus is at the same time flat, finite and locally indistinguishable from an infinitely flat plane universe. – Rexcirus May 10 '19 at 11:10

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?

• 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? – Nathan BeDell Jul 8 '14 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 '14 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. – Nathan BeDell Jul 8 '14 at 15:39
• @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 '14 at 15:43
• "Now, think of a flat, finite universe: Is it possible to maintain that all places are the same?" Why doesn't the 3-torus have that property? – Venge Jul 9 '14 at 2:52

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.

• 3-torus is not isotropic. – F. Jatpil May 13 '19 at 7:06

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.

The section in question appears to be:

"The Vardanyan model says that the curvature of the Universe is tightly constrained around 0. In other words, the most likely model is that the Universe is flat. A flat Universe would also be infinite and their calculations are consistent with this too. These show that the Universe is at least 250 times bigger than the Hubble volume. (The Hubble volume is similar to the size of the observable universe.)".

The Daily Galaxy via MIT Technology Review"

Technology review does make the aforementioned quote and says this as well:

"Other estimates depend on a number factors and in particular on the curvature of the Universe: whether it is closed, like a sphere, flat or open. In the latter two cases, the Universe must be infinite.".

Others and myself disagree.

A fairly simple explanation is provided at Wikipedia, all on one page.

Infinite or finite

One of the presently unanswered questions about the universe is whether it is infinite or finite in extent. For intuition, it can be understood that a finite universe has a finite volume that, for example, could be in theory filled up with a finite amount of material, while an infinite universe is unbounded and no numerical volume could possibly fill it. Mathematically, the question of whether the universe is infinite or finite is referred to as boundedness. An infinite universe (unbounded metric space) means that there are points arbitrarily far apart: for any distance d, there are points that are of a distance at least d apart. A finite universe is a bounded metric space, where there is some distance d such that all points are within distance d of each other. The smallest such d is called the diameter of the universe, in which case the universe has a well-defined "volume" or "scale."

With or without boundary

Assuming a finite universe, the universe can either have an edge or no edge. Many finite mathematical spaces, e.g., a disc, have an edge or boundary. Spaces that have an edge are difficult to treat, both conceptually and mathematically. Namely, it is very difficult to state what would happen at the edge of such a universe. For this reason, spaces that have an edge are typically excluded from consideration.

However, there exist many finite spaces, such as the 3-sphere and 3-torus, which have no edges. Mathematically, these spaces are referred to as being compact without boundary. The term compact basically means that it is finite in extent ("bounded") and complete. The term "without boundary" means that the space has no edges. Moreover, so that calculus can be applied, the universe is typically assumed to be a differentiable manifold. A mathematical object that possesses all these properties, compact without boundary and differentiable, is termed a closed manifold. The 3-sphere and 3-torus are both closed manifolds.

Curvature

The curvature of the universe places constraints on the topology. If the spatial geometry is spherical, i.e., possess positive curvature, the topology is compact. For a flat (zero curvature) or a hyperbolic (negative curvature) spatial geometry, the topology can be either compact or infinite. Many textbooks erroneously state that a flat universe implies an infinite universe; however, the correct statement is that a flat universe that is also simply connected implies an infinite universe. For example, Euclidean space is flat, simply connected, and infinite, but the torus is flat, multiply connected, finite, and compact.

In general, local to global theorems in Riemannian geometry relate the local geometry to the global geometry. If the local geometry has constant curvature, the global geometry is very constrained, as described in Thurston geometries.

The latest research shows that even the most powerful future experiments (like SKA, Planck..) will not be able to distinguish between flat, open and closed universe if the true value of cosmological curvature parameter is smaller than 10$$^{−4}$$. If the true value of the cosmological curvature parameter is larger than 10$$^{−3}$$ we will be able to distinguish between these three models even now.

Results of the Planck mission released in 2015 show the cosmological curvature parameter, ΩK, to be 0.000±0.005, consistent with a flat universe.

It is agreed that the universe is flat, or almost so.

Most people seem to disagree that flatness implies that the only size possible is infinite.

A flat piece of paper is not infinite, rolling it into a tube doesn't change it's size or weight.

They would have been in a better position to argue the inverse, that a sphere has a finite surface.

The failure to explain their reasoning is likely because it isn't true and logic doesn't support the statement.

Mihran Vardanyan (Oxford) has 3 papers on arXiv, 2 about the universe.

"How flat can you get? A model comparison perspective on the curvature of the Universe" (20 Apr 2009), by Mihran Vardanyan (Oxford), Roberto Trotta (Imperial College London), and Joe Silk (Oxford)

Page 14: "6 CONCLUSIONS

We have subjected the geometry of the Universe to a detailed scrutiny from a model comparison perspective, performing a three–way model selection with two physically motivated priors. We found that present–day data provide up to moderate evidence in favour of flatness (maximum odds of 66:1) for a specific choice of prior (the Astronomer’s prior) and assuming that dark energy is a cosmological constant. A Curvature scale prior and a relaxation of the assumption on the nature of dark energy weaken this result considerably, giving only inconclusive odds of less than 3:2 in favour of flatness. Correspondingly, the probability that the Universe is infinite lies in the range from 67% to 98%, depending on assumptions. If the Universe is not infinite, we have found a robust lower limit to the number of Hubble spheres, $$N_U \gtrsim 5$$.

"Applications of Bayesian model averaging to the curvature and size of the Universe" (28 Feb 2011), by Mihran Vardanyan (Oxford), Roberto Trotta (Imperial College London), and Joe Silk (Oxford)

Page 1: "The amount of curvature is usually characterized by the curvature parameter Ωκ: if Ωκ < 0 the geometry of spatial sections is spherical (i.e., the Universe is closed) and the Universe has a finite size. If instead Ωκ > 0 the geometry is hyperbolic (i.e., the Universe is open), while for Ωκ = 0 spatial sections are flat. In both the two latter cases, the spatial extent of the Universe is infinite.".

The definition of "spatial extent" is the maximum of the coordinates.

It seems as though he was misquoted.