# What does it mean that the universe is “infinite”?

This question is about cosmology and general relativity.

I understand the difference between the universe and the observable universe. What I am not really clear about is what is meant when I read that the universe is infinite.

• Does it have infinite mass or is it dishomogeneous?
• How can the universe transition from being finite near the big bang and infinite 14 billion years later? Or would an infinite universe not necessarily have a big bang at all?
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Basically, I think the idea that the universe is infinite comes from considerations of the large-scale curvature of spacetime. In particular, the FLRW cosmological model predicts a certain critical density of matter and energy which would make spacetime "flat" (in the sense that it would have the Minkowski metric on large scales). If the actual density is greater than that density, then spacetime is "positively curved," which implies that it is also bounded - that is, that there is a certain maximum distance between any two spacetime points. (I don't know the details of how you get from positive curvature to being bounded) However, if the actual density is not greater than that critical density, there is no bound, which means that for any distance $d$, you could find two points in the universe that are at least that far away. I think that's what it means to be infinite.

Overall, the observations done to date, paired with current theoretical models, are inconclusive as to whether the actual density of matter and energy in the universe is greater than or less than (or exactly equal to) the critical density.

Now, if the universe is in fact infinite in this sense, it still could have had a big bang. The FLRW metric includes a scale factor $a(\tau)$ which characterizes the relative scale of the universe at different times. Specifically, the distance between two objects (due only to the change in scale, i.e. ignore all interactions between the objects) at different times $t_1$ and $t_2$ satisfies

$$\frac{d(t_1)}{a(t_1)} = \frac{d(t_2)}{a(t_2)}$$

Right now, it seems that the universe is expanding, so $a(\tau)$ is getting larger. But if you imagine running that expansion in reverse, eventually you'd get back to a "time" where $a(\tau) = 0$, and at that time all objects would be in the same position, no matter whether space was infinite or not. That's what we call the Big Bang.

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Makes sense, the infinity is only defined in terms of metric, it couldn't be otherwise. –  Sklivvz Dec 14 '10 at 23:34
@Sklivvz: what do you mean infinity is defined in terms of metric? Metric tells you nothing about topology. Take a flat $n$-dimensional Euclidean space. Now roll it up into torus $S^1\times\cdots\times S^1$. You'll find that the new space again admits a flat metric but now it's compact. The same can be done with solutions of Einstein's field equations: you can compactify some dimensions. For the easiest reference, see Kaluza-Klein theory. –  Marek Dec 15 '10 at 14:53
"Most of the observations done to date, paired with current theoretical models, suggest that the density of matter and energy in the universe is less than the critical density, so we conclude that the universe is infinite." Current observations are within error bars of critical density, to a precision of a few percent. Therefore we know nothing about whether the universe is finite or infinite. The error bars are consistent with either a finite universe or an infinite one. –  Ben Crowell Aug 5 '11 at 4:41
"But if you imagine running that expansion in reverse, eventually you'd get back to a "time" where a(τ)=0, and at that time all objects would be in the same position, no matter whether space was infinite or not." I assume that the reason for the quotes on "time" is that you're aware that a big bang singularity isn't a moment in time. Regardless of whether the universe is finite or infinite, there has never been a time when all objects were at the same position. –  Ben Crowell Aug 5 '11 at 4:43
@Ben: (2 comments up) True, that could be worded better. I'll edit. (1 comment up) Exactly, this is basically where the simplified description I'm trying to use breaks down. I figured it was best not to get into the details too much, since the only point I'm trying to make here is to show that, even if the universe is infinite, it doesn't rule out the possibility of a Big Bang. –  David Z Aug 5 '11 at 4:48
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If the basic question is how we define whether the universe is finite or infinite, then the most straightforward answer is that in a finite universe, there is an upper bound on the proper distance (which is defined as the distance between two points measured by a chain of rulers, each of which is at rest relative to the Hubble flow).

"Does it have infinite mass[...]?" -- GR doesn't have a scalar quantity that plays the role of mass (or mass-energy) and that is conserved in all spacetimes. There is no well-defined way to discuss the total mass of the universe. MTW has a nice discussion of this on p. 457.

"[...]or is it dishomogeneous?" -- I don't understand how this relates to the first part of the sentence. You can have homogeneous or inhmogeneous cosmological solutions.

"How can the universe transition from being finite near the big bang and infinite 14 billion years later? Or would an infinite universe not necessarily have a big bang at all?" -- This was asked again more recently, and a good answer was given: How can something finite become infinite?

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