# 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?
• Since when did the universe become infinite? Space could be infinite but matter can't. If it was, the universe would collapse under it's own gravitation instantly. Commented Apr 25, 2016 at 6:10
• In regard to the above comment - surely matter can be infinite if space is infinite Commented Feb 25, 2020 at 21:58
• Also, if we're treating matter as something homogeneous, one half of an infinite amount of matter would be within the outer half of the volume that all of it occupied, and would, consequently, tend to attract the matter located inward from the imaginary boundary between them, toward the other half: The effect would exactly cancel that described by Udit Dey, thereby leaving the universe static, without collapse but also without the dynamism that we actually observe. Commented Jan 24, 2022 at 21:42

## 3 Answers

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, but as suggested by a commenter, look into Myers's theorem if you're curious.) 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.

• Makes sense, the infinity is only defined in terms of metric, it couldn't be otherwise. Commented Dec 14, 2010 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. Commented Dec 15, 2010 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.
– user4552
Commented Aug 5, 2011 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.
– user4552
Commented Aug 5, 2011 at 4:43
• "if the actual density is not greater than that critical density, there is no bound" This is not true. A space of zero or negative constant curvature can be compact (e.g. torus). I think it is fair to characterise such compact topologies as somewhat less "natural", whatever that means, than an infinite space, but one should not imply that they are a mathematical impossibility. Commented Jun 14, 2021 at 8:49

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?

The universe would be infinitely large now if it started out infinitely large.

If the universe is infinitely large it can still expand, in the sense that the distances between the galaxies can get larger over time.

But we have no way to know whether it is infinitely large. It could be finite.

I think it is possible that when the concept of "infinity", which is a mathematical concept of considerable subtlety, is applied to physical things such as number of galaxies, it may be that we do not really know what we are talking about. It may be a useful way to make progress by simplifying certain kinds of calculations. This is a common method in physics: we use a potential well with infinitely high walls, for example, or a wave with perfect frequency and therefore infinite extension, and delta functions and things like that. These are all useful as mathematical techniques but we do not need to think there really could be a well of infinite depth, a wave of infinite length, etc. Similarly, in cosmology, infinity can be a helpful way to simplify away various issues which we think are not central to whatever is being studied.

A final remark. You often see it written down, in this context, that if the universe were homogeneous (on average at large scales) and had flat or negative spatial curvature on average, then it follows mathematically that it would be spatially infinite. This is not true. The intrinsic curvature does not dictate the large scale topology. You can have a finite space with either positive, zero, or negative curvature. However, it is fair to say that when the curvature is zero or negative then the finite (or "compact") spatial topology feels less natural, and involves a loss of isotropy at the largest scales.

• "However, it is fair to say that when the curvature is zero or negative then the finite (or "compact") spatial topology feels less natural, and involves a loss of isotropy at the largest scales." I interpret what the quote says as considering a universe that is finite with respect to mass/energy in a space that may be much larger (possibly infinite) than the space in which there is mass/energy. Is the what you meant?
– Buzz
Commented Jun 16, 2021 at 18:51
• @Buzz we normally guess that the stuff in the universe is distributed roughly uniformly. It would be odd to think it ran out somewhere and then after that was loads of more space. This is not an area of certain knowledge, but a finite spread of stuff and then an infinite amount of further space is not what is normally considered in physical cosmology. We normally think the stuff is spread roughly smoothly throughout whatever space is available. Commented Jun 16, 2021 at 18:58
• Thank you for responding to my comment. I now see I misunderstood what the quote in my comment meant. However, I now do NOT understand "loss of isotropy at the largest scales". Please explain WHY an infinite universe at the largest scales loses isotropy.
– Buzz
Commented Jun 18, 2021 at 14:40
• @buzz no: the infinite case can be isotropic. The finite case can be locally isotropic in all cases, but not globally isotropic in some cases. Commented Jun 18, 2021 at 16:53
• "You can have a finite space with either positive, zero, or negative curvature." I suggest that you might want to state this quote a bit differently. The way it is phrased does not make clear that a flat or negative curvature universe can be finite only by conjecturing that its geometry is not entirely isometric. The conjecture might be true if the finite space is sufficiently larger than the observable universe that he loss of isometry cannot be observed.
– Buzz
Commented Jun 23, 2021 at 14:20