An open universe extends infinitely, like the cartesian plane. A closed universe wraps in on itself like a sphere or a torus (the surface of a doughnut, like Asteroids is played on). The problem is that you have to think of this in a four-dimensional sense, so the forever-expanding model is the one that goes on forever (in the time direction) and the collapsing model is the one that wraps in on itself.
On the contrary, the torus is finite, having only a finite area.
Thanks for the response.
OK. So it looks like I was both right and wrong at the same time. An open universe expands forever (infinite) because it is not going to collapse because there isn’t enough mass.
A closed universe has enough mass to eventually collapse and as a consequence its shape is a torus and is finite.
So all three of these concepts (shape, open or closed, collapse or not) are tightly inter-related?
If I may, I have a follow-up question. In an open universe, can it be said that it is currently finite? Let me explain why I ask this. There has been a certain amount of time that has passed since the Big Bang. The universe has been expanding at some rate. Time multiplied by rate gives distance traveled. Doesn’t this imply that there is a finite volume to the universe, even if it will be expanding forever?
The problem is that there’s no good definition of “current”. You can take a “spacelike 3-slice” of spacetime, which is basically a three-dimensional surface inside spacetime extending “all the way across” and such that no light beam fired from one point can ever hit the surface again. What GR states is that of the unimaginably infinite number of such surfaces through a given point, no one of them is any better than any other for defining “now”. Astronomers tend to use the one that’s pressed as closely to the past light-cone as possible (recieving all their information from light rays), but the equations don’t (and according to the Principle of General Relativity can’t) depend on which one you pick.
That said, all such slices in the best models we have for our universe have a finite volume, IIRC.
Brian Greene’s The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory has been the book of choice to recommend on this subject for many years.
Greene has a new book out, though, that seems to be even more relevant to this discussion, The Fabric of the Cosmos: Space, Time, and the Texture of Reality.
I’ve just started it, but he is concerned with the nature of space itself. Given the impossibility of trying to explain all the concepts needed in a paragraph on a message board, I’d recommend both books to anyone seriously interested in learning more. He starts from basics and builds to the extraordinary so slowly you don’t realize how much the ground has been pulled out from under your feet until you look down.
Hmmm, Mathochist, let me check something here. Yes, all those words are in English. I even know the meaning of each word individually. However, taken as a whole you may as well have been speaking Mandarin.
Sigh. Don’t get me wrong. The fault is mine, not yours.
Thanks Exapno. Your insight that my questions need more than a paragraph or two is spot on. Off to Amazon I go.
Yes, tightly related. For simplicity, let’s first assume trivial topology and no cosmological constant. In that case, the only three possibilities are spherical geometry, flat geometry, and hyperbolic geometry. The spherical geometry corresponds to a universe which will end in a Big Crunch, and any spacelike slice through it at any given time will have finite volume. In this universe, the sum of the angles of any triangle will be greater than 180 degrees (though the difference won’t be noticeable unless the triangle is close to the size of the universe). The hyperbolic geometry will expand forever, and any spacelike slice at any given time will have infinite volume (except at t=0 exactly, where nothing is really well-defined). In this case, the sum of the angles of a triangle will be less than 90 degrees. The flat geometry, where the sum of the angles of a triangle is exactly 180 degrees, is the borderline between these two cases, but qualitatively it’s a lot like the hyperbolic case: Expansion continues forever, and spacelike slices are infinite.
First complication: The Universe might have nontrivial topology, like an Asteroids screen (or something more complicated). In this case, even if it’s flat or hyperbolic, it can still be finite. But a flat or hyperbolic universe still lasts forever.
Second complication: The cosmological constant might not be zero (in fact, we’re quite sure that it’s pretty large). A positive cosmological constant, such as we appear to have, could, if it’s strong enough, cause a spherical universe to expand forever. Similarly, a sufficiently strong negative cosmological constant could cause a flat or hyperbolic universe to recollapse. This would not have any effect on whether the universe is finite or infinite. Although the cosmological constant is currently positive, it seems to vary in time, or even to turn off and on again. We don’t yet understand why this occurs, so it’s quite conceiveable that it could change radically in the future, and throw off any predictions we might make about the fate of the Universe.
Of course, you can potentially have both of these complications, which means that (for instance) you could have a universe which has finite volume and which recollapses, yet has hyperbolic geometry.
There’s an important distinction, by the way, between “the Universe” and “the observable Universe”. The latter is definitely finite, since the Universe is of finite age and the speed of light is finite, so there’s a limit to how far we can see in any direction. But this does not necessarily imply that the Universe as a whole is finite.
Can you say more about this?
Isn’t it true that the speed of the expansion of “the Universe” is also finite? Doesn’t this imply by the same logic used to limit the “observable Universe” that “the Universe” is also finite?
I remember reading a science article around 2000 or so that astronomers were fairly certain the universe had flat geometry, after studying computer models of other galaxies and their movements and some other data I don’t remember. Has this been called into question with more recent developments?
Expansion does not have a speed. Recession velocity is proportional to the distance from us. In an infinite universe, there is no upper limit to the distance and so no upper limit to the recession velocity.
Maybe the best way to think about it is to say that with an open universe, it starts out infinite. You seem to be thinking that at the BB the universe is small, and then expands. For an open universe, that’s not true. It’s infinite at all times.
Now, how does something that’s infinite expand? It can’t get any bigger! But the space between particles/atom/galaxies can grow, so that the density decreases.
This explanation had me ready to begin drinking this morning. (Well, it is St. Patrick’s Day.) OK. Expansion does not have a constant speed, but it must have an upper limit. Unless you’re saying that the upper “limit” of the recession velocity is infinite, in which case doesn’t that make the expansion instantaneous? So when the Big Bang occured, the universe expanded instantaneously to its current infinite size?
This is intriguing and may actually get to the crux of my inability to understand how the universe could be infinite. The bolded text is indeed how I viewed the Big Bang. It seems every explanation has talked about an “explosion” of space-time from an infinitely dense point containing all space-time.
Is your explanation that an open universe starts out as infinite commonly accepted? If so then I believe that the way this is explained to the lay person needs to be rethought. The commonly used metaphors are disturbingly misleading.
Is this concept something like how the infinite set of real numbers is bigger than the infinite set of integers, even though they’re both infinite?
I don’t think that would work. The “proof” of the big bang is basically by running the equations backwards and showing that under certain conditions (which we believe to obtain in “the real world”) the world lines of all particles pass through a single singularity in the manifold. If I had the Phone Book around I’m sure I could provide a better cite.
The only models I know offhand that are like FriendRob describes are the Friedman models, which are “empty” (containing no matter) universes with no cosmological constant.
The data and analysis behind that conclusion is still accepted. The problem is, it’s not possible to actually prove that the Universe is flat. All you can prove is that the curvature is very low, which means that the characteristic radius of curvature is very large compared to what we can see. In fact, the well-regarded inflationary model of the Big Bang predicts that even if the Universe is not flat, the curvature should be on a very large scale, consistent with what we observe. But an exactly flat Universe is an infinitely small target, so unless we have some explicit reason to believe it’s exactly flat, it’s considered extremely unlikely.
And any universe which is infinite at any time is infinite at all times. I find that thinking about graph paper can help explain how it expands. Picture an infinite sheet of graph paper. If the graph paper starts off ruled at a fifth of an inch, but then turns into quarter-inch ruled, it’s reasonable to say that the graph paper expanded, even though it’s still infinite. Now, suppose we have a sheet of graph paper where the size of the squares, in millimeters, is equal to the age of the graph paper, in seconds. At t=0, we would have squares of zero size, which doesn’t make sense. But that’s OK, we already know that we have problems describing t=0. But at any time after t=0, the squares have some nonzero size, and we have an infinite number of them, so the paper is infinite. Incidentally, if you derive Hubble’s Law for this graph paper, you’ll find that it works just as well as it does for the balloon, and every point still looks like the center.
Not really. The jump from the integers to the reals is an infinite jump, comparable to the jump between finite sets and the set of integers. What we’re dealing with here is more like a comparison between the integers and the even integers. In some sense, the integers are a “bigger” set than the even integers, even though they’re both equally infinite.
The infinite sheet of graph paper methaphor is useful. But I have to come back to the Big Bang question. Again, I’ve always envisioned this as a non-infinite “something” at t=0. Can the Universe pre-Big-Bang be infinitely small at the same time it’s infinitely large? Though your statement that “any universe which is infinite at any time is infinite at all times” leads me to the conclusion that I’m missing a fundamental point here somewhere. Maybe it’s because I’m attempting to visualize all this in three dimensional space.
I find it somewhat amusing that we can even talk about the idea that the infinite set of real numbers is infinitely bigger than the infinite set of integers, but the infinite set of integers is not infinitely bigger than the infinite set of even integers.
By the way, I appreciate everyone’s patience as I try to understand how the Universe before the Big Bang is still infinite.
Since the integers and the even integers can be placed in a one-to-one correspondence to one another, and that is the very definition of equal sets, there is no sense in which the integers are a “bigger” set.
Dern iggerant physicalists! 
The center of the universe is in the Dark Tower. We are all counting on the gunslingers.