I am not a mathematician, a physicist or a cosmologist as my question will clearly demonstrate. If someone can explain this in terms I can understand, I’ll be quite grateful.
Last night I was watching The Universe, a show about. . . the universe. In it a scientist stated that in order for the universe to be infinite, it must be flat because if it was curved it would eventually meet itself. Why must that be? Why can’t it curve forever? Why is it impossible for a curve to never, ever end?
In my head I think of the space between 2 and 3. It is finite. However, if you divide the space in between by half and keep dividing by half you can divide the space in between these two numbers for forever. If you take the space in between one part of the curved universe and divide it in half, you can keep dividing it and never reach the end.
Another way I tried to think of it is: what is a flat universe’s infinity mean? It extends forever on a plane of-- universe. Why can’t the universe extend in all directions at all times in every way? If the universe is flat what is above it and below it?
I was trained as a topologist, but not a physicist, so I have handle on the mathematical side of things here. The sticking point seems to be that physicists assume that the universe has **constant ** curvature. If you try to draw a curve with constant curvature, it will meet back up with itself - a circle. You can certainly draw curves that go on indefinitely, but the curvature of these varies from point to point.
Maybe this Wikipedia article will help. There is a description there as to why physicists make the constant curvature assumption - as far as I can tell, it basically comes down to saying that no one has ever found any evidence against this assumption. It does seem to be, however, a fairly reasonable assumption.
I’ve never heard this assumption. Raft, are you suggesting that a parabola (say y=x^2) is a function for an oval if continued infinitely? Or are we saying parabolas and the like are not “curves”?
ETA: I see President Johnny covered my question. It’s the constant curvature that’s the issue.
Maybe a more direct question will help me. Given an infinite amount of space to extend a constant curve, won’t that curve reach toward it’s beginning but never actually touch it? If the space on (or in) which you are curving constantly is infinite, wouldn’t the curve also be infinite?
Actually a parabola is the dividing “point” between ellipses and hyperbolas. Ellipses have eccentricities 0 < e < 1, and hyperbolas have eccentricities e >1. A parabola has an eccentricity of e=1. If you look at the figures here
for example, you’ll see that a parabola is the intersection of a plane and a cone when the plane is parallel to a line defining the surface of the cone. If the plane is slightly less sloped you get an ellipse which does close upon itself. If the plane is more sloped you get a hyperbola which never closes. So in a sense you can think of a parabola as an ellipse that closes upon itself at infinity.
That’s not quite what constant curvature means. Roughly speaking, the curvature measures how fast the slope of the curve is changing.
Try drawing a curve on a sheet of paper, and pick a point on that curve. There is a unique circle which closely approximates the curve at that point. The curvature is related to the radius of that circle. (More precisely, the two are reciprocals). In a curve with varying curvature, different circles are going to be needed at different points on the curve. In a curve with constant curvature, the same circle will always work - which forces the curve to actually be that same circle.
The problem with this is that it doesn’t quite generalize to higher dimensions. Physicists generally deal with “intrinsic” curvature — the curvature that’s inherent in the surface itself — and only rarely talk about extrinsic curvature, which is how a surface curves within some larger space. A rough way of picturing the distinction is that if you could “flatten out” a piece of a surface without any distortion, then it doesn’t have any intrinsic curvature. For example, the surface of a sphere is intrinsically curved (as generations of map-makers have discovered, you can’t represent the Earth on a flat map without some distortion), but the surfaces of a cylinder or a cone are only extrinsically curved. In particular, this means that lines only have extrinsic curvature.
What’s more, I don’t think it’s even correct that “a curved Universe must be finite.” Even under the assumption of constant curvature (see below), you can still have a hyperbolic space with an infinite volume but non-zero curvature. Physicists and mathematicians do draw a distinction between these spaces and spaces that are more like spheres; sphere-like spaces are said to be those with positive curvature, and hyperboloid-like spaces are those with negative curvature. While it’s not true that spaces of constant negative curvature are always finite, but it is true that spaces of constant positive are always finite.
Yup, it’s just an assumption known as the cosmological principle. It’s an off-shoot of the Copernican principle — the postulate that it is highly unlikely that the Earth is at a special, privileged location in the Universe.
That’s the corollary to the question actually posed. If it is infinite and has a constant curvature, won’t it eventually come back around and touch itself? And thus, it’s no longer infinite. Well, a line can make infinitely many revolutions that way, but planets and shit will start banging into each other. So, that’s a bad thing I guess.
Anyway, in order for it to be infinite, it can’t have a constant non-zero curvature. Of course flat in the sense of the universe isn’t flat like a piece of paper on your table.
To be fair, the extrinsic/intrinsic distinction can be somewhat tricky to grasp, since our (faulty) intuition would indicate that all manifolds sit inside an ambient space. I’m not quite sure how physicists deal with the distinction pedagogically, but the differential geometry/topology courses that I’ve taken started with developing an intuition by looking at the extrinsic curvature of curves and surfaces, and eventually introduced intrinsic curvature later on.
I knew this. :smack: My training was in homotopy theory, and most of this geometry stuff is beneath me
I have a strong feeling what I am about to say isn’t sensible mathematically, but I’ll try anyway.
I can not draw a line of constant curvature on a flat surface without having it meet itself, but it seems to me I can if I just go up a dimension, e.g. drawing a line of constant curvature spiralling up an infinitely extended cylinder.
By analogy with the above, would it not be possible for space to have constant positive curvature if you go up a dimension? I guess that would be three-dimensional space “spiralling” through five dimensions, or is that nonsense?
If you’re dealing with intrinsic curvature, you don’t need to worry at all about how many dimensions the “embedding space” has, since there doesn’t necessarily even need to be an embedding space. And I’m pretty sure that it’s true that a space of constant positive intrinsic curvature must necessarily have finite volume.
Which still doesn’t change the fact that constant-negative-curvature spaces can be infinite, and in fact it takes some clever contrivance to arrange for such a space to be finite.
Ah! I think I may be getting a hint. There are assumptions about the ‘universal plane’ as it were that necessitate a constant curve will meet itself. Am I getting this?
No, I think not. I think you’re taking away, and I could be wrong in understanding your understanding, the precise opposite of what the conjecture is. And we need to be clear that it is, at this point in time, a conjecture; not a theory. At the root of it, we simply don’t know what’s going on at the edge of the universe because we can’t see it. We have no data. Until we have something we can point to and say “there it is!”, all we have are hypotheses, but no real working theory.
In short, flat in space isn’t the same as flat like a table top. It’s really hard to detect to some curvature in space anyway because our view of it is so restricted. It’s like deal with curve in the calculus; if we zoom in sufficiently close, a curve looks flat. A table top is objectively flat; we can see all of it. The universe we talk about is really the observable universe. We can’t really say what is or isn’t happening the point at which we can no longer see. It might be a big sphere, or not. We don’t really know.
I immediately thought of limits when I read the OP.
If you take an “infinitely long curve”, it would be flat. Take a curve with (a curvature based on) a radius of 1, and start increasing it, and what you get on a small slab near x=0 is a straighter and straighter line, but never perfectly straight as long as we’re dealing with r<∞, but if it is truly “infinite” than it could be a flat line.
This I understand. What got me going was the conjecture that an infinite universe must be flat because if it was curved, it would be finite because it would meet itself. Since we do not know the nature of what is outside of our observable universe, I wondered how this conclusion was reached.
If we can detect curvature (which, according to the show I watched, we haven’t) then that would mean the universe is finite. My question is-- why would you come to that conclusion? What’s to say that this curve does not persist for infinity?
The answer-- I thought-- was because of the assumptions we have about the nature of the time/space/place in which this universe exists. If we work with these assumptions, then a constant curve will eventually meet itself.
But now you’ve told me I’m wrong. Dang! I thought I had the universe figured out!
I think you’re close to understanding but are missing an important piece. What troubles me is your third paragraph, particularly the second sentence: “If we work with these assumptions, then a constant curve will eventually meet itself.” The thing is that any constant curve will eventually meet itself, assumptions be damned. The place where assumptions factor in is where we postulate that the universe must be constantly curved, if it is curved at all. If in fact the universe is constantly curved, then unless it is flat, it must meet itself and thus be finite.
If you accept that any constantly curved surface cannot be infinite, then you understand the whole thing. Trying to figure out why astrophysicsts conclude that the universe must be constantly curved is a separate issue that I’m sure requires a lot of specialized knowledge.
