One often hears that space is “curved”, and that if you continue traveling straight in one direction, you will end up back where you started. The common analogy is that of the surface of the earth; to a casual observer it looks superficially like an infinite flat plane, but it is in fact curved, and if you travel in one direction long enough (~25K miles), you do indeed end up back where you started.
Question:
Is the curvature of the universe some kind of supposition, or is it a robust theory that is well-supported by observable facts? If the latter, what are those supporting facts/observations?
The Universe is saddle shaped but most of us (well, me) struggle with that. An hyperbolic paraboloid where parallel lines meet. The balloon analogy is easier to imagine but the saddle is more correct.
I’ll leave it to others far more clever than I to explain further.
It should also be noted that not all “curved” universes allow return. It has to also be closed. In a toroidal universe (which is both closed and curved) you can travel in a straight line all you like, but will never return to your initial point (or so I was informed).
My personal theory (which I came up with when I was about seven) is that the universe is shaped like a ring, only in a demension we can’t perceive - so yeah, you end up back where you started.
Pac-man is not necessarily toroidal (but not inconsistent with it either) because it can be played on a cylinder. The game Asteroids, on the other hand, is toroidal. The difference is the difference between two-dimensional space and three-dimensional space. You can think of a two-dimensional torus by connecting opposite sides of a square. The three-dimensional torus is obtained in a similar way, except by identifying opposite sides of a cube. I couldn’t find any good animation of this in a quick search though.
To answer your question, yes, a torus can be flat. In fact, if you don’t bother with the folding, but just accept that a spaceship can move off one side then appear on the other (as in Asteroids or Pac-man), that is flat.
It is possible to do so in a toroidal universe, but there are some straight lines that will return. In the two dimensional torus, a straight line which doesn’t return will get arbitrarily close to your starting point, however, so although it doesn’t exactly return, for all intents and purposes, it does.
Locally, space is certainly curved, positively in some places and negatively in others. This is what causes gravity. Globally, space appears, to the best of our ability to measure, to be flat, though it’s possible that it’s just curved on a scale too large for us to notice (positively or negatively, we can’t say). If it’s positively curved (analogous to a sphere, in higher dimensions), then it certainly closes in on itself, and you can go “around the Universe” to get back to where you started. If it’s flat or negatively curved and has trivial topology, then it does not close in on itself. If it’s flat or negatively curved and has some other topology, then it might still close in on itself. A toroidal universe would be one example of a universe that is flat (i.e., uncurved) but still closes in on itself; there are also other possible topologies with similar effects.
Yes it is, but what we call curved is just a facsimile of what the shape really is, because, we can neither see nor think it in proper perspective with out primitive brains.
Need some ignorance fighting here. Am I misunderstanding what “toroidal” means? Doesn’t that mean donut-shaped (or whatever is sort of equivalent to that in higher dimensions)? If so, how is that not curved? Does toroidal mean something else?
What’s stopping a universe from having any of various other shapes? Why not hyberboloid or cylindrical or something even weirder? Whatever the overall shape of the Big U, is there some necessary reason for it, or does it just so happen?
Toroidal does mean donut-shaped, and it’s not curved in the same sense that a cylinder is not curved: you can cut it open, and flatten it out; you can’t do that with a sphere, for instance – it possesses intrinsic curvature.
Let me try to correct some misinformation in some of the above posts. First, parallel lines do not meet in a hyperbolic (saddle-shaped) space. Instead a line (that is, geodesic or shortest distance) can have many parallels through the same point not on it. Infinitely many, in fact. Second, while a torus in any dimension does have a flat geometry, it also has lots of non-flat geometries. It all depends how you measure distance on it. As observed, you can cut it open and flatten it out to get a flat geometry and you cannot do that with a sphere. But an ordinary inner tube is certainly curved and flattening it changes the distance function on it. Although it does not change which lines are straight. If you travel straight on a torus you could come back to your starting point, but must come arbitrarily close to it, as noted above. This would mean, if the 4-dimensional universe were toroidal, that you would come arbitrarily close to the same time as well as place. This doesn’t seem plausible. In any case this is all speculative.
AFAIK, the only thing that can affect the shape of the universe is mass. I think that near a mass, the curvature is negative (saddle-shaped), but my memory could be failing me on that point. I don’t think there is any known way of having a positive curvature.
Saying something is a torus isn’t actually a statement about its curvature at all, but a statement about its topology. The surface of a donut is a torus because it has one hole. The surface of a coffee cup is also a torus, even though it has a radically different shape than a donut, because it also has one hole. A torus will always have an average curvature of zero, but that doesn’t necessarily mean that it’s zero everywhere on it: The surface of a donut, for instance, is negatively curved toward the hole, but positively curved on the outer edge.
Does knowing that there’s more universe out there than what we can see or detect make determining the curvature of the universe next to impossible, or are there other more subtle, non-direct ways that might help resolve it?
Well, you do have to assume that on the largest scales, the Universe is more-or-less uniform, so that the curvature we measure in our observable region of space is about the same as that beyond what we can observe. This is probably a decent assumption, though, since there’s no known phenomenon that could cause irregularities on such large scales.
