I read something, I think it was in either A Brief History of Time by Stephen Hawking or The Elegant Universe by Brian Greene.
The suggegtion was that the universe might indeed be “finite, but unbounded”. This means it would be, as the OP described, a lot like the surface of the Earth. If you fly in one direction long enough, you get back where you started. This corresponds, in my understanding, to the universe being ‘positively curved’, like the surface of a ball or a balloon. A table is flat- it’s neutrally curved. Something which curves differently in one direction than in the other is negatively curved. A horse-saddle is negatively curved; it curves inward moving fore-and-aft, and outward moving side-to-side.
They suggested that the universe was so large, however, that it would take longer to cross it (even at the speed of light) than the amount of time that has elapsed since the universe began. So you couldn’t look through a telescope and see the back of your head by looking around the whole of the universe.
Now, you and I are probably both convinced that nothing can move faster than light. So the universe can’t be larger than twice the distance light could have traveled since the Big Bang; the radius of the universe could be no larger than c times the age of the universe. But I just suggested that the universe is so wide that light wouldn’t have had time to cross it since the time began? How can this be?
Whatever it was I was reading suggested, first, that the universe can grow at a rate larger than the speed of light. Nothing’s moving faster than light, but space is expanding that fast. The expansion of the continuum itself is not restricted by the problem of accelerating mass beyond lightspeed. (That’s said to require infinite energy.)
Secondly, it suggested that a major reason that the universe is so very large today is because it underwent an inflationary period in it’s early history, where it rapidly grew to many, many times it’s previous size.
All in all, the idea was that the universe might well ‘wrap around on itself’. It might be ‘finite, but unbounded’ and ‘positively curved’. But it’s so big that we could never see most of it, and certainly it’s too far to go all the way around the universe to see oursleves. I think the analogy used was that the visible portion of the universe is to the whole universe as a single proton is to the surface of the earth.
So, none of the stars in the sky are actually images of our own galaxy, because light from our own galaxy hasn’t had time to get to us (at least not going the long way). And contrary to I am Sparticus’ suggestion, none of the stars would have to look blue-shifted because:
-
They aren’t coming toward us. Think of a finite, unbounded universe as the the surface of a balloon. Blow the balloon up- this is how big the universe is today. Draw a bunch of dots on the balloon- these are galaxies in the universe. Now, inflate the universe some more- you’ll see that no matter which direction you measure, all dots are farther from all others. They’re getting farther apart- in the universe, this would show up as a redshift, not a blueshift.
-
They’re so far away anyway that we couldn’t see them from the backside, even if seeing them from that side would give us a blueshift.
So, while I don’t think the experts are sure, yet, it seems it may still be possible for the universe to ‘wrap around’ to itself. The main implication is that while it isn’t infinite, it doesn’t have an edge. (Just like the surface of the Earth has a fixed surface area, but no edge for you to walk off of.)
[The main weakness in my analogies, above, is comparing the universe, which is three-dimensional in extent, to the surface of the Earth, which is two-dimensional. You have to imagine that the properties of the surface of a sphere can be generalized to three dimensions. So flying in any direction in space would bring you back where you started, just like you can fly an airplane around the world going North-South, East-West, or whatever.]