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There are some reasons why cosmologists don't like the idea of a universe that loops back on itself, one is that such a universe can't be both isotropic (the sa

There are some reasons why cosmologists don't like the idea of a universe that loops back on itself, one is that such a universe can't be both isotropic (the same in every direction) and homogeneous (the same at every point) [edit: while also being flat, as observations seem to indicate it is]. To get a rough idea of why, imagine taking a 1km by 1km square and identifying the opposite sides, you now have a homogeneous space (every point is equivalent to every other point), but it isn't isotropic because some directions are special. If you put a rock on the ground and walk due east, you'll have to walk 1km to reach the rock again (assuming you start in what used to be the center of the square it takes 500m to where the edge used to be, and another 500m from the edge back to your starting point). On the other hand if you walk south east you have to walk sqrt(2) km to get back to your rock (sqrt(2)/2 km to get to what used to be the south east corner, and another sqrt(2)/2 km to get back to where you started). So although the torus space is homogeneous there are traces of the fact that it used to be a square, embedded in the fact that some directions are special (the 4 cardinal directions have the shortest distance to get back to where you started and the 4 intercardinal directions have the longest). Cosmologists think this lack of isotropy is essentially ugly, and don't like the idea of living in a universe where some directions are special. Aside from just not liking the idea (which isn't very scientific) its also relevant that the universe looks pretty isotropic when do observations, we emphatically don't see traces of the sort of anisotropy you'd see in a toroidal universe anywhere.

drdeca 4y ago
What’s wrong with it being (topologically) a 3-sphere ? That seems plenty symmetric to me.
- eigenket 4y ago
  Theres nothing massively wrong with it, it's a perfectly plausible model. The one thing that isn't particularly nice is that spheres have intrinsic curvature, essentially if you draw two parallel lines on a sphere they will eventually touch. We can go and look at astronomical data and see if the universe has any intrinsic curvature that we can see. People did this and it turns out that from all the astronomical data we have the universe looks incredibly, spectacularly flat. No curvature at all that we can detect. This doesn't mean it isn't a sphere, but it means that if it is a sphere it's a really big one. Much much bigger than the observable universe.
  - deepsun 4y ago
    To be pedantic, there is curvatures, see lensing effect of large masses. It's not global, though.
  - drdeca 4y ago
    Ahhh, yes parts of that sound familiar. So, the reason we can (at least pretty much) ~~rule out~~ a 3-sphere [EDIT: I didn’t read carefully, and missed the “it could be a really really big 3-sphere so that the curvature is close enough to zero” part], is because you can’t have a flat sphere, But you can have a flat torus (or some other shapes that “wrap around”), but we have different reasons to disbelieve those shapes (the “looks the same in any direction” and “looks the same in every position” expectations). Cool, thanks!

drdeca 4y ago

What’s wrong with it being (topologically) a 3-sphere ? That seems plenty symmetric to me.

eigenket 4y ago
Theres nothing massively wrong with it, it's a perfectly plausible model. The one thing that isn't particularly nice is that spheres have intrinsic curvature, essentially if you draw two parallel lines on a sphere they will eventually touch. We can go and look at astronomical data and see if the universe has any intrinsic curvature that we can see. People did this and it turns out that from all the astronomical data we have the universe looks incredibly, spectacularly flat. No curvature at all that we can detect. This doesn't mean it isn't a sphere, but it means that if it is a sphere it's a really big one. Much much bigger than the observable universe.
- deepsun 4y ago
  To be pedantic, there is curvatures, see lensing effect of large masses. It's not global, though.
- drdeca 4y ago
  Ahhh, yes parts of that sound familiar. So, the reason we can (at least pretty much) ~~rule out~~ a 3-sphere [EDIT: I didn’t read carefully, and missed the “it could be a really really big 3-sphere so that the curvature is close enough to zero” part], is because you can’t have a flat sphere, But you can have a flat torus (or some other shapes that “wrap around”), but we have different reasons to disbelieve those shapes (the “looks the same in any direction” and “looks the same in every position” expectations). Cool, thanks!