# Universe as 2d sphere surface - a round trip?

If Universe's analogy is 2D surface of the sphere, could we come around in circle? Say on Earth you decided to travel straight in one direction, after completing one full circle you'd find yourself at exact spot where you started.

Now, if we up that up by 1 dimension and apply it to our Universe, would the same thing happen if we pointed the rocket in a straight line? I.e. would the rocket come back to earth?

• If you believe the universe has finite volume (like a sphere has finite surface area) and your rocket has non-zero volume and minimum non-zero speed, the rocket must eventually revisit some point in space, simply because the total amount of volume "visited" by the rocket grows indefinitely. – barrycarter Dec 29 '14 at 21:58

This is another case of taking an analogy too far (the other one I can think of is assuming that the rubber sheet analogy indicates that there is some extra spatial dimension$^1$). In this case, I'm assuming you're talking about the "balloon" analogy - the universe is expanding, like a balloon, but since there's no center, all bits of matter are on the surface of the balloon. This naturally leads to the idea that if you travel in one direction, you'll eventually come back to where you started. This is, unfortunately, not true.

Scientists often talk about the shape of the universe - how it is curved on a local or global scale. There are three possible categories, depending on one parameter, $\Omega$:

• Positive curvature $\to$ $\Omega>1$
• Zero curvature $\to$ $\Omega=1$
• Negative curvature $\to$ $\Omega<1$

This diagram gives a good visual of each curvature: I should add that these are merely examples of universes (more accurately, manifolds) with each $\Omega$. There are other cases with these curvatures, and so other universe where you can start at one point and end at the same point, as in a 3-torus.

All of these shapes could exist. However, in our universe, the global curvature is one where $\Omega \approx 1$ - a flat universe. The measurements that essentially clinch this are from the WMAP probe, which has analyzed the CMB and determined the age of the universe and its curvature. Measurements show, to a decent degree of accuracy, that our universe is flat - and so if we started in one direction, we would never return to where we started. Check out the NASA page for a full rundown of WMAP's results.

$^1$ Good questions dealing with this can be found here, here and here; John Rennie's answer on Physics is especially good.

• Yes I meant the expanding balloon analogy. It's hard to connect the baloon one with Ω=1 here, and think it describes the same universe. If it expands endlessly, like on this sheet here, does that mean that balloon in expanding so fast, we could never made a round trip? – Ska Dec 29 '14 at 15:46
• @Ska I don't think it implies that, but I'm not positive. – HDE 226868 Dec 30 '14 at 18:22
• $\Omega = 1$ does not imply that the universe is spatially infinite, e.g., it could still be a $3$-torus. However, $\Omega = 1$ with the additional assumptions of isotropy and homogeneity do imply that the universe is spatially infinite, so in that sense this answer is good but incomplete. – Stan Liou Jan 4 '15 at 16:21