# Questions regarding the popular "finite but unbounded" universe

Regarding the finite but unbounded universe, according to my understanding some people come up with this idea because of the limitation of tools to detect what's "outside" the cosmic horizon (or "boundary") of the observable universe so they assumed the universe is finite but unbounded. Is not it?

I have two assumptions about what are possibly there outside the cosmic horizon of the observable universe a.k.a finite but unbounded universe, first it is absolute nothing and second there are something all the way through infinity. But if we assumed there is absolutely nothing outside the finite but unbounded universe, is it logical to assumed something can resided in nothing?

If we define our terms precisely in order to have better comprehension, I think the universe as a whole is infinite if we defined universe as everything including what is beyond this and that ad infinitum.

The point of my question is, if there is indeed such thing as finite but unbounded universe, why is it possible for it to exist in absolute nothing?

• Possible duplicate of How can we know there is more than just the observable universe Jan 18 '19 at 11:49
• No, it's not just "maybe there are other observable universes". When we are talking about observable universe, we tend to think of universe as finite but unbounded which could be loop-like shape, spherical, etc. It could be anything. And one more thing, I'm not only talking about the finite but unbounded universe, I'm talking about what is outside of it if there is such thing as finite but unbounded universe. Jan 18 '19 at 11:52

Your confusion comes from the fact that when you think of a universe which has the Topology if a sphere/torus/what-have you (something compact — i.e. “finite but unbounded”), you’re thinking of this object being embedded in some higher-dimensional space (for instance, one usually thinks of the 2-sphere as being embedded in three-dimensions).

Imagine being a “flatlander” — a two dimensional person living in a two dimensional world. Now assume this flatlander lives on a sphere. He can go around the sphere and come back to where he started. This is in effect a consequence of the geometry of his world. When doing his mathematics to write this down, he could say that his world is embedded in a higher-dimensional space. However, he could also describe his world completely without ever referring to this “extra” dimension, by using the intrinsic geometry of his world as a starting point.

Basically, we tend to think of the universe in terms of an intrinsic (we live on the sphere) as opposed to an extrinsic (the sphere is embedded in a higher-dimension) point-of-view. The drawback of working with an extrinsic description is that there is no physical interpretation of the space around the embedded sphere, which is the root of your misunderstanding. If you work in the intrinsic description, there is no need to appeal to an extra dimension, and there is nothing “outside” the universe.

I also want to point out that the idea of a compact universe doesn’t come from physicists trying to resolve our inability to observe outside our physical universe. Rather, the compact spherical (positive-curvature) model comes from the Einstein equations applied to large-scale cosmology, if the energy density of the universe is above a certain number (which we have reason to believe it isn’t). In addition, if the energy density corresponds exactly to this “critical” density, space would be flat and unbounded on large scales, and if the mass density is less than thus critical density, the universe would have an unbounded “saddle” (hyperbolic) shape at large scales. This is an actual prediction of General Relativity, and not an ad-how explanation/model.

• "This is an actual prediction of General Relativity" - No, it's not. It is a prediction specifically of the Friedman model that has more holes in it than a slice of a Swiss cheese. General relativity is much more general than that and allows for very different cosmological models besides the Friedman solution. Jan 18 '19 at 16:53
• It is a prediction of general relativity if one assumes large-scale homogeneity and isotropy, which is well-observed. Jan 18 '19 at 19:53
• You didn't answer the main point of my question. My question is if the universe is indeed finite but unbounded regardless whether it is spherical, torus, etc. How is it even possible for this "universe" to exist in absolute nothing? Is it possible for something to reside in nothing? If we define universe as everything including the "area" beyond the finite but unbounded universe, universe is truly infinite. I define infinite as that which has no boundary/shape/etc. Jan 19 '19 at 3:49
• @MohamedObeidallah You seem to confuse "finite" with "bounded". These are different concepts. There is no such a place as "beyond", so what you describe as "nothing" does not exist. Also your definition of "infinite" is incorrect. For example, a 3-torus is flat with no shape or boundary, but not infinite. Furthermore, nothing is infinite, not even in your imagination. You may think you can imagine infinite, but you cannot, it would be only a self deception. The concept of infinite is much more problematic than anything else in your question. You may want to study some differential geometry. Jan 19 '19 at 5:08
• @BobKnighton "if one assumes large-scale homogeneity and isotropy" - This is what they print in textbooks, but it is not true. The scale factor metric makes stronger assumptions than just that. There are many ways to define a different metric with homogeneity and isotropy. Jan 19 '19 at 5:10

Regarding the finite but unbounded universe, according to my understanding some people come up with this idea because of the limitation of tools to detect what's "outside" the cosmic horizon (or "boundary") of the observable universe so they assumed the universe is finite but unbounded. Is not it?

No, you're misunderstanding. These are three different things:

Finite but unbounded universe: This is a universe whose spatial topology is that of a sphere. It wraps around.

Cosmic horizon: This is the limit on how far we can see right now because light has had a finite time to propagate.

Boundary: A manifold with boundary contains points that do not have a topological neighborhood of points surrounding them. This breaks the equivalence principle and the standard formulation of general relativity.

• Finite but unbounded universe: This is a universe whose spatial topology is that of a sphere It is quite possible to have “finite but unbounded” universe with spatial topology of a torus ($T_3$) and zero spatial curvature or with a variety of compact hyperbolic manifolds and negative spatial curvature. Jan 18 '19 at 19:00
• You didn't answer the question. The main point of my question is if the universe is finite but unbounded, whether it is spherical or torus or whatever, is it even possible there is absolutely nothing outside? Here, I define something or object as that which has closed boundary. Everything has closed boundary, whether it is an atom, molecule, etc. That which has closed boundary is necessarily finite. I define infinite as that which has no shape/boundary/etc. If we take the universe as everything or whole including beyond that this ad infinitum, it is infinite. Jan 19 '19 at 3:47