Curvature and the size of the Universe

[CassiopeiaA]

I always have trouble understanding this concept. Please correct me if I am wrong somewhere in my line of thought.

Is the universe finite or infinite? Using the famous balloon analogy, we can think of it as a finite expanding universe. But in that analogy the balloon has a curvature. Does the universe have a curvature? According to the most recent observations the universe is almost flat. So, how come a flat universe be finite? For our assumption of a homogeneous and isotropic universe shouldn't a flat universe be infinite?

 

[phinds]

Is the universe finite or infinite?

Yes, it is one or the other but we have no idea which

Using the famous balloon analogy, we can think of it as a finite expanding universe. But in that analogy the balloon has a curvature. Does the universe have a curvature?

I recommend the link in my signature. You misunderstand the balloon analogy, as do so many people.

 

[mfb]

Is the universe finite or infinite?

We don't know.

Curvature measurements are consistent with zero, but experimentally you can never rule out a small curvature, positive or negative.

 

[rootone]

Flat and infinite is the default hypothesis (it seems to me), given no evidence to the contrary.

 

[CassiopeiaA]

Yes, it is one or the other but we have no idea whichI recommend the link in my signature. You misunderstand the balloon analogy, as do so many people.

Ah I see it now where I was mistaken. But still, the question about universe being finite or infinite remains in my head.

 

[phinds]

Ah I see it now where I was mistaken. But still, the question about universe being finite or infinite remains in my head.

If you can find a definitive answer I guarantee you a Nobel prize.

 

[Chalnoth]

Flat and infinite is the default hypothesis (it seems to me), given no evidence to the contrary.

I don't see why that should be. Making the universe infinite creates its own issues, such as how could any event possibly produce a universe of infinite extent.

The simplest idea is probably the the universe has spherical topology overall, but that the effective radius of that sphere is so much larger than the observable universe that we can't measure its radius of curvature. This has the benefit that as the universe has a finite spatial extent, it's conceivable that there could have been some event which formed it. With an infinite universe, it's much more difficult to come up with plausible origination scenarios.

But I'm not sure that should be the default. I don't think we need a default in this case. Best to just say we don't know the overall shape, and we should keep open minds to a variety of potential shapes.

 

[CassiopeiaA]

[QUOTE="Chalnoth, post: 5732514, member: 162258"The simplest idea is probably the the universe has spherical topology overall, but that the effective radius of that sphere is so much larger than the observable universe that we can't measure its radius of curvature. This has the benefit that as the universe has a finite spatial extent, it's conceivable that there could have been some event which formed it. With an infinite universe, it's much more difficult to come up with plausible origination scenarios.[/QUOTE]
Isn't the topology of the universe also a debatable question? And if we say universe has a "finite spatial extent", the next question should be "what is universe expanding into". Which is quite absurd, given that spacetime was created only at the Big Bang.

 

[Chalnoth]

Isn't the topology of the universe also a debatable question? And if we say universe has a "finite spatial extent", the next question should be "what is universe expanding into". Which is quite absurd, given that spacetime was created only at the Big Bang.

I think this is addressed in my last paragraph in that post.

 

[Comeback City]

If you can find a definitive answer I guarantee you a Nobel prize.

And then we here at the forums will claim partial credit of the prize for our helpful guidance and motivation...

 

[CassiopeiaA]

And then we here at the forums will claim partial credit of the prize for our helpful guidance and motivation...

No problem. I will cite all of your usernames in my groundbreaking paradigm shifting paper : "Can you really get the whole world in your hand?"

 

[newjerseyrunner]

I don't see why that should be. Making the universe infinite creates its own issues, such as how could any event possibly produce a universe of infinite extent.

The simplest idea is probably the the universe has spherical topology overall, but that the effective radius of that sphere is so much larger than the observable universe that we can't measure its radius of curvature. This has the benefit that as the universe has a finite spatial extent, it's conceivable that there could have been some event which formed it. With an infinite universe, it's much more difficult to come up with plausible origination scenarios.

But I'm not sure that should be the default. I don't think we need a default in this case. Best to just say we don't know the overall shape, and we should keep open minds to a variety of potential shapes.

Spherical in how many dimensions? Do you envision a hypersphere in which I see no way around having time loop back on itself and break causality.

Or you do imagine a sphere moving through time giving it more of a 4-string shape? That would have its own problems, either the string would be infinite in the time direction, which loops back to the original problem, or time will end, which has its own problems.

 

[Flatland]

I still don't understand why the Universe can't have 0 curvature and still be finite and unbounded.

 

[PeterDonis]

I still don't understand why the Universe can't have 0 curvature and still be finite and unbounded.

It can if it has non-trivial spatial topology--for example, a flat 3-torus. Mathematically, such possibilities are perfectly consistent; but we have no evidence for them.

 

[Khashishi]

If we assume that the universe is globally spatially isotropic, then it can't have 0 curvature and be finite and unbounded. Global spatial isotropy implies global spatial homogeneity (but not the other way around). A flat 3-torus is globally homogeneous but not isotropic. For example, if you make a 3-torus by gluing together the sides of a cube, the distance around the universe back to your starting point is longer diagonally than horizontally. It turns out that global spatial isotropy is a very stringent condition which eliminates all possible shapes except for a few. I think these are the 3-sphere, projective space, 3-hyperbolic space, and Euclidean space.

 

[PeterDonis]

if you make a 3-torus by gluing together the sides of a cube, the distance around the universe back to your starting point is longer diagonally than horizontally

You're correct that such a universe would not be globally isotropic. But if the "distance around the universe" were large enough, we would not be able to detect this--i.e., it would still be possible for our observable universe to be isotropic.

 

[CassiopeiaA]

A flat 3-torus is globally homogeneous but not isotropic. For example, if you make a 3-torus by gluing together the sides of a cube, the distance around the universe back to your starting point is longer diagonally than horizontally.

I thought isotropy means uniformity in all directions. For example, if I know density variations in one direction, it should be same in all the other directions. What does this has to do with the distance?

 

[mfb]

It can look isotropic in a small region, even with a global geometry that is not isotropic.

Stupid example: If you are on the surface of a cube, and only see a very small area around you, every direction looks the same - a flat surface. But globally it is not flat, as there are corners and edges.

A torus of sufficient size would look exactly like an infinite, flat universe to us local observers, although it is not infinite, and not isotropic.

 

[Khashishi]

But globally it is not flat, as there are corners and edges.

Actually, the edges are flat and you wouldn't be able to see them. The corners would appear as point defects.

 

[mfb]

I was thinking about the cube in our 3D world. Seen as 2D manifold, the edges are flat, sure.

 

[PeterDonis]

The corners would appear as point defects.

In a flat 3-torus, they wouldn't. Any local piece of the flat 3-torus would look just like Euclidean 3-space. The only way to see the global anisotropy would be to construct lines that circumnavigated the 3-torus in different directions, all returning to the same point, and show that they had different lengths.