Topology of the Universe and infinities

[Arman777]

There are couple things that keep me questioning about the nature of the universe.
Let me start from the begining.

Big Bang happened and our universe was created, and from now on let us suppose that the universe is infinite in size. Later on, the universe expands and after a time we can see only a region of this universe (which we call observable universe). From current observations (within our ability to measure) we can see that, the portion that we see is spatially flat. Since we assumed the universe in infinite in size (at the begining) it doesn't directly mean that universe cannot have a spherical or torus or other kind of finite size geometry ?

Other possibility is that when the Big Bang happened universe has finite size, since I think only in this case we can discuss the possiblity of the finite size geometry models for universe ?

Are these ideas true ? If it's true then the question (what is the topology of the universe) can be reduced to, does the universe has finite or infinite size after singularity ?

Also, in both cases universe is finite in timewise since it has a starting point ?

 

[mathman]

The questions you are raising, particularly finit or infinite universe, are not answerable with current knowledge.

 

[Arman777]

The questions you are raising, particularly finit or infinite universe, are not answerable with current knowledge.

Yeah I know, wish they were answerable though. I am more concerned about the ideas that I said, are they true ? Like the approach to the problem.

In both cases yes there's no way that we can tell which one is true.

 

[PeterDonis]

Since we assumed the universe in infinite in size (at the begining) it doesn't directly mean that universe cannot have a spherical or torus or other kind of finite size geometry ?

If the universe is spatially infinite, it can't have a geometry that can only be spatially finite.

However, it is possible for the universe to be spatially flat without being spatially infinite, if it has, for example, a flat torus geometry. This is considered very unlikely, but it's logically possible.(Note that a spatially flat universe can't have a spherical geometry; there's no way for a spherical geometry to be spatially flat.)

 

[Arman777]

If the universe is spatially infinite, it can't have a geometry that can only be spatially finite.

However, it is possible for the universe to be spatially flat without being spatially infinite, if it has, for example, a flat torus geometry. This is considered very unlikely, but it's logically possible.(Note that a spatially flat universe can't have a spherical geometry; there's no way for a spherical geometry to be spatially flat.)

Well yes it's possible. Why not the spherical geometry cannot be a spatially flat ? To simplfy the question let's think 2D positively curved space. In the surface If we take an infinitesimal portion that corresponds to observable universe, and we measure the curvature wouldn't that be flat ?

 

[kimbyd]

Well yes it's possible. Why not the spherical geometry cannot be a spatially flat ? To simplfy the question let's think 2D positively curved space. In the surface If we take an infinitesimal portion that corresponds to observable universe, and we measure the curvature wouldn't that be flat ?

Spherical geometry simply isn't flat. You can tell this pretty easily by proving that straight lines will always intersect on a spherical surface.

Spherical geometry could be so close to flat that its curvature would be undetectable, if you only have access to a limited portion of the spherical surface (in simple terms, if the radius of curvature is much larger than the size of the region you can observe).

 

[Arman777]

Spherical geometry simply isn't flat. You can tell this pretty easily by proving that straight lines will always intersect on a spherical surface.

Spherical geometry could be so close to flat that its curvature would be undetectable, if you only have access to a limited portion of the spherical surface (in simple terms, if the radius of curvature is much larger than the size of the region you can observe).

I agree and I was talking about the second case. Which from the PeterDonis post, I understood as it can "never" have spherical geometry. But its possible that it can have. In a such way that we can't detect the curvature.

 

[PeterDonis]

In the surface If we take an infinitesimal portion that corresponds to observable universe, and we measure the curvature wouldn't that be flat ?

No. It would be curved, but the curvature would not be measurable by us.

 

[Orodruin]

As a curiosity, you could make the three-dimensional sphere flat by selecting an appropriate connection, since it is parallelizable. However, this connection will not be the Levi-Civita connection of the standard metric.

 

[kimbyd]

As a curiosity, you could make the three-dimensional sphere flat by selecting an appropriate connection, since it is parallelizable. However, this connection will not be the Levi-Civita connection of the standard metric.

Interesting. I did not know this. Looking into it leads me to this:
https://en.wikipedia.org/wiki/Parallelizable_manifold

Apparently the parallelizable spheres are $S^0$ (a point), $S^1$ (a circle), $S^3$, and $S^7$. All other-dimensional spheres cannot be parallelized. These four can because of the behavior of "normed division algebras" (real numbers, complex numbers, quaternions, and octonions).

Mathematics is fascinatingly weird sometimes.

 

[Arman777]

but the curvature would not be measurable by us.

Yes since it's not measurable for us it will be flat. I know that in reality its not flat.

But for us, the living creatures on that infinitesimal surface of the 2D sphere, the curvature would seem flat. In a 2D sphere a 2D creature cannot measure the curvature unless whe travels and comes to the point where he started.

For example we can't proof that Earth is flat just by looking 1m^2 around us.

Earth is not flat. But for creatures living on it seems flat (in infinitesimal distances)

I am not saying different things then you. Why No ? it makes me confused.

 

[Arman777]

So, I understand the general idea. Thanks for your help

 

[PeterDonis]

I am not saying different things then you.

You might not mean to, but the language you are using seems self-contradictory:

since it's not measurable for us it will be flat. I know that in reality its not flat.

It's flat, but it's not flat? You're contradicting yourself. This is why I said "no".

A better way to say it would be that it seems flat in a small enough region, because we can't measure the curvature. As you do later on in your post. If you consistently used the word "seems" there would be no problem.

 

[Arman777]

Okay, thanks again.