Can the space (or else measurable) be actually infinite?

[theBorg]

The (most popular) flat model of Universe is space-infinite. How the infinity is measured? Can you give me references to the papers about the actual infinity of space?

[Alex K]

It doesn't have to be space-infinite to be flat. Standard cosmology works exactly the same if it is a spatially flat but finite manifold, e.g. a 3-torus, as long as all the radii are a bit larger than the observable universe, so I doubt somewhat that there are serious papers empirically arguing for actual infinteness. There are however papers looking for effects of *finiteness* in the cosmic microwave background, but so far no evidence has turned up, with means no more than that the universe is probably at least a bit larger than the observable universe - a conclusion that doesn't surprise anyone, really.

[Arkilogue]

It doesn't have to be space-infinite to be flat. Standard cosmology works exactly the same if it is a spatially flat but finite manifold, e.g. a 3-torus, as long as all the radii are a bit larger than the observable universe, so I doubt somewhat that there are serious papers empirically arguing for actual infinteness. There are however papers looking for effects of *finiteness* in the cosmic microwave background, but so far no evidence has turned up, with means no more than that the universe is probably at least a bit larger than the observable universe - a conclusion that doesn't surprise anyone, really.

Good day to you Alex! What is a 3-torus and does it predict the same negative/saddle like curvature of space as a "normal" 3d torus?

[Alex K]

It doesn't have to be space-infinite to be flat. Standard cosmology works exactly the same if it is a spatially flat but finite manifold, e.g. a 3-torus, as long as all the radii are a bit larger than the observable universe, so I doubt somewhat that there are serious papers empirically arguing for actual infinteness. There are however papers looking for effects of *finiteness* in the cosmic microwave background, but so far no evidence has turned up, with means no more than that the universe is probably at least a bit larger than the observable universe - a conclusion that doesn't surprise anyone, really.

Good day to you Alex! What is a 3-torus and does it predict the same negative/saddle like curvature of space as a "normal" 3d torus?

What you show here is an embedding of a 2-torus into 3D space (and then shown as a projection onto our screens, but that's not important, the important thing is that we are dealing with, basically, a tire or donut). These embeddings are somewhat misleading because they look curved, They do however not reflect a saddle-like negative overall curvature of the underlying manifold. The apparent curvature is entirely an artifact of the attempt to show the 2-torus as embedded in 3D space, which forces you to distribute the vanishing curvature of the torus itself unevenly, with the outside of the embedded torus having positive curvature, the inside negative curvature. The two actually cancel exactly and the overall curvature is zero. The cosmology example I gave assumes a homogeneous torus in which this curvature is evenly distributed and vanishes everywhere, something that cannot be visualised without messing up that crucial property.
A better way to define and think about a homogeneous 2-torus (or 3-torus), which makes its intrinsic flatness more obvious, goes thusly:

Imagine a square or rectangular patch (of paper or whatever), and now imagine the opposite edges are identified as being the same. Then, if you leave on the right, you enter at the same height on the left, and if you leave through the bottom, you enter at the top, etc. This space is, in fact, topologically a 2-torus, and from the construction which starts with a flat square of paper, it is clear that it is a flat space. Only when you now try to embed this object entirely into 3D space by actually glueing the identified edges together, you are forced to deform it and have localized negative and positive curvature, which is however only a curvature of the embedding, not of the underlying manifold, the torus, itself, which can be thought of as flat.

For the purposes of cosmology, when people say "3-torus", they mean a flat 3D-space of this type, where you start with a cube or cuboid, where opposite faces are identified as the same, and where you always reenter on the opposite side if you "leave" it.

[robvalue]

Sure, why not.

Maybe time runs in a loop too. That puts the kaibosh on all these silly regression arguments.

[Arkilogue]

Good day to you Alex! What is a 3-torus and does it predict the same negative/saddle like curvature of space as a "normal" 3d torus?

What you show here is an embedding of a 2-torus into 3D space (and then shown as a projection onto our screens, but that's not important, the important thing is that we are dealing with, basically, a tire or donut). These embeddings are somewhat misleading because they look curved, They do however not reflect a saddle-like negative overall curvature of the underlying manifold. The apparent curvature is entirely an artifact of the attempt to show the 2-torus as embedded in 3D space, which forces you to distribute the vanishing curvature of the torus itself unevenly, with the outside of the embedded torus having positive curvature, the inside negative curvature. The two actually cancel exactly and the overall curvature is zero. The cosmology example I gave assumes a homogeneous torus in which this curvature is evenly distributed and vanishes everywhere, something that cannot be visualised without messing up that crucial property.
A better way to define and think about a homogeneous 2-torus (or 3-torus), which makes its intrinsic flatness more obvious, goes thusly:

Imagine a square or rectangular patch (of paper or whatever), and now imagine the opposite edges are identified as being the same. Then, if you leave on the right, you enter at the same height on the left, and if you leave through the bottom, you enter at the top, etc. This space is, in fact, topologically a 2-torus, and from the construction which starts with a flat square of paper, it is clear that it is a flat space. Only when you now try to embed this object entirely into 3D space by actually glueing the identified edges together, you are forced to deform it and have localized negative and positive  curvature, which is however only a curvature of the embedding, not of the underlying manifold, the torus, itself, which can be thought of as flat.

For the purposes of cosmology, when people say "3-torus", they mean a flat 3D-space of this type, where you start with a cube or cuboid, where opposite faces are identified as the same, and where you always reenter on the opposite side if you "leave" it.

So Asteroids™ got it right?

---

Sure, why not.

Maybe time runs in a loop too. That puts the kaibosh on all these silly regression arguments.

Are you familiar with the Ouroboros?

[Alex K]

Maybe, if you make it 3D

[Arkilogue]

Maybe, if you make it 3D

So the universe is imagined to repeat in all directions after a certain distance?

That's almost as mind warping as a Clifford torus...

[Alex K]

Maybe, if you make it 3D

So the universe is imagined to repeat in all directions after a certain distance?

That's almost as mind warping as a Clifford torus...

Yea, I'm not saying it's true, but it is a theoretical possibility that doesn't contradict any observations...

[Arkilogue]

Yea, I'm not saying it's true, but it is a theoretical possibility that doesn't contradict any observations...

But why reduce 3d spaces to 2d-manifolds? Can it not be 3d modeled in a way that does not contradict observation?