Understanding the Flat Geometry and Curvature of the Universe

[Alltimegreat1]

I'm having trouble understanding the terms "flat geometry of the universe" and "baseline curvature of the universe." How can a 3D universe be flat?

 

[jfizzix]

It is flat in the sense that it has no curvature.

2D geometry can be curved as well. For example, if you draw a triangle on the surface of a sphere (a 2D surface), its interior angles will add up to a number larger than 180 degrees. This is a consequence of the curvature of the space.

Flat space is called Euclidean space, where the interior angles of a triangle add up to 180 degrees, and parallel lines never intersect.

3D space can be flat (i.e., Euclidean) in the sense that parallel lines never meet and keep the same distance from each other.
3D space, doesn't have to be flat though.
If we apply the rules on non-euclidean spaces to 3D, you can get things like parallel lines at one point eventually bending toward each other and intersecting.

The geometry of the Universe usually refers to space and time as a peculiar kind of 4-dimensional system called "space-time". Where spacetime is flat, we see that it obeys the rules of euclidean geometry and spatial relativity. However, the theory of General Relativity says that mass/energy curves or warps spacetime. That means near a massive body, the laws of space and time are non-euclidean.

 

[Orodruin]

3D space can be flat (i.e., Euclidean) in the sense that parallel lines never meet and keep the same distance from each other.

Flat is not the same thing as Euclidean.

 

[Chalnoth]

Flat is not the same thing as Euclidean.

For nearly all intents and purposes, it is.

It's possible for a flat space to wrap back on itself, but this definitely doesn't occur for our observable universe. So as far as we can tell, our universe is very close to Euclidean on large scales.

 

[Orodruin]

For nearly all intents and purposes, it is.

Definitely not. For the application to cosmology, yes, but flattness and being a Euclidean space are different mathematical concepts.

 

[Chalnoth]

Definitely not. For the application to cosmology, yes, but flattness and being a Euclidean space are different mathematical concepts.

Perhaps, but that distinction is quite subtle.

Euclidean space and flat space have all of the same measurable properties, except for the possibility that a flat space can potentially wrap back on itself.

 

[Orodruin]

Perhaps, but that distinction is quite subtle.

Euclidean space and flat space have all of the same measurable properties, except for the possibility that a flat space can potentially wrap back on itself.

This is not true either. Flatness is a local concept although it is perfectly possible to have completely flat spaces which do not wrap back on themselves and are not Euclidean. In particular, spaces with non-zero torsion come to mind.

As long as you work in a Riemannian setting with the Levi-Civita connection you might have a point, but the original statement was a blanket statement without any qualifiers and as such is incorrect.

 

[Alberto87]

how can a flat space wrap back on itself?

 

[Orodruin]

how can a flat space wrap back on itself?

A cylinder is the most usual example of such a space. It is flat, but has a cyclic direction.

Note that we are talking about intrinsic curvature here, which is a property of the space itself, not of its embedding into a higher-dimensional space. Extrinsic curvature is something different.

 

[Alberto87]

A cylinder is the most usual example of such a space. It is flat, but has a cyclic direction.

Note that we are talking about intrinsic curvature here, which is a property of the space itself, not of its embedding into a higher-dimensional space. Extrinsic curvature is something different.

Ah ok, but it only wraps back in one direction, the second direction cannot, am I right?
It´s not possible for a flat space to be finite and unlimited, isn´t it?
Is the torus a flat space time?

 

[Orodruin]

Ah ok, but it only wraps back in one direction, the second direction cannot, am I right?
It´s not possible for a flat space to be finite and unlimited, isn´t it?
Is the torus a flat space time?

It can wrap around in all directions. A torus is an example - whether it is flat or not depends on the metric you put on it.

 

[lavinia]

Ah ok, but it only wraps back in one direction, the second direction cannot, am I right?
It´s not possible for a flat space to be finite and unlimited, isn´t it?
Is the torus a flat space time?

The function $(x,y) \rightarrow 2^{-1/2}(cos(x), sin(x),cos(y),sin(y))$ maps the Euclidean plane into a flat torus in $R^4$. The square [0.2π] x [0,2π] in the plane is bent around in both directions and its opposite edges are identified. A square with opposite edges identified is a torus. The function also preserves the Euclidean metric on the plane so the torus is flat. One can think of it as a cylinder made in both directions.

You are right that you can not wrap the cylinder into a flat torus in 3 dimensions. In 3d one would have to stretch the cylinder and this would create curvature. In 4 dimensions one does not need to do any stretching.