What Does It Mean for the Universe to Be Flat?

[albroun]

Sorry quick question from the ignorant (of maths or physics) but what does it mean to say that the universe is flat? I assume this does not mean it only has two dimensions! I do understand that it means that it has neither negative nor positive curvature, but what exactly does it mean in positive terms?

 

[Nabeshin]

Presumably you've seen pictures such as this:
http://atramateria.com/wp-content/uploads/2010/12/geometry_illustration1.jpg

The key thing to note here is the triangles, which is a good way of describing the curvature of a surface. A flat universe implies the following two (well, it implies a lot more than these two, but these ones are particularly simple) basic geometric facts:

1) The angles in a triangle add to exactly 180 degrees,
2) The circumference of a circle is exactly 2πr.

So quite physically, if you make a triangle or a circle in a flat universe, these facts will be true. Of course, when it's not flat, neither of these is the case (as evidenced in the picture).

 

[albroun]

Presumably you've seen pictures such as this:
http://atramateria.com/wp-content/uploads/2010/12/geometry_illustration1.jpg

The key thing to note here is the triangles, which is a good way of describing the curvature of a surface. A flat universe implies the following two (well, it implies a lot more than these two, but these ones are particularly simple) basic geometric facts:

1) The angles in a triangle add to exactly 180 degrees,
2) The circumference of a circle is exactly 2πr.

So quite physically, if you make a triangle or a circle in a flat universe, these facts will be true. Of course, when it's not flat, neither of these is the case (as evidenced in the picture).

Yes I am familiar with these diagrams. I think I see that saying that the universe is flat is not describing the shape of the universe in the way we would normally think. In otherwords, to say, for example, that the shape of the universe is spherical, does not literally mean it is ball-shaped! Or would it?

 

[Nabeshin]

In otherwords, to say, for example, that the shape of the universe is spherical, does not literally mean it is ball-shaped! Or would it?

Actually, it would (a hypersphere, to be precise).

 

[albroun]

So what then is the shape of a flat universe - a hypercube?

 

[Nabeshin]

So what then is the shape of a flat universe - a hypercube?

No, a flat universe is infinite. Just think of the normal R^3, or euclidian space (the analog of the cartesian plane in 3D) http://en.wikipedia.org/wiki/Euclidean_space

 

[albroun]

The universe is also supposed to be expanding at an accelerating rate. How does this work with the idea that the universe is infinite? (sorry have no maths background whatsoever!)

 

[Nabeshin]

The universe is also supposed to be expanding at an accelerating rate. How does this work with the idea that the universe is infinite? (sorry have no maths background whatsoever!)

This is a question that gets asked a lot: Basically, how can an infinite universe expand?

I don't have time to write up a detailed answer right now (I've done it a few times before), so here's a link to a similar thread and you can read responses from there: https://www.physicsforums.com/showthread.php?t=78659

There are tons of similar threads on PF.

 

[phinds]

Sorry quick question from the ignorant (of maths or physics) but what does it mean to say that the universe is flat? I assume this does not mean it only has two dimensions! I do understand that it means that it has neither negative nor positive curvature, but what exactly does it mean in positive terms?

I think the answer you are looking for is this: A closed U falls back in on itself (the big crunch). An open U is finite but unbounded; in some models, if you travel far enough in a straight line you get back to where you started (not actually doable w/o FTL). This is some kind of 3-D analogy to the Earth's surface which is also finite but unbounded. A flat U is infinite and you can travel in a straight line forever.

I personally have trouble getting my head around the open U, but I think that's caused I was raised in 3D.

 

[Chalnoth]

No, a flat universe is infinite. Just think of the normal R^3, or euclidian space (the analog of the cartesian plane in 3D) http://en.wikipedia.org/wiki/Euclidean_space

Not necessarily. A flat universe can also wrap back on itself just fine. For a simple example, take the video game asteroids, which is completely flat, but when you exit one side of the screen, you come back on the other.

In the end, I think most everybody expects that our measurement of the flatness of our universe to just be a local effect. The universe is so vastly larger than the small patch of it that we can see that we can't really say at all what the overall shape is. We just know that any spatial curvature that does exist is much, much larger in scale than our visible universe.

 

[Chalnoth]

I think the answer you are looking for is this: A closed U falls back in on itself (the big crunch). An open U is finite but unbounded; in some models, if you travel far enough in a straight line you get back to where you started (not actually doable w/o FTL).

This is only the case if you don't have dark energy. With dark energy, the picture changes, and the spatial curvature alone doesn't determine the future fate nearly as much. Instead, if the universe is allowed to expand long enough that the dark energy becomes dominant (as is the case with our universe), it is the dark energy and not the curvature that determines the universe's eventual fate.

 

[Nabeshin]

Not necessarily. A flat universe can also wrap back on itself just fine. For a simple example, take the video game asteroids, which is completely flat, but when you exit one side of the screen, you come back on the other.

Ah you're right, I forgot the good example of the torus :).

 

[Passionflower]

Sorry quick question from the ignorant (of maths or physics) but what does it mean to say that the universe is flat?

It means it is a universe that has no matter (neither mass not energy) in it. A Minkowski spacetime is flat.

Our universe is obviously not flat as it contains matter.

Often people are confused by the term conformally flat and mistake it for flat (e.g. a conformally flat spacetime is not flat).

 

[George Jones]

It means it is a universe that has no matter (neither mass not energy) in it. A Minkowski spacetime is flat.

Our universe is obviously not flat as it contains matter.

Often people are confused by the term conformally flat and mistake it for flat (e.g. a conformally flat spacetime is not flat).

This is not how the term "flat universe" is used in cosmology. In cosmology, "flat universe" means that space is flat, not spacetime. Space is the three-dimensional hypersurface that results when cosmic time is held fixed. The spacetime metric induces a spatial metric on this three-dimensional hypersurface. If the curvature tensor generated by this induced metric vanishes, then we say that space is flat, hence "flat universe". The curvature tensor for the induced metric can vanish even when the curvature tensor for the spacetime metric does not vanish.

 

[Chalnoth]

This is not how the term "flat universe" is used in cosmology. In cosmology, "flat universe" means that space is flat, not spacetime. Space is the three-dimensional hypersurface that results when cosmic time is held fixed. The spacetime metric induces a spatial metric on this three-dimensional hypersurface. If the curvature tensor generated by this induced metric vanishes, then we say that space is flat, hence "flat universe". The curvature tensor for the induced metric can vanish even when the curvature tensor for the spacetime metric does not vanish.

Though technically, space is curved due to the presence of structure (stars, galaxies, galaxy clusters, etc.). It's only that on average on large scales space is flat.

 

[George Jones]

Though technically, space is curved due to the presence of structure (stars, galaxies, galaxy clusters, etc.). It's only that on average on large scales space is flat.

Spacetime curvature is caused by this, but I don't think that this is true for space. The curvature of space also depends on how the hypersurfaces are chosen. Space in the Milne universe is not flat.

 

[Chalnoth]

Spacetime curvature is caused by this, but I don't think that this is true for space. The curvature of space also depends on how the hypersurfaces are chosen. Space in the Milne universe is not flat.

Yes, it is very true that spatial flatness is a coordinate-dependent thing. I didn't feel like getting into the nitty-gritty details of this coordinate-dependence, because I think it largely just confuses things.

But what we mean when we talk about spatial flatness is that on average, on large scales, when we make use of FRW coordinates, space is quite flat in our observable universe. On small scales, it isn't, due to local structure. This is a consistent and correct way of describing our universe. There are potentially other ways of doing it as well where the term "flatness" takes on a different meaning, but I didn't think it would be useful to delve to deep into that.

 

[Tanelorn]

Also, wouldn't flatness be a relative term? I thought I lived in a non flat region until I spent a few years in the Rockies :)

 

[phinds]

This is only the case if you don't have dark energy. With dark energy, the picture changes, and the spatial curvature alone doesn't determine the future fate nearly as much. Instead, if the universe is allowed to expand long enough that the dark energy becomes dominant (as is the case with our universe), it is the dark energy and not the curvature that determines the universe's eventual fate.

Thank you. I did not realize that. So the dark energy doesn't change the curvature, just the distance between objects on a large scale and a closed U will NOT experience a big crunch?

 

[Chalnoth]

Thank you. I did not realize that. So the dark energy doesn't change the curvature, just the distance between objects on a large scale and a closed U will NOT experience a big crunch?

Well, the dark energy certainly has an effect on the curvature. It's just that you can have a universe that is either open or closed with or without dark energy.

 

[Chalnoth]

Also, wouldn't flatness be a relative term? I thought I lived in a non flat region until I spent a few years in the Rockies :)

Yes, flatness is relative. Basically, when talking about the curvature of our universe, one way of describing it is by talking about the radius of curvature. If our universe was closed with uniform overall curvature, this radius would be the radius of the circle that makes up our universe (e.g. the radius of curvature of the Earth's surface is just the radius of the Earth).

What we know right now is that the radius of curvature of our observable region is much larger than our observable region.

 

[Khashishi]

Not necessarily. A flat universe can also wrap back on itself just fine. For a simple example, take the video game asteroids, which is completely flat, but when you exit one side of the screen, you come back on the other.

In the end, I think most everybody expects that our measurement of the flatness of our universe to just be a local effect. The universe is so vastly larger than the small patch of it that we can see that we can't really say at all what the overall shape is. We just know that any spatial curvature that does exist is much, much larger in scale than our visible universe.

I think if you restrict yourself to isotropic, homogeneous manifolds, then a flat universe is infinite. The torus is not isotropic, since going off diagonally, it will take longer to return home. Of course, we don't really know if the universe is isotropic or homogeneous

 

[Chalnoth]

I think if you restrict yourself to isotropic, homogeneous manifolds, then a flat universe is infinite. The torus is not isotropic, since going off diagonally, it will take longer to return home. Of course, we don't really know if the universe is isotropic or homogeneous

Well, right. You have to break one of those two assumptions, or both, to have a flat universe wrap back on itself. But there's no reason to believe that either assumption holds globally anyway.