Atheist

[Polymath257]

You lost me on flat torus

LOL...it is a rather counter-intuitive notion, isn't it?

Think of it like this. Suppose that flat, euclidean space is filled with cubes packed together. Each cube has exactly the contents of every other cube. So, if you move in one direction, you get a repeating sequence forever. Now, just think of one cube being all there is, but connected to itself by opposite faces.

This is topologically the same as a three dimensional torus with volume equal to that of the cube. But it is also flat in the sense of having zero curvature.

 

[LuisDantas]

I respect the beliefs of Atheists. I just have a question. Do they think the universe is finite? Thanks

Honestly, I have so little to go on, that I am not sure that I even have a hunch.

 

[ChristineM]

LOL...it is a rather counter-intuitive notion, isn't it?

Think of it like this. Suppose that flat, euclidean space is filled with cubes packed together. Each cube has exactly the contents of every other cube. So, if you move in one direction, you get a repeating sequence forever. Now, just think of one cube being all there is, but connected to itself by opposite faces.

This is topologically the same as a three dimensional torus with volume equal to that of the cube. But it is also flat in the sense of having zero curvature.

It must curve to connect to itself. Otherwise it doesnt fit in my head

Also where is the confused frube when you need it

 

[Polymath257]

It must curve to connect to itself. Otherwise it doesnt fit in my head

Grin, which is why I suggested filling up space with identical cubes. Each cube is 'flat', but if each is identical to every other, that is identical to having the topology of a torus. So we get a flat torus.

 

[ChristineM]

Grin, which is why I suggested filling up space with identical cubes. Each cube is 'flat', but if each is identical to every other, that is identical to having the topology of a torus. So we get a flat torus.

No, i dont see it, i have played with building blocks with the kids. They dont make torus's unless there is a wedge shaped gap between each block?

So help me out here, could you draw it??? ;-)


P.s. i edited my previous post after you picked it up

 

[Polymath257]

No, i dont see it, i have played with building blocks with the kids. They dont make torus's unless there is a wedge shaped gap between each block?

So help me out here, could you draw it??? ;-)


P.s. i edited my previous post after you picked it up

OK, let's go down one dimension for a two dimensional torus.

First imagine a strip of paper that is infinitely long, but of finite width. Then glue the opposite ends together to get a cylinder. Even though it *looks* like that cylinder is 'curved', it is *flat* according to the method you used above to determine curvature. Two parallel lines will stay the same distance apart, angles for triangles will add up to 180 degrees, etc.

That cylinder is *flat* in the sense used in cosmology.

Now, do the same thing with a cube: glue together one pair of opposite sides to make a finite cylinder. But now, also glue together the other pair of opposite sides. To do this and maintain flatness needs to go into four dimensions, but the same basic ideas hold. Parallel lines stay the same distance apart and angles for triangles add up to 180 degree.

This gives a flat torus.

 

[ChristineM]

OK, let's go down one dimension for a two dimensional torus.

First imagine a strip of paper that is infinitely long, but of finite width. Then glue the opposite ends together to get a cylinder. Even though it *looks* like that cylinder is 'curved', it is *flat* according to the method you used above to determine curvature. Two parallel lines will stay the same distance apart, angles for triangles will add up to 180 degrees, etc.

That cylinder is *flat* in the sense used in cosmology.

Now, do the same thing with a cube: glue together one pair of opposite sides to make a finite cylinder. But now, also glue together the other pair of opposite sides. To do this and maintain flatness needs to go into four dimensions, but the same basic ideas hold. Parallel lines stay the same distance apart and angles for triangles add up to 180 degree.

This gives a flat torus.

The curvature of the universe was measured with a triangle. And in a curve angles of a triangle dont add to 180 degrees



in 3d, maybe the 4th d is the problem.

I bet you wish all students were like me ;-)

 

[Polymath257]

The curvature of the universe was measured with a triangle. And in a curve angles of a triangle dont add to 180 degrees

View attachment 33172

in 3d, maybe the 4th d is the problem.

I bet you wish all students were like me ;-)

Do you see how the angles on the cylinder add up to 180?

Take any triangle on that cylinder. Then unfold the cylinder. The triangle doesn't change and neither do the angles. So they add up to 180.

That means the cylinder is *flat*, not curved, in the sense we are interested in.

And yes, I love students who want to understand!

 

[ChristineM]

Do you see how the angles on the cylinder add up to 180?

Take any triangle on that cylinder. Then unfold the cylinder. The triangle doesn't change and neither do the angles. So they add up to 180.

That means the cylinder is *flat*, not curved, in the sense we are interested in.

And yes, I love students who want to understand!

I will refer back to.



I simply cannot get the angles of a triangle on a curve cannot add up to 180 deg. Drawn when flat then curved the sides distort and angles change, same happens if drawn when curved, it distorts when straightened.

 

[Polymath257]

I will refer back to.

View attachment 33173

I simply cannot get the angles of a triangle on a curve cannot add up to 180 deg. Drawn when flat then curved the sides distort and angles change, same happens if drawn when curved, it distorts when straightened.

Not on a cylinder. Try it. Remember to use geodesics.

 

[ChristineM]

Not on a cylinder. Try it. Remember to use geodesics.

Ahhh, i think the penny has dropped

Dont think it relates to the universe thought unless its one very big cylinder

 

[Polymath257]

Ahhh, i think the penny has dropped

Dont think it relates to the universe thought unless its one very big cylinder

But you can do the same thing in the other direction and get a *flat* torus. The geodesics still 'unwrap' to give straight lines and angles add up to 180 degrees.

For the universe, we would have a *very* large cube (or even a parallelogram) with opposite sides identified. This would give a flat universe with a finite volume. The 'repeat' distance would be larger than the observable universe as far as we can tell.

Before the flatness was established, there was a report that certain aspects of the background radiation matched a finite universe with negative curvature based on an icosahedron. That has been shown wrong, though.

 

[nPeace]

I still at trying to get a cube end to end though.

 

[Regiomontanus]

I mean the space the universe is inflating into is infinite as far as science can tell

No. The universe is not "inflating" into anything. Space, the entire universe, is expanding (in cosmology inflation is a specific thing, a theory within the theory, so one should use "expand" rather that "inflate"). There is no "space" outside of the universe that we know of.

 

[Polymath257]

I mean the space the universe is inflating into is infinite as far as science can tell

OK, this is a misunderstanding of what the expansion means. It is space itself that is expanding. The only thing it is expanding 'into' is the future: it is expanding into the future.

This is very literally the case in GR, by the way. Space is simply a cross section of spacetime. A later time slice is 'locally larger' in the sense that 'fixed' points will move away from each other.

 

[ChristineM]

No. The universe is not "inflating" into anything. Space, the entire universe, is expanding (in cosmology inflation is a specific thing, a theory within the theory, so one should use "expand" rather that "inflate"). There is no "space" outside of the universe that we know of.

Ok, i will tell the cosmology department at the perimeter institute that they are wrong.

What it is inflating into is unknown and cannot be measured due to the restrictions of the speed of light so to make the claim there is no space outside the universe is rather bold

 

[ChristineM]

OK, this is a misunderstanding of what the expansion means. It is space itself that is expanding. The only thing it is expanding 'into' is the future: it is expanding into the future.

This is very literally the case in GR, by the way. Space is simply a cross section of spacetime. A later time slice is 'locally larger' in the sense that 'fixed' points will move away from each other.

And how is that known?

 

[Polymath257]

And how is that known?

Well, the main way is by the fact that GR seems to be a good description. In this, the universe is described as spacetime with the spatial cross sections getting larger with later time cross sections.

But relativity in general is based on the four dimensional viewpoint. It's sort of like asking for x and y coordinates. Until you set up the coordinate system, this makes no sense. Both space and time are part of the coordinate system (proper time is a different thing).

 

[ChristineM]

Well, the main way is by the fact that GR seems to be a good description. In this, the universe is described as spacetime with the spatial cross sections getting larger with later time cross sections.

But relativity in general is based on the four dimensional viewpoint. It's sort of like asking for x and y coordinates. Until you set up the coordinate system, this makes no sense. Both space and time are part of the coordinate system (proper time is a different thing).

So assuming a.multiverse, where are the other universes?

 

[Revoltingest]

Well, the main way is by the fact that GR seems to be a good description. In this, the universe is described as spacetime with the spatial cross sections getting larger with later time cross sections.

But relativity in general is based on the four dimensional viewpoint. It's sort of like asking for x and y coordinates. Until you set up the coordinate system, this makes no sense. Both space and time are part of the coordinate system (proper time is a different thing).

Let's imagine....
- A point is specified outside of spacetime.
Call it "Christine".
- The outermost element (light, particle, etc) will pass this point.
Call it "Poly".

Before Poly passes Christine, does nothing exist
in Christine's environs....no space....no vacuum of
space....no virtual particles doing what they do?

Is the answer knowable, theoretical, or speculation?

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