Lionizing Lyin Eyes.

[Polymath257]

Thanks for the responses. Good to know you are still around also. To add to the confusion ...

Something that has always concerned me when I read about two-dimensional spaces. An example is the idea of "flatland", where a three dimensional sphere passes through their two dimensions. They supposedly "see" first a dot, then a circle that expands to a maximum size, then shrinks to a dot again and disappears. That seems wrong if we are talking about the two-dimensional inhabitants of flatland, who see ... what? They would need a third dimension to see the object as a circle, which they don't have. Putting aside the thought that a two-dimensional thing can't actually exist as we do, as it would be of infinite thinness, how do they tell what shape it is? they would be aware of something in their world when they approach the edge of the circle and would be unable to pass through it. So they try to map the shape of this object, but how? they can't draw a map because it would be a similarly incomprehensible object to them, viewed only from the side. No way to look "down" on it.

I'm wondering if that relates to your determining curvature internally?

Second thoughts. They could perhaps measure distance between two points by the time taken to travel between them. How do they record the position of a point though?

Go up one dimension: how do we know when we are looking at a ball? That is entirely analogous to a 2D 'person' looking at a sphere.

If a 4D 'person' pushed a 4D sphere through our 3D space, we would first see a point. Then it would grow as a sphere up to some maximum size (the equator) and then shrink again to a point.

Remember our 3D space has 'infinite thinness' in the 4th direction.

 

[Alien826]

Go up one dimension: how do we know when we are looking at a ball? That is entirely analogous to a 2D 'person' looking at a sphere.

If a 4D 'person' pushed a 4D sphere through our 3D space, we would first see a point. Then it would grow as a sphere up to some maximum size (the equator) and then shrink again to a point.

Remember our 3D space has 'infinite thinness' in the 4th direction.

OK, but do you think a 2-D being could actually "see" anything at all? Don't we need at least three dimensions to even exist? What is the volume of a 2-D square? 2X2X0=0? Can something with zero volume even exist, let alone function? And photons don't have zero volume, do they? They don't have to have vision as we know it but there has to be some medium of transition, no? Yes I know it's all handled perfectly in mathematics, even 1-D objects in string theory (which I don't understand at all, so this sentence may be totally wrong).

By the way, I once saw an argument for the non-existence of a 2D being which argued that its GI tract would involve it falling half. I didn't buy that as it assumed a particular form of life which is not universal. On the other hand though I had my own reason for saying it couldn't exist, as above.

 

[Heyo]

First, when discussing positive or negative curvature, we are usually talking about the *spatial* cross sections, not spacetime in general.

So, imagine a trumpet shape and time corresponding to the axis of the trumpet. The spatial cross sections are circles that expand as you go outward along the trumpet (i.e, later in time). In this model, space is positively curved (the circles) but spacetime is negatively curved (the trumpet itself).

If, instead, you have a cylinder instead of a trumpet shape, the 'space' cross sections would be positively curved and the spacetime itself would technically be flat (imagine unrolling the cylinder--it becomes a flat piece). So, no, you do not need corresponding negative curvature anywhere to get flatness.

I have thought a bit about your model. You have reduced the spacial dimensions to 1for the time dimension to be Cartesian. Now there can be at least three possibilities, the trumpet - which relates to accelerating expansion, the cone which relates to a constant expansion, and the "teardrop" in which space will ultimately contract. The cylinder is not an option as it relates to a steady state universe.
The trumpet has two underclasses, one where the curvature becomes infinite (goes towards a pole* like the tangent function) and the one where it doesn't.
All three models have positively curved space and only one has negatively curved time. (I have to think more about that, my intuition fails at the moment.)
Do I understand the geometry right?

*In German the line at ½ π is called a "Pol" = engl. pole, but I don't know if that the mathematical correct name.

 

[Polymath257]

I have thought a bit about your model. You have reduced the spacial dimensions to 1for the time dimension to be Cartesian. Now there can be at least three possibilities, the trumpet - which relates to accelerating expansion, the cone which relates to a constant expansion, and the "teardrop" in which space will ultimately contract. The cylinder is not an option as it relates to a steady state universe.
The trumpet has two underclasses, one where the curvature becomes infinite (goes towards a pole* like the tangent function) and the one where it doesn't.
All three models have positively curved space and only one has negatively curved time. (I have to think more about that, my intuition fails at the moment.)
Do I understand the geometry right?

*In German the line at ½ π is called a "Pol" = engl. pole, but I don't know if that the mathematical correct name.

Yes, you basically have the options for a one dimensional space in a two dimensional spacetime. For three dimensional space in a four dimensional spacetime, there are more options. At that point, it is probably best to learn the mathematics and see what general relativity actually says.

One difference between the pi/2 latitude lines and what happens in GR is that the pole is a coordinate singularity and the BB is actually a geometric singularity.

 

[Heyo]

One difference between the pi/2 latitude lines and what happens in GR is that the pole is a coordinate singularity and the BB is actually a geometric singularity.

As in a Penrose diagram? I can work with that, I just need to understand how the different models relate to one another - and I fear that the maths may be beyond my abilities.