Curvature at the center of a spherical universe

[Grinkle]

If the universe is very large relative to the observable universe and it is spherical, and the observable universe is well away from the outside region of the sphere, more towards the center, is spacetime approaching flat for the observable universe?

I always assumed the answer is yes, but then I started thinking that maybe a spherical universe is curved everywhere, more accurately pictured as a layer of spherical surfaces wrapped around each other, and one could see curvature regardless of where along the radius one was.

Maybe one can define multiple spherical geometries with different kinds of interior curvatures?

 

[Orodruin]

There is no "center" of the spherical universe. The "spherical" refers only to the geometry of the three-dimensional space itself. There is no higher-dimensional space in which this needs to be embedded.

 

[Grinkle]

The "spherical" refers only to the geometry of the three-dimensional space itself.

Is the curvature constant throughout that geometry? And the larger the universe the flatter, everywhere?

 

[Orodruin]

Is the curvature constant throughout that geometry? And the larger the universe the flatter, everywhere?

Yes. It is completely analogous to the surface of a ball, i.e., a 2-sphere, just one more dimension.

 

[kimbyd]

If the universe is very large relative to the observable universe and it is spherical, and the observable universe is well away from the outside region of the sphere, more towards the center, is spacetime approaching flat for the observable universe?

I always assumed the answer is yes, but then I started thinking that maybe a spherical universe is curved everywhere, more accurately pictured as a layer of spherical surfaces wrapped around each other, and one could see curvature regardless of where along the radius one was.

Maybe one can define multiple spherical geometries with different kinds of interior curvatures?

A spherical universe has the same curvature everywhere.

To visualize the spherical universe concept, imagine an ant on a (white) billiard ball.

What location on the surface of the billiard ball would be considered to be the center of the surface? Clearly no place, because there isn't any location on the billiard ball that is special.

Furthermore, the curvature at every point on the surface is identical. There is no location which is flatter than any other.

 

[Erk]

Allow me to introduce a related notion if I may.

Let's take the case of a single spherical universe with absolutely nothing else outside of it. If the sphere is perfect then the only location that anything can be within it, must be the absolute center. In other words, no matter where we are, we're always in the center of the universe.

It doesn't matter how big the universe is, and it doesn't matter where we think we are or where anything else seems to be, and here's the reason why.

In the case I just described, there is no possible reference points other than the center. There is no point outside of it, the inside surface has no point that is different than any other, and there is no basis for determining up and down, left and right, and back and forth.

So, attempt to move your mind from center and closer to the sphere. Every step it takes it's also taking the exact same step further away from the sphere in the opposite direction. "Hey Joe! Where you goin'? Well, I'm headed towards the sphere. No you're not. Turn around. You are also moving equally further from it. Actually Joe, you haven't moved at all." As someone once said, "it's merely a very persistent illusion."

 

[kimbyd]

Allow me to introduce a related notion if I may.

Let's take the case of a single spherical universe with absolutely nothing else outside of it. If the sphere is perfect then the only location that anything can be within it, must be the absolute center. In other words, no matter where we are, we're always in the center of the universe.

It doesn't matter how big the universe is, and it doesn't matter where we think we are or where anything else seems to be, and here's the reason why.

In the case I just described, there is no possible reference points other than the center. There is no point outside of it, the inside surface has no point that is different than any other, and there is no basis for determining up and down, left and right, and back and forth.

So, attempt to move your mind from center and closer to the sphere. Every step it takes it's also taking the exact same step further away from the sphere in the opposite direction. "Hey Joe! Where you goin'? Well, I'm headed towards the sphere. No you're not. Turn around. You are also moving equally further from it. Actually Joe, you haven't moved at all." As someone once said, "it's merely a very persistent illusion."

This is a confusing description, because you're talking about a a different situation.

A universe with spherical geometry is a completely different concept than what you've described, which is just a three-dimensional ball. A three-dimensional ball isn't a curved surface, because it isn't a surface at all.

The analog to our universe is not the whole ball, but rather the surface, which is sometimes called a 2-sphere. The surface of a ball has the geometry of a 2-sphere. But the 2-sphere doesn't include any notions about what is outside of the surface. A universe described by a 2-sphere has no center. You could, in order to visualize the 2-sphere universe, draw it as a 2-dimensional surface embedded in 3-dimensional space. That specific visualization would have a center, but it would be specific to the visualization, not a property of the 2-sphere itself.

 

[Ibix]

That specific visualization would have a center, but it would be specific to the visualization, not a property of the 2-sphere itself.

And just to emphasise, the centre @kimbyd is talking about here isn't in the universe. It's in the bits outside the universe that you added when you did the visualisation - which don't exist in reality.

 

[Erk]

This is a confusing description, because you're talking about a a different situation.

A universe with spherical geometry is a completely different concept than what you've described, which is just a three-dimensional ball. A three-dimensional ball isn't a curved surface, because it isn't a surface at all.

The analog to our universe is not the whole ball, but rather the surface, which is sometimes called a 2-sphere. The surface of a ball has the geometry of a 2-sphere. But the 2-sphere doesn't include any notions about what is outside of the surface. A universe described by a 2-sphere has no center. You could, in order to visualize the 2-sphere universe, draw it as a 2-dimensional surface embedded in 3-dimensional space. That specific visualization would have a center, but it would be specific to the visualization, not a property of the 2-sphere itself.

I'm speaking of the surface only. The ball is the accumulation of all that is inside the surface and it together with the surface would equal the universe I just described.

 

[PeterDonis]

Thread closed for moderation.

 

[PeterDonis]

I'm speaking of the surface only. The ball is the accumulation of all that is inside the surface and it together with the surface would equal the universe I just described.

I'm not sure if you are trying to describe a 3-sphere, which is one possible model for the spatial geometry of the universe (though not a likely one given our best current data), and are just confused about what that model actually says, or whether you are trying to describe a model in which the universe is just an ordinary 3-ball, i.e., the 3-manifold interior of a 2-sphere, in which case this is not a viable model for the universe as a whole, because the spatial geometry of the universe does not have a boundary, and a 3-ball does.

In either case, what you are posting doesn't really make sense and discussion doesn't seemed to have helped your understanding, so I have cleaned up the thread to avoid confusing other readers.

 

[PeterDonis]

And since the OP question has been answered, this thread will remain closed.