Is the Hyperboloid Model Truly Homogeneous and Isotropic?

[quantum123]

Hi:
The principle states that : the universe is homogeneous and isotropic.
Then we have three solutions often depicted as
i) Sphere
ii) Plane
iii) Hyperboloid

I can understand that the sphere and plane is homogeneous and isotropic, but the iii) does not seems to be. There seems to be a minimum point (violates homogeneity). There seems to a preferred direction -> towards and away from the horse head if you imagine it as a horse saddle. Did I miss anything here?

 

[markvp]

According to the third option, 3-dimensional space is still spherical, but not when you add the time dimension. You can imagine a similar universe as follows: take a circle, so it's a 1-dimensional universe, and let it expand with time, faster than linear. You can plot this variation in time, which gives you a horn-shaped time-space. At every point this horn is hyperboloid, because the curvature is negative in one direction and positive in another direction.

There is also a saddle-point, but where it is depends on how you hold the horn. But if you hold the time-axis vertical or horizontal, there is no saddle-point (a hyperboloid doesn't have a minimum, by the way).

 

[hellfire]

The usual third option in the FRW geometries is a hyperbolic space with constant negative curvature at every point in space. This is a homogeneous and isotropic three-dimensional space that has no preferred direction. I am not sure whether the two-dimensional surface of a horse saddle a perfect analogy to visualize it.

 

[Garth]

It's not, you have to imagine the 'saddle point' at every event in the hyperbolic space-time. Somewhat difficult to imagine!

On the other hand a hyperbolic space has circles with circumferences C > 2$\pi$R, areas greater than $\pi$R² and parallel lines that diverge.

Now that is something that can be measured and therefore 'imagined'...

Garth