The Cosmological Principle

  • Thread starter quantum123
  • Start date
  • #1
305
1

Main Question or Discussion Point

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?
 

Answers and Replies

  • #2
1
0
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).
 
  • #3
hellfire
Science Advisor
1,047
1
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.
 
Last edited:
  • #4
Garth
Science Advisor
Gold Member
3,574
105
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[itex]\pi[/itex]R, areas greater than [itex]\pi[/itex]R2 and parallel lines that diverge.

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

Garth
 
Last edited:

Related Threads on The Cosmological Principle

  • Last Post
Replies
3
Views
783
Replies
82
Views
10K
  • Last Post
Replies
6
Views
3K
Replies
50
Views
3K
  • Last Post
6
Replies
136
Views
18K
Replies
3
Views
469
Replies
1
Views
2K
Replies
10
Views
3K
Replies
4
Views
3K
Replies
17
Views
315
Top