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The Cosmological Principle

  1. Jan 24, 2007 #1
    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?
  2. jcsd
  3. Jan 24, 2007 #2
    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).
  4. Jan 28, 2007 #3


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    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: Jan 28, 2007
  5. Jan 28, 2007 #4


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    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'...

    Last edited: Jan 28, 2007
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