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Bianchi Models and Kantowski-Sachs

  1. May 26, 2014 #1
    What are main differencies between Bianchi models and Kantowski-Sachs models.
    Can anyone give mathematical definition clearly and possibly simple form?
     
  2. jcsd
  3. May 26, 2014 #2

    Bill_K

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    Both Bianchi and Kantowski-Sachs cosmologies are homogeneous and isotropic. Hiomogeneous means that for each space section, there's a symmetry operation that maps any point P into any other point Q. Bianchi cosmologies are simply transitive, meaning that there's only one symmetry mapping P into Q. Consequently for Bianchi types the symmetry groups are 3-parameter groups. There are nine different types, corresponding to the nine different types of 3-parameter Lie algebras.
    [/PLAIN] [Broken]
    Kantowski-Sachs
    is an example of a homogeneous, isotropic universe with a symmetry group that is multiply transitive. It has a 4-parameter group: SO(3) x R, rotations on a sphere and translations along an axis.
     
    Last edited by a moderator: May 6, 2017
  4. May 26, 2014 #3
    Thanks for your reply.
    I think you want to say homogeneous and anisotropic spaces.
     
  5. May 27, 2014 #4
    Thanks Bill_K,

    Can you express more explicitly the term 3-parameters and 4-parameters groups please?

    thanks.
     
  6. May 27, 2014 #5

    Bill_K

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    Yes, I meant ANisotropic, in both places! :redface: Thanks!

    The number of parameters in a group is the number of variables you need to specify to pick out a particular group element. For example the usual group of 3-d rotations acting on a sphere S - you can rotate the North pole P into any other point Q on the sphere, and you must give the latitude and longitude of Q - that's two parameters. But after doing that, you can still rotate through some angle ψ about Q, keeping Q fixed. That's a total of three parameters. And the group action on S is multiply transitive - since ψ can be anything, there are many group elements that map P into Q.

    The space section of a Bianchi cosmology has a 3-parameter symmetry group, because you can map P into any other point Q, and this time Q needs three coordinates to label it - x, y, z for example. And there is no further symmetry, you can't rotate about Q due to the anisotropy. So the action is simply transitive.

    The Kantowski-Sachs cosmology has one more parameter - there's a symmetry axis at each point, so you can hold Q fixed and rotate through angle ψ about this axis - making it multiply transitive.
     
  7. May 27, 2014 #6
    Thanks Bill_K,
    Your answer is very usefull for me. I understood many of your reply :)
     
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