Bianchi Models and Kantowski-Sachs

[utku]

What are main differencies between Bianchi models and Kantowski-Sachs models.
Can anyone give mathematical definition clearly and possibly simple form?

 

[Bill_K]

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]
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.

 

[utku]

Thanks for your reply.
I think you want to say homogeneous and anisotropic spaces.

 

[utku]

Thanks Bill_K,

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

thanks.

 

[Bill_K]

I think you want to say homogeneous and anisotropic spaces.

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

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

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.

 

[utku]

Thanks Bill_K,
Your answer is very usefull for me. I understood many of your reply :)