# Spatially homogeneous models

1. May 19, 2014

### befj0001

In a spatially homogeneous model, spacetime is filled with a one-parameter set of invariant hypersurfaces H(t). Spatial homogeneity means that the metric on each H(t) is described in terms of constants. Meaning that the metric becomes a function of time only.

I guess that this means that given an isometry group (belonging to the Bianchi classes) one have to choose a set of three 1-forms such that the metric depends on time only? That is, all the Bianchi models can be written in the form where ds^2 is given by:

ds^2 = -dt^2 + g_ij(t)w^ïw^j, where w^i is the set of forms determined by the isometry group such that that the metric becomes a function of t alone.

Could someone clearify this? What do the forms really mean? In Biachi I the forms are given by dx,dy,dz which makes sence. For then the metric will only depend on time since in Biachi I, the isometry group is the group of translations along the spatial coordinate axes.

2. May 19, 2014

### Matterwave

You are free to choose any bases you want in General relativity (as long as they are "good" bases, i.e. linearly independent). You don't have to choose a bases such that g=g(t) alone. You might want to choose such a basis because it manifestly shows the invariance of g, but, for example, if you use spherical coordinates on your hypersurfaces, then g may depend on your spherical coordinates. That's not to say that those symmetries are no longer present, just that they aren't manifest in your coordinate system (Killing's equation is independent of the coordinate system, but is easiest to solve in coordinate systems where g is independent of some coordinates).

3. May 20, 2014

### Bill_K

If you haven't already, you should take a look at the book, "Homogeneous Relativistic Cosmologies", by Michael Ryan and Lawrence Shepley. This is the best reference on the subject.