Models and symmetries 129not a viable cosmology because it has no redshifts, but it laid the foundation for
the discovery of the expanding FLRW models.
TheLRS case q = 1(⇒ r = 5) is the G ̈odel stationary rotating
universe [60], also with no redshifts. This model was important because of
the new understanding it brought as to the nature of time in general relativity
(see [68, 124]). It is a model in which causality is violated (there exist closed
timelike lines through each spacetime point) and there exists no cosmic time
function whatsoever.
The anisotropic modelsq=0(⇒r=4) are all known, but are interesting
only for the light they shed on Mach’s principle; see [101].
3.5.2.2 Spatially homogeneous universes
These models withs=3 are the major models of theoretical cosmology, because
they express mathematically the idea of the ‘cosmological principle’: all points
of space at the same time are equivalent to each other [6].
Theisotropic case q=3(⇒r= 6 )is the family of FL models, the standard
models of cosmology, with the comoving RW metric form:
ds^2 =−dt^2 +S^2 (t)(dr^2 +f^2 (r)(dθ^2 +sin^2 θdφ^2 )), ua=δa 0. (3.75)Here the space sections are of constant curvatureK=k/S^2 and
f(r)=sinr,r,sinhr (3.76)if the normalized spatial curvaturekis+ 1 , 0 ,−1 respectively. The space sections
are necessarily closed ifk=+ 1.
TheLRS case q=1(⇒r= 4 )is the family of Kantowski–Sachs universes
[13,80] plus the LRS orthogonal [45] and tilted [77] Bianchi models. The simplest
are the Kantowski–Sachs family, with comoving metric form
ds^2 =−dt^2 +A^2 (t)dr^2 +B^2 (t)(dθ^2 +f^2 (θ)dφ^2 ), ua=δa 0 , (3.77)wheref(θ)is given by (3.76).
Theanisotropic case q=0(⇒r= 3 )is the family of Bianchi universes
with a group of isometriesG 3 acting simply transitively on spacelike surfaces.
They can be orthogonal or tilted. The simplest class is the Bianchi type I family,
with an Abelian isometry group and metric form:
ds^2 =−dt^2 +A^2 (t)dx^2 +B^2 (t)dy^2 +C^2 (t)dz^2 , ua=δ 0 a. (3.78)The family as a whole has quite complex properties; these models are discussed
in the following section.