124 Cosmological models
a Fermi-propagated (physically non-rotating) basis along the fundamental flow
lines. Finally, the quantitiesaαandnαβ =n(αβ)determine the nine spatial
rotation coefficients. In terms of these quantities, the commutator equations (3.52)
applied to any functionftake the form
[e 0 ,eα](f)= ̇uαe 0 (f)−[^13 'δαβ+σαβ+αβγ(ωγ+γ)]eβ(f), (3.65)
[eα,eβ](f)= 2 αβγωγe 0 (f)+[ 2 a[αδγβ]+αβδnδγ]eγ(f). (3.66)3.4.3 Complete set
The full set of equations for a gravitating fluid can be written in tetrad form, using
the matter variables, the rotation coefficients (3.57) and the tetrad components
(3.50) as the primary variables. The equations needed are the conservation
equations (3.27), (3.28) and all the Ricci equations (3.61) and Jacobi identities
(3.60) for the tetrad basis vectors, together with the tetrad equations (3.50) and
the commutator equations (3.53). This gives a set of constraints and a set of first-
order evolution equations, which include the tetrad form of all the 1+3covariant
equations given earlier, based on the chosen vector field. For a prescribed set of
equations of state, this gives the complete set of relations needed to determine the
spacetime structure. One has the option of including or not including the tetrad
components of the Weyl tensor as variables in this set; whether it is better to
include them or not depends on the problem to be solved (if they are included,
there will be more equations in the corresponding complete set, for we must then
include the full Bianchi identities). The full set of equations is given in [41, 55],
and see [25, 118] for the use of tetrads to study locally rotationally symmetric
spacetimes, and [45, 128] for the case of Bianchi universes.
Finally, when tetrad vectors are chosen uniquely in an invariant way
(e.g. as eigenvectors of a non-degenerate shear tensor), then—because they are
uniquely defined from 1+3 covariant quantities—all the rotation coefficients
are covariantly defined scalars, so these equations are all equations for scalar
invariants. The only times when it is not possible to define unique tetrads in
this way is when the spacetimes are isotropic or locally rotationally symmetric
(these concepts are discussed later).
3.5 Models and symmetries
3.5.1 Symmetries of cosmologies
Symmetries of a space or a spacetime (generically, ‘space’) are transformations of
the space into itself that leave the metric tensor and all physical and geometrical
properties invariant. We deal here only with continuous symmetries, characterized
by a continuous group of transformations and associated vector fields [24].