MODERN COSMOLOGY

Tetrad description 123

of vanishing torsion the relations for an orthonormal tetrad that are the analogues
of the usual Christoffel relation:

γabc=−(abc−acb), abc=^12 (gadγdcb−gbdγdca+gcdγdab). (3.59)

This shows that the rotation coefficients and the commutation functions are each
just linear combinations of the other.
Any set of vectors however must satisfy theJacobi identities:

[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0 ,

which follows from the definition of a commutator. Applying this to the basis
vectorsea,ebandecgives the identities

e[a(γdbc])+γe[abγdc]e= 0 , (3.60)

which are the integrability conditions that theγabc(xi)are the commutation
functions for the set of vectorsea.
If we apply the Ricci identities to the tetrad basis vectorsea, we obtain the
Riemann curvature tensor components in the form

Rabcd=∂c(abd)−∂d(abc)+aecebd−aedebc−abeγecd. (3.61)

Contracting this onaandc, one obtains the EFE in the form

Rbd=∂a(abd)−∂d(aba)+aeaebd−adeeba=Tbd−^12 Tgbd+λgbd.
(3.62)
It is not immediately obvious that this is symmetric, but this follows because
(3.60) impliesRa[bcd]= 0 ⇒Rab=R(ab).

3.4.2 Tetrad formalism in cosmology

In detailed studies of families of exact non-vacuum solutions, it will usually be
advantageous to use an orthonormal tetrad basis, because the tetrad vectors can be
chosen in physically preferred directions. For a cosmological model we choose an
orthonormal tetrad with the timelike vectore 0 chosen to be either the fundamental
4-velocity fieldua,or the normalsnato surfaces of homogeneity when they
exist. This fixing implies that the initial six-parameter freedom of using Lorentz
transformations has been reduced to a three-parameter freedom of rotations of
the spatial frame{eα}. The 24 algebraically independent rotation coefficients can
then be split into (see [25, 45, 53]):

α 00 = ̇uα,α 0 β=^13 'δαβ+σαβ−αβγωγ,αβ 0 =αβγ
γ(3.63)

αβγ = 2 a[αδβ]γ+γδ[αnδβ]+^12 αβδnδγ. (3.64)

The first two sets contain the kinematical variables for the chosen vector field.
The third is the rate of rotation
αof the spatial frame{eα}with respect to