MODERN COSMOLOGY

Tetrad description 121

3.4 Tetrad description

The 1+3 covariant equations are immediately transparent in terms of representing
relations between 1+3 covariantly defined quantities with clear geometrical
and/or physical significance. However, they do not form a complete set of
equations guaranteeing the existence of a corresponding metric and connection.
For that we need to use a full tetrad description. The equations determined will
then form a complete set, which will contain as a subset all the 1+3covariant
equations just derived (albeit presented in a slightly different form) [53, 55]. First
we summarize a generic tetrad formalism, and then describe its application to
cosmological models (cf [25, 92]).

3.4.1 General tetrad formalism

Atetradis a set of four linearly independent vector fields{ea},a= 0 , 1 , 2 ,3,
which serves as a basis for spacetime vectors and tensors. It can be written in
terms of a local coordinate basis by means of thetetrad components eai(xj):

ea=eai(xj)

∂

∂xi

⇔ea(f)=eai(xj)

∂f

∂xi

, eai≡ea(xi), (3.50)

(the latter stating that theith component of theath tetrad vector is just the
directional derivative of theith coordinatexiin the directionea). This relation
can be thought of as just a change of vector basis, leading to a change of tensor
components of the standard tensorial form:

Tabcd=eaiebjeckedlTijkl

with an obvious inverse, where the inverse componentseai(xj)(note the placing
of the indices!) are defined by

eaieaj=δij⇔eaiebi=δba. (3.51)

However, this is a change from an integrable basis to a non-integrable one, so
the non-tensorial relations (specifically the form of the metric and connection
components) differ slightly from when coordinate bases are used. A change of
one tetrad basis to another will also lead to transformations of the standard tensor
form for all tensorial quantities: ifea=λaa
′
(xi)ea′is a change of tetrad basis
with inverseea′=λa′a(xi)ea(in the case of orthonormal bases, each of these
matrices representing a Lorentz transformation), then

Tabcd=λa′aλb′bλcc

′

λdd

′

Ta

′b′

c′d′.

Again the inverse is obvious. Thecommutation functionsrelated to the tetrad are
the quantitiesγabc(xi)defined by thecommutators[ea,eb]of the basis vectors:

[ea,eb]=γcab(xi)ec⇒γabc(xi)=−γacb(xi). (3.52)