MODERN COSMOLOGY

122 Cosmological models

It follows (apply this relation to the coordinatexi) that in terms of the tetrad
components,

γabc(xi)=eai(ebj∂jeci−ecj∂jebi)=− 2 ebiecj∇[ieaj]. (3.53)

These quantities vanish iff the basis{ea}is a coordinate basis: that is, there exist
coordinatesxisuch thatea=δai∂/∂xi,iff

[ea,eb]= 0 ⇔γabc= 0.

Themetric tensor componentsin the tetrad form are given by

gab=gijeaiebj=ea·eb. (3.54)

The inverse equation

gij(xk)=gabeai(xk)ebj(xk) (3.55)

explicitly constructs the coordinate components of the metric from the (inverse)
tetrad componentseai(xj). We can raise and lower tetrad indices by use of the
metricgaband its inversegab.In the case of an orthonormal tetrad,

gab=diag(− 1 ,+ 1 ,+ 1 ,+ 1 )=gab, (3.56)

showing by (3.54) that the basis vectors are unit vectors orthogonal to each
other. Such a tetrad is defined up to an arbitrary position-dependent Lorentz
transformation.
Theconnection componentsabcfor the tetrad are defined by the relations

∇ebea=cabec⇔cab=eciebj∇jeai, (3.57)

i.e. it is thec-component of the covariant derivative in theb-direction of the
a-vector. It follows that all covariant derivatives can be written out in tetrad
components in a way completely analogous to the usual tensor form, for example

∇aTbc=ea(Tbc)−dbaTdc−dcaTbd,

where for any function f,ea(f)= eai∂f/∂xi is the derivative of f in the
directionea. In the case of an orthonormal tetrad, (3.56) shows thatea(gbc)=0;
hence applying this relation to the metric tensor,

∇agbc= 0 ⇔(ab)c= 0 , (3.58)

—the connection components are skew in their first two indices, when we use
the metric to raise and lower the first indices only, and are called ‘Ricci rotation
coefficients’ or justrotation coefficients. We obtain from this and the assumption