Given a fixed base point and a tetrad
, we assign
to a neighbouring point
the four coordinates
If we move the point to
, the new coordinates change to
, so that
It is interesting to note that the Jacobian of
the transformation of Equation (105),
, is given by
, where
is the determinant of the metric in the
original coordinates, and
is the van Vleck determinant
of Equation (95
). This result follows
simply by writing the coordinate transformation in the form
and computing the product
of the determinants. It allows us to deduce that in the RNC, the
determinant of the metric is given by
We now would like to invert Equation (105) in order to express
the line element
in terms of the
displacements
. We shall do this approximately, by
working in a small neighbourhood of
. We recall the
expansion of Equation (89
),
and in this we substitute the frame decomposition
of the Riemann tensor, , and the tetrad decomposition of the parallel
propagator,
, where
is the dual tetrad at
obtained by parallel
transport of
. After some algebra we
obtain
where we have used Equation (103). Substituting this
into Equation (105
) yields
We are now in a position to calculate the metric
in the new coordinates. We have , and in this we substitute
Equation (109
). The final result is
, with
which is the standard transformation law for tensor components.
It is obvious from Equation (110) that
and
, where
is the connection
compatible with the metric
. The Riemann normal
coordinates therefore provide a constructive proof of the local
flatness theorem.