Given a fixed base point and a tetrad , we assign to a neighbouring point the four coordinatesRiemann normal coordinates (RNC), and they are such that , or
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 Eq. (8.3), , is given by , where is the determinant of the metric in the original coordinates, and is the Van Vleck determinant of Eq. (7.2). 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 RNC, the determinant of the metric is given by
We now would like to invert Eq. (8.3) 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 Eq. (6.8),
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 Eq. (8.1). Substituting this into Eq. (8.3) yields
We are now in a position to calculate the metric in the new coordinates. We have , and in this we substitute Eq. (8.7). The final result is , with
which is the standard transformation law for tensor components.
It is obvious from Eq. (8.8) that and , where is the connection compatible with the metric . The Riemann normal coordinates therefore provide a constructive proof of the local flatness theorem.
Living Rev. Relativity 14, (2011), 7
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