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 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.
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