Given a fixed base point and a tetrad , we assign to a neighbouring point the four coordinates

where is the dual tetrad attached to . The new coordinates are called Riemann normal coordinates (RNC), and they are such that , or Thus, is the squared geodesic distance between and the base point . It is obvious that is at the origin of the RNC, where .If we move the point to , the new coordinates change to , so that

The coordinate transformation is therefore determined by , and at coincidence we have the second result follows from the identities and .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

It is easy to show that the geodesics emanating from are straight coordinate lines in RNC. The proper volume of a small comoving region is then equal to , and this is smaller than the flat-spacetime value of if , that is, if the geodesics are focused by the spacetime curvature.

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

and this is easily inverted to give This is the desired approximate inversion of Eq. (8.3). It is useful to note that Eq. (8.7), when specialized from the arbitrary coordinates to , gives us the components of the dual tetrad at in RNC. And since in RNC, we immediately obtain the components of the parallel propagator: .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

The quantities appearing in Eq. (8.8) are the frame components of the Riemann tensor evaluated at the base point , and these are independent of . They are also, by virtue of Eq. (8.4), the components of the (base-point) Riemann tensor in RNC, because Eq. (8.9) can also be expressed aswhich 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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