Riemann normal coordinates, Fermi reference system and the geodesic deviation equation

We obtain the integral formulae for computing the tetrads and metric compon ents in Riemann normal coordinates and the Fermi coordinate system of an ob server in arbitrary motion. Our approach admits an essential enlarging of t he range of validity of these coordinates. The results obtained are applied to the geodesic deviation in the held of a weak plane gravitational wave a nd the computation of the plane-wave metric in Fermi normal coordinates.