Fermi coordinates for weak gravitational fields

We derive the Fermi coordinate system of an observer in arbitrary motion in an arbitrary weak gravitational field valid to all orders in the geodesic distance from the world line of the observer. In flat space-time this leads to a generalization of Rindler space for arbitrary acceleration and rotation. The general approach is applied to the special case of an observer resting with respect to the weak gravitational field of a static mass distribution. This allows us to make the correspondence between general relativity and Newtonian gravity more precise.