  (Some Latex problems should be removed in this version) Fermi coordinates
(FC) are supposed to be the natural extension of Cartesian coordinates for an
arbitrary moving observer in curved space-time. Since their construction cannot
be done on the whole space and even not in the whole past of the observer we
examine which construction principles are responsible for this effect and how
they may be modified. One proposal for a modification is made and applied to
the observer with constant acceleration in the two and four dimensional
Minkowski space. The two dimensional case has some surprising similarities to
Kruskal space which generalize those found by Rindler for the outer region of
Kruskal space and the Rindler wedge. In perturbational approaches the
modification leads also to different predictions for certain physical systems.
As an example we consider atomic interferometry and derive the deviation of the
acceleration-induced phase shift from the standard result in Fermi coordinates.
