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 or even in the whole past of the observer we examine which construction principles are responsible for this effect and how they may be modified. A 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 shows 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 also leads 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.
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