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  • $\begingroup$ Your entire answer is praising a reference. Won't you answer more clearly the question? The user asks about Cartesian coordinates. The grid of which you speak looks like Cartesian? And what about curved space? How looks the grid in it? $\endgroup$ Commented Mar 2, 2015 at 22:31
  • $\begingroup$ Well, the reference frames of spacetime differ from those of Newtonian mechanics in that the Minkowski spacetime is pseudo-Euclidean, not Euclidean, i.e., minus signs show up along the diagonal of the metric. Of course, one may choose to not use Cartesian coordinates and still have a reference frame. Regarding their motion, reference frames only move or rotate relative to other reference frames. So there is a concept of a rigid set of axes that can move or rotate and then the coordinates of physical systems described by the frame will change. $\endgroup$ Commented Mar 3, 2015 at 4:03