He introduces curved spacetime by stating that in a non-inertial reference-frame, the space-time interval ds in curved, since it is described by a general quadratic form:

ds[itex]^{2}[/itex] = g[itex]_{ik}[/itex].dx[itex]^{i}[/itex].dx[itex]^{k}[/itex]

That much is obvious to me sinceif one simply takes ds[itex]^{2}[/itex] and graphs it it would sketch a curve. However, it is then stated that the

*coordinates*are curvilinear.

Why is that? Why cant the above equation simply represent a curve, but in orthogonal Cartesian coordinates?