# Curved-spacetime, but why curved coordinates?

1. Nov 7, 2011

### Master J

I have just started self-studying General Relativity with Landau-Lifgarbagez' Classical theory of Fields, supplementing it with a bit of extra math here and there ( I have an experimental degree).
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$^{2}$ = g$_{ik}$.dx$^{i}$.dx$^{k}$

That much is obvious to me sinceif one simply takes ds$^{2}$ 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?