This is driving me crazy. Consider a two-dimensional spacetime, with coordinates (t,x). If this is a flat spacetime, we can just imagine a regular-old two-dimensional plane. On that plane I could just as easily map a Cartesian/Euclidean coordinate system as a hyperbolic system of coordinates. The coordinate system is a choice. Now, suppose I have a flat rubber sheet. I map a Euclidean coordinate chart to the sheet. Now I bend the sheet so that it has constant negative curvature. Now, I'm "forced" into a hyperbolic coordinate system because of the geometric shape of the space. The Euclidean coordinates I drew onto the sheet have become warped. I now don't have a choice of coordinate systems. If spacetime is really flat, why are we forced into a hyperbolic geometry?