Approaching general relativity: introducing curvilinear coordinate systems

Abstract

Cartesian coordinate systems have straight coordinate lines orthogonal to each other. The coordinate basis vector fields are constant, and all the vectors of each field are unit vectors with the same direction. This is the simplest coordinate system that one can imagine. It seems strange that it can be advantageous to introduce coordinate systems with coordinate curves which are not straight lines, so-called curvilinear coordinate systems. But in fact some objects of investigation have certain symmetries, for example cylindrical or spherical symmetry that makes it advantageous to introduce curvilinear coordinates. In a flat space, however, every coordinate system can be transformed into a Cartesian system, so that in principle one could solve every problem with reference to Cartesian coordinates.