Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 5; Problem 5.7.
Another example of a non-rectangular 2-d curved coordinate system. This time we have a parabolic bowl with equation
where and is a constant. We use the two coordinates (as defined here) and the azimuthal angle .
Using the same technique as with the sphere, we consider infinitesimal displacements along the constant curves. The curve of constant is a parabola, while the curve of constant is a circle at height . The tangents to the two curves at a given point are always perpendicular, so the metric will be diagonal. To find the diagonal components, consider an infinitesimal displacement . We have
and our job is to find the lengths of the two basis vectors.
Consider first a displacement along . As increases, we move a distance up the side of the parabolic bowl. This displacement consists of a horizontal increment of size and a vertical increment of size . By Pythagoras, the total displacement is . The length of is therefore .
A displacement along is a displacement around a circle of constant , so here , giving the length of as . The metric is then