Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 5; Problem 5.6.
The non-rectangular coordinate systems (semi-log and sinusoidal) we’ve looked at so far have all been flat, so it’s time to look at one in curved space. We’ll use the surface of a sphere, but rather than the usual spherical coordinates we’ll use a slight variation. We keep the azimuthal angle but use as the second coordinate the quantity which is the distance along the surface of the sphere measured from the north pole. If the radius of the sphere is , then in terms of normal spherical coordinates, .
Curves of constant are the usual lines of longitude, while curves of constant are lines of latitude. 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 two basis vectors.
The displacement along is just , so is a unit vector. A displacement along depends on the radius of the constant curve. In spherical coordinates, this is , so in our new coordinate system we get the displacement as . Therefore the magnitude of is . The metric tensor is thus