Reference: Moore, Thomas A., A General Relativity Workbook, University Science Books (2013) – Chapter 9; Problem 9.3.
As a (very) simple example of calculating distance in a curved space, we’ll consider the metric for the surface of a sphere:
For any given sphere, is a constant and represents the radius of the sphere when viewed in three dimensions. In the two-dimensional surface of the sphere, however, is just a parameter which governs distances on the surface.
If we start at the north pole, where , and move along a curve of constant to a point then and
Similarly, if we restrict the curve to constant , then and if we move between and :
Although these results might seem obvious, that’s because we’re used to analyzing this situation in 3-d and we have an ‘obvious’ intepretation of , and . Looking at the problem in 2-d, and are the coordinates of a point in curved space, and is just a parameter that determines how curved the space is.