Spherical metric: distance in 2-d curved space

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:

$\displaystyle ds^{2}=R^{2}d\theta^{2}+R^{2}\sin^{2}\theta d\phi^{2} \ \ \ \ \ (1)$

For any given sphere, ${R}$ is a constant and represents the radius of the sphere when viewed in three dimensions. In the two-dimensional surface of the sphere, however, ${R}$ is just a parameter which governs distances on the surface.

If we start at the north pole, where ${\theta=0}$, and move along a curve of constant ${\phi}$ to a point ${\theta=\theta_{0}}$ then ${d\phi=0}$ and

 $\displaystyle ds^{2}$ $\displaystyle =$ $\displaystyle R^{2}d\theta^{2}\ \ \ \ \ (2)$ $\displaystyle ds$ $\displaystyle =$ $\displaystyle Rd\theta\ \ \ \ \ (3)$ $\displaystyle s$ $\displaystyle =$ $\displaystyle R\int_{0}^{\theta_{0}}d\theta\ \ \ \ \ (4)$ $\displaystyle$ $\displaystyle =$ $\displaystyle R\theta_{0} \ \ \ \ \ (5)$

Similarly, if we restrict the curve to constant ${\theta}$, then ${d\theta=0}$ and if we move between ${\phi=0}$ and ${\phi=\phi_{0}}$:

 $\displaystyle ds^{2}$ $\displaystyle =$ $\displaystyle R^{2}\sin^{2}\theta d\phi^{2}\ \ \ \ \ (6)$ $\displaystyle s$ $\displaystyle =$ $\displaystyle R\left(\sin\theta\right)\phi_{0} \ \ \ \ \ (7)$

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 ${R}$, ${\theta}$ and ${\phi}$. Looking at the problem in 2-d, ${\theta}$ and ${\phi}$ are the coordinates of a point in curved space, and ${R}$ is just a parameter that determines how curved the space is.