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.

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