General Relativity: An Introduction for Physicists · Problem 2.8

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · curve-length-parametrisation

Problem

A curve on the surface of a 2-sphere of radius

a

is defined parametrically by

\theta = u

\phi = 2u - \pi

, where

0 \leq u \leq \pi

. Sketch the curve and show that its total length is

L = a\int_0^\pi \sqrt{1 + 4\sin^2 u}\,du.

Show that, in general, the length of a curve is independent of the parameter used to describe it.