General Relativity: An Introduction for Physicists · Problem 3.20

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · geodesics-2-space-line-element

Problem

In the 2-space with line element

ds^2 = \frac{dr^2 + r^2 d\theta^2}{r^2 - a^2} - \frac{r^2\, dr^2}{(r^2 - a^2)^2},

where

r > a

, show that the differential equation for the geodesics may be written as

a^2\left(\frac{dr}{d\theta}\right)^2 + a^2 r^2 = K r^4,

where

K

is a constant such that

K = 1

if the geodesic is null. By setting

r\, d\theta/dr = \tan\phi

, show that the space is mapped onto a Euclidean plane in which

(r, \phi)

are taken as polar coordinates and the geodesics are mapped to straight lines.