General Relativity: An Introduction for Physicists · Problem 13.11

The Kerr geometry — Hobson, Efstathiou & Lasenby

Physics · kerr-equatorial-geodesic-equations

Problem

Show that the geodesic equations for particle motion in the equatorial plane of the Kerr geometry may be written in Boyer–Lindquist coordinates as

\dot{t} = \frac{1}{\Delta}\left[\left(r^2 + a^2 + \frac{2\mu a^2}{r}\right)k - \frac{2\mu a}{cr}h\right],

\dot{\phi} = \frac{1}{\Delta}\left[\frac{2\mu ac}{r}k + \left(1 - \frac{2\mu}{r}\right)h\right],

\dot{r}^2 = (c^2k^2 - \epsilon^2) + \frac{2\epsilon^2\mu}{r} + \frac{a^2(c^2k^2 - \epsilon^2) - h^2}{r^2} + \frac{2\mu(h - ack)^2}{r^3},

where

\epsilon^2 = c^2

for a massive particle and

\epsilon = 0

for a photon. Verify that these equations reduce to the Schwarzschild case in the limit

a \to 0