General Relativity: An Introduction for Physicists · Problem 13.20

The Kerr geometry — Hobson, Efstathiou & Lasenby

Physics · kerr-photon-orbit-rdot-squared

Problem

For a photon orbit in the equatorial plane of the Kerr geometry, show that

\dot{r}^2 = \frac{(r^2 + a^2)^2 - a^2\Delta}{c^2 r^4}\left[c^2k - V_+(r)\right]\left[c^2k - V_-(r)\right],

where

V_{\pm}(r) = \frac{2\mu r a \pm r^2 \Delta^{1/2}}{(r^2 + a^2)^2 - a^2\Delta}\, ch = \left[\omega \pm \left(\omega^2 - \frac{g^{\phi\phi}}{g^{tt}}\right)^{1/2}\right]h.