General Relativity: An Introduction for Physicists · Problem 13.13

The Kerr geometry — Hobson, Efstathiou & Lasenby

Physics · kerr-circular-orbit-angular-velocity

Problem

For a Kerr black hole, using the Boyer–Lindquist coordinates show that, for a particle in circular orbit in the

\theta = \pi/2

plane, the coordinate angular velocity

\Omega = d\phi/dt

satisfies

\Omega = \frac{c\mu^{1/2}}{a\mu^{1/2} \pm r^{3/2}}.

This is the Kerr-metric analogue to

\Omega^2 = GM/r^3

for the Schwarzschild metric. Here the plus sign corresponds to prograde orbits, the minus to retrograde orbits.