General Relativity: An Introduction for Physicists · Problem 13.15

The Kerr geometry — Hobson, Efstathiou & Lasenby

Physics · kerr-circular-orbit-stability

Problem

Show that the values of the parameters

k

and

h

for a circular orbit of coordinate radius

r = r_c

, given in (13.54) and (13.55) respectively, satisfy the requirements

V_{\rm eff}(r_c; h, k) = \tfrac{1}{2}c^2(k^2 - 1) \qquad \text{and} \qquad \left.\frac{dV_{\rm eff}}{dr}\right|_{r=r_c} = 0.

Show further that for the orbit to be stable one requires

r_c^2 - 6\mu r_c - 3a^2 \mp 8a\sqrt{\mu r_c} = 0.