General Relativity: An Introduction for Physicists · Problem 13.19

The Kerr geometry — Hobson, Efstathiou & Lasenby

Physics · kerr-circular-photon-orbit

Problem

For a circular photon orbit of coordinate radius r=rcr = r_c in the Kerr geometry, show that rc=2μ{1+cos⁡[23cos⁡−1(±aμ)]},r_c = 2\mu\left\{1 + \cos\left[\frac{2}{3}\cos^{-1}\left(\pm\frac{a}{\mu}\right)\right]\right\}, b=3μrc−a,b = 3\sqrt{\mu r_c} - a, where the upper sign in the first equation corresponds to retrograde orbits and the lower sign to prograde orbits. Hence show that, for an extreme Kerr black hole (a=μa = \mu), rc=4μr_c = 4\mu for a retrograde orbit and rc=μr_c = \mu for a prograde orbit.

Related theory

Browse the problem library