Spacetime and Geometry · Problem 6.2

Perturbation Theory and Gravitational Radiation — Sean Carroll

Physics — General Relativity · kerr-black-hole-photon-orbits

Problem

Consider the orbits of massless particles, with affine parameter

\lambda

, in the equatorial plane of a Kerr black hole. (a) Show that

\left(\frac{dr}{d\lambda}\right)^2 = \frac{\Sigma^2}{\rho^4}(E - LW_+(r))(E - LW_-(r)), \tag{6.121}

where

\Sigma^2 = (r^2+a^2)^2 - a^2\Delta(r)\sin^2\theta

E

and

L

are the conserved energy and angular momentum, and you have to find expressions for

W_\pm(r)

. (b) Using this result, and assuming that

\Sigma^2 > 0

everywhere, show that the orbit of a photon in the equatorial plane cannot have a turning point inside the outer event horizon

r_+

. This means that ingoing light rays cannot escape once they cross

r_+

, so it really is an event horizon.