Angular momentum in general relativity. II. Perturbations of a rotating black hole

Abstract

The definition of angular momentum proposed in part I of this series is investigated when applied to rotating black holes. It is shown how to use the formula to evaluate the angular momentum of a stationary black hole. This acts as a description of a background space on which the effect of first matter and then gravitational perturbations is considered. The latter are of most interest and the rate of change of angular momentum, dJ/dt, is found as an expression in the shear induced in the event horizon by the perturbation and in its time integral. Teukolsky’s solutions for the perturbed componentΨ0of the Weyl tensor are then used to find this shear and hence to give an exact answer for dJ/dt. One of the implications of the result is a direct verification of Bekenstein’s formula relating in a simple way the rate of change of angular momentum to the rate of change of mass caused by a plane wave. A more general expression is also given for dM/dt. Considering only stationary perturbations, it is shown how to generalize the definition of angular momentum so as to include information about its direction as well. Three problems are particularly discussed-a single moon, two or more moons and a ring of matter causing the perturbation - since they provide illustrations of all the main features of the black hole’s behaviour. In every case it is found that the black hole realigns its axis of rotation so that the final configuration is axisymmetric if possible; otherwise it slows down completely to reach a static state. This is entirely in agreement with Hawking’s theorem that a stationary black hole must be either static or axisymmetric.