The interaction between a weak magnetic field and a black hole

Abstract

The problem investigated is: what happens to a rotating black hole sunk in a vacuum magnetic field, constantly aligned at angle γ to its rotation axis far from the hole? The Newman-Penrose quantities Φ 0 and ρ -2 Φ 2 , which describe the radiation parts of the external field, are obtained as solutions to the Teukolsky equation with appropriate boundary conditions. From these two quantities the complete distant asymptotic form of the electromagnetic field is constructed via the four-vector potential A i , it by using the method of Chandrasekhar. Changes in the angular momentum of the hole are calculated. The component perpendicular to the field decreases exponentially with time according to the law J ⊥ = ( J ⊥ ) initial exp (- t ⊤ -1 ), Where ⊤ -1 =16/3 π G 2 c -5 (mass of hole) x (distant magnetic field energy-density), while the component parallel to the field remains constant. No energy emerges from the hole, kinetic rotational energy instead transforming into irreducible mass. This is precisely the outcome known from study of the slowly rotating hole. Extension of the result is of astrophysical relevance, since a real black hole may be rotating relatively fast. And it is of some theoretical interest that terms of second and higher order in angular momentum make no difference to the spin-down behaviour.