## Abstract

A method due to Chrzanowski, involving horizon multipole moments, is applied to the problem of a black hole perturbed by an enclosing, distant, spinning, spherical shell of matter. The hole, of mass $M$ and angular momentum $J=aM$, is at the centre of the shell, their respective axes of rotation differing by an angle $\zeta $. The matter-distribution on the shell is axisymmetric about its axis of rotation, but otherwise arbitrary, except that the total mass of the shell is small in comparison with $M$. The energy-momentum tensor of such a shell has been previously found by Bass & Pirani. Using their expression, we calculate the spin-down law for the black hole, correct to leading order in the inverse of the shell's radius, and to second order in its angular velocity. The solution may be expressed in terms of the 'electric' and 'magnetic' components $E_{\alpha \beta}$ and $B_{\alpha \beta}$ of the Weyl tensor $C_{ijkl}$ as calculated at the centre of the shell, in the absence of the black hole. For, denoting by $J_{\|}$ and $J_{\perp}$ the components of $J$ parallel and perpendicular, respectively, to the direction of spin of the shell, we have always $\frac{\text{d}J_{\|}}{\text{d}t}=0$ and $\frac{1}{J_{\perp}}\frac{\text{d}J_{\perp}}{\text{d}t}=-\frac{4}{15}M^{3}(E_{\alpha \beta}E^{\alpha \beta}+B_{\alpha \beta}B^{\alpha \beta})(1-\frac{3}{4}\tilde{a}^{2}+\frac{15}{4}\tilde{a}^{2}$ sin$^{2}\zeta)$, where $\tilde{a}=a/M$. This law is of theoretical interest. It shows points both of similarity to, and of difference from, the known laws describing the response of a black hole to (uniform) scalar and electromagnetic fields.