The gravitational perturbations of the Kerr black hole. III. Further amplifications

Abstract

The present paper is devoted to an amplification of the solution of the Newman-Penrose equations considered in the two earlier papers of this series. The principal amplification consists in showing that the function Ψ, in terms of which the metric perturbations are most simply expressed (and which was thought to require quadratures), besides being separable in its variables, is expressible directly in terms of the Teukolsky functions (and eliminates the need for quadratures). It is further shown that the completion of the solution for the metric perturbations requires the consideration of four additional equations which follow from four Ricci identities (not hitherto considered); and the solution of these equations is found. It is also pointed out that while the perturbation in the Weyl scalar, Ψ₂, can be set equal to zero, it cannot be deduced to be zero. (The contrary statement in the earlier paper arose from an error of a factor 2 in one of the equations.) Numerical verification of some of the principal equations and identities of the theory is provided. An important aspect of the analysis contained in this paper is the emergence of several crucial identities among the Teukolsky functions which one might despair of verifying directly.