On perturbations of a Kerr black hole

Abstract

Gravitational perturbations of a Kerr black hole are analyzed using the Newman‐Penrose formalism. Teukolsky has obtained decoupled wave equations for the perturbed Weyl tensor components ψ₀ and ψ₄. In this paper we prove that for well‐behaved perturbations ψ₀ and ψ₄ uniquely determine each other, i.e., ψ₀ = 0 if and only if ψ₄ = 0. Then we solve the Kerr perturbation equations with ψ₀ = ψ₄ = 0 and show that the only well‐behaved solutions are the trivial perturbations to other Kerr solutions via an infinitesmal change in the mass and angular momentum parameters. These results prove that either of the quantities ψ₀ or ψ₄ alone uniquely specifies the nontrivial part of a gravitational perturbation of a Kerr black hole. Consequences of this result are discussed.