On perturbations of a Kerr black hole

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.