A conserved energy integral for perturbation equations in the Kerr-de Sitter geometry

An analytic proof of mode stability of the Kerr black hole was provided by Whiting. In his proof, the construction of a conserved quantity for the uns table mode was crucial. We extend the method of this analysis to the Kerr-d e Sitter geometry. The perturbation equations of massless fields in the Ker r-de Sitter geometry can be transformed into Heun's equations, which have f our regular singularities. In this paper we investigate differential and in tegral transformations of solutions of these equations. Using these, we con struct a conserved quantity for unstable modes in the Kerr-de Sitter geomet ry, and we find that this quantity cannot bound the magnitudes of the time derivative of perturbations.