Conserved currents for Kerr and orthogonality of quasinormal modes

Abstract

We introduce a bilinear form for Weyl scalar perturbations of Kerr. The formis symmetric and conserved, and we show that, when combined with a suitablerenormalization prescription involving complex r integration contours,quasinormal modes are orthogonal in the bilinear form for different (l, m, n).These properties are not in any straightforward way consequences of standardproperties for the radial and angular solutions to the decoupled Teukolskyrelations and rely on the Petrov type D character of Kerr and its t-$\phi$reflection isometry. Finally, we show that quasinormal mode excitationcoefficients are given precisely by the projection with respect to our bilinearform. We believe that these properties can make our bilinear form useful to setup a framework for nonlinear quasinormal mode coupling in Kerr. We include ageneral discussion on conserved local currents and their associated localsymmetry operators for metric and Weyl perturbations of Kerr. In particular, weobtain an infinite set of conserved, local, gauge invariant currents associatedwith Carter's constant for metric perturbations, containing 2n + 9 derivatives.<br