Conserved currents for a Kerr black hole and orthogonality of quasinormal modes

We introduce a bilinear form for Weyl scalar perturbations of Kerr. The form is symmetric and conserved, and we show that, when combined with a suitable renormalization prescription involving complex r integration contours, quasinormal modes are orthogonal in the bilinear form for different ol, m, n thorn . These properties are apparently not evident consequences of standard properties for the radial and angular solutions to the decoupled Teukolsky relations and rely on the Petrov type D character of Kerr and its t-phi reflection isometry. We show that quasinormal mode excitation coefficients are given precisely by the projection with respect to our bilinear form. These properties can make our bilinear form useful to set up a framework for nonlinear quasinormal mode coupling in Kerr. We also provide a general discussion on conserved local currents and their associated local symmetry operators for metric and Weyl perturbations, identifying a collection containing an increasing number of derivatives.

Automated summary

We introduce a symmetric and conserved bilinear form for Weyl scalar perturbations of Kerr. By combining this form with a suitable renormalization prescription involving complex r integration contours, we show that quasinormal modes are orthogonal in the bilinear form for different ol, m, n thorn. These properties rely on the Petrov type D character of Kerr and its t-phi reflection isometry, and can be used to set up a framework for nonlinear quasinormal mode coupling in Kerr.