# Conserved currents for Kerr and orthogonality of quasinormal modes

### Abstract

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 $(l, m, n)$. These properties are not in any straightforward way 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. Finally, we show that quasinormal mode excitation coefficients are given precisely by the projection with respect to our bilinear form. We believe that these properties can make our bilinear form useful to set up a framework for nonlinear quasinormal mode coupling in Kerr. We include a general discussion on conserved local currents and their associated local symmetry operators for metric and Weyl perturbations of Kerr. In particular, we obtain an infinite set of conserved, local, gauge invariant currents associated with Carter’s constant for metric perturbations, containing $2n + 9$ derivatives.

##### Stephen R. Green
###### Nottingham Research Fellow

I am a theoretical physicist studying gravitational waves, based at the University of Nottingham. My main interests are in black hole perturbation theory and applying probabilistic machine-learning methods to analyze LIGO data.