Conserved currents for Kerr and orthogonality of quasinormal modes

TL;DR

This paper introduces a conserved bilinear form for Kerr perturbations, demonstrating orthogonality of quasinormal modes and enabling precise calculation of excitation coefficients, which could facilitate nonlinear mode coupling analysis.

Contribution

The paper presents a novel conserved bilinear form for Kerr Weyl scalar perturbations, establishing mode orthogonality and precise excitation coefficient computation, based on Kerr's Petrov type D structure.

Findings

Quasinormal modes are orthogonal under the new bilinear form.

Excitation coefficients are given by the bilinear form projection.

The bilinear form is conserved and symmetric, aiding nonlinear analysis.

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 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.