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 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-$\varphi$ 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.

14+11 pages, 2 figures. v3: fixed typos in Eq. (55) compared to the version accepted to PRD

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• Relativistic perturbation theory for black-hole boson clouds
• Hyperboloidal Approach to Quasinormal Modes
• Transient dynamics of quasinormal mode sums
• Metric perturbations of Kerr spacetime in Lorenz gauge: Circular equatorial orbits
• The Confluent Heun functions in Black Hole Perturbation Theory: a spacetime interpretation
• Quasi-normal mode expansions of black hole perturbations: a hyperboloidal Keldysh's approach
• Dynamical quasinormal mode excitation
• Linear vs. nonlinear modeling of black hole ringdowns
• Flux-balance laws for spinning bodies under the gravitational self-force
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• Transients in black hole perturbation theory
• QNM orthogonality relations for AdS black holes
• Numerical investigation of the late-time tails of the solutions of the Fackerell-Ipser equation
• Spin-2 Green's Functions on Kerr in Radiation Gauge
• Nonlinear Ringdowns as Sources and Detectors of Quantum Gravitational Waves
• Homogeneous Symmetry Operators in Kerr--NUT--AdS Spacetimes
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• Relativistic frequency shifts in gravitational waves from axion clouds
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