Conserved Currents and Bilinear Forms for Quasinormal Modes in the Hyperboloidal Framework

Explore the mathematical framework for analyzing quasinormal modes in black hole spacetimes through this 22-minute conference lecture from the Erwin Schrödinger International Institute. Delve into the late-time gravitational-wave signals from binary black hole mergers, which are dominated by quasinormal ringing - a spectrum of damped oscillations with complex frequencies that depend solely on the remnant black hole's mass and spin. Learn how these modes create a unique spectral fingerprint of the final black hole and discover why current observations already suggest the presence of overtones. Examine the challenges in developing a comprehensive framework for quasinormal mode interactions, including the non-orthogonal nature of Kerr QNMs and the divergence issues of their spatial wavefunctions at both the bifurcation sphere and spatial infinity. Understand the complications these properties create for constructing canonical inner products on standard Cauchy hypersurfaces and projecting onto QNMs to study nonlinear mode mixing. Investigate the innovative solution of bilinear forms that function as inner-product analogues for Weyl scalars, constructed from conserved currents and symmetry operators. Analyze their key properties on hyperboloidal slices using the Schwarzschild spacetime as a model example, providing insights into the connection between linear superposition models and fully nonlinear descriptions of gravitational wave signals.