An indispensable tool in the study and characterisation of the dynamics of black holes is their spectrum of Quasinormal Modes (QNMs). These are solutions to the wave equation arising when general relativity is considered perturbatively at linear order, and they determine how small perturbations evolve over time, capturing their “ringdown” behaviour. As such, QNMs have received a lot of attention in the literature. In astrophysics, the detection of QNMs in gravitational wave experiments would allow precise measurements of the mass and spin of black holes — through the so-called black hole spectroscopy programme — as well as new tests of general relativity. Similarly, QNMs also serve as indicators of black hole instabilities: a single unstable mode signals exponentially growing perturbations leading to a new equilibrium configuration. In addition, QNMs also play an instrumental role in semiclassical gravity, e.g. in the context of Hawking radiation. The study of second order QNMs is currently at the forefront of research.
A defining property of a black hole —and arguably the most intriguing one — is its event horizon, through which energy dissipates. This dissipative nature of black holes has a direct imprint on the properties of QNMs. Specifically, QNM eigenfunctions are neither orthogonal nor complete, while the QNM frequencies are highly sensitive to small perturbations, resulting in spectral instability. These features substantially complicate the interpretation of QNMs and, in fact, in certain contexts question the validity of their use. To-date, we have only explored the tip of the iceberg in terms of the implications of these properties, especially in dynamical settings, where the non-orthogonality of QNMs can give rise to short-term, transient phenomena.