QNMs

Quasi-normal modes (QNMs) are characteristic oscillations of dissipative systems described by linear perturbations about a stationary background, such as a black hole or a compact star. They are solutions to wave equations with boundary conditions that permit energy to leak away, yielding complex eigenfrequencies. The real part of a QNM frequency sets the oscillation rate, while the imaginary part determines the exponential damping due to energy loss.

In general relativity, the prototypical QNMs arise from perturbations of black holes. For a Schwarzschild black hole these modes are obtained from the Regge-Wheeler and Zerilli equations; for rotating (Kerr) black holes the Teukolsky equation applies. The boundary conditions require purely ingoing waves at the event horizon and purely outgoing waves at infinity, leading to a discrete spectrum of complex frequencies. In Kerr spacetimes, the spectrum is primarily determined by the black hole’s mass and spin, reflecting the no-hair property: the late-time ringing can, to a good approximation, reveal remnant parameters without regard to the details of the initial perturbation.

Observationally, QNMs govern the ringdown phase of gravitational-wave signals from binary mergers. Measuring QNM frequencies and damping times allows estimation of the remnant mass and spin and enables tests of general relativity, including the no-hair theorem. Multi-mode analyses can probe deviations from the Kerr spectrum.

Calculation methods include Leaver’s continued fraction technique, time-domain perturbation theory, and numerical relativity simulations. Beyond black holes, QNMs also describe oscillations of neutron stars and other compact objects, and appear in diverse dissipative systems such as optical resonators.