Gravitational wave signals from binary black hole mergers encode the remnant black hole's quasinormal mode frequencies — the characteristic oscillation frequencies of the perturbed Kerr spacetime. These frequencies depend on the black hole's mass and spin. If the mass and spin change after the merger — through accretion, superradiance, or phase transitions — the frequencies shift, and the shift produces a dephasing of the gravitational wave signal over long observation times.
The paper derives analytical expressions for this long-term dephasing caused by phase transitions in the black hole's environment. A phase transition — for instance, in surrounding matter, in a dark matter halo, or in a superfluid condensate near the horizon — changes the effective potential that the quasinormal modes oscillate in. The change is sudden (on the dynamical timescale of the transition) but its observable effect — the accumulated phase difference — grows linearly with observation time.
The dephasing is potentially measurable: for next-generation gravitational wave detectors with observation windows of hours to days, phase transitions in the black hole's environment produce dephasing of order radians, well above detection threshold. The signal is distinctive — a phase drift that grows linearly in time, as opposed to the power-law decay of the quasinormal modes themselves.
Long-term gravitational wave observation becomes a probe not of the black hole but of its environment. The black hole is the oscillator; the environment sets the frequency; phase transitions change the frequency; and the accumulated dephasing records the history of changes. The gravitational wave is a clock, and the clock's drift measures what happened around it.