Drop a pebble into still water — ripples spread at once. Now strike a large church bell: first you hear a sharp clang, and only then a low hum swells. A black hole after a collision behaves more like a bell than a pond. Its 'ringing' is gravitational waves, and within them there is a short delay before the main signal, which is called the fast response. The authors of a new study looked into the mathematical 'kitchen' of this phenomenon and found an intriguing order there.

They studied the equation that describes how space around a rotating black hole (Kerr black hole) quivers. In this equation, as in a complex clockwork, there are several gears: one describes a wave running from the horizon, another a wave coming from infinity. To assemble the full picture, you need to know how these gears mesh. It turned out that each 'gear' individually has simple mathematical features, poles, at strictly defined frequencies, which are called Matsubara frequencies. But when the pieces are put together, these poles neatly cancel out, like numerator and denominator in a fraction. The final signal contains no infinities, but the very existence of such hidden poles explains why the fast response looks the way it does.

Furthermore, the researchers found that in the parts of the final signal responsible for individual 'chunks' of the wave, strong zero-frequency singularities hide. Their strength depends on how fast the hole spins. Fortunately, in the full response of the black hole, all these infinities also cancel each other out. This is like several musicians in an orchestra hitting a wrong note, but together it gets lost in the overall sound. Understanding such internal compensation helps build more accurate models of gravitational-wave signals that LIGO and Virgo detectors catch. After all, it is from the shape of this 'ringing' that we learn the mass and spin of the colliding black holes.