One formula for the overtones of spinning black holes

When two black holes merge, the new one rings like a bell before settling down. The tones of that ringing, its quasinormal modes, are fixed by just its mass and spin. Measuring them is how gravitational-wave astronomy tests general relativity.

Born from the forecast

Singularity predicted that Kolmogorov-Arnold networks, a recent kind of neural network that learns simple one-dimensional functions instead of opaque weights, would reach general relativity. We asked a concrete question: can such a network help find a formula where today there are only tables?

The question

The standard fitting formulas cover the fundamental tone and one or two overtones, one mode at a time, and lose accuracy as the black hole spins close to its maximum. We wanted a single expression that covers many tones and overtones at once, all the way to near-extremal spin.

What we found

The damping of each tone is tied to a physical quantity, the instability of light orbiting the black hole, and the formula learns only a small correction on top of it. That is what lets it generalise to modes it never saw. Every comparison was written down in advance and run once; in all of them the formula was more accurate than the established fits calibrated separately on each mode.

Why it matters

Fast and accurate formulas for the ringing tones are a basic ingredient of the models used to read gravitational-wave signals. One expression that holds across many overtones and near-maximal spin makes those models simpler and more robust.

Limits

• It covers the dominant family of modes (prograde, l = m). A dedicated variant covers the l = m = 2 mode with errors of 1.2% and 1.0%.
• It is a fit guided by physics, not a derivation from first principles.