A black hole is like a giant whirlpool in a river — space and time swirl around it and get pulled inward. Mathematically, this whirlpool is described by the Kerr solution, but it has remained a mystery why perturbations on such a background split into independent parts. Ripples from a thrown stone spread in circles without mixing with waves from the wind.

It turns out the Kerr metric has a secret twin. If you change the scale, like putting a different coordinate grid on the whirlpool, the resulting geometry becomes Kähler. This is a special type of surface where curvature is simple and harmonious. In such geometry, there are natural 'tracks' for signals that run parallel to each other.

The key trick is that on a Kähler background, self-dual 2-forms are transported by a special derivative. This preserves their decomposition into parts. These 2-forms are mathematical objects that describe, for example, electromagnetic waves. Operators acting on such waves do not mix components, as if each string on a guitar sounded on its own, even if you plucked them all at once. The authors showed that the famous Teukolsky operator for perturbations of any spin arises from the ordinary Laplacian on the Kähler twin via a simple similarity transformation. For electromagnetic waves, it's even clearer: Maxwell's equations in vacuum reduce to the equation dδF = 0, where δ is the codifferential of the Kähler metric, and it automatically splits. Thus, hidden geometry explains why equations on a black hole are solved piecewise rather than turning into an indecipherable mess.