A top spun to its limit changes the air flow around it — the air swirls differently than around a barely spinning top. Black holes in some extended theories of gravity behave similarly. When a hole spins fast enough, the curved spacetime near its poles acquires new properties. A region emerges that acts as a trap for the scalar field — a special field not part of ordinary gravity.

In this work, the scientists considered a model where the scalar field couples to curvature through the Gauss-Bonnet term — a particular combination of curvatures that does not affect Einstein's equations in four dimensions but does influence the extra field. Often such a field arises from an instability: like a ball on a hilltop, ready to roll down at the slightest nudge. But here the mechanism is different. There is no instability; the field grows only through nonlinear effects when the spin exceeds the threshold χ=0.5 — half the maximum possible spin.

The scientists constructed full solutions that include the field's backreaction on the geometry. It turned out that scalarized black holes do not exist everywhere, but only in a wedge on the mass-spin diagram: at low masses and high spins. Near the upper boundary of the wedge, where the spin is almost maximal, the scalar 'hair' becomes very strong. Near the lower boundary — at χ→0.5 — the field is barely noticeable. This contrasts sharply with spontaneous scalarization, where solutions line up in a narrow band.

This nonlinear mechanism resembles a quiet backwater by the bank of a fast river. The current itself creates the conditions, and water suddenly begins to swirl into a vortex — not because someone pushed it, but because the flow rearranged itself. Here, the role of the current is played by the rotation of spacetime, and the vortex is the growing scalar field.