Two spinning black holes held apart by their own hair

Normally, two black holes can't just hover near each other — gravity pulls them together and nothing classical stops the collapse. This work shows that a self-interacting scalar field, synchronized with the black holes' spin, can provide the missing repulsive force to maintain equilibrium. Adding quartic self-interactions reshapes the solution landscape in unexpected ways: ergoregions undergo topological transitions from single to double tori, and repulsive self-interactions make horizons 'hairier' but not heavier — only attractive self-interactions can increase the horizon mass fraction.