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Researchers show that fast rotation alone can spur a black hole to grow a nontrivial scalar field, even when the ordinary linear trigger for such growth is absent. The work studies a version of Einstein-scalar-Gauss-Bonnet (EsGB) gravity, a theory that adds a single extra scalar field and a special curvature term (the Gauss–Bonnet invariant) to Einstein’s equations. By choosing a coupling between the scalar and curvature that removes the usual linear instability, the authors isolate a genuinely nonlinear route to “scalarization”—the appearance of scalar hair around a rotating (Kerr) black hole.
The team first held the spacetime geometry fixed and evolved the scalar field on a Kerr black hole background. They found that sufficiently rapid rotation alters the Gauss–Bonnet invariant near the horizon so that a negative region appears close to the poles. For negative values of the coupling parameter used in the study, this negative region acts like a geometric trapping zone and creates an effective potential that can support nonlinear growth of the scalar field. The authors identify a clear threshold in dimensionless spin, χ = 0.5: below that value no trapping region forms near the poles, while above it the region appears and can support growth.
Because their coupling was chosen so that small perturbations are governed by the ordinary massless wave equation, there is no linear, or “tachyonic,” instability of the bald Kerr solution. In practice this means whether scalar hair grows depends on the size of the initial perturbation. In time-evolution examples on fixed backgrounds, low-spin cases saw all perturbations decay back to zero. For high-spin cases (χ above 0.5), sufficiently large initial wavepackets led to sustained growth and saturation of the scalar field, demonstrating a nonlinear scalarization process that requires a finite perturbation to get started.
Next the authors solved the full coupled field equations, allowing the scalar to deform the spacetime (backreaction). They constructed stationary, axisymmetric scalarized black hole solutions and mapped their domain of existence. Unlike earlier spin-induced spontaneous scalarization studies—where scalarized solutions form a narrow line—these nonlinear solutions occupy a finite wedge in the spin–mass plane: they lie at relatively low mass and high spin. Toward the high-spin edge the scalar hair becomes stronger and the solutions approach a near-extremal (very fast rotating) state, while toward the low-spin boundary the scalar field fades and approaches the threshold χ→0.5.
These results show that rotation can trigger scalar hair by purely nonlinear means, not only through linear instabilities. Important caveats apply. The effect depends on model choices: the particular coupling function used (which enforces no linear instability) and a negative curvature coupling were essential in these calculations. The nonlinear growth also requires a sufficiently large initial perturbation, so not every rapidly spinning black hole would scalarize automatically. The numerical results shown use specific parameter values (for example a large coupling parameter in the authors’ runs) and focus on a limited region of parameter space. The paper presents a proof of principle rather than a definitive statement about astrophysical black holes or their observable signatures.