Spin-Induced Scalarization in Black Holes

• Spin-induced scalarization is a threshold instability in rotating compact objects that leads to scalar hair formation when curvature or electromagnetic invariants become negative.
• The mechanism is analyzed linearly through perturbations on Kerr or Kerr–Newman spacetimes and nonlinearly, revealing critical spin thresholds around a/M ≈ 0.5 and distinct phase transitions.
• Extensions include spin-charge induced scalarization, environmental influences like magnetic fields and dark matter, and dynamical effects in mergers, impacting tests of gravity and black hole uniqueness.

Spin-induced scalarization is a strong-field threshold phenomenon in which rotation destabilizes a scalar-free compact-object solution and drives the formation of scalar hair. In the black-hole setting, which provides the cleanest realization, the mechanism is usually formulated by linearizing a nonminimally coupled scalar field on a Kerr or Kerr–Newman background and showing that spin changes the sign structure of the curvature or electromagnetic invariant entering the scalar’s effective mass squared. When a sufficiently negative near-horizon region appears, the trivial scalar configuration becomes tachyonically unstable; nonlinear effects are then expected to quench the growth and yield a stationary scalarized branch. In the literature this mechanism is most developed in scalar–Gauss-Bonnet models for Kerr black holes, and in Einstein–Maxwell–scalar models for charged rotating black holes, where it can become genuinely spin-charge induced (Doneva et al., 2022, Dima et al., 2020, 2206.12074).

1. General mechanism

Scalarization is commonly described as a phase-transition-like instability of a bald solution. In the generic language adopted by the review literature, the linearized scalar perturbation satisfies an equation of the form

$$g_{\rm eff}^{\mu\nu}\nabla^{(0)}_\mu\nabla^{(0)}_\nu\delta\varphi-\mu_{\rm eff}^2\,\delta\varphi+{\rm NLC}=0,$$

and the onset of scalarization is associated with $\mu_{\rm eff}^2<0$ in a sufficiently large region outside the horizon (Doneva et al., 2022). The nonlinear terms then regulate the instability and permit a new equilibrium with nontrivial scalar profile.

Spin-induced scalarization differs from compactness-induced scalarization in that the trigger is not primarily the overall compactness or mass scale of the object, but the rotation-induced restructuring of the source invariant. In scalar–Gauss-Bonnet models this invariant is $\mathcal G$; in Einstein–Maxwell–scalar models it is the Maxwell invariant $\mathcal F=F_{\mu\nu}F^{\mu\nu}$. The scalar perturbation then acquires an effective mass squared proportional to one of these invariants. In the Gauss-Bonnet case one has, schematically, $$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$, while in the EMS case with weak-field expansion $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$ the linearized equation is $\square\phi=\mu_{\rm eff}^2\phi$ with $\mu_{\rm eff}^2=-\alpha\mathcal F$ (Dima et al., 2020, 2206.12074).

The essential role of spin is therefore geometric. Rotation can make an invariant change sign or become sufficiently negative in localized exterior regions even when the corresponding nonrotating solution is stable. In the Kerr black-hole case this effect is tied to the sign structure of $\mathcal G$; in Kerr–Newman it can also be tied to the angular dependence of $\mathcal F$. This makes the onset problem model dependent, but the underlying instability logic is uniform across the literature (Doneva et al., 2022).

2. Kerr black holes in scalar–Gauss-Bonnet gravity

The earliest systematic analyses of black-hole spin-induced scalarization were carried out in quadratic scalar–Gauss-Bonnet theories, where the scalar perturbation on Kerr feels a position-dependent effective mass generated by the Gauss-Bonnet invariant. For Kerr,

$\mu_{\rm eff}^2<0$0

so $\mu_{\rm eff}^2<0$1 is no longer positive definite. The central result is that regions with $\mu_{\rm eff}^2<0$2 appear outside the horizon only once the dimensionless spin is sufficiently large, around $\mu_{\rm eff}^2<0$3, and this is what enables tachyonic growth for the coupling sign that would leave Schwarzschild stable (Dima et al., 2020, Herdeiro et al., 2020).

The linear instability was studied both through mode-coupled $\mu_{\rm eff}^2<0$4 evolutions and direct $\mu_{\rm eff}^2<0$5 time-domain simulations. The latter confirmed that the instability is strongest in the $\mu_{\rm eff}^2<0$6 sector, that the growth time decreases as $\mu_{\rm eff}^2<0$7 and the Gauss-Bonnet coupling increase, and that the critical spin approaches $\mu_{\rm eff}^2<0$8 in the large-coupling limit (Doneva et al., 2020). The original Kerr analysis also emphasized that the instability is tachyonic rather than superradiant: it is dominated by $\mu_{\rm eff}^2<0$9 modes, can operate on very short timescales, and disappears if the sign structure of the effective mass term is removed (Dima et al., 2020).

Several refinements of the basic Kerr picture were subsequently developed. Adding a scalar mass suppresses the instability by increasing the minimum Gauss-Bonnet coupling required for growth and shrinking the unstable region in the $\mathcal G$0 plane, but it does not change the critical minimum value of the black-hole angular momentum at which scalarization becomes possible (Doneva et al., 2020). Conversely, external magnetic fields do not lower the Kerr spin threshold in the near-horizon regime. For Kerr–Melvin and Kerr–Newman–Melvin spacetimes with $\mathcal G$1, the negative-$\mathcal G$2 region moves to larger values of the dimensionless spin $\mathcal G$3, so the magnetic field works against spin-induced scalarization rather than facilitating it (Annulli et al., 2022).

3. Nonlinear scalarized Kerr solutions and phase structure

The linear onset problem motivated the construction of fully backreacted scalarized Kerr black holes. In quadratic EsGB models with negative coupling, stationary scalarized solutions were obtained with both even and odd scalar parity under equatorial reflection. Their domain of existence is bounded by threshold Kerr solutions on one side and by “critical” families on the other, where the near-horizon expansion ceases to admit real coefficients (Berti et al., 2020). These solutions can violate the Kerr rotation bound $\mathcal G$4; the odd-parity branch does so already for $\mathcal G$5, while the even-parity branch does so marginally for $\mathcal G$6 (Berti et al., 2020).

A complementary nonlinear construction showed that the scalarized Kerr branch appears near $\mathcal G$7, extends roughly over $\mathcal G$8, and reaches slightly above the Kerr bound with $\mathcal G$9 (Herdeiro et al., 2020). In that model the scalarized solutions are the nonlinear completion of the spin-triggered tachyonic instability, whereas black holes below the onset remain Kerr. For the quadratic coupling studied in (Berti et al., 2020), the scalarized black holes are entropically favored over Kerr at fixed mass and angular momentum, and their horizon area can exceed that of Kerr by about $\mathcal F=F_{\mu\nu}F^{\mu\nu}$0 near the Kerr bound.

Later work broadened the concept of spin-induced scalarization beyond the standard linear-bifurcation picture. In an EsGB model with

$\mathcal F=F_{\mu\nu}F^{\mu\nu}$1

one has $\mathcal F=F_{\mu\nu}F^{\mu\nu}$2 and $\mathcal F=F_{\mu\nu}F^{\mu\nu}$3, so scalar-free Kerr is linearly stable and there is no linear zero mode. Nevertheless, sufficiently rapid rotation produces a negative near-horizon region of $\mathcal F=F_{\mu\nu}F^{\mu\nu}$4 near the poles once $\mathcal F=F_{\mu\nu}F^{\mu\nu}$5, creating a geometric trapping zone for genuinely nonlinear scalar growth. The resulting fully backreacted scalarized black holes occupy a finite low-mass high-spin wedge in the $\mathcal F=F_{\mu\nu}F^{\mu\nu}$6 plane rather than the narrow bifurcation band familiar from spontaneous scalarization (Lai et al., 30 Apr 2026).

A further extension appears in cubic scalar–Gauss-Bonnet theory with $\mathcal F=F_{\mu\nu}F^{\mu\nu}$7, where $\mathcal F=F_{\mu\nu}F^{\mu\nu}$8 and the instability is again nonlinear. At large spin, curvature-induced and spin-induced scalarization can coexist at fixed sign of the coupling, producing a phase diagram with regions in which Kerr, curvature-induced scalarized black holes, and spin-induced scalarized black holes all coexist for the same global charges. The transitions between these branches can be continuous or discontinuous, and the coexistence region represents a particularly strong breaking of black-hole uniqueness (Eichhorn et al., 5 Mar 2026).

4. Kerr–Newman black holes and spin-charge induced scalarization

For charged rotating black holes, spin-induced scalarization acquires additional structure because the scalar may couple to the Maxwell invariant rather than to $\mathcal F=F_{\mu\nu}F^{\mu\nu}$9. In EMS theory the Kerr–Newman background yields

$$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$0

so the instability is controlled jointly by the coupling sign, the charge $$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$1, and the rotation parameter $$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$2 (Lai et al., 2022, Lai et al., 2022).

For negative coupling, analytic and numerical studies found a genuine spin-induced regime. In the strong-coupling limit $$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$3, the threshold condition reduces to $$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$4 with $$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$5, giving the onset bound

$$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$6

Time-domain hyperboloidal evolutions then yielded threshold curves $$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$7 separating bald Kerr–Newman black holes from scalarized ones (Lai et al., 2022). Fully nonlinear stationary constructions later showed that for $$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$8 the scalarized Kerr–Newman solutions exist only in a narrow region of the $$m_{\rm eff}^2\sim-(\lambda^2/4)f''(\phi_0)\mathcal G$$9 plane, bounded by a bifurcation line and a critical line, and that the scalar sector carries only a tiny fraction of the total mass, with $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$0 for $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$1 and $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$2 for $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$3. In that regime the scalarized solutions coexist with linearly stable, entropically favored Kerr–Newman black holes and are not themselves entropically preferred (Cheng et al., 2 Jun 2025).

For positive coupling, the literature identifies a distinct spin-charge induced mechanism. In the analytic EMS treatment of (2206.12074), the weak-field expansion $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$4 implies $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$5, so scalarization with $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$6 requires $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$7. In Kerr–Newman this occurs only when both spin and charge are nonzero, because neutral Kerr has $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$8 and nonspinning Reissner–Nordström has $f(\phi)=1-\alpha\phi^2+\mathcal O(\phi^4)$9. Writing

$\square\phi=\mu_{\rm eff}^2\phi$0

the onset follows from $\square\phi=\mu_{\rm eff}^2\phi$1, leading to the physically relevant root $\square\phi=\mu_{\rm eff}^2\phi$2 and the universal threshold

$\square\phi=\mu_{\rm eff}^2\phi$3

The scalarized region begins near the poles, with allowed angular domain $\square\phi=\mu_{\rm eff}^2\phi$4 (2206.12074).

Time-domain HFM evolutions of the positive-coupling EMS model reached a different numerical characterization. They produced a three-dimensional onset surface $\square\phi=\mu_{\rm eff}^2\phi$5 and reported that there is no lower bound on the rotational parameter $\square\phi=\mu_{\rm eff}^2\phi$6, while high rotation enhances spontaneous scalarization and the only upper bound is the horizon existence condition $\square\phi=\mu_{\rm eff}^2\phi$7 (Lai et al., 2022). This tension reflects a model-sensitive part of the current literature: the analytic and numerical onset criteria are not presented in identical terms. The addition of a scalar potential sharpens the parameter dependence further. In EMS theory with $\square\phi=\mu_{\rm eff}^2\phi$8, the effective mass becomes

$\square\phi=\mu_{\rm eff}^2\phi$9

and the scalar mass suppresses the instability, raising the threshold coupling and shrinking the unstable region; the threshold curve then depends on $\mu_{\rm eff}^2=-\alpha\mathcal F$0, $\mu_{\rm eff}^2=-\alpha\mathcal F$1, $\mu_{\rm eff}^2=-\alpha\mathcal F$2, and $\mu_{\rm eff}^2=-\alpha\mathcal F$3 under the bound $\mu_{\rm eff}^2=-\alpha\mathcal F$4 (Luo et al., 8 Apr 2026).

Charged rotating black holes in Gauss-Bonnet theory display yet another variant. In ESTGB theory with quadratic scalar–GB coupling, the Kerr–Newman instability is controlled by the joint effect of $\mu_{\rm eff}^2=-\alpha\mathcal F$5, but scalarization is not purely spin driven: in the infinite-coupling limit one finds $\mu_{\rm eff}^2=-\alpha\mathcal F$6 when $\mu_{\rm eff}^2=-\alpha\mathcal F$7, whereas in the vanishing-spin limit one still has $\mu_{\rm eff}^2=-\alpha\mathcal F$8, so sufficiently highly charged Reissner–Nordström black holes can scalarize even at $\mu_{\rm eff}^2=-\alpha\mathcal F$9 (Zhang et al., 2022).

5. Environmental and dynamical extensions

Spin-induced scalarization is not restricted to isolated stationary black holes. Environmental fields can either compete with it or replace spin as the operative trigger. In magnetized black-hole spacetimes modeled locally by Melvin backgrounds, the near-horizon effect of the magnetic field for $\mathcal G$0 is suppressive: the critical spin required for scalarization increases with $\mathcal G$1, which the literature interprets geometrically through the competition between rotation-induced oblateness and magnetic-field-induced prolateness of the horizon. At the same time, a sufficiently strong magnetic field can by itself generate the negative-GB region needed for GB$\mathcal G$2 scalarization in Schwarzschild–Melvin, with the explicit threshold $\mathcal G$3 (Annulli et al., 2022).

An analogous replacement of spin by matter environment occurs in extended scalar-tensor-Gauss-Bonnet theory with perfect fluid dark matter. In the GB$\mathcal G$4 regime, vacuum Schwarzschild does not scalarize, but the PFDM deformation of the metric can create a negative effective mass region outside the horizon. Analytical and numerical work found a critical threshold

$\mathcal G$5

so dark matter can induce the same tachyonic mechanism that vacuum solutions realize only through rotation (Tang et al., 21 Apr 2025).

Binary dynamics makes the role of spin time dependent. In quadratic scalar–Gauss-Bonnet gravity with negative coupling, decoupling-limit simulations of quasi-circular mergers showed both dynamical scalarization and dynamical descalarization. Initially unscalarized black holes can merge into a remnant with $\mathcal G$6 and scalarize after merger, whereas initially scalarized progenitors with $\mathcal G$7 can produce a remnant with $\mathcal G$8 that descalarizes because the final spin lies below the threshold. Near threshold, scalar effects may remain hidden through inspiral and even merger, appearing only in late ringdown; this is the “stealth” regime (Elley et al., 2022).

Hyperbolic encounters extend this picture. In the decoupling limit of quadratic sGB gravity, close passages can produce temporary dynamical scalarization even when the isolated black holes cannot sustain hair. More strikingly, changes in the individual spin magnitudes during scattering can permanently move a black hole across the scalarization threshold. The literature refers to these permanent post-encounter transitions as spin-up scalarization and spin-up descalarization (Pardoe et al., 29 Jun 2026).

6. Mathematical status and theoretical implications

The nonlinear mathematical status of spin-induced scalarization is more delicate than the linear onset analysis alone suggests. Full $\mathcal G$9 nonlinear evolutions of isolated Kerr black holes in scalar–Gauss-Bonnet theory, performed with a modified CCZ4 formulation, show that hyperbolicity can be lost during the phase of strongest scalar growth, when the scalar field and especially its gradients become large. The breakdown is diagnosed through the effective metric of the physical gravitational modes and persists under changes of gauge parameters, which supports the interpretation that the loss of hyperbolicity is dominated by the physical sector rather than by gauge artifacts (Doneva et al., 2023).

Those same evolutions also sharpen the EFT interpretation of the theory. The weak-coupling condition

$\mathcal F$0

is already strongly violated by the time hyperbolicity is lost, so the pathological phase lies well outside the regime where the sGB model should be trusted as an effective theory (Doneva et al., 2023). This does not invalidate the onset calculations, but it constrains how far nonlinear evolutions can be interpreted within the same truncated theory.

Broader model-building work points in a similar direction. Two-scalar constructions can accommodate both curvature-induced and spin-induced scalarization and thereby violate black-hole uniqueness, but they do not naturally isolate scalarization to only one mass range such as supermassive black holes. In particular, EFTs obtained by integrating out a heavy second scalar do not generate the needed sign structure or hierarchy of scales for the proposed supermassive-only mechanism (Thaalba et al., 26 Jun 2025).

Taken together, the literature presents spin-induced scalarization as a robust instability mechanism with several distinct realizations. In Kerr scalar–Gauss-Bonnet models it is the archetypal example of rotation-driven spontaneous scalarization above a threshold near $\mathcal F$1. In Kerr–Newman spacetimes it can become spin-charge induced, with the Maxwell invariant rather than the Gauss-Bonnet invariant providing the tachyonic trigger. Environmental fields, dark matter, mergers, and scattering encounters can either suppress the effect or activate analogous dynamical transitions. The remaining central questions concern the stability of the scalarized branches, the true nonlinear endpoint in EFT-consistent regimes, and the observational imprint of spin-triggered hair in gravitational-wave and electromagnetic probes (Doneva et al., 2022).