Black-Hole Scalarization Mechanisms

• Black-hole scalarization is a mechanism in which general-relativistic black holes develop nontrivial scalar-field configurations once a critical parameter, such as curvature or spin, is exceeded.
• The instability is analyzed via scalar perturbation equations near the horizon, where a negative effective mass squared triggers a tachyonic mode and signals the onset of scalarization.
• Studies reveal diverse phase structures and coupling dependencies, with the nonlinear evolution and thermodynamic analysis offering insights into the stability and unique properties of hairy black holes.

Searching arXiv for the cited scalarization literature to ground the article in current papers. Black-hole scalarization is a strong-field, nonperturbative mechanism in which a bald general-relativistic or GR-like black-hole solution becomes unstable and develops a nontrivial scalar-field configuration once a control parameter—such as curvature, charge, spin, temperature, or coupling strength—crosses a threshold. In the literature it is repeatedly described as phase-transition-like: the scalar-free branch is stable in one regime, but a tachyonic or genuinely nonlinear instability can produce a new branch of hairy black holes whose existence and stability depend on the coupling sector and on nonlinear saturation (Doneva et al., 2022, Blázquez-Salcedo et al., 2021).

1. Core mechanism and defining structure

The standard formulation begins with a theory in which the scalar-free black hole remains an exact solution, typically enforced by a coupling choice such as $\dot F(0)=0$ or $f'(0)=0$. Linearization around the bald branch then yields a scalar perturbation equation of the schematic form

$(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$

with the effective mass determined by curvature invariants or gauge-field invariants. Scalarization occurs when $\mu_{\rm eff}^2$ becomes negative enough in an exterior region to support a bound state or tachyonic mode, so that the trivial scalar configuration is destabilized and a hairy branch bifurcates from it (Doneva et al., 2022, Blázquez-Salcedo et al., 2021).

In the best-known Einstein-scalar-Gauss-Bonnet setting, the trigger is the Gauss-Bonnet invariant

$$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$

or equivalently $R_{\rm GB}^2$ in alternative notation. For a quadratic coupling, the linearized scalar equation produces an effective mass of the form

$m_{\rm eff}^2=-\eta R_{\rm GB}^2,$

so a sufficiently positive Gauss-Bonnet curvature near the horizon can drive a tachyonic instability. For Schwarzschild,

$R_{\rm GB}^2=\frac{48M^2}{r^6},$

which makes the near-horizon region the natural site of the onset (Blázquez-Salcedo et al., 2021, Doneva et al., 2022).

The threshold problem is therefore an eigenvalue problem: one seeks zero-frequency scalar clouds that are regular at the horizon and decay appropriately at infinity. These zero modes mark the onset of instability, and their nonlinear continuation produces scalarized black holes. The fundamental branch is usually nodeless, while higher-node branches correspond to excited solutions and are frequently unstable. This structure recurs across asymptotically flat, AdS, higher-dimensional, charged, and rotating settings, although the control invariant and nonlinear completion differ from model to model (Doneva et al., 2022, Astefanesei et al., 2020).

2. Curvature-induced scalarization in Gauss-Bonnet and related theories

The prototypical realization is Einstein-scalar-Gauss-Bonnet gravity, where the scalar couples nonminimally to the Gauss-Bonnet invariant and the Schwarzschild or Kerr family survives as the bald branch. The review literature emphasizes that the onset is universal at linear order: the sign and magnitude of the quadratic term in the coupling function determine whether the scalar acquires a negative effective mass squared on the GR background, while the nonlinear part of the coupling controls saturation, branch structure, and stability (Blázquez-Salcedo et al., 2021, Doneva et al., 2022).

A central distinction is between coupling functions that merely trigger the instability and those that also stabilize the scalarized solution. For the exponential coupling

$$F(\phi)=\frac{\lambda^2}{12}\left(1-e^{-3\phi^2/2}\right),$$

the $n=0$ branch is reported to have higher entropy than Schwarzschild and to be mode-stable over the region studied up to a hyperbolicity limit. By contrast, for the simpler quadratic coupling the fundamental static scalarized black holes are unstable, and the branch is short. This coupling dependence is one of the defining features of the subject (Blázquez-Salcedo et al., 2021).

The effective-field-theory analysis of self-interactions sharpens this point. In the minimal model

$f'(0)=0$0

the scalar mass term shifts the onset threshold by making scalarization harder as $f'(0)=0$1 increases, while the quartic self-interaction is sufficient to stabilize scalarized black holes without relying on higher-order Gauss-Bonnet terms as the primary quenching mechanism. The resulting stable solutions occupy a finite mass interval, and only the nodeless branch is stable (Macedo et al., 2019).

A complementary extension is the combined Gauss-Bonnet–Ricci model

$f'(0)=0$2

Here the effective mass is

$f'(0)=0$3

For vacuum GR black holes, $f'(0)=0$4, so the Ricci coupling does not trigger the onset of scalarization by itself. Once hair develops and the full nonlinear solution has $f'(0)=0$5, however, the Ricci term materially reshapes the existence domain, scalar charge, and likely stability. In that sense it is a nonlinear reshaper rather than the linear trigger of black-hole scalarization (Antoniou et al., 2021).

3. Beyond spontaneous scalarization: nonlinear and dynamical mechanisms

A major development is the demonstration that black-hole scalarization need not originate in a linear tachyonic instability. In Einstein-scalar-Gauss-Bonnet gravity, coupling functions satisfying

$f'(0)=0$6

eliminate the linear effective mass term, so the Schwarzschild solution is linearly stable: $f'(0)=0$7 Nevertheless, numerical evolutions show that sufficiently large perturbations can trigger nonlinear growth if the coupling contains terms higher than $f'(0)=0$8. For the explicit example

$f'(0)=0$9

one obtains scalarized Schwarzschild black holes even though there is no linear bifurcation from the bald branch. The parameter $(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$0 controls the branch structure more strongly than $(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$1, and the resulting solutions satisfy the first law numerically (Zhang et al., 2024).

The rotating counterpart exhibits the same conceptual split between tachyonic and purely nonlinear scalarization. For the Kerr background evolved in the decoupling limit, the coupling

$(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$2

satisfies $(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$3, so there is no tachyonic instability. Scalarization then requires a finite-amplitude seed, and there is no probe limit in which the black hole scalarizes with zero charge. The bald and hairy Kerr families are separated by a gap, and they connect only when both $(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$4 and $(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$5 (Doneva et al., 2022).

Nonlinear dynamical scalarization also appears in charged systems. In Einstein-Maxwell-scalar theory with

$(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$6

the choice $(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$7 and $(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$8 leaves the Reissner–Nordström black hole linearly stable, but nonlinear accretion of a scalar pulse can nevertheless drive the system either to a bald RN black hole or to a scalarized charged black hole, depending on an initial-data parameter $(\Box-\mu_{\rm eff}^2)\,\delta\phi=0,$9. Near the critical value $\mu_{\rm eff}^2$0, the evolution is attracted to a metastable critical solution and displays the logarithmic residence-time scaling

$\mu_{\rm eff}^2$1

together with mass scaling

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

This is a threshold phenomenon of type-I-like form, but separating two black-hole end states rather than dispersion and collapse (Zhang et al., 2021).

A related fully nonlinear process occurs in Einstein-Maxwell-dilaton theory, where a tiny initial dilaton perturbation is amplified by the Maxwell-dilaton coupling and the system settles into a hairy black hole. The horizon scalar follows a damped oscillatory stage,

$\mu_{\rm eff}^2$3

followed by exponential saturation,

$\mu_{\rm eff}^2$4

For an initial naked singularity, the evolution can generate horizons dynamically and end in a stable hairy black hole. This suggests that scalarization can act as a nonlinear energy-transfer process rather than only as a bifurcation of a stationary bald solution (Zhang et al., 2021).

4. Rotation, charge, and encounter-driven scalarization

Rotation introduces a qualitatively distinct control parameter because the Gauss-Bonnet invariant of Kerr can change sign outside the horizon. In the linear time-domain analysis of spin-induced scalarization, the perturbation equation on Kerr develops a negative region only for sufficiently rapid rotation, and the critical threshold approaches

$\mu_{\rm eff}^2$5

at large coupling. The instability is strongest for the axisymmetric $\mu_{\rm eff}^2$6 mode, and the time evolutions support the interpretation that unstable Kerr black holes evolve toward scalarized rotating black holes (Doneva et al., 2020).

This spin channel can coexist with the purely nonlinear mechanism. In the decoupling-limit study of nonlinear Kerr scalarization, spin widens the region of parameter space supporting scalarized stationary states, but the tachyonic and non-tachyonic couplings remain sharply distinguishable: with tachyonic onset the final state is insensitive to arbitrarily small seeds, whereas with purely nonlinear onset the initial pulse must exceed a threshold amplitude (Doneva et al., 2022).

Charged black holes supply an alternative scalarization channel through the electromagnetic sector. In the Einstein–Euler–Heisenberg-scalar theory with

$\mu_{\rm eff}^2$7

the magnetically charged Einstein–Euler–Heisenberg black hole can scalarize for positive coupling $\mu_{\rm eff}^2$8, and the theory admits infinitely many branches labeled by the node number $\mu_{\rm eff}^2$9. The $$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$0 branch is stable against radial perturbations, whereas the $$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$1 branch is unstable. A notable feature is that the onset behavior differs between $$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$2 and $$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$3, reflecting the role of the nonlinear electrodynamics sector (Zhang et al., 9 Oct 2025).

The same broad logic extends to dynamical binaries. In quadratic scalar Gauss-Bonnet gravity, numerical-relativity evolutions in the decoupling limit show that hyperbolic black-hole encounters can trigger temporary dynamical scalarization even when the individual holes are initially unscalarized. For negative coupling and spinning holes, close encounters can also induce permanent spin-up scalarization or spin-up descalarization by changing the post-encounter spin across the scalarization threshold. This establishes that scalar hair in binaries can be created, erased, or made permanent by orbital dynamics rather than by isolated-black-hole parameters alone (Pardoe et al., 29 Jun 2026).

5. Topology, asymptotics, and extended phase structure

The asymptotic structure and horizon geometry substantially modify the onset and thermodynamics of scalarization. Probe analyses in Einstein-scalar-Gauss-Bonnet theory show that AdS backgrounds favor scalarization more than asymptotically flat space: for spherical horizons the near-horizon potential well is deeper in AdS, so the threshold coupling is smaller than in flat space. In topological AdS black holes the ease of scalarization is ordered as

$$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$4

while the de Sitter case is problematic in the probe setup because regularity at the cosmological horizon conflicts with the desired asymptotic decay (Guo et al., 2020).

A fully backreacted AdS analysis is given by the Einstein–Maxwell–charged-scalar model with quartic self-interaction,

$$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$5

in which the metric takes the form

$$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$6

with $$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$7 corresponding to spherical, planar, and hyperbolic horizons. In the extended phase space,

$$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$8

The principal result is that scalarization occurs at low temperature in all three topologies, but the detailed phase structure depends strongly on horizon geometry and on pressure (Zhao et al., 22 Nov 2025).

The spherical topology is singled out as structurally richer. Even with $$\mathscr{G}=R^{\mu\nu\rho\sigma}R_{\mu\nu\rho\sigma}-4R^{\mu\nu}R_{\mu\nu}+R^2,$$9, without extra nonlinear self-interaction, the spherical Reissner–Nordström–AdS case already exhibits unstable scalarized branches and a broad low-pressure scalarized regime extending to much higher temperatures than in the planar and hyperbolic cases. By contrast, the planar and hyperbolic topologies require the quartic self-interaction to generate the unstable branches associated with zeroth-order behavior. This suggests that spherical topology itself supplies the ingredients needed for the unstable branch structure that in the other topologies must be assisted by explicit nonlinear terms (Zhao et al., 22 Nov 2025).

The extended-phase-space analysis is performed in the canonical ensemble using

$R_{\rm GB}^2$0

which allows the scalarized branches to be classified as stable, metastable, or unstable. As pressure increases, the system is reported to evolve through the sequence

$R_{\rm GB}^2$1

and the condensate can undergo a transition from first-order style to cave-of-wind style. In this AdS setting, scalarization is therefore not only an instability problem but also a thermodynamic phase-structure problem controlled by temperature, pressure, back-reaction, self-interaction, and topology (Zhao et al., 22 Nov 2025).

6. Extensions, thermodynamics, and current status

Scalarization is not confined to the standard four-dimensional curvature-Gauss-Bonnet setup. In higher-dimensional scalar-tensor theory, electrovacuum black holes can scalarize because Maxwell theory is no longer conformally invariant for $R_{\rm GB}^2$2, so the Reissner–Nordström background has $R_{\rm GB}^2$3. This yields scalar clouds and fully nonlinear scalarized branches already in the simplest $R_{\rm GB}^2$4 model, with explicit constructions in $R_{\rm GB}^2$5. In even dimensions, coupling the scalar to the critical Euler density $R_{\rm GB}^2$6 produces the higher-dimensional analogue of Gauss-Bonnet scalarization and again leads to zero modes and scalarized Schwarzschild–Tangherlini black holes (Astefanesei et al., 2020).

Alternative geometric formulations also support scalarization. In Gauss-Bonnet teleparallel gravity, the four-dimensional Gauss-Bonnet invariant decomposes as

$R_{\rm GB}^2$7

and the scalar can couple separately to the teleparallel Gauss-Bonnet torsion invariant $R_{\rm GB}^2$8 and the boundary term $R_{\rm GB}^2$9. The resulting asymptotically flat scalarized black holes can display non-monotonic metric functions and non-monotonic scalar profiles, behavior explicitly identified as absent from previously known static scalarized black-hole solutions in the standard Einstein-Gauss-Bonnet case (Bahamonde et al., 2022).

There are likewise matter-sector generalizations. Scalarization can be triggered by the breakdown of scale invariance through a standard nonminimal coupling $m_{\rm eff}^2=-\eta R_{\rm GB}^2,$0 when quantum corrections or non-conformal matter render the Ricci scalar nonzero. In that framework, electro-vacuum black holes remain unscalarized as long as $m_{\rm eff}^2=-\eta R_{\rm GB}^2,$1, but effective-field-theory or matter corrections producing $m_{\rm eff}^2=-\eta R_{\rm GB}^2,$2 can generate a discrete set of scalar-cloud eigenvalues and entropically favored scalarized branches (Herdeiro et al., 2019). Regular black holes supported by nonlinear electrodynamics admit a general scalarization framework based on the P-dual formalism, and in the worked Balart–Vagenas example scalarized and scalar-free branches coexist in a non-uniqueness domain where the scalarized solutions are entropically preferred. The associated shadow and scalar quasi-normal-mode deviations are reported at the percent level, with less than $m_{\rm eff}^2=-\eta R_{\rm GB}^2,$3 deviations for small charge-to-mass ratios (Contreras et al., 3 Nov 2025).

Thermodynamics is central throughout the subject. Depending on the model, entropy receives Wald-type corrections from Gauss-Bonnet or nonminimal scalar couplings; in several settings the scalarized branch is entropically favored even when its horizon area is smaller than that of the bald solution (Antoniou et al., 2021, Herdeiro et al., 2019). At the same time, the literature repeatedly emphasizes that thermodynamic preference does not by itself settle dynamical viability. The current open issues include cosmological viability, well-posedness and hyperbolicity in parts of parameter space, the full perturbative stability of rotating scalarized black holes, and the nonlinear completion of binary dynamics (Doneva et al., 2022).

An additional constraint comes from effective-field-theory model building. The proposal that only supermassive black holes scalarize while stellar-mass ones remain GR-like was tested against two-scalar effective-field-theory constructions and found not to arise naturally from integrating out a heavy scalar: the induced $m_{\rm eff}^2=-\eta R_{\rm GB}^2,$4 term has the wrong sign and is parametrically suppressed. Even keeping both scalars dynamical leads generically to curvature-induced and spin-induced scalarization rather than to a natural supermassive-only mechanism (Thaalba et al., 26 Jun 2025).

Taken together, these results define black-hole scalarization as a broad class of bifurcation phenomena rather than a single mechanism. The common element is the appearance of a nontrivial scalar branch once the bald black hole crosses a threshold set by curvature, gauge fields, spin, topology, or nonlinear amplitude. The differences lie in what supplies the effective instability, how the nonlinear completion saturates it, and whether the endpoint is continuously connected to the bald family, separated by a gap, dynamically selected, or embedded in a richer thermodynamic phase diagram (Doneva et al., 2022, Zhao et al., 22 Nov 2025).