Scalarized Charged Black Holes

• Scalarized charged black holes are modified solutions of Einstein's equations where traditional charged black holes acquire scalar hair via nonminimal couplings.
• They exhibit discrete branching, novel horizon geometries, and phase transitions driven by critical coupling parameters and charge-to-mass ratios.
• Their rich phenomenology provides practical insights into observational deviations, affecting black hole shadows, gravitational wave emissions, and thermodynamic properties.

Scalarized charged black holes constitute a family of solutions to Einstein’s field equations in which standard charged black holes (such as Reissner–Nordström, C-metric, or Bardeen spacetimes) become unstable or modified due to nonminimally coupled scalar fields. In these models, scalar “hair” arises either through spontaneous scalarization—triggered by a tachyonic instability induced by the electromagnetic, gravitational, or nonlinear electrodynamics sector—or via the inclusion of nonminimal couplings to curvature or gauge invariants. Scalarized charged black holes feature rich phenomenology: discrete branching structures indexed by node number, non-trivial horizon topology, regularization of conical singularities, new thermodynamic and critical behaviors, and implications for astrophysical signatures beyond General Relativity.

1. Theoretical Frameworks and Scalarization Mechanisms

Scalarization in charged black holes is realized in several classes of extended gravity theories:

• Einstein–Maxwell–Scalar (EMS) theories employ nonminimal couplings between a scalar field $\phi$ and electromagnetic invariants, typically using functions such as $f(\phi) = \exp(\alpha\phi^2)$ or $f(\phi) = 1 + \alpha\phi^2$. The direct coupling modulates the effective mass of the scalar and can result in a tachyonic instability of the “bald” black hole (Myung et al., 2018, Myung et al., 2019).
• Conformal and Gauss–Bonnet couplings incorporate spacetime curvature, as in the Einstein-scalar-Gauss–Bonnet (EsGB) and EMCS (Einstein–Maxwell–Conformally coupled Scalar) models, where nonminimal scalar couplings to the Ricci scalar or Gauss–Bonnet invariant trigger spontaneous scalarization. The conformal case ensures a traceless energy–momentum tensor for the scalar and enables black holes even in spacetime with constant Ricci scalar (0907.0219).
• Nonlinear Electrodynamics and Horndeski couplings further enrich the landscape, allowing multiple scalarization channels (e.g. via nonlinear electromagnetic terms or vector-tensor Horndeski interactions) (Brihaye et al., 2020, Myung, 22 Jul 2025).
• Planar, AdS, and cavity settings extend scalarization to non-compact or boundary-modified spacetimes, producing critical phenomena and phase transitions reminiscent of condensed matter systems (Guo et al., 2021, Promsiri et al., 2023, Yao, 2021).

The onset of scalarization is generically controlled by the sign and magnitude of the coupling parameter ($\alpha$, $\beta$, $\gamma$, etc.), the charge-to-mass ratio of the black hole, and, for models with a cosmological constant or AdS boundary, the scale set by $L$.

2. Branch Structure and Bifurcation Analysis

Scalarized charged black holes emerge through discrete bifurcations from scalar-free solutions. The key features are:

• Node structure: Scalarized solutions are indexed by an integer $n \geq 0$, where each $n$ corresponds to the number of radial nodes (zero crossings) of the scalar field profile. $n=0$ defines the fundamental branch, while $n\geq1$ yields “excited” branches (Myung et al., 2018, Myung et al., 2019, Zou et al., 2020).
• Bifurcation thresholds: Each branch bifurcates from the parent black hole at a critical value of the coupling parameter $\alpha_n(Q)$ that depends on charge-to-mass ratio, boundary conditions, and the coupling function. The fundamental branch (n=0) bifurcates at the lowest critical value, e.g., $\alpha_{n=0}\simeq8.019$ for certain parameters in the EMS theory (Myung et al., 2018).
• Bands and existence domains: In multi-parameter theories (e.g., including scalar mass), bands of existence in $(\alpha, Q)$ or $(\alpha, \beta)$ space arise. In quadratic EsGB models, charged black holes admit two branches for high $Q$—one for positive and one for negative quadratic coupling—due to a sign change in the Gauss–Bonnet invariant near the horizon (Brihaye et al., 2019).

This branching is underpinned by solving spectral eigenvalue problems for scalar perturbations in a black hole background, often recast as Schrödinger-type equations with effective potentials.

3. Horizon Structure, Regularity, and Global Properties

Scalarized charged black holes exhibit several modifications to the horizon geometry and global structure:

• Cohomogeneity-two and non-Einstein horizons: In conformally coupled systems, event horizons may lack maximal symmetry and instead be cohomogeneity-two, neither Einstein nor homogeneous, with richer curvature structure than their Reissner–Nordström or Schwarzschild counterparts (0907.0219).
• Removal of conical singularities: Scalar backreaction can regularize singular features (such as conical singularities in the C-metric), resulting in a metric where the angular deficits are precisely eliminated by scalar-induced modifications (0907.0219).
• Critical splitting: For certain couplings (e.g., quartic), a critical value of the black hole charge induces a splitting of spacetime into two disconnected regions: an interior part with scalar hair and an outer extremal Reissner–Nordström geometry. The spacetime thus develops an infinite throat and a degenerate horizon, mirroring patterns in non-Abelian monopole and “hairy” black hole systems (Blázquez-Salcedo et al., 2020).

Asymptotically, the spacetime may exhibit standard (A)dS or flat behavior, depending on the cosmological constant and the boundary conditions.

4. Stability and Dynamical Endpoints

The stability of scalarized charged black holes is critically determined by the node number and the coupling constants:

Branch	Stability	Typical Instability Mode	Thermodynamic Preference
$n = 0$	Dynamically stable	None (QNMs: all $\mathrm{Im}(\omega) < 0$)	Preferred vs. bald BH
$n \geq 1$	Dynamically unstable	Radial ($l = 0$) “s-mode” scalar (GL type)	Not dynamically realized

• Fundamental branch ($n=0$): Linear perturbation theory (scalar, vector, tensor) and full quasinormal mode analyses show this branch is typically dynamically and thermodynamically stable (Myung et al., 2018, Myung et al., 2019, Zou et al., 2020).
• Excited branches ($n \geq 1$): These exhibit at least one growing “s-mode” under radial perturbations, signaling instability akin to the Gregory–Laflamme instability for black strings. They are not endpoints of the scalarization process.
• Novel critical instabilities: In the approach to extremality and for high charge, certain branches can develop curvature singularities outside the horizon, marking a breakdown of the solution in the strong-coupling limit (Brihaye et al., 2019).
• Mixed scalarization: When both spontaneous and nonlinear mechanisms are present (e.g., in $f(\phi)=\exp(-\alpha\phi^2-\beta\phi^4)$), the spontaneous (tachyonic) instability dominates unless nonlinear terms are extremely large; “counter scalarization” (wrong-sign coupling constants) acts like a mass term and can quench scalarization entirely (Belkhadria et al., 2023).

The endpoint of the instability in all cases is the $n=0$ branch with the same charge as the original RN solution, confirming its dynamical viability.

5. Thermodynamic, Critical, and Phase Structure

Scalarized charged black holes display rich thermodynamic behavior and critical phenomena:

• Entropy and free energy: The fundamental scalarized branch is typically entropically and energetically preferred over the scalar-free solution, as found in micro-canonical, canonical, and grand-canonical ensembles (Guo et al., 2021, Promsiri et al., 2023, Yao, 2021).
• Phase transitions: In AdS spacetimes or cavity models, scalarization induces second-order (continuous) and zeroth-order (discontinuous) phase transitions, with reentrant sequences (scalar-free $\to$ hairy $\to$ scalar-free) reminiscent of those in condensed matter (e.g., the conductor–superconductor transition), controlled by temperature $T$ and electric charge $Q$ (Guo et al., 2021, Promsiri et al., 2023, Yao, 2021).
• Smarr-like relations: Generalized Smarr formulae include contributions from the action parameter (e.g., Gauss–Bonnet coupling) or NED terms, linking mass, entropy, temperature, and generalized chemical potentials (Myung, 22 Jul 2025).
• Critical points: Davies points—where the specific heat diverges—are associated with, but not in general coincident with, the onset of scalarization-induced phase transitions (Myung, 22 Jul 2025).

Thermodynamic analysis consistently reveals that only the fundamental (nodeless) scalarized black hole is a possible endpoint of real-world physical processes.

6. Extensions: Nontrivial Topologies, Rotating and Planar Solutions

The scope of scalarization encompasses a broad range of geometrical and dynamical extensions:

• Planar and AdS black holes: Scalarization persists in planar AdS black holes, with similar discrete branching, stability, and phase structure, and the scalar-hairy phase is favored at low temperature both canonically and grand-canonically (Promsiri et al., 2023). This mirrors holographic superconductor models.
• Rotating black holes: Analyses indicate that scalarization induced by curvature terms (e.g., via the Gauss–Bonnet invariant) is robust even up to near-extremal spin, with the scalar charge comparable for rapidly rotating and static solutions, provided the coupling parameter is sufficiently large to avoid breakdown of the effective description (Staykov et al., 19 Mar 2025).
• Bardeen and nonlinear electromagnetic backgrounds: In regular (Bardeen) black holes, scalarization is again triggered via tachyonic instability from nonminimal coupling to nonlinear electrodynamics, producing infinite branches; only the fundamental branch is stable (Zhang et al., 18 Sep 2024).
• Quasi-topological, vector-tensor, and Horndeski couplings: More exotic couplings (e.g., quasi-topological terms, Horndeski-type interactions) permit multiple distinct scalarization channels, influence asymptotic charges, and generate new examples of stable scalarized charged solutions with nontrivial scalar and vector charge interrelations (Brihaye et al., 2020, Myung et al., 2020).

7. Observational Signatures and Implications

Scalarized charged black holes provide potential observable deviations from classical General Relativity:

• Astrophysical implications: Non-GR horizon structure, modified shadow radius, gravitational wave emission, and scalar field tails may all constitute observable differences relative to standard charged black holes. Recent parameter studies suggest that mass bounds can be derived from shadow observations, with less stringent bounds on certain coupling parameters (Myung, 22 Jul 2025).
• Dynamical processes: Thermodynamic preference, energy extraction mechanisms (such as Penrose-type processes from scalar charge), and decay channels via emission of scalar waves are characteristic dynamical phenomena (Gao et al., 2019).
• Universality: Across a wide range of models—Einstein–Maxwell–Scalar, Gauss–Bonnet, Horndeski, and beyond—the dominance of the fundamental $n=0$ branch and the emergence of scalar hair at threshold couplings reflect universal aspects of black hole scalarization, with implications for no-hair conjectures and beyond (Myung et al., 2018, Myung et al., 2019, Belkhadria et al., 2023).

These aspects frame scalarized charged black holes as essential objects for precision tests of gravity in the strong-field regime, both theoretically and observationally.