Scalar-Hairy Black Holes

• Scalar-hairy black holes are solutions in Einstein gravity with nontrivial scalar field profiles that alter local geometry and thermodynamic properties.
• They are constructed using diverse scalar potentials and couplings, yielding modified horizon structures, phase transitions, and stability features.
• These configurations serve as theoretical probes for extensions of general relativity and have implications for gravitational wave signatures and observational tests.

Scalar-hairy black hole solutions describe a broad class of nontrivial black hole metrics supported by coexisting scalar fields, leading to modifications of both local and global properties compared to conventional vacuum or charged (Schwarzschild–Tangherlini, Reissner–Nordström) solutions. These configurations can be constructed in arbitrary space-time dimensions, diverse scalar potentials (including those derived from supergravity), and both neutral and charged sectors. They frequently exhibit new thermodynamic and geometric behavior, modifications in horizon structure, and domain walls or naked singularities in special parameter regimes. Scalar hair is defined rigorously via the presence of a nontrivial scalar profile outside the horizon without violating regularity or asymptotic conditions.

1. Theoretical Frameworks and Scalar Potentials

Scalar-hairy black holes typically arise in Einstein gravity coupled to a scalar field via the action

$$S = \int d^Dx\,\sqrt{-g}\,\left( R - \tfrac12 (\partial\phi)^2 - V(\phi) \right),$$

with $V(\phi)$ governing the nontrivial scalar “hair.” In general dimensions, explicit closed-form potentials have been constructed which admit both asymptotically flat and (A)dS solutions (Feng et al., 2013). For instance, one family parametrizes

$$\phi(r) = \sqrt{\frac{2(D-3)}{D-2}}\,p\, \ln H_1(r),\quad H_1(r)=1+\frac{q_1}{r^{D-3}},$$

subject to $p^2+v^2=1$, and

$$V(\phi) = -\frac12(D-2)(D-3)g^2 e^{-\lambda\phi} -...,$$

with parametric dependence on $g$, $p$ (see (Feng et al., 2013) eqs.(21,32)). For $a=0$, the potential further admits a superpotential form,

$$V = [W'(\phi)]^2 - \frac{D-1}{2(D-2)} W^2(\phi),\quad W(\phi) = c_+e^{\mu\phi} + c_-e^{-\mu\phi}.$$

Hair can also be supported by inverted Mexican-hat (quartic) potentials, $V(\phi) = -\Lambda \phi^4 + \mu \phi^2$, yielding nontrivial hair bifurcating from Schwarzschild at the onset of instability (Chew et al., 2023).

In the presence of non-canonical or non-minimal (e.g., Gauss–Bonnet, Maxwell–scalar) couplings, the action generalizes with terms like $\phi \mathcal G$ (Gauss–Bonnet density) (Hunter et al., 2020), or scalar-dependent kinetic and potential terms, enabling broader families of solutions.

2. Exact Solutions and Geometric Structure

Scalar-hairy black holes admit static, spherically symmetric metrics of the general form

$$ds^2 = -H(r)^{-1} f(r) dt^2 + H(r)^{1/(D-3)} \left( \frac{dr^2}{f(r)} + r^2 d\Omega_{D-2}^2 \right),$$

with metric function $f(r)$ and “harmonic” function $H(r)$ (see (Feng et al., 2013) eq.(5)). In specific models, $f(r)$ is built from nested hypergeometric functions and polynomial terms. The scalar field profile generally falls as $\phi \sim q / r^{D-3}$ (neutral solutions), and may include charge-dependent or parametrically “fixed” branches in the presence of gauge fields.

For charged solutions (Einstein–Maxwell–scalar), the scalar can couple linearly or nonlinearly (e.g., exponential or quartic in the field) to $F^2$. The corresponding metric, scalar, and gauge profiles are determined by coupled ODEs, solved analytically in special cases or numerically otherwise. Charged hairy solutions arise in models with, e.g., $f(\phi) = 1 + \alpha \phi^4$ for the Maxwell coupling, exhibiting bifurcating “hot” and “cold” branches in the phase diagram (Blázquez-Salcedo et al., 2020).

Notably, for $a=0$ in the general potential, the horizon may disappear leaving a domain wall solution with a naked singularity at $r=0$, and the spacetime ceases to be a regular black hole (Feng et al., 2013).

3. Thermodynamics and Scalar Hair

Scalar-hairy black holes possess modified thermodynamic properties compared to their hairless counterparts. The ADM mass,

$$M = \frac{(D-2)\omega_{D-2}}{16\pi} (pq + aq^{D-2}),$$

temperature,

$T = \frac{f'(r_0)\, H(r_0)^{-1/(D-2)}}{4\pi},$

and Bekenstein–Hawking entropy,

$$S = \frac{\omega_{D-2}}{4} r_0^{D-2} H(r_0)^{(D-3)/(D-2)},$$

are parameterized by hair and remain related through $dM = TdS$, ensuring compliance with the first law (Feng et al., 2013). In several models, the first law is not altered by hair, i.e., there is no independent scalar charge; rather, hair is “secondary,” fixed by other physical charges or couplings (Hunter et al., 2020).

In cases with irreducible scalar charges (continuous hair), the area–mass relation can be modified such that, for fixed ADM mass, the horizon area is strictly smaller than Schwarzschild (flat case), and it can be tuned to be arbitrarily small (Rao et al., 2024). In contrast, some models yield branches with positive specific heat, at odds with classical thermodynamic instabilities associated with Schwarzschild black holes (negative $dM/dT$) (Kleihaus et al., 2013).

Phase transitions can occur between hairy and non-hairy phases. In AdS settings, coexistence of multiple branches and swallow-tail free energy behavior underpin first-order transitions with critical points in $(T,\mu)$ space (Guo et al., 27 Dec 2025). In higher-curvature or exotic theories, “$\lambda$-lines” (continuous transition lines) or isolated critical points may arise, indicating novel thermodynamic structure (Dykaar et al., 2017).

4. Stability, Energy Conditions, and Hair Classification

The stability of scalar-hairy black holes is highly model-dependent. Linear perturbation studies reveal that many branches with nontrivial hair are radially unstable: the master Schrödinger equation for scalar perturbations generically admits negative modes ($\omega^2<0$), signifying dynamical instability (Chew et al., 2023Kleihaus et al., 2013). However, in models with spontaneous symmetry breaking in the Gauss–Bonnet sector, hairless black holes become unstable beyond critical couplings and evolve to stable hairy solutions in a broken symmetry phase via second-order phase transitions (Hyun et al., 2024).

Energy condition analysis reveals frequent violations of the weak energy condition (WEC) in regions near or outside the horizon, especially in models with negative quartic potentials or with hair supporting domain walls or naked singularities. In certain branches, naked singularities exist with positive mass and full energy condition compliance (Rao et al., 2024). In others, axi-symmetric secondary hair can exist without violating classical energy conditions (Radu et al., 2011).

Scalar hair is characterized as “secondary” or "fixed" if it is parameterically determined (e.g., by mass, charge, or coupling constants) rather than being a freely adjusted integration constant. In many models, no Noether charge is associated with the hair, and the first law remains unmodified (1312.53742010.10312Karakasis et al., 2023). Scalar charge may appear as a parameter in the asymptotic expansion, distinct from thermodynamic, topological, or electromagnetic charges.

5. Supergravity, Higher Dimensions, and String Theory Embedding

Several scalar-hairy black hole families admit embeddings into higher-dimensional supergravities. For $a=0$, the scalar potential arises directly from gauged supergravity and permits consistent “uplift” to string/M-theoretic settings. The $D$-dimensional solutions can be reinterpreted as spherical M-branes or D3-branes, with metric uplift formulas such as

$$ds_{10}^2 = H^{-\frac{1}{2}}(-f dt^2 + d\vec{x}_3^2) + H^{+\frac{1}{2}} \left( \frac{dr^2}{f} + r^2 d\Omega_5^2 \right)$$

used to capture warped compactifications (Feng et al., 2013).

Charged generalizations involve extending the action to multiple $U(1)$ sectors with dilaton couplings, matching those in supergravity STU-models, and the scalar field can participate in the low-energy effective dynamics of string theory compactifications (Feng et al., 2013).

6. Extensions: Rotating, Topological and Exotic Solutions

Scalar hair has been demonstrated in rotating settings, multiscalar theories, and with non-trivial horizon topology. In tensor–multiscalar theories, scalarization is linked to superradiant instabilities, demanding rotation and harmonically varying scalar fields. The presence of a Killing vector field in target space modulates the existence and domain structure of rotating hairy black holes, leading to complex phase diagrams (e.g., non-uniqueness strips and bifurcating curves in $(M, J)$ parameter space) (Collodel et al., 2020). The horizon geometry can be finely tuned and exhibits deformation compared to standard Kerr or Myers–Perry metrics.

In Einstein–Gauss–Bonnet gravity, higher curvature terms further enrich the solution space. Scalar-hairy configurations exist with extremal (AdS$_2\times S^3$) near-horizon limits and mass gaps relative to vacuum black holes (Brihaye et al., 2015). Hyperbolic horizon scalar-hairy solutions, as constructed in AdS$_4$, lack spherical or planar analogues but exhibit unique phase transitions and third-order critical points in thermodynamic landscapes (Ren, 2019).

7. Physical and Observational Implications

Scalar-hairy black holes yield ringdown signatures (altered quasinormal mode spectra), potential deviations in shadow and light ring phenomenology, and modifications of horizon area and geometry measurable through gravitational wave and electromagnetic probes (Chew et al., 2023Rao et al., 2024Collodel et al., 2020). Although many hairy solutions are nonlinearly or linearly unstable, certain symmetry-broken or topological charge branches persist as stable end-states (Hyun et al., 2024Radu et al., 2011). Energy condition violations or secondary hair constrain astrophysical relevance, but provide robust theoretical laboratories for exploring extensions of general relativity and high-energy embeddings.

Tables of parameter dependencies, solution types, and phase-transition behaviors are commonplace in the literature, organizing characteristics such as:

Model Type	Hair Character	Thermodynamics	Stability
Einstein–Scalar (flat/AdS)	Secondary	Area $<A_S(M)$	Unstable in quartic
Gauss–Bonnet–Scalar	Fixed by couplings	Modified $S$	Symmetry-breaking: stable
Multiscalar (rotating)	Synchronized	$J>J_{Kerr}$	Superradiant formation
String/M-theory uplift	Supergravity	Domain walls	Naked singularities

• "Scalar Hairy Black Holes in General Dimensions" (Feng et al., 2013)
• "Scalar Hairy Black Holes with Inverted Mexican Hat Potential" (Chew et al., 2023)
• "Hairy Black Holes with Arbitrary Small Areas" (Rao et al., 2024)
• "Analytic solutions of neutral hyperbolic black holes with scalar hair" (Ren, 2019)
• "Novel Hairy Black Hole Solutions in Einstein-Maxwell-Gauss-Bonnet-Scalar Theory" (Hunter et al., 2020)
• "Axially symmetric static scalar solitons and black holes with scalar hair" (Kleihaus et al., 2013)
• "Critical solutions of scalarized black holes" (Blázquez-Salcedo et al., 2020)
• "Scalar Field Perturbation of Hairy Black Holes in EsGB theory" (Hyun et al., 2024)
• "Scalar-hairy AdS Black Hole in the Einstein-Maxwell-Scalar Theory: first-order phase transition with a critical point" (Guo et al., 27 Dec 2025)
• "Rotating tensor-multiscalar black holes with two scalars" (Collodel et al., 2020)
• "Hairy black holes in scalar extended massive gravity" (Tolley et al., 2015)
• "Black Holes with Scalar Hairs in Einstein-Gauss-Bonnet Gravity" (Brihaye et al., 2015)
• "Black-Hole Solutions with Scalar Hair in Einstein-Scalar-Gauss-Bonnet Theories" (Antoniou et al., 2017)
• "On black holes with scalar hairs" (Gao et al., 2021)
• "Scalar hairy black holes and solitons in a gravitating Goldstone model" (Radu et al., 2011)
• "Hairy black holes in cubic quasi-topological gravity" (Dykaar et al., 2017)
• "New Charged Black Holes with Conformal Scalar Hair" (0907.0219)
• "Scalar hairy black holes in Einstein-Maxwell-conformally coupled scalar theory" (Zou et al., 2019)
• "Black Holes with Scalar Hair in Three Dimensions" (Karakasis et al., 2023)
• "Hairy black holes in N=2 gauged supergravity" (Faedo et al., 2015)