Spherically Symmetric Dyonic Solutions in Gravity

Updated 11 November 2025

Spherically symmetric dyonic solutions are static gravitational configurations exhibiting both electric and magnetic charges, unifying black hole, soliton, and regular black-bounce phenomena.
They emerge from Einstein–Maxwell, nonlinear electrodynamics, and Yang–Mills frameworks, where gauge fields and scalar couplings enforce spherical symmetry and horizon regularity.
These solutions are pivotal in holography, thermodynamics, and quantization studies, offering insights into black hole hair, phase transitions, and extensions of no‐hair theorems.

Spherically symmetric dyonic solutions are static or stationary configurations in four-dimensional (or higher-dimensional) gravitational theories coupled to nonlinear or linear gauge fields, in which both electric and magnetic charges are present and the entire spacetime admits SO(3) spatial symmetry. They provide a unified framework for studying generalizations of the Reissner–Nordström (RN) black hole and solitons in a range of contexts: including Einstein–Maxwell–scalar models, Einstein–Yang–Mills (EYM) theories, (non)linear electrodynamics, various supergravity truncations, and regular black-bounce geometries. Dyonic solutions play a central role in holography, the paper of black hole hair, and the analysis of quantization/discreteness phenomena for horizon data.

1. Theoretical Frameworks and Universal Features

Spherically symmetric dyonic solutions arise in gravitational theories with at least one U(1) (or non-Abelian) gauge sector, generically described by an action of the form

$S = \int d^4x \sqrt{-g} \left[ \frac{R}{16\pi G} - \frac14 F_{\mu\nu}F^{\mu\nu} + \mathcal{L}_{\mathrm{matter}} \right],$

where $F_{\mu\nu}$ is the gauge field strength and $\mathcal{L}_{\mathrm{matter}}$ includes scalar fields, higher-order terms, or general nonlinear electrodynamics (NED) Lagrangians depending on $F$ , $\mathcal{F} \equiv F_{\mu\nu}F^{\mu\nu}$ , or more general invariants.

The generic static, spherically symmetric metric is

$ds^2 = -A(r) dt^2 + A(r)^{-1} dr^2 + r^2 d\Omega_2^2.$

The gauge field comprises an electric potential $A_t(r)$ and a fixed magnetic monopole configuration $A_{\varphi}(r,\theta) = q_m \cos\theta$ (for U(1)), or non-Abelian analogs (Shepherd et al., 2015, Baxter, 2018, Baxter, 2019).

Spherically symmetric dyons satisfy

Nonzero electric and magnetic charge, generically $q_e$ and $q_m$ ;
Regular Maxwell (or generalized) field equations, possibly requiring scalar multiplets (in the U(1) case) for consistency with spherical symmetry if $q_m\neq0$ (Herdeiro et al., 15 Jun 2024, Kunz et al., 31 Jul 2024);
Boundary conditions: regular horizon (black holes) or regular origin (solitons/bounces), and prescribed fall-off at infinity (e.g., asymptotic flatness or AdS).

2. U(1) Dyonic Black Holes with Scalar and Hair

Static spherically symmetric solutions in Einstein–Maxwell–scalar models are governed by field equations descending from actions such as

$S = \int d^4x \sqrt{-g} \left[\frac{R}{16\pi G} - \frac14 F_{\mu\nu}F^{\mu\nu} - \frac12(D_{\mu}\Phi)(D^{\mu}\Phi^*) - U(|\Phi|^2) \right].$

For $q_m \neq 0$ , a single charged scalar cannot be made spherically symmetric due to Dirac quantization and regularity constraints; a minimal multiplet (e.g., $n=2$ ) is required, with angular dependence cancelling the Dirac string in $T_{\mu\nu}$ (Herdeiro et al., 15 Jun 2024).

Key features include:

The resonance condition at the horizon, $\omega = q V(r_h)$ , is essential for nontrivial scalar hair (synchronization) (Herdeiro et al., 15 Jun 2024, Kunz et al., 31 Jul 2024).
The existence of a finite mass gap between hairy and bald (RN) dyons: $M_\text{hairy}(q_e,q_m) - M_\text{RN}(q_e,q_m) > 0$ (Herdeiro et al., 15 Jun 2024).
Hairy dyonic black holes require both a horizon (which regularizes the $q_m^2/r^4$ energy density) and nonzero electric charge (required to satisfy the resonance).
When $q_m=0$ , a horizonless (boson star) limit exists; for $q_m\neq 0$ , no solitonic limit exists due to the singular behavior of the magnetic core—only trivial (vanishing scalar) solutions survive as $r_h \rightarrow 0$ (Herdeiro et al., 15 Jun 2024).
There are parameter bands for the hairy solutions (frequency detuning $\omega$ or chemical potential $\mu_{ch}$ ) with two branches, connected at a minimum and separated from RN in generic cases, except in special massless-scalar limits where bifurcation can occur directly from RN (Kunz et al., 31 Jul 2024).

3. Dyonic Solutions in Nonlinear Electrodynamics

Dyonic configurations in NED coupled to GR satisfy generalized Einstein and field equations with $L(f)$ depending on the single invariant $f = F_{\mu\nu}F^{\mu\nu}$ (Bronnikov, 2017, Yang, 2022, Kruglov, 2019). The static dyonic field framework is:

Electric charge encoded via $r^2 L_f F^{tr} = q_e$ ;
Magnetic charge via $F_{\theta\phi} = q_m \sin\theta$ ;
The system reduces to (i) an algebraic equation for $f(r)$ (often transcendental in $L_f$ ), (ii) explicit expressions for $E(r)$ , (iii) a quadrature for the mass function $M(r)$ .

Notable features:

For standard Born–Infeld-type Lagrangians, closed-form dyonic solutions can be obtained (e.g., using hypergeometric functions), exhibiting regular energy density in the core for appropriate parameter regimes (Bronnikov, 2017, Kruglov, 2019, Yang, 2022).
Corrections to Coulomb’s law and RN metric appear at $O(r^{-6})$ or higher orders and vanish identically in the self-dual case $|q_e|=|q_m|$ (Kruglov, 2019, Bronnikov, 2017).
Finite-energy dyonic solutions exist only when a “mixed” invariant (involving $E \cdot B$ ) is present, as in generalized BI models, restoring electric–magnetic duality (Yang, 2022).
Stability and thermodynamic analysis reveal curves of second-order phase transitions in the $(M, Q_e, Q_m)$ parameter space, tied to the heat capacity diverging at critical horizon radii (Kruglov, 2019).

4. Dyonic Black Holes and Solitons in Yang–Mills and Supergravity Sectors

Spherically symmetric dyons in EYM and supergravity manifest an interplay between gauge structure, scalar couplings, and horizon regularity:

In EYM AdS ( $\mathfrak{su}(N)$ ) theories, generic solutions are labeled by $N-1$ electric and $N-1$ magnetic functions. For large $|\Lambda|$ , there exist stable, nodeless (in magnetic gauge fields) dyonic black holes, whereas for $|\Lambda| \to 0$ , all node sectors appear. The electric functions are always monotonic and node-free (Shepherd et al., 2015).
In the $\mathfrak{su}(\infty)$ limit, the infinite number of magnetic and electric functions at infinity leads to solutions characterized by a countably infinite sequence of global charges, thus violating finite-hair versions of the no-hair theorem (Baxter, 2018, Baxter, 2019).
Dyonic dilaton black holes exhibit both analytic (Toda-related, existing only at special couplings) and discrete (non-integrable, specified by quantized horizon data for general coupling) solution classes. Only at special $a$ does a continuous family exist; for generic $a$ , regularity at both horizons quantizes the dilaton’s value at the horizon (Davydov, 2017).

In N=1 supergravity and its extensions, dyonic extremal black holes admit a moduli space determined by attractor flow equations, with AdS $_2 \times S^2$ near-horizon geometries and scalar profiles interpolating between boundary and horizon critical points (Gunara et al., 2010).

5. Regular and Non-Singular Dyonic Solutions

Regular dyonic geometries—those with a smooth bounce or infinite throat replacing the central singularity—can be constructed using specific NED, scalar, and metric profiles:

Black-bounce metrics of the Simpson–Visser type allow for a throat of radius $a = \sqrt{Q_e^2+Q_m^2}$ ; both the metric function and Kretschmann scalar remain finite everywhere for $a > 0$ (Junior et al., 18 Feb 2025). This can be realized with either nonlinear or linear (Maxwell) Lagrangians, with the scalar field and potential engineered by an inverse method.
There exists a threshold structure: for $a < 2M$ , two horizons exist; $a=2M$ gives an extremal horizon at $r=0$ ; $a>2M$ describes a traversable wormhole. In the standard Maxwell case, singularity persists since the throat radius vanishes.
These constructions generalize to a broader class of scalar–NED couplings, including phantom scalars or higher curvature corrections (Junior et al., 18 Feb 2025).

6. Thermodynamics, Hair, and Quantization Phenomena

Dyonic black holes are a central testing ground for several theoretical phenomena:

Thermodynamics demonstrates the necessity of generalized first laws. In AdS and gauged supergravity, one finds that the usual first law is violated unless the charges are aligned or vanish (e.g., $Q=P$ or $Q=0$ ) and that an additional conjugate pair $(X, Y)$ , associated to scalar hair and quantifying bondary-data breaking of asymptotic symmetries, must be included (Lu et al., 2013).
In the presence of complex scalar multiplets, the resonance (synchronization) condition for scalar hair at the horizon is critical for regular solutions; for nonzero magnetic charge, this requires an explicit angular dependence in the scalar ansatz to maintain spherical symmetry (Herdeiro et al., 15 Jun 2024).
Quantization of hair data arises in dyonic dilaton black holes: for generic coupling, only discrete—and, for each node number, isolated—values of horizon data produce regular, two-horizon solutions, illustrating a new kind of spectral quantization in classical general relativity (Davydov, 2017).

7. Implications, Applications, and Open Problems

Spherically symmetric dyonic solutions have broad significance:

They serve as building blocks for higher-dimensional black hole constructions, numerical relativity, and analytic studies of field theory/gravity correspondence and holographic matter.
In the AdS/CFT correspondence, dyonic black holes model dual phases with both charge and magnetic flux, and in EYM AdS, the infinite-hair solutions provide a new class of order parameters in large N CFTs, potentially relevant for exotic condensed matter analogs (Baxter, 2019, Baxter, 2018).
Black-bounce dyons provide consistent, nonsingular alternatives to classical singularities, with tunable horizon and bounce structure, and explicit matter sourcing providing insight into regularization mechanisms (Junior et al., 18 Feb 2025).
The stability, quantization, and attractor structure of dyonic solutions remain key questions, especially in the context of non-Abelian hair, infinite charge sequences, and supergravity embeddings (Shepherd et al., 2015, Baxter, 2018, Gunara et al., 2010).

A plausible implication is that the interplay of gauge symmetry, horizon regularity, and scalar/gauge field content in dyonic solutions serves as a diagnostic for both uniqueness and novel classification phenomena in gravitational theory, advancing the understanding of black hole microstates, no-hair conjectures, and semiclassical regularity.