Dyonic Black Holes

Updated 31 August 2025

Dyonic black holes are solutions carrying both electric and magnetic charges that alter global conserved quantities and horizon structures.
They serve as testing grounds for nonlinear electrodynamics, supergravity, and holographic duality, offering insights into phase transitions and stability.
Recent studies extend the traditional first law by incorporating scalar hair and nonlinear parameters to capture the rich thermodynamic behavior.

A dyonic black hole is a solution to the coupled gravitational and gauge field equations in which the black hole carries both electric and magnetic charge. These solutions are central to the paper of classical and quantum properties of black holes, tests of no-hair theorems, holographic dualities, and the interplay between gauge symmetry, scalar or non-Abelian hair, and black hole thermodynamics. The dyonic attribute fundamentally alters both the global conserved charges and the structure of the horizon, and leads to rich thermodynamic and geometric behavior. Dyonic black holes serve as controlled testing grounds for nonlinear electrodynamics, supergravity, higher-dimensional gravity, and scalar–gauge–gravity systems in a variety of spacetime asymptotics, especially asymptotically anti-de Sitter (AdS) backgrounds relevant for holography.

1. Foundational Models and Solution Structure

The simplest realization of dyonic black holes is found in Einstein–Maxwell theory, where the electric and magnetic global charges act as independent parameters and the gauge field $A_\mu$ supports both nonvanishing electric ( $Q$ ) and magnetic ( $P$ ) flux. The canonical spherically symmetric dyonic black hole is the Reissner–Nordström (RN) solution generalized to include both $Q$ and $P$ : $ds^2 = -f(r)\,dt^2 + f(r)^{-1} dr^2 + r^2 (d\theta^2 + \sin^2\theta\, d\phi^2), \qquad f(r) = 1 - \frac{2M}{r} + \frac{Q^2 + P^2}{r^2}.$ The Maxwell field is $F = (Q/r^2)\, dt\wedge dr + P\sin\theta \, d\theta \wedge d\phi$ .

This picture generalizes in several directions:

Inclusion of a cosmological constant (particularly, negative for AdS asymptotics) modifies $f(r)$ and brings dyonic solutions into the AdS/CFT context.
Nonlinear electrodynamics (e.g., Born–Infeld) introduces corrections to both Maxwell equations and their energy–momentum contribution, leading to new features such as regularized central fields, modified singularity structure, and loss of electric–magnetic duality (Li et al., 2016, Panahiyan, 2018, Kruglov, 2019, Kruglov, 2019).
Non-Abelian generalizations (Einstein–Yang–Mills), and scalar–gauge–gravity systems permit dyonic solutions with nontrivial self-interaction of the matter sector, scalar field “hair,” and gauge field “hair” beyond global charges (Baxter, 2015, Meessen et al., 2017, Priyadarshinee et al., 2021).

2. Thermodynamics and the Extended First Law

Canonical First Law and Its Modifications

In Einstein–Maxwell (and simple generalizations), the first law of black hole thermodynamics for dyonic black holes takes the form: $dM = T dS + \Phi_Q dQ + \Phi_P dP,$ where $\Phi_Q$ , $\Phi_P$ are the electrostatic and magnetostatic (horizon) potentials.

However, in gauged supergravity and models with coupled scalars (e.g., dilaton), this law generically fails when both $Q,P\neq 0$ . For spherically or planar-symmetric dyonic AdS black holes, the first law closes only in special cases—either $Q=0$ or $P=0$ or $Q=P$ (sometimes the “self-dual” point)—or in the ungauged limit ( $g=0$ ). For generic charges, an extra thermodynamic pair must be introduced: $dM = T dS + \Phi_Q dQ + \Phi_P dP + X dY$ where $(X, Y)$ are conjugate variables associated with scalar “hair” breaking part of the asymptotic symmetry (Lu et al., 2013).

This phenomenon persists in more complicated settings: in minimal supergravity, the inclusion of Chern–Simons couplings and magnetic fields yields extra “bulk” terms in the on-shell action. The proper free energy must include non-local contributions (such as $Q_{cs}$ ), so that Legendre transforms and the quantum statistical relation (Euclidean action) yield the correct first law and Smarr relations (Cai et al., 1 Oct 2024).

Extended Phase Space and Nonlinearities

When treating the cosmological constant $\Lambda$ ( $P=-\Lambda/8\pi$ ) and nonlinear couplings (e.g., Born–Infeld parameter $b$ , higher-derivative couplings $a$ , $b$ ) as thermodynamic variables, further generalized first laws and Smarr formulas appear: $dM = T dS + \Phi_e dq + \Phi_m dp + V dP + \cdots$ with additional potentials ( $\chi_a$ , $\chi_b$ ) conjugate to the nonlinear parameters (Croney et al., 6 Jun 2025, Li et al., 2016). These extensions enable the paper of critical phenomena and phase transitions in dyonic backgrounds.

3. Hair, Symmetry Breaking, and Boundary Conditions

A distinctive feature of dyonic black holes in supergravity, scalar–gauge–gravity, and non-Abelian models is the emergence of “hair.” In models with a scalar field $\phi$ , the nontrivial fall-off or boundary conditions of $\phi$ are tied to the presence of both $Q$ and $P$ . For instance, in AdS $_4$ dyonic solutions, the asymptotic expansion of $\phi$ deviates from the $\phi\sim 1/r^\Delta$ behavior expected for pure AdS. The subleading term encodes the scalar hair and breaks some of the AdS isometries, implying that the gravitational background cannot be characterized by mass, electric, and magnetic charges alone. The extra pair $(X,Y)$ in the first law captures the missing thermodynamic degree of freedom associated with this scalar mode (Lu et al., 2013).

In non-Abelian Einstein–Yang–Mills or Einstein–Maxwell–scalar models, global charges are supplemented (or replaced) by configuration parameters of the gauge field or the scalar. The “no-hair” conjecture is explicitly violated, and regular, nodeless (and thus likely stable) configurations exist in open sets of parameter space (Baxter, 2015, Meessen et al., 2017, Priyadarshinee et al., 2021).

In supergravity (e.g., $U(1)^4$ gauged SUGRA), boundary conditions for the gauge fields become nontrivial: the Fefferman–Graham expansion allows electric and magnetic boundary conditions to be chosen (Dirichlet, Neumann, mixed), but only certain discrete families (such as $P^I=\pm Q_I$ ) lead to a well-defined symplectic structure and integrable mass (Chow et al., 2013).

4. Nonlinear Electrodynamics, Extremal Limits, and Singularity Structure

Nonlinear extensions (Born–Infeld, logarithmic, or polynomial in field invariants $F$ , $G$ ) introduce several qualitative new features:

Electric–magnetic duality is generically broken: the thermodynamics and field configurations depend asymmetrically on $Q$ and $P$ (Panahiyan, 2018, Kruglov, 2019, Kruglov, 2019, Mignemi, 2021).
Relaxed or improved singularity structure: Born–Infeld-type models and their generalizations regularize the divergences of $E(r)$ and of curvature invariants at the origin, so that the Kretschmann invariant diverges no faster than in Schwarzschild, and under critical conditions (e.g., $M=E_{\mathrm{em}}$ ) the singularity can be further relegated or even removed in the pure magnetic case (Yang, 2022).
Altered extremal limits: Additional $r^{-6}$ or higher-order terms in $f(r)$ can remove the standard double-root structure of $f(r)$ at extremality, leading to non-monotonic $M$ vs. $r_+$ relations and, in some parameter ranges, “jumps” in horizon location. No local extremal limit arises for certain charge/coupling values (Croney et al., 6 Jun 2025).
The field equations for $F_{rt}$ can become algebraic of degree higher than one (e.g., cubic), and analytic tractability is typically limited to special charge cases or truncations (magnetic-only, vanishing nonlinear coefficients).

5. Global Structure, Stability, and Multipole Solutions

Dyonic black holes support a variety of global configurations and stability properties:

Multicenter solutions: In non-Abelian (SEYM) and even Abelian theories, stationary multicenter dyonic black holes can exist, subject to regularity and positivity (mass, entropy, superadditivity) constraints. Uniquely, in SEYM the relative locations of non-Abelian centers can be arbitrary; no bubble or force-balance equations are required (Meessen et al., 2017).
Symmetry properties: Dyonic solutions can break discrete symmetries of the black hole, notably north–south reflection symmetry in the presence of a magnetic monopole (dyonic KN), leading to asymmetric “hairy” black holes with observable consequences in electromagnetic lensing and black hole shadows (Cunha et al., 28 Oct 2024).
Thermodynamic stability is model-dependent. Phase structures range from unique stable thermal phases (in the linear limit) to intricate multi-branch, multi-reentrant phase diagrams (in nonlinear electrodynamics) with up to five turning points, van der Waals–like transitions, triple points, and reentrant transitions (Panahiyan, 2018, Croney et al., 6 Jun 2025).

Superradiant instabilities provide a mechanism for dynamical transition from symmetric to asymmetric (hairy) dyonic black holes in the presence of complex scalar fields; the presence of $Q_m$ enhances the instability due to modified boundary conditions and the structure of scalar quasi-bound states (Cunha et al., 28 Oct 2024, Huang et al., 2018).

6. Two-Potential Formalism and Dirac Strings

A rigorous treatment of dyonic charge in curved backgrounds requires the two-potential (Schwinger/Dirac) formalism, in which electric and magnetic charges are represented by dual gauge potentials $A_\mu$ and $B_\mu$ : $F=dA$ , $*F=dB$ . This framework renders the field strengths and all physical quantities regular, and relegates the Dirac string to an auxiliary, non-physical artifact of certain gauge representations.

Calculations of invariant quantities (e.g., Komar mass, Smarr formula) must distinguish between “auxiliary” contributions from string singularities and “physical” contributions fully encoded by $A_t$ (electric) and $B_t$ (magnetic) components. Only with the symmetric (“string-free”) representation of the electromagnetic energy–momentum tensor are auxiliary string terms absent and all physical quantities horizon-based, confirming the physical irrelevance of the Dirac string (Ramírez-Valdez et al., 2023, García-Compeán et al., 2023). This is essential when analyzing multipole and stationary charged (Kerr–Newman–dyon) solutions, and fundamental for maintaining the symmetry between electric and magnetic sectors.

7. Higher Dimensions, Topology, and Extended Theories

Dyonic black holes have been systematically generalized to higher dimensions, diverse topologies (spherical, planar, hyperbolic, and more exotic, such as Thurston geometries), and within extended field content:

Einstein–Born–Infeld and Einstein–Maxwell–Chern–Simons theories admit dyonic black holes whose thermodynamics and phase structures are richer than their Maxwell counterparts. The extended phase space allows Smarr-like relations that balance the scaling weights of horizons, charge, and coupling constants (Li et al., 2016, Cai et al., 1 Oct 2024).
Thurston-type horizon geometries yield dyonic black holes supporting Lifshitz and hyperscaling-violating asymptotics. In such cases, the Hamiltonian (boundary-term) formalism yields a proportionality between electric and magnetic charges and requires care in the derivation of the first law (Bravo-Gaete et al., 2017).
In the AdS/CFT context, dyonic solutions with planar horizons serve as duals to finite charge-density, finite magnetic field states in strongly coupled field theories, especially relevant for holographic models of transport.
Topological, non-Abelian, and non-spherical horizon dyonic black holes may provide insights into no-hair theorem violations, strongly coupled gauge/gravity duality, and large $N$ generalizations (Baxter, 2015, Meessen et al., 2017).

Dyonic black holes encapsulate a broad class of charged black hole solutions that demonstrate the intersection of gauge symmetries, scalar dynamics, nonlinearity, and gravitational thermodynamics. Their paper elucidates the consequences of charge duality and asymmetry, the nuances of black hole phase structure, the role of scalar and gauge “hair,” and the technical requirements of regularity, quantization, and symmetry at both the geometric and physical level. The field continues to develop, intersecting with quantum gravity, holography, and the phenomenology of black holes in both theoretical and astrophysical contexts.