On gravitating dyonic configurations in nonlinear electrodynamics

Abstract: We consider static, spherically symmetric configurations of nonlinear electromagnetic fields with Lagrangians $L(f)$, where $f = F_{μν} F^{μν}$, in general relativity (GR) and other metric theories of gravity. The corresponding exact solutions are well known in the framework of GR in cases where only an electric charge ($q_e$) or a magnetic charge ($q_m$) are present, but only a few solutions in particular examples of $L(f)$ are known for dyonic systems with both nonzero $q_e$ and $q_m$. We study the properties of such systems in the special case of equal electric and magnetic charges and, assuming a correct Maxwell limit of $L(f)$, show that there always exists such a configuration of the electromagnetic field that the invariant $f$ is zero in the whole space. It leads to the existence of the corresponding families of solutions both in GR in the presence of other sources of gravity (like fluids or scalar fields) and in a wide range of extended theories of gravity (e.g., scalar-tensor and $F(R)$ gravity), in which the nonlinear electromagnetic field behaves in an especially simple manner.

• The paper presents analytic and numerical dyonic solutions in various NED models, uncovering conditions under which standard Maxwell results emerge.
• It demonstrates that when q_e^2 equals q_m^2, the field invariant f vanishes, yielding a self-dual configuration that mimics classical Einstein-Maxwell solutions.
• The methodology extends to alternate gravity frameworks, offering a foundation for further exploration of symmetry extensions and numerical benchmarks.

Gravitating Dyonic Configurations in Nonlinear Electrodynamics

Background and Motivation

Nonlinear electrodynamics (NED), with its origins in the works of Born-Infeld and Heisenberg-Euler, introduces generalizations of Maxwell's theory by allowing electromagnetic Lagrangians $L(f)$ to be nonlinear in the field invariant $f = F_{\mu\nu} F^{\mu\nu}$. The coupling of such NED models to general relativity (GR) or its extensions has produced spacetime configurations of significant physical interest, including regular black holes and soliton-like objects. While static, spherically symmetric solutions with either pure electric or pure magnetic charges have been extensively analyzed for arbitrary NED models, dyonic solutions (with both electric and magnetic charges present) are much less tractable. This technical gap is critical, as dyonic configurations can probe the full structure of a given NED and provide insight into both field-theoretic and spacetime properties.

Dyonic Solutions in NED-Gravity

Equations of Motion and Structure

Considering a static, spherically symmetric spacetime, with the standard metric parametrization,

$$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$

the field equations for an NED Lagrangian $L(f)$ constrained by spherical symmetry yield an invariant of the form:

$$f = \frac{2}{r^4(u)}\left(q_m^2 - \frac{q_e^2}{L_f^2}\right),$$

where $q_e$ and $q_m$ are the electric and magnetic charges and $L_f = dL/df$. For an arbitrary $L(f)$, obtaining exact solutions for $f(r)$ is generally not possible; analytic solutions exist only for specific models.

The associated stress-energy tensor of NED in this setup, entering the Einstein or other gravitational equations, is given by:

$f = F_{\mu\nu} F^{\mu\nu}$0

The symmetry properties of the stress-energy tensor and the spacetime imply that the metric function $f = F_{\mu\nu} F^{\mu\nu}$1 can be determined via:

$f = F_{\mu\nu} F^{\mu\nu}$2

where $f = F_{\mu\nu} F^{\mu\nu}$3 denotes the energy density.

Explicit Examples

The paper systematically discusses explicit dyonic solutions in several well-known NED models:

• Maxwell Theory ($f = F_{\mu\nu} F^{\mu\nu}$4): Recovers the Reissner-Nordström-(A)dS solution, with electric and magnetic charges entering symmetrically.
• Born-Infeld-type Theories: For example, the truncated Born-Infeld theory allows closed-form expressions for $f = F_{\mu\nu} F^{\mu\nu}$5 and $f = F_{\mu\nu} F^{\mu\nu}$6, though general dyonic solutions yield singular centers except when $f = F_{\mu\nu} F^{\mu\nu}$7.
• Arcsin and Logarithmic Electrodynamics: These lead to quadratic (or higher-degree) algebraic equations for $f = F_{\mu\nu} F^{\mu\nu}$8, typically producing two branches of solutions; one always satisfies $f = F_{\mu\nu} F^{\mu\nu}$9 for equal charge magnitudes, while the other may not be physically relevant.
• Rational Electrodynamics: Results in quartic equations for $$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$0, again showing $$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$1 as a solution for the self-dual case.

In each of these models, the analytic structure remains tractable only in select cases; otherwise, numerical approaches are necessary.

Self-Duality and General Property for $$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$2

The central analytical result is the identification of a universal solution for any NED Lagrangian with a correct Maxwell limit ($$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$3 at small $$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$4): if $$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$5, then there always exists a solution with $$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$6 everywhere in space. This is shown by Taylor expanding $$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$7 near $$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$8 and analyzing the field equations, which admit $$ds^2 = e^{2\gamma(u)} dt^2 - e^{2\alpha(u)} du^2 - r^2(u) \, d\Omega^2,$$9 as a root independently of all higher-order nonlinearities. In this branch, $L(f)$0 and the electromagnetic field's contribution to the stress-energy is Maxwellian, so the solution reduces to that of standard Einstein-Maxwell theory with $L(f)$1.

This result is robust—it is not limited to GR or electrovacuum backgrounds. The field equations retain this self-dual solution in any metric theory of gravity, including scalar-tensor and $L(f)$2 models, for static, spherically symmetric configurations. It applies equally in the presence of additional sources such as fluids or scalar fields, with the caveat that S-duality may not extend to matter couplings without additional constraints.

Implications and Discussion

Theoretical Consequences

The ubiquity of the $L(f)$3 solution for equal-charged dyons implies that, at least for spherically symmetric setups, NED corrections become "invisible" for this configuration regardless of the details of the Lagrangian, provided the Maxwellian limit is met. This is a nontrivial restriction on possible observable signatures of NED in such systems, as standard gravitational and electromagnetic diagnostics cannot distinguish them from their Maxwellian counterparts when $L(f)$4.

The result can be leveraged in scalar-tensor and extended gravity theories, due to the universal structure of the field equations under conformal rescaling, provided the spacetime region considered does not involve conformal continuation features such as those found in certain wormhole geometries.

Practical Implications and Future Directions

From a practical standpoint, this observation enables immediate generalization of many existing Einstein-Maxwell solutions (charged fluids, black holes, wormholes) to arbitrary NEDs for the self-dual dyonic sector, with only trivial modifications. This offers a template for solution construction in more complex gravitational frameworks and can be used to set initial conditions or benchmarks in numerical studies.

Further investigation is required in several directions:

• Other Spacetime Symmetries: The extension to planar, hyperbolic, and cylindrical symmetry is straightforward in some cases but problematic in others, particularly for axially symmetric rotating solutions.
• Higher Dimensions: Self-dual solutions may persist in even-dimensional spacetimes, but their properties (e.g., regularity, stability) can qualitatively differ.
• Theories with Two Invariants $L(f)$5: Models depending on both $L(f)$6 and $L(f)$7 (e.g., full Born-Infeld and ModMax theories) present richer duality structures. Here, the existence and properties of $L(f)$8 solutions—and the possibility of regular dyonic black holes—require independent analysis, as the second invariant breaks the simplicity present in $L(f)$9 models.

Conclusion

This work systematizes the construction and analysis of static, spherically symmetric dyonic solutions in NED coupled to gravity, highlighting both technical challenges and a remarkable general property: for any physically admissible NED (with proper Maxwell limit), there always exists a self-dual solution with $$f = \frac{2}{r^4(u)}\left(q_m^2 - \frac{q_e^2}{L_f^2}\right),$$0 for $$f = \frac{2}{r^4(u)}\left(q_m^2 - \frac{q_e^2}{L_f^2}\right),$$1, indistinguishable from Maxwell theory in the same symmetry class. The result is generic across GR and various extended theories of gravity, with implications for both solution-generation techniques and the interpretation of NED effects in gravitational contexts. Open questions remain concerning symmetry extensions, regularity of centers for two-invariant NEDs, and the physical distinction between different dyonic branches.