Generalized Nonlinear Electrodynamics (GNED)

Updated 24 November 2025

GNED is a framework extending Maxwell’s theory with nonlinear Lagrangians that incorporate quantum corrections and effective string theory actions.
It features nonlinear constitutive relations, multiple effective metrics for wave propagation, and predicts phenomena such as vacuum birefringence and point-charge regularization.
GNED models are vital for exploring black hole physics, holography, and cosmology while satisfying key symmetry constraints like duality and conformal invariance.

Generalized Nonlinear Electrodynamics (GNED) encompasses a broad class of Lagrangian-based extensions of Maxwell’s theory, in which the field equations follow from actions depending nonlinearly on the electromagnetic invariants. GNED frameworks arise from quantum corrections (e.g., Euler–Heisenberg), effective descriptions in string theory (notably Born–Infeld-type actions), and the systematic pursuit of novel symmetry principles (such as conformal or duality invariance). Key properties of GNED include nonlinear constitutive relations, multiple effective metrics for wave propagation, complex causal structure, and a wide range of physical phenomena—including regularization of point-charge self-energies, vacuum birefringence, and application to black hole solutions and condensed matter systems.

1. Action Principle and Lagrangian Structure

GNED is formulated in terms of a Lagrangian density $\mathcal{L}(X, Y)$ (or $\mathcal{L}(S, P)$ ), which depends smoothly on the two scalar electromagnetic invariants

$X = F_{ab}F^{ab} \,,\qquad Y = F_{ab}{}^{*}\!F^{ab}\,,\quad \text{with }\;{}^{*}\!F^{ab} \equiv \frac{1}{2}\,\epsilon^{abcd}F_{cd}$

or, equivalently,

$S = -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\,,\qquad P = -\frac{1}{4}F_{\mu\nu}\tilde{F}^{\mu\nu}$

for metric signature $(+, -, -, -)$ . The action in vacuum is

$S = \int d^4x\,\sqrt{-g}\;\mathcal{L}(X, Y)$

Examples include:

Born–Infeld: $\mathcal{L}_{\text{BI}} = \beta^2\left[ 1 - \sqrt{1 + X/(2\beta^2) - Y^2/(16\beta^4)} \right]$
Euler–Heisenberg (QED): $\mathcal{L}_\text{EH} = -\frac14 X + \alpha(X^2 + \frac74 Y^2)/90\pi E_c^2 + \cdots$
ModMax: $\mathcal{L}_{\text{ModMax}} = \cosh\gamma\,S + \sinh\gamma\,\sqrt{S^2 + P^2}$

The generic expansion about the Maxwell limit is: $\mathcal{L} \to -S + O(S^2, P^2)$ ensuring consistency with classical electrodynamics at weak fields (Sorokin, 2021).

2. Field Equations, Constitutive Relations, and Stress–Energy

Varying the action with respect to $A_\mu$ yields generalized Maxwell equations: $\nabla_\mu\left(\mathcal{L}_X F^{\mu\nu} + \mathcal{L}_Y\,{}^{*}F^{\mu\nu}\right) = 0$ with $\mathcal{L}_X = \partial\mathcal{L}/\partial X$ and $\mathcal{L}_Y = \partial\mathcal{L}/\partial Y$ . The homogeneous Bianchi identity $\nabla_{[\lambda}F_{\mu\nu]}=0$ persists.

The stress–energy tensor has the general structure: $T_{\mu\nu} = 4\mathcal{L}_X\,F_\mu{}^\lambda F_{\nu\lambda} + 4\mathcal{L}_Y\,F_\mu{}^\lambda\,{}^{*}F_{\nu\lambda} - g_{\mu\nu}\,\mathcal{L}$ or, equivalently (Sorokin, 2021),

$T^{\mu\nu} = \eta^{\mu\nu}\,\mathcal{L} - F^{\mu}{}_{\rho} G^{\nu\rho}$

with $G^{\mu\nu} = -2(\mathcal{L}_S F^{\mu\nu} + \mathcal{L}_P\,\tilde{F}^{\mu\nu})$ .

The constitutive relations connecting $\mathbf{D},\,\mathbf{H}$ to $\mathbf{E},\,\mathbf{B}$ are, in 3+1: $D^i = -\frac{\partial\mathcal{L}}{\partial E_i}\,,\quad H^i = -\frac{\partial\mathcal{L}}{\partial B_i}$ and in general can be highly nonlinear functions of the field invariants (Gaete et al., 2014, Kruglov, 2021).

3. Wave Propagation, Effective Metrics, and Causality

Linearized perturbations around a background field propagate according to a quartic Fresnel equation, whose factorization generically yields two Lorentzian effective metrics. For high-frequency waves, the characteristic surfaces are determined by: $P^*(k) = [g_1^{ab}\,k_a k_b]\cdot[g_2^{ab}\,k_a k_b] = 0$ with

$g_1^{ab} = a\,g^{ab} + b_1 F^a{}_c F^{bc},\qquad g_2^{ab} = g^{ab} + (b_2/a) F^a{}_c F^{bc}$

where $(a, b_1, b_2)$ encode the nonlinear response and depend on derivatives of $\mathcal{L}$ with respect to the invariants (Abalos et al., 2015). Generic GNED thus predicts birefringence: two light-cones for orthogonal polarization states (Melo et al., 2014).

Hyperbolicity and thus the well-posedness of the initial value problem require a nontrivial intersection $C_{g_1} \cap C_{g_2} \neq \emptyset$ of the two effective cones. Explicit algebraic conditions, such as $\alpha_1 \beta_2 > 0$ , delineate the physically admissible parameter region (Abalos et al., 2015).

Uniqueness results:

Only Maxwell theory ( $\mathcal{L} = cF$ ) yields two cones both conformal to Minkowski (Melo et al., 2014).
Only Born–Infeld admits a single effective metric with no birefringence, provided Maxwell limit and analyticity are imposed (Melo et al., 2014).

4. Physical Phenomena: Self-Energy, Birefringence, and Static Potentials

A central result in GNED is the regularization of point-charge fields. For Born–Infeld-type Lagrangians, the electric field of a point charge $Q$ is

$E_{\text{BI}}(r) = \frac{Q}{\sqrt{(4\pi r^2)^2 + (Q/\beta)^2}}$

implying a finite maximum $E(0)=\beta$ at the origin, and the electrostatic self-energy is likewise finite (Gaete et al., 2014).

Vacuum birefringence arises generically in GNED except for Born–Infeld and ModMax. Given a background magnetic field $B_0$ , the two orthogonal polarizations obey different refractive indices $n_\perp, n_\parallel$ ; e.g., for a BI-like model with $p=3/4$ : $n_\perp = \left[1+\frac{2B_0^2}{\beta^2}\right]^{3/4},\quad n_\parallel = \left[1+\frac{B_0^2}{\beta^2}\right]^{-1/4}$ (Gaete et al., 2014). In contrast, BI with $p=1/2$ and ModMax do not exhibit birefringence due to enhanced symmetry.

Corrections to the static potential between charges emerge at subleading orders; e.g., the leading-order nonlinear correction in BI-like/exponential models is $V(L) \propto 1/L - c/L^5$ (Gaete et al., 2014). In the model of (Gaete et al., 2017) a linear (confining) potential arises, a phenomenon not seen in Born–Infeld or Euler–Heisenberg theories.

5. Symmetry Constraints: Duality and Conformal Invariance

Electric–magnetic duality invariance and conformal symmetry have been used to single out unique GNED classes. The ModMax theory is the only known (analytic, Lagrangian) GNED that is both conformal and SO(2) duality invariant. Its Lagrangian is

$\mathcal{L}_{\text{ModMax}} = \cosh\gamma\,S + \sinh\gamma\,\sqrt{S^2+P^2}$

and its field equations are second-order, with stress tensor identically traceless and automatically satisfying the Gaillard–Zumino self-duality condition (Kosyakov, 2020, Sorokin, 2021).

More generally, any conformal-invariant Lagrangian must be homogeneous of degree one in $F_{\mu\nu}$ and can be expressed as

$L(S, P) = -\frac{1}{2\mu_0}S + S\int^{u} M(w)dw$

where $u = P/(cS)$ is the unique conformal-invariant ratio (Duplij et al., 2019).

The Courant–Hilbert approach, as extended in (Babaei-Aghbolagh et al., 21 Nov 2025), produces the full class of duality-invariant GNED by introducing a generating function $\ell(\tau)$ satisfying particular differential constraints, whose perturbative expansion encompasses ModMax, generalized Born–Infeld (GBI), and self-dual logarithmic NED. Duality invariance is characterized by $\partial_U L \cdot \partial_V L = -1$ in variables $U, V$ (Babaei-Aghbolagh et al., 21 Nov 2025).

6. Applications: Gravity Couplings, Black Holes, and Holography

GNED models contribute fundamentally to black hole physics, cosmology, and condensed matter via holographic dualities.

Magnetically charged black holes have been constructed in generalized ModMax and GBI models, revealing regularity at the origin for parameter ranges satisfying $\sigma < 1$ . The spacetime metric interpolates between Reissner–Nordström (asymptotically) and nonsingular cores ( $f(0)=1$ ). Phase transitions and thermodynamic properties, including heat capacity and Hawking temperature, show rich structure, with regions of local and global stability tightly controlled by GNED parameters (Kruglov, 2022, Babaei-Aghbolagh et al., 21 Nov 2025).
In AdS/CFT and AdS/CMT, the GNED stress–energy tensor modifies the dual charge response and conductivity, reflecting the specifics of the underlying nonlinear electrodynamics (Sorokin, 2021).
Cosmologically, GNED can drive accelerated expansion or produce a non-singular bounce, depending on the equation of state encoded in $\mathcal{L}(S, P)$ (Sorokin, 2021).

Bekenstein-type entropy/energy bounds apply in nonlinear contexts, with Born–Infeld saturating all the inequalities but not the rigidity (maximal) configurations of Maxwell (Peñafiel et al., 2017).

7. Model Classification and Physical Constraints

Meaningful GNED models must satisfy multiple criteria:

Maxwell limit: $\mathcal{L}\to -S$ at small fields.
Causality: hyperbolicity and no superluminal propagation, which translates into inequalities on derivatives of the Lagrangian (e.g., $\ell'(\tau)\geq 1$ , $\ell''(\tau)\geq 0$ for the Courant–Hilbert generating function (Babaei-Aghbolagh et al., 21 Nov 2025)).
Duality and/or conformal invariance, depending on physical motivation.
Absence of pathological static configurations, verified by energy/charge angular momentum inequalities (Peñafiel et al., 2017).
Birefringence structure: BI and ModMax are uniquely non-birefringent; generic GNED is birefringent except in these cases (Abalos et al., 2015, Melo et al., 2014).

Recent generalizations include multi-parameter extensions (generalized ModMax, exponential, logarithmic, and fractional CH flows), non-Lagrangian but conformal models (Duplij et al., 2019), and models with spontaneous Lorentz violation, which lead to a Goldstone-mode structure in the permeability tensor (Escobar et al., 2013).

Selected Table: GNED Model Classes and Defining Properties

Model	Lagrangian $\mathcal{L}$	Symmetry
Maxwell	$-S$	Linear, conformal, dual
Born–Infeld	$b^2(1-\sqrt{1+F/b^2-G^2/b^4})$	Duality, causal
Euler–Heisenberg	$-S+\kappa(4S^2+7P^2)+\dots$	Quantum, not conformal
ModMax	$\cosh\gamma\,S + \sinh\gamma\,\sqrt{S^2+P^2}$	Conformal, dual
Gen. ModMax/GBI	see above (4-param. forms)	Duality, causal
Exp/log/power-law NED	Various	Model dependent

This taxonomy encapsulates the core Lagrangian representatives, their closed-form structure, and key symmetry properties (Sorokin, 2021, Kosyakov, 2020, Duplij et al., 2019, Babaei-Aghbolagh et al., 21 Nov 2025, Kruglov, 2022).

Generalized Nonlinear Electrodynamics, by unifying symmetry principles, causal structure, and a broad range of field-theoretic phenomena, provides a comprehensive framework for both foundational and phenomenological exploration in gauge field theory, gravity, and quantum matter. Advances in laboratory experiments (e.g., high-precision vacuum birefringence, light–light scattering) and in black hole thermodynamics will differentiate among possible GNED scenarios in the near future (Gaete et al., 2014, Babaei-Aghbolagh et al., 21 Nov 2025).