Frozen ModMax Sector in Electrodynamics

• Frozen ModMax Sector is a constrained branch of ModMax electrodynamics where nonlinearities remain but are fixed to constant invariant configurations.
• It simplifies to Maxwell-like equations under purely electric and constant-invariant conditions, facilitating analytic black hole and dilatonic solutions.
• The framework preserves duality, enabling integrable Ernst formulations that modify thermodynamics and optical observables in gravitational settings.

Frozen ModMax Sector denotes a constrained branch of ModMax electrodynamics in which the nonlinear theory remains present but its invariant dependence is locked to a special configuration, so that the constitutive equations simplify and, in some settings, become Maxwell-like up to constant renormalizations. The recent literature suggests that the expression is not a single canonical definition but a family of closely related restrictions. In direct usage, it refers to the purely electric branch with $\mathcal P=0$, to stationary Einstein–ModMax–scalar regimes with constant coefficients $v=v_0$ and $w=w_0$, and to rotating dilatonic branches with $\mathcal F/\mathcal G=\mathrm{const}$; related papers use analogous notions such as on-shell Maxwell-equivalent backgrounds or bosonic truncations, but not always the same terminology (Al-Badawi et al., 13 May 2026, Bixano et al., 18 May 2026, Bixano et al., 27 Mar 2026, Barrientos et al., 3 Jun 2025, Bandos et al., 2021).

1. Core definition and formal criteria

In the black-hole literature, the phrase is explicitly clarified as not meaning that ModMax electrodynamics disappears. Rather, it means that the field configuration probes only a restricted branch of the theory, so the genuinely nonlinear, duality-rich structure is not fully active. For the convention used in the higher-curvature quintessence solution, the ModMax Lagrangian depends on the invariants

$$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$

with

$$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$

The purely electric restriction imposes $\mathcal P=0$, and the field equation reduces to

$$\partial_\mu\!\left(e^{-\gamma}\sqrt{-g}\,F^{\mu\nu}\right)=0.$$

In the generalized Einstein–ModMax–scalar formulation, freezing is expressed differently: one defines $\tan\Theta=\mathtt Y/\mathtt X$, together with

$$w(\Theta)=\cosh\gamma+\cos\Theta\,\sinh\gamma,\qquad v(\Theta)=\sin\Theta\,\sinh\gamma,$$

and the frozen regime is the constant-angle branch in which $v=v_0$0 and $v=v_0$1 cease to vary across spacetime (Al-Badawi et al., 13 May 2026, Bixano et al., 27 Mar 2026).

Setting	Freezing condition	Resulting structure
Purely electric black-hole branch	$v=v_0$2	Maxwell-like equations with rescaled charge
Ernst-like Einstein–ModMax–scalar regime	$v=v_0$3	sectorial Harrison maps survive
Rotating dilatonic nonlinear branch	$v=v_0$4	exact solutions on a fixed nonlinear branch

This suggests that “frozen” is best understood as a structural qualifier: the nonlinear electrodynamics is retained, but its functional freedom is reduced to constant coefficients or a single fixed branch. The shared feature is not trivialization, but analytic rigidity.

2. Purely electric freezing and Maxwell-like reduction

The clearest realization of the frozen sector is the purely electric black hole in constant-curvature $v=v_0$5 gravity with quintessence. The gauge potential is chosen as

$v=v_0$6

so the electric field is Coulomb-like. In this branch, the only surviving ModMax imprint is the effective charge renormalization

$v=v_0$7

The resulting solution therefore behaves as a Maxwell-charged black hole with effective charge $v=v_0$8, embedded in $v=v_0$9 gravity and quintessence. The same paper emphasizes that the dominant qualitative effects in thermodynamics and optics are then controlled mainly by higher-curvature and quintessence terms, while ModMax shifts the electromagnetic sector through this multiplicative factor (Al-Badawi et al., 13 May 2026).

An analogous simplification appears in Einstein–ModMax theory for static, axisymmetric spacetimes. There, the integrable sectors are the purely electric and purely magnetic ones, both characterized by $w=w_0$0. In those branches, the field equations reduce to Maxwell-like form with a constant factor $w=w_0$1, and the reduced system admits Ernst-like potentials. This is precisely the regime in which generalized Harrison transformations can be defined without generating forbidden electric–magnetic mixing. The same restriction is central to the construction of charged black holes, Melvin universes, and black diholes in Einstein–ModMax and Einstein–dilaton–ModMax theory (Bokulić et al., 22 Jul 2025).

A common misconception is that Maxwell-like reduction implies loss of all nonlinear content. The papers do not support that interpretation. What is removed is the spacetime variation of the nonlinear constitutive structure; what remains is a constant deformation of couplings, charges, or effective potentials.

3. Constant-invariant branches and intrinsically nonlinear rotating sectors

A second major meaning of the frozen sector arises in stationary Einstein–ModMax–scalar systems, where the constitutive tensor is written as

$w=w_0$2

If $w=w_0$3, then the ModMax angle $w=w_0$4 is constant, hence $w=w_0$5, $w=w_0$6, and the derivative couplings

$w=w_0$7

vanish. The field equations then retain only constant algebraic deformations through $w=w_0$8. In this form, the Weyl–Ernst system admits exact rotating families with $w=w_0$9. The same work stresses that this frozen branch is not fully Maxwell-like when the scalar is present; only after switching off the scalar can one linearly redefine the frozen system into a Maxwell-type one (Bixano et al., 27 Mar 2026).

The rotating dilatonic branch of Einstein–ModMax-type gravity sharpens this point. There, the defining nonlinear-sector condition is

$\mathcal F/\mathcal G=\mathrm{const}$0

equivalently a constant ratio of electromagnetic invariants. This freezes the constitutive relation to a fixed nonlinear branch and yields an exact stationary, axisymmetric solution with both $\mathcal F/\mathcal G=\mathrm{const}$1 and $\mathcal F/\mathcal G=\mathrm{const}$2 nonzero, a nontrivial dilaton, and a genuine gravitomagnetic structure. The crucial integrability condition is

$\mathcal F/\mathcal G=\mathrm{const}$3

In the parametrization used there, the Maxwell limit would require $\mathcal F/\mathcal G=\mathrm{const}$4, which occurs only at the isolated coupling $\mathcal F/\mathcal G=\mathrm{const}$5. The familiar low-energy string value $\mathcal F/\mathcal G=\mathrm{const}$6 and Kaluza–Klein value $\mathcal F/\mathcal G=\mathrm{const}$7 therefore remain intrinsically nonlinear. In the prolate sector, the paper identifies a genuine black-hole regime with horizon at $\mathcal F/\mathcal G=\mathrm{const}$8, NEC-satisfying exterior, and singularities hidden behind the event horizon (Bixano et al., 15 Apr 2026).

The contrast between these two frozen branches is significant. One can freeze ModMax and still obtain a Maxwell-equivalent theory after further truncation, or freeze it into a branch that has no generic Maxwell analogue at all.

4. Ernst structures, hidden symmetries, and Harrison maps

The frozen sector has a precise symmetry-theoretic role in real potential-space formulations. For stationary, axisymmetric Einstein–Maxwell–scalar and Einstein–ModMax–scalar systems, the real potentials are

$\mathcal F/\mathcal G=\mathrm{const}$9

with

$$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$0

In the ModMax case the electromagnetic block is deformed by $$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$1 and $$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$2, and the frozen regime is exactly

$$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$3

The decisive result is that coexistence of electric and magnetic Harrison transformations imposes

$$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$4

which selects precisely the frozen ModMax sector. On this subspace, the Ehlers symmetry remains in the gravito-rotational sector, while the Harrison generators survive with constant renormalizations

$$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$5

The same analysis identifies a discrete electric–magnetic map

$$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$6

and interprets the quadrature for the Weyl function $$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$7 through the affine-geodesic Hamiltonian of the target space (Bixano et al., 18 May 2026).

In Einstein–ModMax without scalar coupling, the sector-preserving Harrison transformations are given explicitly on the Ernst variables. For the purely electric frozen sector,

$$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$8

and there is an analogous magnetic transformation with $$\mathcal S=\frac{\mathbb F}{2},\qquad \mathcal P=\frac{\widetilde{\mathbb F}}{2},$$9 inserted in the appropriate electromagnetic term. These maps rederive the charged ModMax black hole from Schwarzschild, generate ModMax Melvin universes, and produce black diholes in equilibrium once the external magnetic field is tuned to cancel the conical excess (Bokulić et al., 22 Jul 2025).

The symmetry content therefore identifies the frozen sector as the integrable core of several Einstein–ModMax systems. Outside it, the full nonlinear theory is generally too unconstrained for standard Ernst machinery.

5. Thermodynamics, phase structure, and optical observables

In the higher-curvature quintessence black hole, the frozen purely electric branch leads to a thermodynamics that is Maxwell-like in structure but modified in coefficients. The horizon radius is defined by $$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$0, and Wald entropy replaces the area law: $$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$1 The first law is written as

$$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$2

The heat capacity serves as the local stability diagnostic, and the paper reports sign changes and zeros that indicate possible phase transitions. Thermodynamic geometry then supplies a consistency check: the Ruppeiner curvature scalar diverges exactly where the heat capacity vanishes, whereas the Weinhold curvature does not display the correspondence as cleanly. The Helmholtz and Gibbs free energies remain positive and smooth enough for the authors to identify overall global stability despite local unstable intervals (Al-Badawi et al., 13 May 2026).

The same frozen sector controls the optical analysis. For null geodesics,

$$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$3

and the shadow radius for an observer at finite radius $$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$4 is

$$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$5

Exact photon-sphere radii are derived for several values of the quintessence state parameter. The parameter trends are unambiguous: increasing $$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$6, $$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$7, or $$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$8 enlarges the photon sphere and shadow, while increasing $$\mathcal L_{\rm MM}=\frac12\left(\cosh\gamma\,\mathcal S-\sinh\gamma\sqrt{\mathcal P^2+\mathcal S^2}\right).$$9 shrinks them. Quintessence produces a stronger optical effect than the charge. Because ModMax enters only through the effective charge factor, its role is to shift the size and critical locations of optical and thermodynamic features rather than to generate qualitatively new structures.

This is a characteristic signature of frozen ModMax phenomenology: the nonlinear electrodynamics remains observable, but predominantly through renormalized couplings rather than through new functional dependence on the invariants.

6. Related constructions, adjacent meanings, and recurrent misconceptions

Several papers use closely related ideas without always using the exact phrase. In holographic Einstein–AdS models, the static dyonic black-hole backgrounds satisfy a proportionality relation between the electromagnetic invariants, so the ModMax action can be recast as a Maxwell action with an effective Newton constant $\mathcal P=0$0. The background geometry, Euclidean on-shell action, and thermodynamics are then essentially Maxwell-like. However, the nonlinear parameter $\mathcal P=0$1 reappears in Hall conductivity, Nernst signal, and axion-induced DC transport, and in the strong-coupling limit $\mathcal P=0$2 the geometry reduces to AdS–Schwarzschild while the gauge field becomes a purely magnetic stealth configuration and the Hall angle vanishes. This is a background-level frozen sector, not a complete decoupling of ModMax (Barrientos et al., 3 Jun 2025).

In superModMax, the paper does not name a frozen sector, but its closest analogue is the bosonic truncation

$\mathcal P=0$3

For $\mathcal P=0$4, the auxiliary-field equation has the unique solution $\mathcal P=0$5, so the supersymmetric theory collapses exactly to bosonic ModMax. A further Volkov–Akulov reparametrization absorbs the higher-derivative photino sector into a constrained goldstino multiplet, leaving a bosonic ModMax action in a goldstino-dependent metric (Bandos et al., 2021). Higher-derivative deformations of ModMax and superModMax isolate another related constrained subsector: deformations are restricted to duality-invariant, Weyl-invariant composite scalars, auxiliary fields are eliminated algebraically, and the supersymmetric $\mathcal P=0$6-field again has the on-shell solution $\mathcal P=0$7 in perturbation theory (Kuzenko et al., 2024).

By contrast, the Galilean-covariant “ModMax cousin” is explicitly presented not as a frozen truncation of relativistic ModMax but as a separate Newton–Cartan/Galilean-conformal construction built from Galilean electromagnetic invariants (Banerjee et al., 2022). Another adjacent case is the Lorentz-violating Kalb–Ramond black hole with perfect-fluid dark matter, where the static purely electric ModMax contribution enters through the screened combination $\mathcal P=0$8, shifting extremality, temperature, heat-capacity poles, and free energy. That behavior resembles a Maxwell-like reduction of the electric sector, but the paper frames it as screening rather than as a frozen sector (Belchior et al., 20 May 2026).

The main misconception addressed across this literature is therefore straightforward: frozen does not mean absent. It means restricted, on-shell degenerate, or fixed to a branch where the nonlinear constitutive freedom is no longer spacetime-dependent. Depending on the model, that can yield a Maxwell-equivalent effective description, a symmetry-preserving integrable subsystem, or an intrinsically nonlinear branch with no generic Maxwell continuation.