Generalized Einstein-ModMax-ScalarField theories and new exact solutions

Abstract: We present a generalized Ernst-type framework for stationary, axisymmetric spacetimes in which a scalar field is coupled to the electrodynamic field, with a particular focus on the ModMax theory. Our approach relies on the Weyl stationary-axisymmetric ansatz and explicitly allows for a nonzero rotational metric function, $ω\neq 0$. The resulting setup is broad enough to encompass wide classes of scalar couplings, including dilatonic and phantom-like sectors, and can be tailored to specific models such as Einstein-ModMax, Kaluza-Klein theories, low-energy string-inspired scenarios, entanglement relativity and related generalizations. Within this scheme, we derive two novel families of exact rotating solutions in the sector where the electromagnetic invariants obey $\mathcal F/\mathcal G=\mathrm{constant}$. This regime is particularly significant for ModMax, as it preserves genuinely nonlinear features while still admitting an analytically manageable description.

• The paper introduces a generalized Ernst framework for stationary, axisymmetric spacetimes coupling scalar fields with ModMax nonlinear electrodynamics.
• It presents three distinct families of exact rotating solutions, including geon-type, real harmonic, and phase-modulated electromagnetic potentials.
• The analysis reveals that nonlinear ModMax effects only manifest with active scalar fields, confirming triviality in the frozen sector without dynamical scalars.

Generalized Einstein-ModMax-ScalarField Theories and Rotating Exact Solutions

Introduction and Motivation

The paper "Generalized Einstein-ModMax-ScalarField theories and new exact solutions" (2603.26073) develops a comprehensive formalism for stationary axisymmetric spacetimes where a scalar field couples to the nonlinear electromagnetic ModMax theory. ModMax, uniquely among nonlinear generalizations of Maxwell electrodynamics in four dimensions, preserves both conformal and continuous electric-magnetic duality invariance. This qualifies ModMax as a highly constrained and structurally rich nonlinear theory, distinguishing it from traditional nonlinear models such as Born–Infeld. The prior catalog of exact self-gravitating solutions in Einstein–ModMax theory is largely confined to static or highly symmetric configurations, rendering rotating, scalar-coupled solutions scarce.

The authors present a generalized Ernst-type framework incorporating the Weyl ansatz for stationary, axisymmetric metrics with nonzero rotation $\omega \neq 0$. The formalism explicitly encompasses wide classes of scalar couplings including dilatonic and phantom sectors and interpolates between Einstein–ModMax, Kaluza–Klein, string-inspired, and entanglement relativity scenarios through the scalar coupling $\alpha_0$.

ModMax Nonlinear Electrodynamics and Scalar Coupling

ModMax extensions maintain two stringent symmetries: conformal invariance and duality invariance. The metric Lagrangian in the Einstein frame is: $$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$ where $\mathfrak{L}_{MM}$ is the ModMax electromagnetic Lagrangian parameterized by the deformation $\gamma$, and $\epsilon_0$ distinguishes dilaton ($+1$) from phantom ($-1$).

The approach enables the study of a broad class of scalar–electrodynamic theories, with $\alpha_0$ governing the strength and type of scalar coupling (e.g. $\alpha_0=0$ for EMM, $\alpha_0$0 for string-inspired, $\alpha_0$1 for KK, $\alpha_0$2 for entanglement relativity).

Generalized Ernst Formalism and Potential Structure

Following and extending the Ernst potential methods [Bixano:2026xum], the field content is recast in terms of five potentials $\alpha_0$3 representing the gravitational, rotational, electric, magnetic, and scalar degrees of freedom. The field equations are derived via Euler–Lagrange methods, yielding a compact system that exposes the ModMax nonlinear structure through auxiliary constants $\alpha_0$4 and $\alpha_0$5 (related to the deformation parameter $\alpha_0$6 and the ratio $\alpha_0$7 of electromagnetic invariants).

A Newman–Penrose-type null coframe is introduced for the potential space, facilitating the reduction of the field equations and enabling construction of exact invariant scalars. When the electromagnetic invariants ratio is held constant (the "frozen sector"), the resulting potential space becomes maximally symmetric and conformally flat, and the field equations simplify to allow analytic integration.

Exact Rotating Solutions: Construction and Analysis

Maxwell versus Frozen ModMax Sector

A central result is that, without an active scalar ($\alpha_0$8, $\alpha_0$9), the frozen ModMax sector reduces trivially to Maxwell theory via a global redefinition of potentials, amounting to rescalings and shifts. The electromagnetic four-potential is normalized by $$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$0 and the rotational potential absorbs the nonlinear terms. This demonstrates that nonlinear ModMax corrections require both nonzero deformation as well as a nontrivial scalar field to manifest genuinely new phenomenology beyond the Maxwell sector.

New Classes of Rotating Solutions

Three distinct families of exact solutions are constructed in the frozen ModMax sector:

First Class (Geon-type)

This class corresponds to complex-valued potential variables and recovers solutions studied previously in dilatonic Einstein–Maxwell theory, now extended to ModMax. The metric is flat ($$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$1), while the scalar field, electromagnetic, and rotational potentials are nontrivial functions of a harmonic function $$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$2. Critically, even with constant $$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$3, the rotational metric function $$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$4 is nontrivial due to the intricate coupling between $$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$5 and $$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$6. This describes non-gravitating geon-type compact objects with electromagnetic and scalar structure.

Second Class (Real-valued Potentials)

Solutions in this class employ real harmonic functions and yield explicit scaling relations for potentials, governed by the parameter $$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$7. In Maxwell theory ($$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$8), magnetic and rotational structures vanish identically, reducing to static purely electric states. In contrast, ModMax nonlinearity ($$\mathfrak{L} = \sqrt{-g}\left( -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi} \mathfrak{L}_{MM} \right)$$9) activates nontrivial magnetic and rotational structure, manifesting the genuinely nonlinear phenomenology.

Third Class (Phase-modulated Electromagnetic Potential)

Here, the electromagnetic potential is endowed with a phase, generating solutions dual to the second class. When $\mathfrak{L}_{MM}$0, rotational and magnetic potentials trivialize; with $\mathfrak{L}_{MM}$1, the phase-modulation produces new rotating, magnetized scalar–electrodynamic backgrounds.

Kerr–Newman Solutions in ModMax

The authors recover the Kerr–Newman solution in both Maxwell and frozen ModMax theory. In the ModMax case, the solution is equivalent to introducing a rescaling of electromagnetic quantities, confirming that nonlinear effects become trivial without scalar dynamics.

Strong Numerical and Structural Results

• The paper provides explicit analytic forms for all new solution classes, demonstrating algebraic solvability within the frozen sector.
• It rigorously proves the triviality of the ModMax theory without scalar fields, establishing equivalence to Maxwell via global redefinitions.
• Nonlinear effects in rotating solutions are only nontrivial when scalar fields are dynamically active. This is substantiated through analytic forms and the examination of symmetries.
• Analysis of the invariant scalars demonstrates maximal symmetry and conformal flatness in the frozen ratio sector.

Implications, Applications, and Future Directions

The formalism enables systematic exploration of nonlinear scalar–electrodynamics in a context previously inaccessible to analytic solutions. The explicit construction of exact rotating solutions with scalar and nonlinear electromagnetic structure is relevant for the study of:

• Nonlinear black hole solutions and their global charges in generalizations of Einstein–Maxwell theory
• Rotating geon-type solutions and wormholes in nonlinear electrodynamics with scalar couplings
• Compact objects in Kaluza–Klein and string-inspired models with duality-preserving nonlinear fields
• Model-building in theoretical high-energy physics and cosmology involving nontrivial scalar–electrodynamic backgrounds

Further developments may include extension to non-frozen sectors, analysis of stability and physical properties (e.g., energy conditions, horizon structure), and generalization to higher-dimensional or AdS backgrounds. The compact potential formalism is amenable to symmetry-based solution-generating techniques and may facilitate classification of nonlinear gravitating objects in scalar–ModMax systems.

Conclusion

This paper presents a rigorous extension of the Ernst potential machinery to the nonlinear, duality-invariant ModMax electrodynamics coupled to scalar fields, focusing on rotating axisymmetric geometries. The generalized formalism establishes the algebraic structure and integrability properties of stationary, axisymmetric solutions, produces several new analytic families—including geon, phase-modulated, and scaling solutions—and clarifies the circumstances under which ModMax nonlinearities yield nontrivial phenomenology. The results provide a foundation for systematic investigation of scalar–electrodynamic nonlinear gravitational objects and set the stage for further analytic and theoretical advances in extended gravitation models (2603.26073).