Exact rotating dilatonic branch in ModMax electrodynamics without Maxwell analogue

Abstract: We present a novel class of rotating dilatonic solutions within the framework of Einstein-ModMax-type gravity. The configuration belongs to the nonlinear sector characterized by $\mathcal F/\mathcal G=\mathrm{const}$ and carries nontrivial electric and magnetic potentials, with both $A_t$ and $A_\varphi$ turned on, together with a nontrivial gravitomagnetic structure. We show that this solution does not admit continuation to the Maxwell framework of our parametrization, so it is intrinsically tied to the nonlinear ModMax regime. It includes both a NUT geometry and a NUT-free asymptotically flat limit, and it is valid for a broad class of dilatonic couplings, including the low-energy string and Kaluza-Klein cases. Moreover, in the prolate sector we identify a genuine black-hole regime in which the exterior region satisfies the null energy condition while the curvature singularity remains hidden behind the event horizon. These results provide an exact rotating dilatonic ModMax configuration with no Maxwell analog and a physically well-behaved exterior black-hole sector.

• The paper presents the first exact rotating dilatonic black hole solutions with no Maxwell analogue in the nonlinear ModMax framework.
• It leverages an axisymmetric Papapetrou metric ansatz and dual electromagnetic potentials to decouple monopole and rotational charges.
• The solutions satisfy the null energy condition and reveal robust event horizons that encapsulate all curvature singularities.

Exact Rotating Dilatonic Solutions in Nonlinear ModMax Electrodynamics without Maxwell Limit

Introduction

The paper "Exact rotating dilatonic branch in ModMax electrodynamics without Maxwell analogue" (2604.13490) presents a novel class of exact solutions within the Einstein–ModMax–dilaton theoretical framework. ModMax electrodynamics, as the unique nonlinear, one-parameter extension of Maxwell theory in four dimensions that upholds both conformal invariance and continuous electric-magnetic duality, has attracted considerable attention due to its potential for exploring strong-field regimes unamenable to linear electromagnetism.

A persistent obstacle in the field has been the construction of rotating, self-gravitating solutions with a nontrivial scalar (dilaton) field in genuinely nonlinear electrodynamics—that is, solutions not continuously connected to their Maxwell counterparts. This work achieves such a construction by leveraging a previously developed analytic framework, extending the solution space beyond previously known static, accelerating, or NUT (Newman-Unti-Tamburino) solutions.

Theoretical Framework

The starting point is the Einstein–ModMax–dilaton action in the Einstein frame, with a dilatonic scalar $\phi$ and a nonlinear electromagnetic Lagrangian:

$$\mathfrak{L} = \sqrt{-g}\left[ -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi}\mathfrak{L}_{MM} \right]$$

where

$$\mathfrak{L}_{MM} = \mathcal{F}\cosh\gamma + \sqrt{\mathcal{F}^2+\mathcal{G}^2}\sinh\gamma$$

$\mathcal{F}=F_{\mu\nu}F^{\mu\nu}$ and $\mathcal{G}=F_{\mu\nu}*F^{\mu\nu}$ are the usual Maxwell invariants and their dual; $\gamma$ is the ModMax deformation parameter. The dilatonic coupling parameter $\alpha_0$ selects theories of physical interest, such as Kaluza-Klein ($\alpha_0=1$), low-energy string ($\alpha_0= \sqrt{1/2}$), and others.

The field equations are highly nonlinear, and their solution requires a carefully chosen ansatz both for the metric and the gauge sector. The construction utilizes an axisymmetric, stationary metric of the Papapetrou class, and a two-potential electromagnetic field with both electric ($A_t$) and magnetic ($$\mathfrak{L} = \sqrt{-g}\left[ -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi}\mathfrak{L}_{MM} \right]$$0) components, as well as a nontrivial dilaton.

Nonlinear Solution without Maxwell Limit

The principal achievement is the explicit analytic construction of a rotating, dilatonic spacetime with nonzero electric and magnetic potentials, subject to the constraint $$\mathfrak{L} = \sqrt{-g}\left[ -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi}\mathfrak{L}_{MM} \right]$$1, which is inaccessible in standard Maxwell theory except at an isolated, non-physical point in the space of dilatonic couplings ($$\mathfrak{L} = \sqrt{-g}\left[ -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi}\mathfrak{L}_{MM} \right]$$2). Thus, these solutions are irreducibly nonlinear and have no Maxwell analog for the physically relevant dilatonic couplings.

Key properties of the solution:

• Both NUT (non-asymptotically flat, with gravitomagnetic charge) and NUT-less, asymptotically flat configurations are admitted by suitable parameter choices.
• In the prolate branch, an event horizon forms at a constant coordinate surface, and the exterior region obeys the null energy condition (NEC). All curvature singularities are enclosed by the event horizon under a broad range of parameters.
• The solution’s conserved charges (mass, NUT charge, angular momentum, electric/magnetic fluxes) can be calculated in closed analytic form and exhibit a clean separation: monopole-type charges are solely controlled by a single integration constant ($$\mathfrak{L} = \sqrt{-g}\left[ -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi}\mathfrak{L}_{MM} \right]$$3), while the angular momentum is independently governed by another ($$\mathfrak{L} = \sqrt{-g}\left[ -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi}\mathfrak{L}_{MM} \right]$$4).
• The solution remains nontrivial even when the entire monopole sector is deactivated, revealing a purely rotational, asymptotically flat geometry with nonvanishing ModMax dipole moments.

A crucial and bold feature of this result is the incompatibility with any Maxwellian limit for all commonly studied dilatonic couplings (such as Kaluza-Klein and low-energy string theory). The NEC is satisfied over the full black-hole exterior domain for $$\mathfrak{L} = \sqrt{-g}\left[ -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi}\mathfrak{L}_{MM} \right]$$5, and the configuration respects the weak cosmic censorship conjecture in this regime.

Singularity Structure and Horizon Analysis

The curvature invariants (Ricci and Kretschmann scalars) reveal both ring and surface singularities. In the prolate branch, the event horizon at $$\mathfrak{L} = \sqrt{-g}\left[ -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi}\mathfrak{L}_{MM} \right]$$6 (in prolate spheroidal coordinates) encapsulates all singularities for suitable parameter ranges, fulfilling classical regularity requirements outside the event horizon. In limiting cases, specifically $$\mathfrak{L} = \sqrt{-g}\left[ -\mathcal{R} + 2\epsilon_0(\nabla\phi)^2 + e^{-2\alpha_0\phi}\mathfrak{L}_{MM} \right]$$7, singularities may touch the horizon at the poles, causing violations of the NEC at those points.

The explicit construction of the horizon and verification of the NEC are carried out through rigorous evaluation of the stress-energy tensor and its contractions with null vectors, using the full analytic expressions, and following established conventions for conserved quantities.

Implications and Future Directions

The existence of this intrinsically nonlinear, rotating, asymptotically flat black hole family in Einstein–ModMax–dilaton theory is a significant extension of the exact solution landscape. The demonstrated clean decoupling of monopolar and rotational degrees of freedom is structurally non-generic, offering a sharp tool for exploring the interplay between nonlinearity, rotation, and scalar fields in classical gravity.

These results motivate several further lines of inquiry:

• Extension to higher-derivative or multi-parameter nonlinear electrodynamics models, exploiting the methodology developed.
• Dynamical and perturbative stability analysis, which could clarify astrophysical and holographic applications.
• Investigation of the quantum properties and possible observable signatures (shadows, lensing) associated with these non-Maxwellian black holes.
• Application as attractor geometries for string- or Kaluza-Klein inspired compactifications with ModMax-like gauge sectors.
• Exploration of the extremal and ultrarepulsive parameter regimes where the horizon structure undergoes further bifurcations.

Furthermore, this work reinforces the necessity of nonlinear extensions of electrodynamics—such as ModMax—for accessing solution sectors fundamentally inaccessible within the linear Maxwell regime.

Conclusion

This paper supplies the first exact, rotating, dilatonic solution in the self-gravitating ModMax nonlinear electrodynamics, with no Maxwell analog for physically canonical dilaton-gauge couplings. The prolate sector contains genuine black holes with event horizons cloaking all singularities under broad conditions, while the structure of charges illustrates an atypical decoupling between monopolar and rotational invariants. These findings deepen the analytic understanding of nonlinear electromagnetic-gravitational systems and open various avenues for both theoretical and phenomenological advancement in modified gravity and high-energy regimes (2604.13490).