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Generalized Einstein-ModMax-ScalarField theories and new exact solutions

Leonel Bixano, Tonatiuh Matos

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.

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, . 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 . This regime is particularly significant for ModMax, as it preserves genuinely nonlinear features while still admitting an analytically manageable description.

Paper Structure

This paper contains 22 sections, 1 theorem, 95 equations.

Table of Contents

  1. Introduction
  2. Convention for the ModMax sector.
  3. Potentials method
  4. Definition of the potentials
  5. Space of potentials
  6. New ModMax Anzat
  7. Subspace projection
  8. Kerr–Newman solution in the frozen ModMax framework
  9. Kerr Newman solution in Maxwell framework
  10. Kerr Newman potentials in Frozen ModMax framework
  11. First class of solutions
  12. Second class of solutions
  13. Third class of solutions
  14. Conclusions
  15. Acknowledgements
  16. ...and 7 more sections

Key Result

Theorem 1

Consider the frozen sector of ModMax theory characterized by $v = v_0, w = w_0$, which corresponds to the case of constant $\mathcal{F}/\mathcal{G}$. Let us further assume that the scalar field is turned off, i.e. $\mathtt{z}_i = 0$, and that $\alpha_0^2 = 0$, which implies that $\kappa = \kappa_0$ Therefore, the definition of the frozen potentials reduces precisely to the standard Maxwell-like t

Theorems & Definitions (2)

  • Theorem 1: Trivialization of the frozen ModMax theory without scalar fields to a Maxwell-type theory
  • proof