Potential Space Symmetries in Ernst-like Formulations of Einstein-Maxwell/ModMax-Scalar field Theories

TL;DR

This paper explores the symmetries and transformations of Ernst-like potentials in Einstein-Maxwell-Scalar and Einstein-ModMax-Scalar theories, analyzing their algebraic structures, invariants, and solution-generating techniques.

Contribution

It characterizes the full symmetry structure, including hidden and sectorial symmetries, and applies these to generate and analyze solutions in modified gravity theories.

Findings

Identified exact visible symmetries and their Lie algebra in potential space.

Characterized hidden symmetries acting on invariant subspaces.

Derived quadratures and solution-generating transformations for specific sectors.

Abstract

We complete the visible, hidden, sectorial, and discrete symmetries of Ernst-like potential spaces in stationary, axisymmetric Einstein-Maxwell-Scalar Field (EMSF) and Einstein-ModMax-Scalar Field (EMMSF) theories. In the real potential space \((f,\epsilon,\psi,\chi,\kappa)\), we determine the exact visible symmetries and their solvable Lie algebra. We characterize the hidden symmetries on invariant subspaces: Ehlers acts in the gravito-rotational sector, while electric and magnetic Harrison transformations act in static electromagnetic sectors. In the frozen EMMSF regime, \(v=v_0,\ w=w_0\), we show how EMSF sectorial transformations are deformed in ModMax theory. We also show that coexistence of electric and magnetic sectorial Harrison transformations imposes \(d w=0\) and \(d[(v^2+w^2)/w]=0\), selecting precisely the frozen ModMax sector. We study the Hamiltonian formulation, Noether charges, and Casimir invariants of the sectorial algebras. In harmonic branches of the \(A,B,C\) one-forms, the affine geodesic energy is constant, so the quadrature for \(k\) is controlled by the affine-geodesic Hamiltonian. The functions \(\omega\) and \(A_\varphi\) follow from Noether charges along the Killing directions of \(\epsilon\) and \(\chi\), and are written using dual harmonic functions. We examine the electric and magnetic Lewis-Weyl-Papapetrou frames and their discrete map, which sends \(\kappa\mapsto\kappa^{-1}\), \(\psi\mapsto\chi\), \(\chi\mapsto-\psi\), and \(\epsilon\mapsto\epsilon-\psi\chi\). Finally, we apply the sectorial transformations to harmonic scalar--acuum Weyl seeds with independent gravitational and scalar harmonics. Frozen-ModMax Harrison maps generate charged branches, while Ehlers generates the gravito-rotational branch. For these solutions we give the final quadratures for \(k\), \(\omega\), and \(A_\varphi\).