Unified Ernst potentials formalism for Einstein-Maxwell scalar fields theories

Abstract

In the present work, we develop a unified Ernst-type formulation for stationary and axisymmetric Einstein-Maxwell systems coupled to a dilatonic or phantom scalar field, with arbitrary scalar coupling for the Einstein-Maxwell theory. We derive the corresponding five-dimensional potential-space metric and establish an explicit relation between the standard Ernst potentials and the complex one-forms (A, B, C). The degenerate limit \(\alpha _0=0\) is treated separately, distinguishing pure Einstein-Maxwell theory from Einstein-Maxwell gravity coupled to an independent free scalar field. We then project the covariant field equations onto an auxiliary two-dimensional maximally symmetric space with scalar curvature \(\mathcal {R}_{\Sigma }=-8\sigma \), and derive the corresponding extended system. Within the non-flat branch and the ansatz considered, we obtain an obstruction to non-constant real scalar configurations in the absence of the electromagnetic sector. The Kerr, Kerr-Newman, Ker-Dilaton in Kaluza-Klein theory, and Bonnor/black-dihole sectors are recovered as consistency checks and applications of the construction. Finally, we introduce a Newman-Penrose-like semi-null coframe for the potential space and compute its connection, curvature forms, and scalar invariants, showing that the target space has constant scalar curvature but is neither conformally flat nor maximally symmetric.