The starting point is a Lagrangian in the Einstein frame that contains gravity, a Maxwell field, and a real scalar field phi. The parameter epsilon zero distinguishes the dilatonic case, where epsilon zero equals plus one, from the phantom case, where epsilon zero equals minus one. The coupling constant alpha zero takes the values one, the square root of three, and one over two times the square root of three, corresponding respectively to the low-energy effective superstring theory, Kaluza-Klein theory, and entanglement relativity. When alpha zero vanishes, the exponential coupling between the scalar and the Maxwell field reduces to one, and the theory becomes Einstein-Maxwell gravity supplemented by a free, minimally coupled, massless scalar field. Pure Einstein-Maxwell theory is recovered only when the scalar field is additionally held constant.

Bixano and Matos restrict attention to spacetimes that are stationary and axisymmetric, admitting two Killing vectors associated with time translations and rotations. They adopt the Weyl ansatz metric, in which the metric functions f, omega, and k depend only on the Weyl coordinates rho and z, together with an axisymmetric four-potential ansatz for the electromagnetic field. Two mutually orthogonal differential operators, D and its rotated partner, allow the full four-dimensional field equations to be recast as a reduced system of five equations for the metric functions and the electromagnetic potentials. The scalar coupling enters the reduced Maxwell and gravitational equations through the combination kappa squared, defined as the exponential of minus two alpha zero phi, while alpha zero and epsilon zero appear directly in the reduced Klein-Gordon equation.

The heart of the construction is the introduction of five potentials, denoted Y, which serve as coordinates on an abstract potential space. These are the gravitational potential f, the rotational potential epsilon, the electric potential psi, the magnetic potential chi, and the scalar potential kappa. The existence of two of these potentials is equivalent to fulfilling one of the Maxwell equations and one of the Einstein equations, a hallmark of the Ernst approach. Substituting the potential definitions back into the field equations yields a closed system of five equations, which can be derived from a Lagrangian density on the potential space. From that Lagrangian, the authors read off the line element of the corresponding five-dimensional potential-space metric, expressed entirely in terms of the potentials f, epsilon, psi, chi, and kappa.

A subtle point concerns the degenerate limit alpha zero equals zero. Because the scalar variable kappa is defined through an exponential involving alpha zero, the transformation between phi and kappa becomes invertible only when alpha zero is nonzero. Taking the straightforward limit of a regular scalar field gives kappa tending to one, but this does not mean the scalar field is constant; it only indicates that kappa ceases to encode information about the scalar. The authors therefore formulate the alpha zero equals zero sector directly in terms of phi, showing that the potential-space metric factorizes locally into the direct product of the Einstein-Maxwell potential space and a separate scalar direction. When alpha zero equals zero and the scalar field is non-constant, the theory contains an independent free scalar degree of freedom that does not interact with the electromagnetic field but still contributes to gravity; pure Einstein-Maxwell theory emerges only when the scalar gradient vanishes.

The authors then establish the explicit correspondence with the classical Ernst formalism. They identify the Ernst potential epsilon, the Ernst twist potential h, and the Ernst electromagnetic complex potential Phi in terms of the five potentials, and derive the generalized Ernst equations for the pair epsilon and Phi in the presence of a scalar field. Upon restricting to the pure Einstein-Maxwell truncation, where alpha zero equals zero and the scalar gradient vanishes, these equations reduce exactly to the well-known Ernst equations for epsilon and Phi. In the extended formulation, the field equations take a remarkably compact form in terms of three complex one-forms A, B, and C, which the authors interpret as abstract pseudo-forces: the combination two f A is analogous to an Ernst pseudo-force, while C may be regarded as a pseudo-force generated by either a dilatonic or a phantom field.

To go further, the authors project the covariant field equations onto an auxiliary two-dimensional maximally symmetric space with coordinates xi and its conjugate, endowed with a metric whose scalar curvature is minus eight sigma. The parameter sigma determines the constant curvature of this auxiliary space, which is locally maximally symmetric by construction, although this property refers exclusively to the auxiliary space and does not require the full five-dimensional potential space to be maximally symmetric. For sigma equal to zero, the resulting system reduces to the flat, purely harmonic equations presented in earlier work by Matos and collaborators. For sigma nonzero, the authors obtain an extended system of equations for the coefficients x, y, and z of the second ansatz, in which the one-forms A, B, and C are expanded in terms of the differentials of xi and its conjugate.

Within this non-flat branch, the authors prove a theorem with a striking physical message. If the scalar-sector coefficients are generated by a real scalar potential of class C squared, and the block of equations governing the scalar sector is decoupled from the electromagnetic sector, either because the electromagnetic coefficients vanish or because alpha zero equals zero, then the only solution compatible with the reality and integrability conditions is the trivial one. For alpha zero nonzero, this implies that the scalar field must be locally constant: within the chosen ansatz, the theorem excludes a non-constant real scalar field when the electromagnetic field is absent. The authors are careful to emphasize that this is not a general no-scalar-hair statement for the full Einstein-Maxwell-scalar system, and that it does not constrain solutions falling outside the ansatz. For alpha zero equal to zero, the vanishing of the scalar-sector coefficients does not imply that phi is constant; the free scalar field remains governed by its decoupled axisymmetric harmonic equation.

As consistency checks and applications, the formalism systematically reproduces a remarkable gallery of known solutions. With neither scalar nor electromagnetic fields, and a chiral ansatz, the extended equations yield the Kerr solution, with the mass parameter emerging as an integration constant and the metric function f recovered in Boyer-Lindquist coordinates. Imposing a chiral ansatz in the electromagnetic sector as well, and working in pure Einstein-Maxwell theory, the authors obtain the Kerr-Newman solution, with the electric and magnetic charges encoded in a complex integration constant and an integrability condition relating the mass, the charges, and the curvature of the auxiliary space. Promoting the real mass parameter to a complex geometrical mass, whose imaginary part is the NUT parameter, carries the rational structure of the potentials over to the dyonic Kerr-Newman-NUT sector, yielding precisely the non-accelerating subset of the Plebanski-Demianski solutions.

The construction also contains rotating dilatonic black holes and dipolar configurations. At the Kaluza-Klein point, where epsilon zero equals plus one and alpha zero squared equals three, the target space reduces to the symmetric coset SL(3,R)/SO(2,1), and the authors use a finite electric Harrison transformation, combined with an electric shift and gravitational and scalar-electromagnetic scalings chosen to restore asymptotic normalization, to generate a charged Zipoy-Voorhees-dilaton solution directly through the global symmetries of the Ernst potential space. The limit of vanishing charging parameter recovers the vacuum Zipoy-Voorhees seed, while unit deformation parameter gives the static electrically charged Kaluza-Klein black hole. The authors likewise reconstruct the potentials of the rotating dyonic Kaluza-Klein black hole with dilaton of Rasheed, and recover the dyonic branch of the Bonnor black-dihole family from the Matos SO(2,1) subclass of the chiral model. Finally, the authors introduce a Newman-Penrose-like semi-null coframe for the potential space and compute its connection and curvature forms, showing that the target space has constant scalar curvature but, because its Weyl invariant is strictly positive, is neither conformally flat nor maximally symmetric, a result that must be distinguished from the auxiliary two-dimensional space, which is maximally symmetric by construction.

Subject of Research: A unified Ernst potentials formalism for stationary axisymmetric Einstein-Maxwell systems coupled to dilatonic or phantom scalar fields, with arbitrary scalar coupling, including a five-dimensional potential-space metric, generalized Ernst equations, and recovery of known black hole solutions.

Article Title: Unified Ernst potentials formalism for Einstein-Maxwell scalar fields theories

Article References: Bixano, L., & Matos, T. (2026). Unified Ernst potentials formalism for Einstein-Maxwell scalar fields theories. General Relativity and Gravitation, 58(10), Article 119. https://doi.org/10.1007/s10714-026-03621-3

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DOI: 10.1007/s10714-026-03621-3

Keywords: Ernst potentials, General relativity, Einstein-Maxwell theory, Scalar fields, Black holes, Kaluza-Klein theory, Exact solutions, Sigma models, General Relativity and Gravitation, Theoretical physics, Unified, Ernst