Potential-space geometry and scalar rigidity of the Plebański-Demiański ansatz without a cosmological constant

Abstract

We analyze stationary, axisymmetric Einstein-Maxwell-dilaton theory with zero cosmological constant in the conformal Carter class, employing a generalized Ernst approach on the space of potentials. We begin by establishing a complete local correspondence between Plebański-Demiański principal coordinates and Weyl data, covering the electric, magnetic, and twist potentials as well as the 1-forms \(A\), \(B\), and \(C\). In the source-free Weyl reduction, requiring the invariant area density to be harmonic singles out the \(Λ=0\) sector, and the principal chiral splitting isolates both electromagnetic misalignment and the part due to acceleration. Next, we obtain the Einstein reconstruction equations for general local EMD configurations in this setting. In a physical Carter domain with an ordinary scalar, the bilinear factor \(\Om=1-\acc pq\) enforces \(C=0\) and \(B_-=0\). With fixed \(\alphao\neq0\), the scalar field equation further rules out the charged PD Coulomb-type 1-form. Lastly, assuming an exact quadratic Stäckel scalar 1-form, the conditions of exactness together with reconstruction restrict the system to a single aligned branch, but consistency with the full potential equations removes this branch for arbitrary smooth Carter structure functions. Consequently, within the quadratic Stäckel class studied here, there is no local nonconstant solution with an ordinary dilaton. This conclusion does not extend to phantom scalars, conformal factors of higher degree, or non-Stäckel 1-forms.