A Class of Einstein-Maxwell Fields Generalizing the Equilibrium Solutions

Abstract

The Einstein-Maxwell fields of rotating stationary sources are represented by the SU(2,1) spinor potential $Ψ_A$ satisfying \[ \nabla \cdot [Θ^{-1}(Ψ_A\nabla Ψ_B-Ψ_B\nabla Ψ_A)]=-2Θ^{-2}\vec{C}\cdot (Ψ_A\nabla Ψ_B-Ψ_B\nabla Ψ_A) \] where $Θ=Ψ^{\dagger }\cdot Ψ$ is the SU(2,1) norm of $Ψ$% . The Ernst potentials are expressed in terms of the spinor potential by $% {\cal E}=\frac{Ψ_1-Ψ_2}{Ψ_1+Ψ_2}$, $Φ=\frac{Ψ_3}{% Ψ_1+Ψ_2}$ . The group invariant vector $\vec{C}=-2i\func{Im}\{Ψ^{\dagger}\cdot \nabla Ψ\}$ is generated exclusively by the rotation of the source, hence it is appropriate to refer to $\vec{C}$ as the {\em swirl} of the field. Static fields have no swirl. The fields with no swirl are a class generalizing the equilibrium ($| e| =m$) class of Einstein-Maxwell fields. We obtain the integrability conditions and a highly symmetrical set of field equations for this class, as well as exact solutions and an open research problem.