A Class of Einstein-Maxwell Fields Generalizing the Equilibrium Solutions

TL;DR

This paper introduces a new class of Einstein-Maxwell fields characterized by a 'swirl' parameter, generalizing static equilibrium solutions, with derived integrability conditions, symmetrical field equations, and exact solutions.

Contribution

It defines a novel class of Einstein-Maxwell fields with non-zero swirl, extending static equilibrium solutions, and provides their integrability conditions, symmetrical equations, and exact solutions.

Findings

Defined the 'swirl' vector as a source rotation measure.

Derived integrability conditions for the new class.

Presented exact solutions within this class.

Abstract

The Einstein-Maxwell fields of rotating stationary sources are represented by the SU(2,1) spinor potential $\Psi_A$ satisfying \[ \nabla \cdot [\Theta ^{-1}(\Psi_A\nabla \Psi_B-\Psi_B\nabla \Psi_A)]=-2\Theta ^{-2}\vec{C}\cdot (\Psi_A\nabla \Psi_B-\Psi_B\nabla \Psi_A) \] where $\Theta =\Psi ^{\dagger }\cdot \Psi $ is the SU(2,1) norm of $\Psi $% . The Ernst potentials are expressed in terms of the spinor potential by $% {\cal E}=\frac{\Psi_1-\Psi _2}{\Psi_1+\Psi_2}$, $\Phi =\frac{\Psi_3}{% \Psi_1+\Psi_2}$ . The group invariant vector $\vec{C}=-2i\func{Im}\{\Psi ^{\dagger}\cdot \nabla \Psi \}$ 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.