Gauge-Theoretical Method in Solving Zero-curvature Equations II--Non-Weyl Class Solutions of the Static Einstein-Maxwell Equations

TL;DR

The paper addresses solving axisymmetric, static Einstein-Maxwell equations beyond the Weyl class, where gravitational and electromagnetic potentials are not functionally related. It advances a gauge-theoretical zero-curvature framework and employs a soliton method to generate solutions from a trivial seed via singular gauge transformations, yielding explicit 2-soliton solutions. The electrostatic sector reproduces Bonnor’s non-Weyl solution describing two electrically charged masses with a dipole moment, while the magnetostatic sector provides a new non-Weyl solution for two magnetically charged masses connected by Dirac strings and possibly with a strut between them; various horizon and Dirac-string configurations are analyzed. The results illuminate the structure of non-Weyl solutions, their horizon and string features, and how equilibrium states differ from Weyl-class equilibria, underscoring the utility of the gauge-theoretical approach for Einstein-Maxwell systems.

Abstract

The gauge-theoretical method introduced in our previous paper is applied to solve the axisymmetric and static Einstein-Maxwell equations. We obtain the solutions of the non-Weyl class, where the gravitational and electric or magnetic potentials are not functionally related. In the electrostatic case, we show that the obtained solution coincides with the solution given by Bonnor in 1979. In the magnetostatic case, we present a solution describing the gravitational field created by two magnetically charged masses. In this solution, we present a case in which the Dirac string does not stretch to spatial infinity but lies between the magnetically charged masses.