Stationary Axisymmetric Fields as Two-Dimensional Geodesics

TL;DR

This paper reformulates Einstein's equations for stationary axisymmetric fields as geodesic equations in a two-dimensional space, enabling new insights into solutions describing dyons and rotating bodies.

Contribution

It introduces a novel geometric approach to Einstein's equations, relating solutions with different physical properties via affine collineations in a reduced two-dimensional space.

Findings

Reformulation of Einstein's equations as geodesic equations

Identification of affine collineations linking solutions

Application to dyon and rotating body solutions

Abstract

Einstein's equations for stationary axisymmetric fields are reformulated as the equations for affine geodesics in a two--dimensional space. The affine collineations of this space are investigated and used to relate explicit solutions of Einstein's equations with different physical properties. Particularly, the solutions describing the exterior fields of a dyon and a slowly rotating body are discussed.