Hidden symmetry of the three-dimensional Einstein-Maxwell equations

TL;DR

This paper reveals hidden symmetries in three-dimensional Einstein-Maxwell equations, showing how to generate solutions using group transformations and connecting these symmetries to known four-dimensional Einstein solutions.

Contribution

It identifies a Heisenberg group symmetry for Einstein-Maxwell fields and links the Einstein-Maxwell-dilaton system to the Geroch transformation via Kaluza-Klein theory.

Findings

Generated new Einstein-Maxwell solutions from known ones.

Discovered the Heisenberg group symmetry including Harrison transformations.

Established the SL(2,R) symmetry relating Maxwell and dilaton fields.

Abstract

It is shown how to generate three-dimensional Einstein-Maxwell fields from known ones in the presence of a hypersurface-orthogonal non-null Killing vector field. The continuous symmetry group is isomorphic to the Heisenberg group including the Harrison-type transformation. The symmetry of the Einstein-Maxwell-dilaton system is also studied and it is shown that there is the $SL(2,{\bf R})$ transformation between the Maxwell and the dilaton fields. This $SL(2,{\bf R})$ transformation is identified with the Geroch transformation of the four-dimensional vacuum Einstein equation in terms of the Ka{\l}uza-Klein mechanism.