Azimuthal electric field in a static rotationally symmetric (2+1)-dimensional spacetime

The fundamental metrics, which describe any static three-dimensional Einstein–Maxwell spacetime (depending only on a unique spacelike coordinate), are found. In this case there are only three independent components of the electromagnetic field: two for the vector electric field and one for the scalar magnetic field. It is shown that we cannot have any superposition of these components of the electric and magnetic fields in this kind of static gravitational field. One of the electrostatic Einstein–Maxwell solutions is related to the magnetostatic solution by a duality mapping, while the second electrostatic gravitational field must be solved separately. Solutions induced by the more general (2+1)-Maxwell tensor on the static cylindrically symmetric spacetimes are studied and it is shown that all of them are also connected by duality mappings.