Solutions of the vacuum Einstein equation having toroidal infinite red-shift surface

Abstract

The author presents a brief description of the simplest algebraic geometric solution of the stationary axisymmetric vacuum Einstein equation. This localized solution has a ring singularity surrounded by the infinite red-shift surface representing the infinite family of converging tori containing one another and the singular ring. So it seems natural to name this solution the 'toron'. It has zero mass and non-zero NUT parameter and is set by three real parameters: site on symmetry axis, diameter of the singular ring and an additional 'NUT density' parameter responsible for the size of the largest torus of infinite red-shift surface. The topological structure of the 'toron' is non-trivial; the singular ring plays the role of the 'branch ring' on its infinite-sheeted Einstein manifold; the Ernst potential on different sheets of the manifold differs by a positive constant factor.