Generalized Weyl solutions

Abstract

It was shown by Weyl that the general static axisymmetric solution of the vacuum Einstein equations in four dimensions is given in terms of a single axisymmetric solution of the Laplace equation in three-dimensional flat space. Weyl's construction is generalized here to arbitrary dimension $D>~4.$ The general solution of the D-dimensional vacuum Einstein equations that admits $D\ensuremath{-}2$ orthogonal commuting non-null Killing vector fields is given either in terms of $D\ensuremath{-}3$ independent axisymmetric solutions of Laplace's equation in three-dimensional flat space or by $D\ensuremath{-}4$ independent solutions of Laplace's equation in two-dimensional flat space. Explicit examples of new solutions are given. These include a five-dimensional asymptotically flat ``black ring'' with an event horizon of topology ${S}^{1}\ifmmode\times\else\texttimes\fi{}{S}^{2}$ held in equilibrium by a conical singularity in the form of a disk.