Generalized Weyl Solutions

TL;DR

This paper generalizes Weyl's method for static axisymmetric solutions of Einstein's equations to higher dimensions, providing explicit new solutions including a five-dimensional black ring with a specific horizon topology.

Contribution

It extends Weyl's classical construction to arbitrary dimensions, offering a systematic way to generate new higher-dimensional vacuum solutions with multiple Killing vectors.

Findings

Derived general solutions in higher dimensions using Laplace equations.

Constructed explicit examples, including a five-dimensional black ring.

Demonstrated equilibrium solutions with conical singularities.

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 $D\ge 4$. The general solution of the D-dimensional vacuum Einstein equations that admits D-2 orthogonal commuting non-null Killing vector fields is given either in terms of D-3 independent axisymmetric solutions of Laplace's equation in three-dimensional flat space or by D-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 x S^2 held in equilibrium by a conical singularity in the form of a disc.