arXiv:hep-th/0110258hep-th
Generalized Weyl Solutions
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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.
> heavy-duty use of trig Trig functions are useful in differential equations, expect them everywhere in differential geometry and geometric analysis. > How was it derived If you mean eqn (15) of "the pdf" <https://arxiv.org/abs/2203.12625> (which is what the author at the link…
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