Stationary axisymmetric one-soliton solutions of the Einstein equations

Abstract

The soliton technique due to V. Belinsky and V. Zakharov, (1980) is applied to find stationary axisymmetric one-soliton solutions of the Einstein equations in vacuum. In order to generate (2n+1)-soliton solutions with physical signature the (unphysical) Euclidean metric is taken as the seed solution. The one-soliton solutions are a family of non-asymptotically flat metrics depending on one parameter and can be considered as being the stationary generalisations of a very simple family of static Weyl metrics. They are Petrov type I metrics except for one of its members, which is Petrov type II and can be simply related to the van Stockum class. The Ernst potential of these solutions and the use of prolate spheroidal coordinates suggest new related families of solutions which are asymptotically flat. One of them contains the Zipoy-Voorhees metric with deformation parameter delta =1/2 as a particular case.