Bäcklund transformations in the Hauser–Ernst formalism for stationary axisymmetric spacetimes

Abstract

It is shown that Harrison’s Bäcklund transformation for the Ernst equation of general relativity is a two-parameter subset (not subgroup) of the infinite-dimensional Geroch group K. We exhibit the specific matrix u(t) appearing in the Hauser–Ernst representation of K for vacuum spacetimes which gives the Harrison transformation. Harrison transformations are found to be associated with quadratic branch points of u(t) in the complex t plane. The coalescence of two such branch points to form a simple pole exhibits in a simple way the known factorization of the (null generalized) HKX transformation into two Harrison transformations. We also show how finite (i.e., already exponentiated) transformations in the B group and nonnull groups of Kinnersley and Chitre can be constructed out of Harrison and/or HKX transformations. Similar considerations can be applied to electrovac spacetimes to provide hitherto unknown Bäcklund transformations. As an example, we construct a six-parameter transformation which reduces to the double Harrison transformation when restricted to vacuum and which generates Kerr–Newman–NUT space from flat space.