Published November 1, 1981 | public
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Bäcklund transformations in the Hauser–Ernst formalism for stationary axisymmetric spacetimes

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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.

Additional Information

Copyright © 1981 American Institute of Physics. (Received 30 March 1981; accepted for publication 26 June 1981) This research has benefited from discussions with William Kinnersley and Terry Lemley. I also wish to thank Isidore Hauser and Frederick J. Ernst for making their results available prior to publication. Supported in part by the National Science Foundation (AST79-22012). [C.M.C. was a] Richard Chase Tolman Research Fellow.

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