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

Cosgrove, Christopher M. (1981) Bäcklund transformations in the Hauser–Ernst formalism for stationary axisymmetric spacetimes. Journal of Mathematical Physics, 22 (11). pp. 2624-2639. ISSN 0022-2488.

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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.

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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.
Issue or Number:11
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ID Code:10708
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Deposited On:04 Jun 2008
Last Modified:03 Oct 2019 00:12

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