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Review: Aspects of Solution-generating Techniques for Space-times with Two Commuting Killing Vectors

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Abstract

Certain aspects of solution-generatingtechniques for spacetimes with two commuting Killingvectors are reviewed. A brief historical introduction tostationary axisymmetric systems is given. The importance of the Homogeneous Hilbert problem associatedwith the equations, unifying the group-theoretic withthe soliton-theoretic approaches, is emphasized. Theformalism of generating functions is introduced, both for vacuum and electrovacuum.Sibgatullin's technique for electrovacuum solutions isrelated to the Hauser Ernst variables and a method byErnst is briefly discussed. The solitonic methods ofBelinsky-Zakharov and Alekseev are reviewed. Their relation isemphasized by an explicit proof, at the level ofgenerating techniques, that the BZ two soliton with twocomplex conjugate poles is isomorphic to the Alekseev one-soliton (restricted to vacuum) with trivialgauge. The Alekseev non-soliton technique is discussed.Some recent developments are brieflydiscussed.

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  1. Panayiotis Kordas
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Kordas, P. Review: Aspects of Solution-generating Techniques for Space-times with Two Commuting Killing Vectors. General Relativity and Gravitation 31, 1941–1984 (1999). https://doi.org/10.1023/A:1026799124832

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